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700 | Polars DataFrame | ü¶úÔ∏èüîó Langchain | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: Polars DataFrame | ü¶úÔ∏èüîó Langchain |
701 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersPolars DataFramePolars DataFrameThis notebook goes over how to load data from a polars | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersPolars DataFramePolars DataFrameThis notebook goes over how to load data from a polars |
702 | notebook goes over how to load data from a polars DataFrame.#!pip install polarsimport polars as pldf = pl.read_csv("example_data/mlb_teams_2012.csv")df.head()<div><style>.dataframe > thead > tr > th,.dataframe > tbody > tr > td { text-align: right;}</style><small>shape: (5, 3)</small><table border="1" class="dataframe"><thead><tr><th>Team</th><th> "Payroll (millions)"</th><th> "Wins"</th></tr><tr><td>str</td><td>f64</td><td>i64</td></tr></thead><tbody><tr><td>"Nationals"</td><td>81.34</td><td>98</td></tr><tr><td>"Reds"</td><td>82.2</td><td>97</td></tr><tr><td>"Yankees"</td><td>197.96</td><td>95</td></tr><tr><td>"Giants"</td><td>117.62</td><td>94</td></tr><tr><td>"Braves"</td><td>83.31</td><td>94</td></tr></tbody></table></div>from langchain.document_loaders import PolarsDataFrameLoaderloader = PolarsDataFrameLoader(df, page_content_column="Team")loader.load() [Document(page_content='Nationals', metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98}), Document(page_content='Reds', metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97}), Document(page_content='Yankees', metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95}), Document(page_content='Giants', metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94}), Document(page_content='Braves', metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94}), Document(page_content='Athletics', metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94}), Document(page_content='Rangers', metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93}), Document(page_content='Orioles', metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93}), Document(page_content='Rays', metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90}), Document(page_content='Angels', metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89}), Document(page_content='Tigers', metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88}), | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: notebook goes over how to load data from a polars DataFrame.#!pip install polarsimport polars as pldf = pl.read_csv("example_data/mlb_teams_2012.csv")df.head()<div><style>.dataframe > thead > tr > th,.dataframe > tbody > tr > td { text-align: right;}</style><small>shape: (5, 3)</small><table border="1" class="dataframe"><thead><tr><th>Team</th><th> "Payroll (millions)"</th><th> "Wins"</th></tr><tr><td>str</td><td>f64</td><td>i64</td></tr></thead><tbody><tr><td>"Nationals"</td><td>81.34</td><td>98</td></tr><tr><td>"Reds"</td><td>82.2</td><td>97</td></tr><tr><td>"Yankees"</td><td>197.96</td><td>95</td></tr><tr><td>"Giants"</td><td>117.62</td><td>94</td></tr><tr><td>"Braves"</td><td>83.31</td><td>94</td></tr></tbody></table></div>from langchain.document_loaders import PolarsDataFrameLoaderloader = PolarsDataFrameLoader(df, page_content_column="Team")loader.load() [Document(page_content='Nationals', metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98}), Document(page_content='Reds', metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97}), Document(page_content='Yankees', metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95}), Document(page_content='Giants', metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94}), Document(page_content='Braves', metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94}), Document(page_content='Athletics', metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94}), Document(page_content='Rangers', metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93}), Document(page_content='Orioles', metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93}), Document(page_content='Rays', metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90}), Document(page_content='Angels', metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89}), Document(page_content='Tigers', metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88}), |
703 | "Payroll (millions)"': 132.3, ' "Wins"': 88}), Document(page_content='Cardinals', metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88}), Document(page_content='Dodgers', metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86}), Document(page_content='White Sox', metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85}), Document(page_content='Brewers', metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83}), Document(page_content='Phillies', metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81}), Document(page_content='Diamondbacks', metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81}), Document(page_content='Pirates', metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79}), Document(page_content='Padres', metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76}), Document(page_content='Mariners', metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75}), Document(page_content='Mets', metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74}), Document(page_content='Blue Jays', metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73}), Document(page_content='Royals', metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72}), Document(page_content='Marlins', metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69}), Document(page_content='Red Sox', metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69}), Document(page_content='Indians', metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68}), Document(page_content='Twins', metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66}), Document(page_content='Rockies', metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64}), Document(page_content='Cubs', metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61}), Document(page_content='Astros', metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55})]# Use lazy load for larger table, which won't read the full table into memoryfor i in loader.lazy_load(): print(i) | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: "Payroll (millions)"': 132.3, ' "Wins"': 88}), Document(page_content='Cardinals', metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88}), Document(page_content='Dodgers', metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86}), Document(page_content='White Sox', metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85}), Document(page_content='Brewers', metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83}), Document(page_content='Phillies', metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81}), Document(page_content='Diamondbacks', metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81}), Document(page_content='Pirates', metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79}), Document(page_content='Padres', metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76}), Document(page_content='Mariners', metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75}), Document(page_content='Mets', metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74}), Document(page_content='Blue Jays', metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73}), Document(page_content='Royals', metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72}), Document(page_content='Marlins', metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69}), Document(page_content='Red Sox', metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69}), Document(page_content='Indians', metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68}), Document(page_content='Twins', metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66}), Document(page_content='Rockies', metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64}), Document(page_content='Cubs', metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61}), Document(page_content='Astros', metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55})]# Use lazy load for larger table, which won't read the full table into memoryfor i in loader.lazy_load(): print(i) |
704 | memoryfor i in loader.lazy_load(): print(i) page_content='Nationals' metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98} page_content='Reds' metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97} page_content='Yankees' metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95} page_content='Giants' metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94} page_content='Braves' metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94} page_content='Athletics' metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94} page_content='Rangers' metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93} page_content='Orioles' metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93} page_content='Rays' metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90} page_content='Angels' metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89} page_content='Tigers' metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88} page_content='Cardinals' metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88} page_content='Dodgers' metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86} page_content='White Sox' metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85} page_content='Brewers' metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83} page_content='Phillies' metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81} page_content='Diamondbacks' metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81} page_content='Pirates' metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79} page_content='Padres' metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76} page_content='Mariners' metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75} page_content='Mets' metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74} page_content='Blue Jays' metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73} page_content='Royals' metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72} page_content='Marlins' metadata={' | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: memoryfor i in loader.lazy_load(): print(i) page_content='Nationals' metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98} page_content='Reds' metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97} page_content='Yankees' metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95} page_content='Giants' metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94} page_content='Braves' metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94} page_content='Athletics' metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94} page_content='Rangers' metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93} page_content='Orioles' metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93} page_content='Rays' metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90} page_content='Angels' metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89} page_content='Tigers' metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88} page_content='Cardinals' metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88} page_content='Dodgers' metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86} page_content='White Sox' metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85} page_content='Brewers' metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83} page_content='Phillies' metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81} page_content='Diamondbacks' metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81} page_content='Pirates' metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79} page_content='Padres' metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76} page_content='Mariners' metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75} page_content='Mets' metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74} page_content='Blue Jays' metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73} page_content='Royals' metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72} page_content='Marlins' metadata={' |
705 | 72} page_content='Marlins' metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69} page_content='Red Sox' metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69} page_content='Indians' metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68} page_content='Twins' metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66} page_content='Rockies' metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64} page_content='Cubs' metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61} page_content='Astros' metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55}PreviousAmazon TextractNextPsychicCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This notebook goes over how to load data from a polars DataFrame. | This notebook goes over how to load data from a polars DataFrame. ->: 72} page_content='Marlins' metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69} page_content='Red Sox' metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69} page_content='Indians' metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68} page_content='Twins' metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66} page_content='Rockies' metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64} page_content='Cubs' metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61} page_content='Astros' metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55}PreviousAmazon TextractNextPsychicCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
706 | Nuclia | ü¶úÔ∏èüîó Langchain | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. ->: Nuclia | ü¶úÔ∏èüîó Langchain |
707 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersNucliaOn this pageNucliaNuclia automatically indexes your unstructured data from any internal | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersNucliaOn this pageNucliaNuclia automatically indexes your unstructured data from any internal |
708 | indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing.The Nuclia Understanding API supports the processing of unstructured data, including text, web pages, documents, and audio/video contents. It extracts all texts wherever they are (using speech-to-text or OCR when needed), it also extracts metadata, embedded files (like images in a PDF), and web links. If machine learning is enabled, it identifies entities, provides a summary of the content and generates embeddings for all the sentences.Setup‚ÄãTo use the Nuclia Understanding API, you need to have a Nuclia account. You can create one for free at https://nuclia.cloud, and then create a NUA key.#!pip install --upgrade protobuf#!pip install nucliadb-protosimport osos.environ["NUCLIA_ZONE"] = "<YOUR_ZONE>" # e.g. europe-1os.environ["NUCLIA_NUA_KEY"] = "<YOUR_API_KEY>"Example‚ÄãTo use the Nuclia document loader, you need to instantiate a NucliaUnderstandingAPI tool:from langchain.tools.nuclia import NucliaUnderstandingAPInua = NucliaUnderstandingAPI(enable_ml=False)from langchain.document_loaders.nuclia import NucliaLoaderloader = NucliaLoader("./interview.mp4", nua)You can now call the load the document in a loop until you get the document.import timepending = Truewhile pending: time.sleep(15) docs = loader.load() if len(docs) > 0: print(docs[0].page_content) print(docs[0].metadata) pending = False else: print("waiting...")Retrieved information‚ÄãNuclia returns the following information:file metadataextracted textnested text (like text in an embedded image)paragraphs and sentences splitting (defined by the position of their first and last characters, plus start time and end time for a video or audio file)linksa thumbnailembedded filesNote: Generated files (thumbnail, extracted embedded files, etc.) are provided | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. ->: indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing.The Nuclia Understanding API supports the processing of unstructured data, including text, web pages, documents, and audio/video contents. It extracts all texts wherever they are (using speech-to-text or OCR when needed), it also extracts metadata, embedded files (like images in a PDF), and web links. If machine learning is enabled, it identifies entities, provides a summary of the content and generates embeddings for all the sentences.Setup‚ÄãTo use the Nuclia Understanding API, you need to have a Nuclia account. You can create one for free at https://nuclia.cloud, and then create a NUA key.#!pip install --upgrade protobuf#!pip install nucliadb-protosimport osos.environ["NUCLIA_ZONE"] = "<YOUR_ZONE>" # e.g. europe-1os.environ["NUCLIA_NUA_KEY"] = "<YOUR_API_KEY>"Example‚ÄãTo use the Nuclia document loader, you need to instantiate a NucliaUnderstandingAPI tool:from langchain.tools.nuclia import NucliaUnderstandingAPInua = NucliaUnderstandingAPI(enable_ml=False)from langchain.document_loaders.nuclia import NucliaLoaderloader = NucliaLoader("./interview.mp4", nua)You can now call the load the document in a loop until you get the document.import timepending = Truewhile pending: time.sleep(15) docs = loader.load() if len(docs) > 0: print(docs[0].page_content) print(docs[0].metadata) pending = False else: print("waiting...")Retrieved information‚ÄãNuclia returns the following information:file metadataextracted textnested text (like text in an embedded image)paragraphs and sentences splitting (defined by the position of their first and last characters, plus start time and end time for a video or audio file)linksa thumbnailembedded filesNote: Generated files (thumbnail, extracted embedded files, etc.) are provided |
709 | extracted embedded files, etc.) are provided as a token. You can download them with the /processing/download endpoint. Also at any level, if an attribute exceeds a certain size, it will be put in a downloadable file and will be replaced in the document by a file pointer. This will consist of {"file": {"uri": "JWT_TOKEN"}}. The rule is that if the size of the message is greater than 1000000 characters, the biggest parts will be moved to downloadable files. First, the compression process will target vectors. If that is not enough, it will target large field metadata, and finally it will target extracted text.PreviousNotion DB 2/2NextObsidianSetupExampleRetrieved informationCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. | Nuclia automatically indexes your unstructured data from any internal and external source, providing optimized search results and generative answers. It can handle video and audio transcription, image content extraction, and document parsing. ->: extracted embedded files, etc.) are provided as a token. You can download them with the /processing/download endpoint. Also at any level, if an attribute exceeds a certain size, it will be put in a downloadable file and will be replaced in the document by a file pointer. This will consist of {"file": {"uri": "JWT_TOKEN"}}. The rule is that if the size of the message is greater than 1000000 characters, the biggest parts will be moved to downloadable files. First, the compression process will target vectors. If that is not enough, it will target large field metadata, and finally it will target extracted text.PreviousNotion DB 2/2NextObsidianSetupExampleRetrieved informationCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
710 | Snowflake | ü¶úÔ∏èüîó Langchain | This notebooks goes over how to load documents from Snowflake | This notebooks goes over how to load documents from Snowflake ->: Snowflake | ü¶úÔ∏èüîó Langchain |
711 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersSnowflakeSnowflakeThis notebooks goes over how to load documents from Snowflakepip install | This notebooks goes over how to load documents from Snowflake | This notebooks goes over how to load documents from Snowflake ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersSnowflakeSnowflakeThis notebooks goes over how to load documents from Snowflakepip install |
712 | how to load documents from Snowflakepip install snowflake-connector-pythonimport settings as sfrom langchain.document_loaders import SnowflakeLoaderQUERY = "select text, survey_id from CLOUD_DATA_SOLUTIONS.HAPPY_OR_NOT.OPEN_FEEDBACK limit 10"snowflake_loader = SnowflakeLoader( query=QUERY, user=s.SNOWFLAKE_USER, password=s.SNOWFLAKE_PASS, account=s.SNOWFLAKE_ACCOUNT, warehouse=s.SNOWFLAKE_WAREHOUSE, role=s.SNOWFLAKE_ROLE, database=s.SNOWFLAKE_DATABASE, schema=s.SNOWFLAKE_SCHEMA,)snowflake_documents = snowflake_loader.load()print(snowflake_documents)from snowflakeLoader import SnowflakeLoaderimport settings as sQUERY = "select text, survey_id as source from CLOUD_DATA_SOLUTIONS.HAPPY_OR_NOT.OPEN_FEEDBACK limit 10"snowflake_loader = SnowflakeLoader( query=QUERY, user=s.SNOWFLAKE_USER, password=s.SNOWFLAKE_PASS, account=s.SNOWFLAKE_ACCOUNT, warehouse=s.SNOWFLAKE_WAREHOUSE, role=s.SNOWFLAKE_ROLE, database=s.SNOWFLAKE_DATABASE, schema=s.SNOWFLAKE_SCHEMA, metadata_columns=["source"],)snowflake_documents = snowflake_loader.load()print(snowflake_documents)PreviousSlackNextSource CodeCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This notebooks goes over how to load documents from Snowflake | This notebooks goes over how to load documents from Snowflake ->: how to load documents from Snowflakepip install snowflake-connector-pythonimport settings as sfrom langchain.document_loaders import SnowflakeLoaderQUERY = "select text, survey_id from CLOUD_DATA_SOLUTIONS.HAPPY_OR_NOT.OPEN_FEEDBACK limit 10"snowflake_loader = SnowflakeLoader( query=QUERY, user=s.SNOWFLAKE_USER, password=s.SNOWFLAKE_PASS, account=s.SNOWFLAKE_ACCOUNT, warehouse=s.SNOWFLAKE_WAREHOUSE, role=s.SNOWFLAKE_ROLE, database=s.SNOWFLAKE_DATABASE, schema=s.SNOWFLAKE_SCHEMA,)snowflake_documents = snowflake_loader.load()print(snowflake_documents)from snowflakeLoader import SnowflakeLoaderimport settings as sQUERY = "select text, survey_id as source from CLOUD_DATA_SOLUTIONS.HAPPY_OR_NOT.OPEN_FEEDBACK limit 10"snowflake_loader = SnowflakeLoader( query=QUERY, user=s.SNOWFLAKE_USER, password=s.SNOWFLAKE_PASS, account=s.SNOWFLAKE_ACCOUNT, warehouse=s.SNOWFLAKE_WAREHOUSE, role=s.SNOWFLAKE_ROLE, database=s.SNOWFLAKE_DATABASE, schema=s.SNOWFLAKE_SCHEMA, metadata_columns=["source"],)snowflake_documents = snowflake_loader.load()print(snowflake_documents)PreviousSlackNextSource CodeCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
713 | Airbyte Shopify | ü¶úÔ∏èüîó Langchain | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: Airbyte Shopify | ü¶úÔ∏èüîó Langchain |
714 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersAirbyte ShopifyOn this pageAirbyte ShopifyAirbyte is a data integration platform for ELT | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersAirbyte ShopifyOn this pageAirbyte ShopifyAirbyte is a data integration platform for ELT |
715 | is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases.This loader exposes the Shopify connector as a document loader, allowing you to load various Shopify objects as documents.Installation‚ÄãFirst, you need to install the airbyte-source-shopify python package.#!pip install airbyte-source-shopifyExample‚ÄãCheck out the Airbyte documentation page for details about how to configure the reader. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases.This loader exposes the Shopify connector as a document loader, allowing you to load various Shopify objects as documents.Installation‚ÄãFirst, you need to install the airbyte-source-shopify python package.#!pip install airbyte-source-shopifyExample‚ÄãCheck out the Airbyte documentation page for details about how to configure the reader. |
716 | The JSON schema the config object should adhere to can be found on Github: https://github.com/airbytehq/airbyte/blob/master/airbyte-integrations/connectors/source-shopify/source_shopify/spec.json.The general shape looks like this:{ "start_date": "<date from which to start retrieving records from in ISO format, e.g. 2020-10-20T00:00:00Z>", "shop": "<name of the shop you want to retrieve documents from>", "credentials": { "auth_method": "api_password", "api_password": "<your api password>" }}By default all fields are stored as metadata in the documents and the text is set to an empty string. Construct the text of the document by transforming the documents returned by the reader.from langchain.document_loaders.airbyte import AirbyteShopifyLoaderconfig = { # your shopify configuration}loader = AirbyteShopifyLoader(config=config, stream_name="orders") # check the documentation linked above for a list of all streamsNow you can load documents the usual waydocs = loader.load()As load returns a list, it will block until all documents are loaded. To have better control over this process, you can also you the lazy_load method which returns an iterator instead:docs_iterator = loader.lazy_load()Keep in mind that by default the page content is empty and the metadata object contains all the information from the record. To create documents in a different, pass in a record_handler function when creating the loader:from langchain.docstore.document import Documentdef handle_record(record, id): return Document(page_content=record.data["title"], metadata=record.data)loader = AirbyteShopifyLoader(config=config, record_handler=handle_record, stream_name="orders")docs = loader.load()Incremental loads‚ÄãSome streams allow incremental loading, this means the source keeps track of synced records and won't load them again. This is useful for sources that have a high volume of data and are updated frequently.To take advantage of this, store the last_state | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: The JSON schema the config object should adhere to can be found on Github: https://github.com/airbytehq/airbyte/blob/master/airbyte-integrations/connectors/source-shopify/source_shopify/spec.json.The general shape looks like this:{ "start_date": "<date from which to start retrieving records from in ISO format, e.g. 2020-10-20T00:00:00Z>", "shop": "<name of the shop you want to retrieve documents from>", "credentials": { "auth_method": "api_password", "api_password": "<your api password>" }}By default all fields are stored as metadata in the documents and the text is set to an empty string. Construct the text of the document by transforming the documents returned by the reader.from langchain.document_loaders.airbyte import AirbyteShopifyLoaderconfig = { # your shopify configuration}loader = AirbyteShopifyLoader(config=config, stream_name="orders") # check the documentation linked above for a list of all streamsNow you can load documents the usual waydocs = loader.load()As load returns a list, it will block until all documents are loaded. To have better control over this process, you can also you the lazy_load method which returns an iterator instead:docs_iterator = loader.lazy_load()Keep in mind that by default the page content is empty and the metadata object contains all the information from the record. To create documents in a different, pass in a record_handler function when creating the loader:from langchain.docstore.document import Documentdef handle_record(record, id): return Document(page_content=record.data["title"], metadata=record.data)loader = AirbyteShopifyLoader(config=config, record_handler=handle_record, stream_name="orders")docs = loader.load()Incremental loads‚ÄãSome streams allow incremental loading, this means the source keeps track of synced records and won't load them again. This is useful for sources that have a high volume of data and are updated frequently.To take advantage of this, store the last_state |
717 | take advantage of this, store the last_state property of the loader and pass it in when creating the loader again. This will ensure that only new records are loaded.last_state = loader.last_state # store safelyincremental_loader = AirbyteShopifyLoader(config=config, stream_name="orders", state=last_state)new_docs = incremental_loader.load()PreviousAirbyte SalesforceNextAirbyte StripeInstallationExampleIncremental loadsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: take advantage of this, store the last_state property of the loader and pass it in when creating the loader again. This will ensure that only new records are loaded.last_state = loader.last_state # store safelyincremental_loader = AirbyteShopifyLoader(config=config, stream_name="orders", state=last_state)new_docs = incremental_loader.load()PreviousAirbyte SalesforceNextAirbyte StripeInstallationExampleIncremental loadsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
718 | Psychic | ü¶úÔ∏èüîó Langchain | This notebook covers how to load documents from Psychic. See here for more details. | This notebook covers how to load documents from Psychic. See here for more details. ->: Psychic | ü¶úÔ∏èüîó Langchain |
719 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersPsychicOn this pagePsychicThis notebook covers how to load documents from Psychic. See here | This notebook covers how to load documents from Psychic. See here for more details. | This notebook covers how to load documents from Psychic. See here for more details. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersPsychicOn this pagePsychicThis notebook covers how to load documents from Psychic. See here |
720 | how to load documents from Psychic. See here for more details.Prerequisites‚ÄãFollow the Quick Start section in this documentLog into the Psychic dashboard and get your secret keyInstall the frontend react library into your web app and have a user authenticate a connection. The connection will be created using the connection id that you specify.Loading documents‚ÄãUse the PsychicLoader class to load in documents from a connection. Each connection has a connector id (corresponding to the SaaS app that was connected) and a connection id (which you passed in to the frontend library).# Uncomment this to install psychicapi if you don't already have it installedpoetry run pip -q install psychicapi [notice] A new release of pip is available: 23.0.1 -> 23.1.2 [notice] To update, run: pip install --upgrade pipfrom langchain.document_loaders import PsychicLoaderfrom psychicapi import ConnectorId# Create a document loader for google drive. We can also load from other connectors by setting the connector_id to the appropriate value e.g. ConnectorId.notion.value# This loader uses our test credentialsgoogle_drive_loader = PsychicLoader( api_key="7ddb61c1-8b6a-4d31-a58e-30d1c9ea480e", connector_id=ConnectorId.gdrive.value, connection_id="google-test",)documents = google_drive_loader.load()Converting the docs to embeddings‚ÄãWe can now convert these documents into embeddings and store them in a vector database like Chromafrom langchain.embeddings.openai import OpenAIEmbeddingsfrom langchain.vectorstores import Chromafrom langchain.text_splitter import CharacterTextSplitterfrom langchain.llms import OpenAIfrom langchain.chains import RetrievalQAWithSourcesChaintext_splitter = CharacterTextSplitter(chunk_size=1000, chunk_overlap=0)texts = text_splitter.split_documents(documents)embeddings = OpenAIEmbeddings()docsearch = Chroma.from_documents(texts, embeddings)chain = RetrievalQAWithSourcesChain.from_chain_type( OpenAI(temperature=0), chain_type="stuff", | This notebook covers how to load documents from Psychic. See here for more details. | This notebook covers how to load documents from Psychic. See here for more details. ->: how to load documents from Psychic. See here for more details.Prerequisites‚ÄãFollow the Quick Start section in this documentLog into the Psychic dashboard and get your secret keyInstall the frontend react library into your web app and have a user authenticate a connection. The connection will be created using the connection id that you specify.Loading documents‚ÄãUse the PsychicLoader class to load in documents from a connection. Each connection has a connector id (corresponding to the SaaS app that was connected) and a connection id (which you passed in to the frontend library).# Uncomment this to install psychicapi if you don't already have it installedpoetry run pip -q install psychicapi [notice] A new release of pip is available: 23.0.1 -> 23.1.2 [notice] To update, run: pip install --upgrade pipfrom langchain.document_loaders import PsychicLoaderfrom psychicapi import ConnectorId# Create a document loader for google drive. We can also load from other connectors by setting the connector_id to the appropriate value e.g. ConnectorId.notion.value# This loader uses our test credentialsgoogle_drive_loader = PsychicLoader( api_key="7ddb61c1-8b6a-4d31-a58e-30d1c9ea480e", connector_id=ConnectorId.gdrive.value, connection_id="google-test",)documents = google_drive_loader.load()Converting the docs to embeddings‚ÄãWe can now convert these documents into embeddings and store them in a vector database like Chromafrom langchain.embeddings.openai import OpenAIEmbeddingsfrom langchain.vectorstores import Chromafrom langchain.text_splitter import CharacterTextSplitterfrom langchain.llms import OpenAIfrom langchain.chains import RetrievalQAWithSourcesChaintext_splitter = CharacterTextSplitter(chunk_size=1000, chunk_overlap=0)texts = text_splitter.split_documents(documents)embeddings = OpenAIEmbeddings()docsearch = Chroma.from_documents(texts, embeddings)chain = RetrievalQAWithSourcesChain.from_chain_type( OpenAI(temperature=0), chain_type="stuff", |
721 | OpenAI(temperature=0), chain_type="stuff", retriever=docsearch.as_retriever())chain({"question": "what is psychic?"}, return_only_outputs=True)PreviousPolars DataFrameNextPubMedPrerequisitesLoading documentsConverting the docs to embeddingsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This notebook covers how to load documents from Psychic. See here for more details. | This notebook covers how to load documents from Psychic. See here for more details. ->: OpenAI(temperature=0), chain_type="stuff", retriever=docsearch.as_retriever())chain({"question": "what is psychic?"}, return_only_outputs=True)PreviousPolars DataFrameNextPubMedPrerequisitesLoading documentsConverting the docs to embeddingsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
722 | Airbyte Hubspot | ü¶úÔ∏èüîó Langchain | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: Airbyte Hubspot | ü¶úÔ∏èüîó Langchain |
723 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersAirbyte HubspotOn this pageAirbyte HubspotAirbyte is a data integration platform for ELT | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersAirbyte HubspotOn this pageAirbyte HubspotAirbyte is a data integration platform for ELT |
724 | is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases.This loader exposes the Hubspot connector as a document loader, allowing you to load various Hubspot objects as documents.Installation‚ÄãFirst, you need to install the airbyte-source-hubspot python package.#!pip install airbyte-source-hubspotExample‚ÄãCheck out the Airbyte documentation page for details about how to configure the reader. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases.This loader exposes the Hubspot connector as a document loader, allowing you to load various Hubspot objects as documents.Installation‚ÄãFirst, you need to install the airbyte-source-hubspot python package.#!pip install airbyte-source-hubspotExample‚ÄãCheck out the Airbyte documentation page for details about how to configure the reader. |
725 | The JSON schema the config object should adhere to can be found on Github: https://github.com/airbytehq/airbyte/blob/master/airbyte-integrations/connectors/source-hubspot/source_hubspot/spec.yaml.The general shape looks like this:{ "start_date": "<date from which to start retrieving records from in ISO format, e.g. 2020-10-20T00:00:00Z>", "credentials": { "credentials_title": "Private App Credentials", "access_token": "<access token of your private app>" }}By default all fields are stored as metadata in the documents and the text is set to an empty string. Construct the text of the document by transforming the documents returned by the reader.from langchain.document_loaders.airbyte import AirbyteHubspotLoaderconfig = { # your hubspot configuration}loader = AirbyteHubspotLoader(config=config, stream_name="products") # check the documentation linked above for a list of all streamsNow you can load documents the usual waydocs = loader.load()As load returns a list, it will block until all documents are loaded. To have better control over this process, you can also you the lazy_load method which returns an iterator instead:docs_iterator = loader.lazy_load()Keep in mind that by default the page content is empty and the metadata object contains all the information from the record. To process documents, create a class inheriting from the base loader and implement the _handle_records method yourself:from langchain.docstore.document import Documentdef handle_record(record, id): return Document(page_content=record.data["title"], metadata=record.data)loader = AirbyteHubspotLoader(config=config, record_handler=handle_record, stream_name="products")docs = loader.load()Incremental loads‚ÄãSome streams allow incremental loading, this means the source keeps track of synced records and won't load them again. This is useful for sources that have a high volume of data and are updated frequently.To take advantage of this, store the last_state property of the loader and | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: The JSON schema the config object should adhere to can be found on Github: https://github.com/airbytehq/airbyte/blob/master/airbyte-integrations/connectors/source-hubspot/source_hubspot/spec.yaml.The general shape looks like this:{ "start_date": "<date from which to start retrieving records from in ISO format, e.g. 2020-10-20T00:00:00Z>", "credentials": { "credentials_title": "Private App Credentials", "access_token": "<access token of your private app>" }}By default all fields are stored as metadata in the documents and the text is set to an empty string. Construct the text of the document by transforming the documents returned by the reader.from langchain.document_loaders.airbyte import AirbyteHubspotLoaderconfig = { # your hubspot configuration}loader = AirbyteHubspotLoader(config=config, stream_name="products") # check the documentation linked above for a list of all streamsNow you can load documents the usual waydocs = loader.load()As load returns a list, it will block until all documents are loaded. To have better control over this process, you can also you the lazy_load method which returns an iterator instead:docs_iterator = loader.lazy_load()Keep in mind that by default the page content is empty and the metadata object contains all the information from the record. To process documents, create a class inheriting from the base loader and implement the _handle_records method yourself:from langchain.docstore.document import Documentdef handle_record(record, id): return Document(page_content=record.data["title"], metadata=record.data)loader = AirbyteHubspotLoader(config=config, record_handler=handle_record, stream_name="products")docs = loader.load()Incremental loads‚ÄãSome streams allow incremental loading, this means the source keeps track of synced records and won't load them again. This is useful for sources that have a high volume of data and are updated frequently.To take advantage of this, store the last_state property of the loader and |
726 | store the last_state property of the loader and pass it in when creating the loader again. This will ensure that only new records are loaded.last_state = loader.last_state # store safelyincremental_loader = AirbyteHubspotLoader(config=config, stream_name="products", state=last_state)new_docs = incremental_loader.load()PreviousAirbyte GongNextAirbyte JSONInstallationExampleIncremental loadsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. | Airbyte is a data integration platform for ELT pipelines from APIs, databases & files to warehouses & lakes. It has the largest catalog of ELT connectors to data warehouses and databases. ->: store the last_state property of the loader and pass it in when creating the loader again. This will ensure that only new records are loaded.last_state = loader.last_state # store safelyincremental_loader = AirbyteHubspotLoader(config=config, stream_name="products", state=last_state)new_docs = incremental_loader.load()PreviousAirbyte GongNextAirbyte JSONInstallationExampleIncremental loadsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
727 | acreom | ü¶úÔ∏èüîó Langchain | acreom is a dev-first knowledge base with tasks running on local markdown files. | acreom is a dev-first knowledge base with tasks running on local markdown files. ->: acreom | ü¶úÔ∏èüîó Langchain |
728 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersacreomacreomacreom is a dev-first knowledge base with tasks running on local markdown | acreom is a dev-first knowledge base with tasks running on local markdown files. | acreom is a dev-first knowledge base with tasks running on local markdown files. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersacreomacreomacreom is a dev-first knowledge base with tasks running on local markdown |
729 | base with tasks running on local markdown files.Below is an example on how to load a local acreom vault into Langchain. As the local vault in acreom is a folder of plain text .md files, the loader requires the path to the directory. Vault files may contain some metadata which is stored as a YAML header. These values will be added to the document’s metadata if collect_metadata is set to true. from langchain.document_loaders import AcreomLoaderloader = AcreomLoader("<path-to-acreom-vault>", collect_metadata=False)docs = loader.load()PreviousDocument loadersNextAirbyte CDKCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | acreom is a dev-first knowledge base with tasks running on local markdown files. | acreom is a dev-first knowledge base with tasks running on local markdown files. ->: base with tasks running on local markdown files.Below is an example on how to load a local acreom vault into Langchain. As the local vault in acreom is a folder of plain text .md files, the loader requires the path to the directory. Vault files may contain some metadata which is stored as a YAML header. These values will be added to the document’s metadata if collect_metadata is set to true. from langchain.document_loaders import AcreomLoaderloader = AcreomLoader("<path-to-acreom-vault>", collect_metadata=False)docs = loader.load()PreviousDocument loadersNextAirbyte CDKCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
730 | RSS Feeds | ü¶úÔ∏èüîó Langchain | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: RSS Feeds | ü¶úÔ∏èüîó Langchain |
731 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersRSS FeedsRSS FeedsThis covers how to load HTML news articles from a list of RSS feed URLs into | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersRSS FeedsRSS FeedsThis covers how to load HTML news articles from a list of RSS feed URLs into |
732 | news articles from a list of RSS feed URLs into a document format that we can use downstream.pip install feedparser newspaper3k listparserfrom langchain.document_loaders import RSSFeedLoaderurls = ["https://news.ycombinator.com/rss"]Pass in urls to load them into Documentsloader = RSSFeedLoader(urls=urls)data = loader.load()print(len(data))print(data[0].page_content) (next Rich) 04 August 2023 Rich Hickey It is with a mixture of heartache and optimism that I announce today my (long planned) retirement from commercial software development, and my employment at Nubank. It’s been thrilling to see Clojure and Datomic successfully applied at scale. I look forward to continuing to lead ongoing work maintaining and enhancing Clojure with Alex, Stu, Fogus and many others, as an independent developer once again. We have many useful things planned for 1.12 and beyond. The community remains friendly, mature and productive, and is taking Clojure into many interesting new domains. I want to highlight and thank Nubank for their ongoing sponsorship of Alex, Fogus and the core team, as well as the Clojure community at large. Stu will continue to lead the development of Datomic at Nubank, where the Datomic team grows and thrives. I’m particularly excited to see where the new free availability of Datomic will lead. My time with Cognitect remains the highlight of my career. I have learned from absolutely everyone on our team, and am forever grateful to all for our interactions. There are too many people to thank here, but I must extend my sincerest appreciation and love to Stu and Justin for (repeatedly) taking a risk on me and my ideas, and for being the best of partners and friends, at all times fully embodying the notion of integrity. And of course to Alex Miller - who possesses in abundance many skills I lack, and without whose indomitable spirit, positivity and friendship Clojure would not have become what it did. I | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: news articles from a list of RSS feed URLs into a document format that we can use downstream.pip install feedparser newspaper3k listparserfrom langchain.document_loaders import RSSFeedLoaderurls = ["https://news.ycombinator.com/rss"]Pass in urls to load them into Documentsloader = RSSFeedLoader(urls=urls)data = loader.load()print(len(data))print(data[0].page_content) (next Rich) 04 August 2023 Rich Hickey It is with a mixture of heartache and optimism that I announce today my (long planned) retirement from commercial software development, and my employment at Nubank. It’s been thrilling to see Clojure and Datomic successfully applied at scale. I look forward to continuing to lead ongoing work maintaining and enhancing Clojure with Alex, Stu, Fogus and many others, as an independent developer once again. We have many useful things planned for 1.12 and beyond. The community remains friendly, mature and productive, and is taking Clojure into many interesting new domains. I want to highlight and thank Nubank for their ongoing sponsorship of Alex, Fogus and the core team, as well as the Clojure community at large. Stu will continue to lead the development of Datomic at Nubank, where the Datomic team grows and thrives. I’m particularly excited to see where the new free availability of Datomic will lead. My time with Cognitect remains the highlight of my career. I have learned from absolutely everyone on our team, and am forever grateful to all for our interactions. There are too many people to thank here, but I must extend my sincerest appreciation and love to Stu and Justin for (repeatedly) taking a risk on me and my ideas, and for being the best of partners and friends, at all times fully embodying the notion of integrity. And of course to Alex Miller - who possesses in abundance many skills I lack, and without whose indomitable spirit, positivity and friendship Clojure would not have become what it did. I |
733 | would not have become what it did. I have made many friends through Clojure and Cognitect, and I hope to nurture those friendships moving forward. Retirement returns me to the freedom and independence I had when originally developing Clojure. The journey continues!You can pass arguments to the NewsURLLoader which it uses to load articles.loader = RSSFeedLoader(urls=urls, nlp=True)data = loader.load()print(len(data)) Error fetching or processing https://twitter.com/andrewmccalip/status/1687405505604734978, exception: You must `parse()` an article first! Error processing entry https://twitter.com/andrewmccalip/status/1687405505604734978, exception: list index out of range 13data[0].metadata['keywords'] ['nubank', 'alex', 'stu', 'taking', 'team', 'remains', 'rich', 'clojure', 'thank', 'planned', 'datomic']data[0].metadata['summary'] 'It’s been thrilling to see Clojure and Datomic successfully applied at scale.\nI look forward to continuing to lead ongoing work maintaining and enhancing Clojure with Alex, Stu, Fogus and many others, as an independent developer once again.\nThe community remains friendly, mature and productive, and is taking Clojure into many interesting new domains.\nI want to highlight and thank Nubank for their ongoing sponsorship of Alex, Fogus and the core team, as well as the Clojure community at large.\nStu will continue to lead the development of Datomic at Nubank, where the Datomic team grows and thrives.'You can also use an OPML file such as a Feedly export. Pass in either a URL or the OPML contents.with open("example_data/sample_rss_feeds.opml", "r") as f: loader = RSSFeedLoader(opml=f.read())data = loader.load()print(len(data)) Error fetching http://www.engadget.com/rss-full.xml, exception: Error fetching http://www.engadget.com/rss-full.xml, exception: document declared as us-ascii, but parsed as utf-8 20data[0].page_content 'The electric vehicle startup | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: would not have become what it did. I have made many friends through Clojure and Cognitect, and I hope to nurture those friendships moving forward. Retirement returns me to the freedom and independence I had when originally developing Clojure. The journey continues!You can pass arguments to the NewsURLLoader which it uses to load articles.loader = RSSFeedLoader(urls=urls, nlp=True)data = loader.load()print(len(data)) Error fetching or processing https://twitter.com/andrewmccalip/status/1687405505604734978, exception: You must `parse()` an article first! Error processing entry https://twitter.com/andrewmccalip/status/1687405505604734978, exception: list index out of range 13data[0].metadata['keywords'] ['nubank', 'alex', 'stu', 'taking', 'team', 'remains', 'rich', 'clojure', 'thank', 'planned', 'datomic']data[0].metadata['summary'] 'It’s been thrilling to see Clojure and Datomic successfully applied at scale.\nI look forward to continuing to lead ongoing work maintaining and enhancing Clojure with Alex, Stu, Fogus and many others, as an independent developer once again.\nThe community remains friendly, mature and productive, and is taking Clojure into many interesting new domains.\nI want to highlight and thank Nubank for their ongoing sponsorship of Alex, Fogus and the core team, as well as the Clojure community at large.\nStu will continue to lead the development of Datomic at Nubank, where the Datomic team grows and thrives.'You can also use an OPML file such as a Feedly export. Pass in either a URL or the OPML contents.with open("example_data/sample_rss_feeds.opml", "r") as f: loader = RSSFeedLoader(opml=f.read())data = loader.load()print(len(data)) Error fetching http://www.engadget.com/rss-full.xml, exception: Error fetching http://www.engadget.com/rss-full.xml, exception: document declared as us-ascii, but parsed as utf-8 20data[0].page_content 'The electric vehicle startup |
734 | 'The electric vehicle startup Fisker made a splash in Huntington Beach last night, showing off a range of new EVs it plans to build alongside the Fisker Ocean, which is slowly beginning deliveries in Europe and the US. With shades of Lotus circa 2010, it seems there\'s something for most tastes, with a powerful four-door GT, a versatile pickup truck, and an affordable electric city car.\n\n"We want the world to know that we have big plans and intend to move into several different segments, redefining each with our unique blend of design, innovation, and sustainability," said CEO Henrik Fisker.\n\nStarting with the cheapest, the Fisker PEAR—a cutesy acronym for "Personal Electric Automotive Revolution"—is said to use 35 percent fewer parts than other small EVs. Although it\'s a smaller car, the PEAR seats six thanks to front and rear bench seats. Oh, and it has a frunk, which the company is calling the "froot," something that will satisfy some British English speakers like Ars\' friend and motoring journalist Jonny Smith.\n\nBut most exciting is the price—starting at $29,900 and scheduled for 2025. Fisker plans to contract with Foxconn to build the PEAR in Lordstown, Ohio, meaning it would be eligible for federal tax incentives.\n\nAdvertisement\n\nThe Fisker Alaska is the company\'s pickup truck, built on a modified version of the platform used by the Ocean. It has an extendable cargo bed, which can be as little as 4.5 feet (1,371 mm) or as much as 9.2 feet (2,804 mm) long. Fisker claims it will be both the lightest EV pickup on sale and the most sustainable pickup truck in the world. Range will be an estimated 230–240 miles (370–386 km).\n\nThis, too, is slated for 2025, and also at a relatively affordable price, starting at $45,400. Fisker hopes to build this car in North America as well, although it isn\'t saying where that might take place.\n\nFinally, there\'s the Ronin, a four-door GT that bears more than a passing resemblance to the Fisker | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: 'The electric vehicle startup Fisker made a splash in Huntington Beach last night, showing off a range of new EVs it plans to build alongside the Fisker Ocean, which is slowly beginning deliveries in Europe and the US. With shades of Lotus circa 2010, it seems there\'s something for most tastes, with a powerful four-door GT, a versatile pickup truck, and an affordable electric city car.\n\n"We want the world to know that we have big plans and intend to move into several different segments, redefining each with our unique blend of design, innovation, and sustainability," said CEO Henrik Fisker.\n\nStarting with the cheapest, the Fisker PEAR—a cutesy acronym for "Personal Electric Automotive Revolution"—is said to use 35 percent fewer parts than other small EVs. Although it\'s a smaller car, the PEAR seats six thanks to front and rear bench seats. Oh, and it has a frunk, which the company is calling the "froot," something that will satisfy some British English speakers like Ars\' friend and motoring journalist Jonny Smith.\n\nBut most exciting is the price—starting at $29,900 and scheduled for 2025. Fisker plans to contract with Foxconn to build the PEAR in Lordstown, Ohio, meaning it would be eligible for federal tax incentives.\n\nAdvertisement\n\nThe Fisker Alaska is the company\'s pickup truck, built on a modified version of the platform used by the Ocean. It has an extendable cargo bed, which can be as little as 4.5 feet (1,371 mm) or as much as 9.2 feet (2,804 mm) long. Fisker claims it will be both the lightest EV pickup on sale and the most sustainable pickup truck in the world. Range will be an estimated 230–240 miles (370–386 km).\n\nThis, too, is slated for 2025, and also at a relatively affordable price, starting at $45,400. Fisker hopes to build this car in North America as well, although it isn\'t saying where that might take place.\n\nFinally, there\'s the Ronin, a four-door GT that bears more than a passing resemblance to the Fisker |
735 | more than a passing resemblance to the Fisker Karma, Henrik Fisker\'s 2012 creation. There\'s no price for this one, but Fisker says its all-wheel drive powertrain will boast 1,000 hp (745 kW) and will hit 60 mph from a standing start in two seconds—just about as fast as modern tires will allow. Expect a massive battery in this one, as Fisker says it\'s targeting a 600-mile (956 km) range.\n\n"Innovation and sustainability, along with design, are our three brand values. By 2027, we intend to produce the world’s first climate-neutral vehicle, and as our customers reinvent their relationships with mobility, we want to be a leader in software-defined transportation," Fisker said.'PreviousrspaceNextRSTCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. | This covers how to load HTML news articles from a list of RSS feed URLs into a document format that we can use downstream. ->: more than a passing resemblance to the Fisker Karma, Henrik Fisker\'s 2012 creation. There\'s no price for this one, but Fisker says its all-wheel drive powertrain will boast 1,000 hp (745 kW) and will hit 60 mph from a standing start in two seconds—just about as fast as modern tires will allow. Expect a massive battery in this one, as Fisker says it\'s targeting a 600-mile (956 km) range.\n\n"Innovation and sustainability, along with design, are our three brand values. By 2027, we intend to produce the world’s first climate-neutral vehicle, and as our customers reinvent their relationships with mobility, we want to be a leader in software-defined transportation," Fisker said.'PreviousrspaceNextRSTCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
736 | Dropbox | 🦜�🔗 Langchain | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Dropbox | 🦜�🔗 Langchain |
737 | Skip to main content🦜�🔗 LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersDropboxOn this pageDropboxDropbox is a file hosting service that brings everything-traditional | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Skip to main content🦜�🔗 LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersDropboxOn this pageDropboxDropbox is a file hosting service that brings everything-traditional |
738 | service that brings everything-traditional files, cloud content, and web shortcuts together in one place.This notebook covers how to load documents from Dropbox. In addition to common files such as text and PDF files, it also supports Dropbox Paper files.Prerequisites​Create a Dropbox app.Give the app these scope permissions: files.metadata.read and files.content.read.Generate access token: https://www.dropbox.com/developers/apps/create.pip install dropbox (requires pip install unstructured for PDF filetype).Instructions​`DropboxLoader`` requires you to create a Dropbox App and generate an access token. This can be done from https://www.dropbox.com/developers/apps/create. You also need to have the Dropbox Python SDK installed (pip install dropbox).DropboxLoader can load data from a list of Dropbox file paths or a single Dropbox folder path. Both paths should be relative to the root directory of the Dropbox account linked to the access token.pip install dropbox Requirement already satisfied: dropbox in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (11.36.2) Requirement already satisfied: requests>=2.16.2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (2.31.0) Requirement already satisfied: six>=1.12.0 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (1.16.0) Requirement already satisfied: stone>=2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (3.3.1) Requirement already satisfied: charset-normalizer<4,>=2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (3.2.0) Requirement already satisfied: idna<4,>=2.5 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (3.4) | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: service that brings everything-traditional files, cloud content, and web shortcuts together in one place.This notebook covers how to load documents from Dropbox. In addition to common files such as text and PDF files, it also supports Dropbox Paper files.Prerequisites​Create a Dropbox app.Give the app these scope permissions: files.metadata.read and files.content.read.Generate access token: https://www.dropbox.com/developers/apps/create.pip install dropbox (requires pip install unstructured for PDF filetype).Instructions​`DropboxLoader`` requires you to create a Dropbox App and generate an access token. This can be done from https://www.dropbox.com/developers/apps/create. You also need to have the Dropbox Python SDK installed (pip install dropbox).DropboxLoader can load data from a list of Dropbox file paths or a single Dropbox folder path. Both paths should be relative to the root directory of the Dropbox account linked to the access token.pip install dropbox Requirement already satisfied: dropbox in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (11.36.2) Requirement already satisfied: requests>=2.16.2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (2.31.0) Requirement already satisfied: six>=1.12.0 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (1.16.0) Requirement already satisfied: stone>=2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from dropbox) (3.3.1) Requirement already satisfied: charset-normalizer<4,>=2 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (3.2.0) Requirement already satisfied: idna<4,>=2.5 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (3.4) |
739 | (from requests>=2.16.2->dropbox) (3.4) Requirement already satisfied: urllib3<3,>=1.21.1 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (2.0.4) Requirement already satisfied: certifi>=2017.4.17 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (2023.7.22) Requirement already satisfied: ply>=3.4 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from stone>=2->dropbox) (3.11) Note: you may need to restart the kernel to use updated packages.from langchain.document_loaders import DropboxLoader# Generate access token: https://www.dropbox.com/developers/apps/create.dropbox_access_token = "<DROPBOX_ACCESS_TOKEN>"# Dropbox root folderdropbox_folder_path = ""loader = DropboxLoader( dropbox_access_token=dropbox_access_token, dropbox_folder_path=dropbox_folder_path, recursive=False)documents = loader.load() File /JHSfLKn0.jpeg could not be decoded as text. Skipping. File /A REPORT ON WILES’ CAMBRIDGE LECTURES.pdf could not be decoded as text. Skipping.for document in documents: print(document) page_content='# � Getting Started with Dropbox Paper\nDropbox Paper is great for capturing ideas and gathering quick feedback from your team. You can use words, images, code, or media from other apps, or go ahead and connect your calendar and add to-dos for projects.\n\n*Explore and edit this doc to play with some of these features. This doc is all yours. No one will see your edits unless you share this doc.*\n\n\n# The basics\n\n**Selecting text** activates the formatting toolbar, where you can apply basic formatting, create lists, and add comments.\n\n[ ] Create to-do lists\n- Bulleted lists\n1. Numbered lists\n\n**Starting a new line** activates the insert toolbar, where you can add media from other apps, links to Dropbox files, photos, and | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: (from requests>=2.16.2->dropbox) (3.4) Requirement already satisfied: urllib3<3,>=1.21.1 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (2.0.4) Requirement already satisfied: certifi>=2017.4.17 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from requests>=2.16.2->dropbox) (2023.7.22) Requirement already satisfied: ply>=3.4 in /Users/rbarragan/.local/share/virtualenvs/langchain-kv0dsrF5/lib/python3.11/site-packages (from stone>=2->dropbox) (3.11) Note: you may need to restart the kernel to use updated packages.from langchain.document_loaders import DropboxLoader# Generate access token: https://www.dropbox.com/developers/apps/create.dropbox_access_token = "<DROPBOX_ACCESS_TOKEN>"# Dropbox root folderdropbox_folder_path = ""loader = DropboxLoader( dropbox_access_token=dropbox_access_token, dropbox_folder_path=dropbox_folder_path, recursive=False)documents = loader.load() File /JHSfLKn0.jpeg could not be decoded as text. Skipping. File /A REPORT ON WILES’ CAMBRIDGE LECTURES.pdf could not be decoded as text. Skipping.for document in documents: print(document) page_content='# � Getting Started with Dropbox Paper\nDropbox Paper is great for capturing ideas and gathering quick feedback from your team. You can use words, images, code, or media from other apps, or go ahead and connect your calendar and add to-dos for projects.\n\n*Explore and edit this doc to play with some of these features. This doc is all yours. No one will see your edits unless you share this doc.*\n\n\n# The basics\n\n**Selecting text** activates the formatting toolbar, where you can apply basic formatting, create lists, and add comments.\n\n[ ] Create to-do lists\n- Bulleted lists\n1. Numbered lists\n\n**Starting a new line** activates the insert toolbar, where you can add media from other apps, links to Dropbox files, photos, and |
740 | other apps, links to Dropbox files, photos, and more.\n\n\n\n\n\n**Add emojis** to your doc or comment by typing `**:**` ****and choosing a character. \n\n# ğŸ‘� ğŸ‘� ğŸ‘� ✅ â�Œ â�¤ï¸� â� 💡 📌\n\n\n# Images\n\n**Selecting images** activates the image toolbar, where you can align images left, center, right or expand them to full width.\n\n\n\n\nPaste images or gifs right next to each other and they\'ll organize automatically. Click on an image twice to start full-screen gallery view.\n\n\n\n\n\n\n\n\n\n\n \n\n\n# Form meets function\n\nYou and your team can create the way you want, with what you want. Dropbox Paper adapts to the way your team captures ideas.\n\n**Add media from apps** like YouTube and Vimeo, or add audio from Spotify and SoundCloud. Files from Google Drive and Dropbox update automatically. Start a new line and choose add media, or drop in a link to try it | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: other apps, links to Dropbox files, photos, and more.\n\n\n\n\n\n**Add emojis** to your doc or comment by typing `**:**` ****and choosing a character. \n\n# ğŸ‘� ğŸ‘� ğŸ‘� ✅ â�Œ â�¤ï¸� â� 💡 📌\n\n\n# Images\n\n**Selecting images** activates the image toolbar, where you can align images left, center, right or expand them to full width.\n\n\n\n\nPaste images or gifs right next to each other and they\'ll organize automatically. Click on an image twice to start full-screen gallery view.\n\n\n\n\n\n\n\n\n\n\n \n\n\n# Form meets function\n\nYou and your team can create the way you want, with what you want. Dropbox Paper adapts to the way your team captures ideas.\n\n**Add media from apps** like YouTube and Vimeo, or add audio from Spotify and SoundCloud. Files from Google Drive and Dropbox update automatically. Start a new line and choose add media, or drop in a link to try it |
741 | and choose add media, or drop in a link to try it out.\n\n\n\n\n\n\n## YouTube\nhttps://www.youtube.com/watch?v=fmsq1uKOa08&\n\n\n[https://youtu.be/fmsq1uKOa08](https://youtu.be/fmsq1uKOa08)\n\n\n\n## SoundCloud\nhttps://w.soundcloud.com/player/?url=https%3A%2F%2Fsoundcloud.com%2Ftycho%2Fspoon-inside-out-tycho-version&autoplay=false\n\n\n[https://soundcloud.com/tycho/spoon-inside-out-tycho-version](https://soundcloud.com/tycho/spoon-inside-out-tycho-version) \n\n\n\n## Dropbox files\nhttps://www.dropbox.com/s/bgi58tkovntch5e/Wireframe%20render.pdf?dl=0\n\n\n\n\n## Code\n\n**Write code** in Dropbox Paper with automatic language detection and syntax highlighting. Start a new line and type three backticks (```).\n\n\n public class HelloWorld { \n public static void main(String[] args) { \n System.out.println("Hello, World");\n }\n }\n\n\n\n## Tables\n\n**Create a table** with the menu that shows up on the right when you start a new line.\n\n| To insert a row or column, hover over a dividing line and click the + | â� |\n| ------------------------------------------------------------------------------------------------------- | ----- |\n| To delete, select rows/columns and click the trash can | â� â� |\n| To delete the entire table, click inside a cell, then click the dot in the top left corner of the table | â� â� â� |\n\n\n\n\n\n# Collaborate with people\n\n**Invite people to your doc** so they can view, comment, and edit. Invite anyone you’d like—team members, contractors, stakeholders—to give them access to your doc.\n\n\n\n\n**Make your docs discoverable to your | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: and choose add media, or drop in a link to try it out.\n\n\n\n\n\n\n## YouTube\nhttps://www.youtube.com/watch?v=fmsq1uKOa08&\n\n\n[https://youtu.be/fmsq1uKOa08](https://youtu.be/fmsq1uKOa08)\n\n\n\n## SoundCloud\nhttps://w.soundcloud.com/player/?url=https%3A%2F%2Fsoundcloud.com%2Ftycho%2Fspoon-inside-out-tycho-version&autoplay=false\n\n\n[https://soundcloud.com/tycho/spoon-inside-out-tycho-version](https://soundcloud.com/tycho/spoon-inside-out-tycho-version) \n\n\n\n## Dropbox files\nhttps://www.dropbox.com/s/bgi58tkovntch5e/Wireframe%20render.pdf?dl=0\n\n\n\n\n## Code\n\n**Write code** in Dropbox Paper with automatic language detection and syntax highlighting. Start a new line and type three backticks (```).\n\n\n public class HelloWorld { \n public static void main(String[] args) { \n System.out.println("Hello, World");\n }\n }\n\n\n\n## Tables\n\n**Create a table** with the menu that shows up on the right when you start a new line.\n\n| To insert a row or column, hover over a dividing line and click the + | â� |\n| ------------------------------------------------------------------------------------------------------- | ----- |\n| To delete, select rows/columns and click the trash can | â� â� |\n| To delete the entire table, click inside a cell, then click the dot in the top left corner of the table | â� â� â� |\n\n\n\n\n\n# Collaborate with people\n\n**Invite people to your doc** so they can view, comment, and edit. Invite anyone you’d like—team members, contractors, stakeholders—to give them access to your doc.\n\n\n\n\n**Make your docs discoverable to your |
742 | your docs discoverable to your team** by adding them to shared folders. Invite-only folders create more privacy.\n\n\n## Comments\n\n**Add comments** on a single character, an entire document, or any asset by highlighting it. **Add stickers** by clicking the 😄 in the message box.\n\n\n## To-dos\n\n**Bring someone’s attention to a comment or to-do** by typing **@** and their name or email address. Reference a doc or folder by typing **+** and its name.\n\n[ ] Mentioning someone on a to-do assigns it to them and sends an email [@Patricia J](http://#)\n[ ] Add a due date by clicking the calendar icon [@Jonathan C](http://#) [@Patricia J](http://#)\n[ ] You can also mention docs [+� Getting Started with Dropbox Paper](http://#)\n\n\n\n# Go mobile\n\nEdit, create, and share Paper docs on Android or iOS phones and tablets. Download the apps in the [App Store](https://itunes.apple.com/us/app/paper-by-dropbox/id1126623662) and [Google Play Store](https://play.google.com/store/apps/details?id=com.dropbox.paper).\n\n\n\n# Help\n\n**Visit the** [**help center**](https://www.dropbox.com/help/topics/paper) for more about Dropbox Paper.\n\n**For more tips,** click the **?** in the bottom right of the screen and choose **Paper guide**.\n\n**Give us feedback** by selecting “Feedback� from the **?** in the bottom right of the screen. We’d love to hear what you think. \n\n' metadata={'source': 'dropbox:///_ Getting Started with Dropbox Paper.paper', 'title': '_ Getting Started with Dropbox Paper.paper'} page_content='# 🥂 Toast to Droplets\n� **Rationale:** Reflection, especially writing, is the key to deep learning! Let’s take a few minutes to reflect on your first day at Dropbox individually, and then one lucky person will have the chance to share their toast.\n\n�� **How to fill out this template:**\n\n- Option 1: You can sign in and then click “Create doc� to make a copy of this template. Fill in the blanks!\n- Option 2: If you don’t know your | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: your docs discoverable to your team** by adding them to shared folders. Invite-only folders create more privacy.\n\n\n## Comments\n\n**Add comments** on a single character, an entire document, or any asset by highlighting it. **Add stickers** by clicking the 😄 in the message box.\n\n\n## To-dos\n\n**Bring someone’s attention to a comment or to-do** by typing **@** and their name or email address. Reference a doc or folder by typing **+** and its name.\n\n[ ] Mentioning someone on a to-do assigns it to them and sends an email [@Patricia J](http://#)\n[ ] Add a due date by clicking the calendar icon [@Jonathan C](http://#) [@Patricia J](http://#)\n[ ] You can also mention docs [+� Getting Started with Dropbox Paper](http://#)\n\n\n\n# Go mobile\n\nEdit, create, and share Paper docs on Android or iOS phones and tablets. Download the apps in the [App Store](https://itunes.apple.com/us/app/paper-by-dropbox/id1126623662) and [Google Play Store](https://play.google.com/store/apps/details?id=com.dropbox.paper).\n\n\n\n# Help\n\n**Visit the** [**help center**](https://www.dropbox.com/help/topics/paper) for more about Dropbox Paper.\n\n**For more tips,** click the **?** in the bottom right of the screen and choose **Paper guide**.\n\n**Give us feedback** by selecting “Feedback� from the **?** in the bottom right of the screen. We’d love to hear what you think. \n\n' metadata={'source': 'dropbox:///_ Getting Started with Dropbox Paper.paper', 'title': '_ Getting Started with Dropbox Paper.paper'} page_content='# 🥂 Toast to Droplets\n� **Rationale:** Reflection, especially writing, is the key to deep learning! Let’s take a few minutes to reflect on your first day at Dropbox individually, and then one lucky person will have the chance to share their toast.\n\n�� **How to fill out this template:**\n\n- Option 1: You can sign in and then click “Create doc� to make a copy of this template. Fill in the blanks!\n- Option 2: If you don’t know your |
743 | the blanks!\n- Option 2: If you don’t know your personal Dropbox login quickly, you can copy and paste this text into another word processing tool and start typing! \n\n\n\n## To my Droplet class:\n\nI feel so happy and excited to be making a toast to our newest Droplet class at Dropbox Basecamp.\n\nAt the beginning of our first day, I felt a bit underwhelmed with all information, and now, at the end of our first day at Dropbox, I feel I know enough for me to ramp up, but still a lot to learn**.**\n\nI can’t wait to explore every drl, but especially drl/(App Center)/benefits/allowance. I heard it’s so informative!\n\nDesigning an enlightened way of working is important, and to me, it means **a lot since I love what I do and I can help people around the globe**.\n\nI am excited to work with my team and flex my **technical and social** skills in my role as a **Software Engineer**.\n\nAs a Droplet, I pledge to:\n\n\n1. Be worthy of trust by **working always with values and integrity**.\n\n\n1. Keep my customers first by **caring about their happiness and the value that we provide as a company**.\n\n\n1. Own it, keep it simple, and especially make work human by **providing value to people****.**\n\nCongrats, Droplets!\n\n' metadata={'source': 'dropbox:///_ Toast to Droplets.paper', 'title': '_ Toast to Droplets.paper'} page_content='APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, Number 1, July 1994, Pages 15-38\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n4 9 9 1\n\nK. RUBIN AND A. SILVERBERG\n\nl u J\n\nAbstract. In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat’s Last Theorem. This report for nonexperts dis- cusses the mathematics involved in Wiles’ lectures, including the necessary background and the mathematical history.\n\n1\n\n] T N . h t a m\n\nIntroduction\n\nOn June 23, 1993, Andrew Wiles wrote on a blackboard, | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: the blanks!\n- Option 2: If you don’t know your personal Dropbox login quickly, you can copy and paste this text into another word processing tool and start typing! \n\n\n\n## To my Droplet class:\n\nI feel so happy and excited to be making a toast to our newest Droplet class at Dropbox Basecamp.\n\nAt the beginning of our first day, I felt a bit underwhelmed with all information, and now, at the end of our first day at Dropbox, I feel I know enough for me to ramp up, but still a lot to learn**.**\n\nI can’t wait to explore every drl, but especially drl/(App Center)/benefits/allowance. I heard it’s so informative!\n\nDesigning an enlightened way of working is important, and to me, it means **a lot since I love what I do and I can help people around the globe**.\n\nI am excited to work with my team and flex my **technical and social** skills in my role as a **Software Engineer**.\n\nAs a Droplet, I pledge to:\n\n\n1. Be worthy of trust by **working always with values and integrity**.\n\n\n1. Keep my customers first by **caring about their happiness and the value that we provide as a company**.\n\n\n1. Own it, keep it simple, and especially make work human by **providing value to people****.**\n\nCongrats, Droplets!\n\n' metadata={'source': 'dropbox:///_ Toast to Droplets.paper', 'title': '_ Toast to Droplets.paper'} page_content='APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, Number 1, July 1994, Pages 15-38\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n4 9 9 1\n\nK. RUBIN AND A. SILVERBERG\n\nl u J\n\nAbstract. In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat’s Last Theorem. This report for nonexperts dis- cusses the mathematics involved in Wiles’ lectures, including the necessary background and the mathematical history.\n\n1\n\n] T N . h t a m\n\nIntroduction\n\nOn June 23, 1993, Andrew Wiles wrote on a blackboard, |
744 | 23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermat’s Last Theorem. His announce- ment came at the end of his series of three talks entitled “Modular forms, elliptic curves, and Galois representations� at the week-long workshop on “p-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motives�.\n\n[\n\n1 v 0 2 2 7 0 4 9 / h t a m : v i X r a\n\nIn the margin of his copy of the works of Diophantus, next to a problem on\n\nPythagorean triples, Pierre de Fermat (1601–1665) wrote:\n\nCubum autem in duos cubos, aut quadratoquadratum in duos quadrato- quadratos, et generaliter nullam in in�nitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.\n\n(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)\n\nWe restate Fermat’s conjecture as follows.\n\nFermat’s Last Theorem. If n > 2, then an +bn = cn has no solutions in nonzero integers a, b, and c.\n\nA proof by Fermat has never been found, and the problem has remained open, inspiring many generations of mathematicians. Much of modern number theory has been built on attempts to prove Fermat’s Last Theorem. For details on the\n\nReceived by the editors November 29, 1993. 1991 Mathematics Subject Classi�cation. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for �nancial support.\n\nc(cid:13)1994 American Mathematical Society 0273-0979/94 $1.00 + $.25 per page\n\n1\n\n2\n\nK. RUBIN AND A. SILVERBERG\n\nhistory of | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermat’s Last Theorem. His announce- ment came at the end of his series of three talks entitled “Modular forms, elliptic curves, and Galois representations� at the week-long workshop on “p-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motives�.\n\n[\n\n1 v 0 2 2 7 0 4 9 / h t a m : v i X r a\n\nIn the margin of his copy of the works of Diophantus, next to a problem on\n\nPythagorean triples, Pierre de Fermat (1601–1665) wrote:\n\nCubum autem in duos cubos, aut quadratoquadratum in duos quadrato- quadratos, et generaliter nullam in in�nitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.\n\n(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)\n\nWe restate Fermat’s conjecture as follows.\n\nFermat’s Last Theorem. If n > 2, then an +bn = cn has no solutions in nonzero integers a, b, and c.\n\nA proof by Fermat has never been found, and the problem has remained open, inspiring many generations of mathematicians. Much of modern number theory has been built on attempts to prove Fermat’s Last Theorem. For details on the\n\nReceived by the editors November 29, 1993. 1991 Mathematics Subject Classi�cation. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for �nancial support.\n\nc(cid:13)1994 American Mathematical Society 0273-0979/94 $1.00 + $.25 per page\n\n1\n\n2\n\nK. RUBIN AND A. SILVERBERG\n\nhistory of |
745 | RUBIN AND A. SILVERBERG\n\nhistory of Fermat’s Last Theorem (last because it is the last of Fermat’s questions to be answered) see [5], [6], and [26].\n\nWhat Andrew Wiles announced in Cambridge was that he could prove “many� elliptic curves are modular, sufficiently many to imply Fermat’s Last Theorem. In this paper we will explain Wiles’ work on elliptic curves and its connection with 1 we introduce elliptic curves and modularity, and Fermat’s Last Theorem. give the connection between Fermat’s Last Theorem and the Taniyama-Shimura Conjecture on the modularity of elliptic curves. In 2 we describe how Wiles re- duces the proof of the Taniyama-Shimura Conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In 4 we show § how the Semistable Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in 5 we describe Wiles’ method of attack on this conjecture. In order to make this survey as acces- sible as possible to nonspecialists, the more technical details are postponed as long as possible, some of them to the appendices.\n\nIn\n\n§\n\n§\n\n3 and §\n\n§\n\nMuch of this report is based on Wiles’ lectures in Cambridge. The authors apol- ogize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not prop- erly acknowledge.\n\nThe ideas Wiles introduced in his Cambridge lectures will have an important influence on research in number theory. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In early December 1993, shortly before this paper went to press, Wiles announced that “the �nal calculation of a precise upper bound for the Selmer group in the semistable | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: RUBIN AND A. SILVERBERG\n\nhistory of Fermat’s Last Theorem (last because it is the last of Fermat’s questions to be answered) see [5], [6], and [26].\n\nWhat Andrew Wiles announced in Cambridge was that he could prove “many� elliptic curves are modular, sufficiently many to imply Fermat’s Last Theorem. In this paper we will explain Wiles’ work on elliptic curves and its connection with 1 we introduce elliptic curves and modularity, and Fermat’s Last Theorem. give the connection between Fermat’s Last Theorem and the Taniyama-Shimura Conjecture on the modularity of elliptic curves. In 2 we describe how Wiles re- duces the proof of the Taniyama-Shimura Conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In 4 we show § how the Semistable Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in 5 we describe Wiles’ method of attack on this conjecture. In order to make this survey as acces- sible as possible to nonspecialists, the more technical details are postponed as long as possible, some of them to the appendices.\n\nIn\n\n§\n\n§\n\n3 and §\n\n§\n\nMuch of this report is based on Wiles’ lectures in Cambridge. The authors apol- ogize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not prop- erly acknowledge.\n\nThe ideas Wiles introduced in his Cambridge lectures will have an important influence on research in number theory. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In early December 1993, shortly before this paper went to press, Wiles announced that “the �nal calculation of a precise upper bound for the Selmer group in the semistable |
746 | bound for the Selmer group in the semistable case� (see 5.4 below) “is not yet § complete as it stands,� but that he believes he will be able to �nish it in the near future using the ideas explained in his Cambridge lectures. While Wiles’ proof of Theorem 5.3 below and Fermat’s Last Theorem depends on the calculation he referred to in his December announcement, Theorem 5.4 and Corollary 5.5 do not. Wiles’ work provides for the �rst time in�nitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit in�nite family).\n\n5.3 and\n\n§\n\n§\n\nNotation. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp, respectively. If F is a �eld, then ¯F denotes an algebraic closure of F .\n\n1. Connection between Fermat’s Last Theorem and elliptic curves\n\n1.1. Fermat’s Last Theorem follows from modularity of elliptic curves. Suppose Fermat’s Last Theorem were false. Then there would exist nonzero integers a, b, c, and n > 2 such that an + bn = cn. It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4) and that a and b are relatively prime. Write down the cubic curve:\n\ny2 = x(x + an)(x\n\nbn).\n\n(1)\n\n−\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n3\n\n1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [27] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular. But the results announced by Wiles imply the following.\n\nIn\n\n1.3 we will see that such curves are elliptic curves, and in\n\n§\n\n§\n\nTheorem 1.1 (Wiles). If A and B are distinct, nonzero, relatively prime integers, and AB(A\n\nB) is divisible by 16, then the elliptic curve\n\n−\n\ny2 = x(x + A)(x + B)\n\nis modular.\n\nbn with | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: bound for the Selmer group in the semistable case� (see 5.4 below) “is not yet § complete as it stands,� but that he believes he will be able to �nish it in the near future using the ideas explained in his Cambridge lectures. While Wiles’ proof of Theorem 5.3 below and Fermat’s Last Theorem depends on the calculation he referred to in his December announcement, Theorem 5.4 and Corollary 5.5 do not. Wiles’ work provides for the �rst time in�nitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit in�nite family).\n\n5.3 and\n\n§\n\n§\n\nNotation. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp, respectively. If F is a �eld, then ¯F denotes an algebraic closure of F .\n\n1. Connection between Fermat’s Last Theorem and elliptic curves\n\n1.1. Fermat’s Last Theorem follows from modularity of elliptic curves. Suppose Fermat’s Last Theorem were false. Then there would exist nonzero integers a, b, c, and n > 2 such that an + bn = cn. It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4) and that a and b are relatively prime. Write down the cubic curve:\n\ny2 = x(x + an)(x\n\nbn).\n\n(1)\n\n−\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n3\n\n1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [27] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular. But the results announced by Wiles imply the following.\n\nIn\n\n1.3 we will see that such curves are elliptic curves, and in\n\n§\n\n§\n\nTheorem 1.1 (Wiles). If A and B are distinct, nonzero, relatively prime integers, and AB(A\n\nB) is divisible by 16, then the elliptic curve\n\n−\n\ny2 = x(x + A)(x + B)\n\nis modular.\n\nbn with |
747 | = x(x + A)(x + B)\n\nis modular.\n\nbn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satis�ed since n 5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribet’s result together imply Fermat’s Last Theorem!\n\nTaking A = an and B =\n\n−\n\n≥\n\n1.2. History. The story of the connection between Fermat’s Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927–1958) posed problems which may be viewed as a weaker version of the following conjecture (see [38]).\n\nTaniyama-Shimura Conjecture. Every elliptic curve over Q is modular.\n\nThe conjecture in the present form was made by Goro Shimura around 1962–64 and has become better understood due to work of Shimura [33–37] and of Andr´e Weil [42] (see also [7]). The Taniyama-Shimura Conjecture is one of the major conjectures in number theory.\n\nBeginning in the late 1960s [15–18], Yves Hellegouarch connected Fermat equa- tions an + bn = cn with elliptic curves of the form (1) and used results about Fer- mat’s Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermat’s Last Theorem could not be mod- ular [11]. Shortly thereafter Ribet [27] proved this, following ideas of Jean-Pierre Serre [32] (see [24] for a survey). In other words, “Taniyama-Shimura Conjecture\n\nFermat’s Last Theorem�. Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermat’s Last Theorem.\n\n⇒\n\n1.3. Elliptic curves.\n\nDe�nition. An elliptic curve over Q is a nonsingular curve de�ned by an equation of the form\n\ny2 + a1xy + a3y = x3 + a2x2 + a4x + a6\n\n(2)\n\nwhere the coefficients ai are integers. The solution ( on the elliptic | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: = x(x + A)(x + B)\n\nis modular.\n\nbn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satis�ed since n 5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribet’s result together imply Fermat’s Last Theorem!\n\nTaking A = an and B =\n\n−\n\n≥\n\n1.2. History. The story of the connection between Fermat’s Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927–1958) posed problems which may be viewed as a weaker version of the following conjecture (see [38]).\n\nTaniyama-Shimura Conjecture. Every elliptic curve over Q is modular.\n\nThe conjecture in the present form was made by Goro Shimura around 1962–64 and has become better understood due to work of Shimura [33–37] and of Andr´e Weil [42] (see also [7]). The Taniyama-Shimura Conjecture is one of the major conjectures in number theory.\n\nBeginning in the late 1960s [15–18], Yves Hellegouarch connected Fermat equa- tions an + bn = cn with elliptic curves of the form (1) and used results about Fer- mat’s Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermat’s Last Theorem could not be mod- ular [11]. Shortly thereafter Ribet [27] proved this, following ideas of Jean-Pierre Serre [32] (see [24] for a survey). In other words, “Taniyama-Shimura Conjecture\n\nFermat’s Last Theorem�. Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermat’s Last Theorem.\n\n⇒\n\n1.3. Elliptic curves.\n\nDe�nition. An elliptic curve over Q is a nonsingular curve de�ned by an equation of the form\n\ny2 + a1xy + a3y = x3 + a2x2 + a4x + a6\n\n(2)\n\nwhere the coefficients ai are integers. The solution ( on the elliptic |
748 | ai are integers. The solution ( on the elliptic curve.\n\n, �\n\n) will be viewed as a point\n\n�\n\n4\n\nK. RUBIN AND A. SILVERBERG\n\nRemarks. (i) A singular point on a curve f (x, y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points.\n\n(ii) Two elliptic curves over Q are isomorphic if one can be obtained from the other by changing coordinates x = A2x′ + B, y = A3y′ + Cx′ + D, with A, B, C, D\n\nQ and dividing through by A6.\n\n∈ (iii) Every elliptic curve over Q is isomorphic to one of the form\n\ny2 = x3 + a2x2 + a4x + a6\n\nwith integers ai. A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots.\n\nExample. The equation y2 = x(x + 32)(x\n\n42) de�nes an elliptic curve over Q.\n\n−\n\n1.4. Modularity. Let H denote the complex upper half plane C : Im(z) > 0 } where Im(z) is the imaginary part of z. If N is a positive integer, de�ne a group of matrices\n\nz\n\n{\n\n∈\n\na b c d\n\nSL2(Z) : c is divisible by N\n\n.\n\nΓ0(N ) =\n\n∈\n\n(z) = az+b The group Γ0(N ) acts on H by linear fractional transformations cz+d . (cid:9) (cid:1) The quotient space H/Γ0(N ) is a (noncompact) Riemann surface. It can be com- pleted to a compact Riemann surface, denoted X0(N ), by adjoining a �nite set of points called cusps. The cusps are the �nitely many equivalence classes of Q �} under the action of Γ0(N ) (see Chapter 1 of [35]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface.\n\na b c d\n\n(cid:8)(cid:0)\n\n(cid:1)\n\n(cid:0)\n\ni\n\n∪{\n\nDe�nition. An elliptic curve E is modular if, for some integer N , there is a holo- morphic map from X0(N ) onto E.\n\nExample. It can be shown that there is a (holomorphic) isomorphism from X0(15) onto the elliptic curve y2 = x(x + 32)(x\n\n42).\n\n−\n\nRemark . There are many equivalent de�nitions of modularity (see II.4.D of [24] and appendix of [22]). In | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: ai are integers. The solution ( on the elliptic curve.\n\n, �\n\n) will be viewed as a point\n\n�\n\n4\n\nK. RUBIN AND A. SILVERBERG\n\nRemarks. (i) A singular point on a curve f (x, y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points.\n\n(ii) Two elliptic curves over Q are isomorphic if one can be obtained from the other by changing coordinates x = A2x′ + B, y = A3y′ + Cx′ + D, with A, B, C, D\n\nQ and dividing through by A6.\n\n∈ (iii) Every elliptic curve over Q is isomorphic to one of the form\n\ny2 = x3 + a2x2 + a4x + a6\n\nwith integers ai. A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots.\n\nExample. The equation y2 = x(x + 32)(x\n\n42) de�nes an elliptic curve over Q.\n\n−\n\n1.4. Modularity. Let H denote the complex upper half plane C : Im(z) > 0 } where Im(z) is the imaginary part of z. If N is a positive integer, de�ne a group of matrices\n\nz\n\n{\n\n∈\n\na b c d\n\nSL2(Z) : c is divisible by N\n\n.\n\nΓ0(N ) =\n\n∈\n\n(z) = az+b The group Γ0(N ) acts on H by linear fractional transformations cz+d . (cid:9) (cid:1) The quotient space H/Γ0(N ) is a (noncompact) Riemann surface. It can be com- pleted to a compact Riemann surface, denoted X0(N ), by adjoining a �nite set of points called cusps. The cusps are the �nitely many equivalence classes of Q �} under the action of Γ0(N ) (see Chapter 1 of [35]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface.\n\na b c d\n\n(cid:8)(cid:0)\n\n(cid:1)\n\n(cid:0)\n\ni\n\n∪{\n\nDe�nition. An elliptic curve E is modular if, for some integer N , there is a holo- morphic map from X0(N ) onto E.\n\nExample. It can be shown that there is a (holomorphic) isomorphism from X0(15) onto the elliptic curve y2 = x(x + 32)(x\n\n42).\n\n−\n\nRemark . There are many equivalent de�nitions of modularity (see II.4.D of [24] and appendix of [22]). In |
749 | (see II.4.D of [24] and appendix of [22]). In some cases the equivalence is a deep result. For Wiles’ 1.7 proof of Fermat’s Last Theorem it suffices to use only the de�nition given in below.\n\n§\n\n§\n\n1.5. Semistability.\n\nDe�nition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singu- lar point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime.\n\nExample. The elliptic curve y2 = x(x + 32)(x isomorphic to y2 + xy + y = x3 + x2 x(x + 42)(x\n\n42) is semistable because it is − 10, but the elliptic curve y2 =\n\n10x\n\n−\n\n−\n\n32) is not semistable (it is not semistable at 2).\n\n−\n\n2 we explain how Wiles shows that his main result on Galois representations (Theorem 5.3) implies the following part of the Taniyama-Shimura Conjecture.\n\nBeginning in\n\n§\n\nSemistable Taniyama-Shimura Conjecture. Every semistable elliptic curve over Q is modular.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n5\n\nProposition 1.2. The Semistable Taniyama-Shimura Conjecture implies Theorem 1.1.\n\nProof. If A and B are distinct, nonzero, relatively prime integers, write EA,B for the elliptic curve de�ned by y2 = x(x + A)(x + B). Since EA,B and E−A,−B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA,B is modular if and only if E−A,−B is modular. If further AB(A B) is divisible by 16, then either EA,B or E−A,−B is semistable (this is easy to check directly; see for example I.1 of [24]). The Semistable Taniyama-Shimura Conjecture now implies that both § EA,B and E−A,−B are modular, and thus implies Theorem 1.1.\n\n−\n\nRemark . In 1.1 we saw that Theorem 1.1 and Ribet’s Theorem together imply Fermat’s Last Theorem. Therefore, the Semistable Taniyama-Shimura Conjecture implies Fermat’s Last Theorem.\n\n§\n\n1.6. Modular forms. In this paper we | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: (see II.4.D of [24] and appendix of [22]). In some cases the equivalence is a deep result. For Wiles’ 1.7 proof of Fermat’s Last Theorem it suffices to use only the de�nition given in below.\n\n§\n\n§\n\n1.5. Semistability.\n\nDe�nition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singu- lar point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime.\n\nExample. The elliptic curve y2 = x(x + 32)(x isomorphic to y2 + xy + y = x3 + x2 x(x + 42)(x\n\n42) is semistable because it is − 10, but the elliptic curve y2 =\n\n10x\n\n−\n\n−\n\n32) is not semistable (it is not semistable at 2).\n\n−\n\n2 we explain how Wiles shows that his main result on Galois representations (Theorem 5.3) implies the following part of the Taniyama-Shimura Conjecture.\n\nBeginning in\n\n§\n\nSemistable Taniyama-Shimura Conjecture. Every semistable elliptic curve over Q is modular.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n5\n\nProposition 1.2. The Semistable Taniyama-Shimura Conjecture implies Theorem 1.1.\n\nProof. If A and B are distinct, nonzero, relatively prime integers, write EA,B for the elliptic curve de�ned by y2 = x(x + A)(x + B). Since EA,B and E−A,−B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA,B is modular if and only if E−A,−B is modular. If further AB(A B) is divisible by 16, then either EA,B or E−A,−B is semistable (this is easy to check directly; see for example I.1 of [24]). The Semistable Taniyama-Shimura Conjecture now implies that both § EA,B and E−A,−B are modular, and thus implies Theorem 1.1.\n\n−\n\nRemark . In 1.1 we saw that Theorem 1.1 and Ribet’s Theorem together imply Fermat’s Last Theorem. Therefore, the Semistable Taniyama-Shimura Conjecture implies Fermat’s Last Theorem.\n\n§\n\n1.6. Modular forms. In this paper we |
750 | Modular forms. In this paper we will work with a de�nition of modularity which uses modular forms.\n\nDe�nition. If N is a positive integer, a modular form f of weight k for Γ0(N ) is C which satis�es a holomorphic function f : H\n\n→\n\nf (γ(z)) = (cz + d)kf (z)\n\na b c d\n\nH,\n\n(3)\n\nΓ0(N ) and z\n\nfor every γ =\n\n∈\n\n∈\n\n(cid:1)\n\n(cid:0)\n\nand is holomorphic at the cusps (see Chapter 2 of [35]).\n\n1 1 0 1\n\nΓ0(N )), so � n=0 ane2πinz, with complex numbers an and it has a Fourier expansion f (z) = (cid:1) . We say f is a cusp form if it with n vanishes at all the cusps; in particular for a cusp form the coefficient a0 (the value at i\n\nA modular form f satis�es f (z) = f (z + 1) (apply (3) to\n\n∈\n\n(cid:0)\n\n0 because f is holomorphic at the cusp i\n\n≥\n\n�\n\nP\n\n) is zero. Call a cusp form normalized if a1 = 1.\n\n� For �xed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (�nite-dimensional) vector space of cusp forms of weight\n\nfor integers m two for Γ0(N ) (see Chapter 3 of [35]). If f (z) =\n\n≥\n\n� n=1 ane2πinz, then\n\nP danm/d2\n\n�\n\ne2πinz\n\n(4)\n\nTmf (z) =\n\nn=1 X\n\n(d,N )=1 d|(n,m)\n\n(cid:0) X\n\n(cid:1)\n\nwhere (a, b) denotes the greatest common divisor of a and b and a b means that a divides b. The Hecke algebra T (N ) is the ring generated over Z by these operators.\n\n|\n\nDe�nition. In this paper an eigenform will mean a normalized cusp form of weight two for some Γ0(N ) which is an eigenfunction for all the Hecke operators.\n\n� n=1 ane2πinz is an eigenform, then Tmf = amf for all m.\n\nBy (4), if f (z) =\n\nP\n\n6\n\nK. RUBIN AND A. SILVERBERG\n\n1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the �nite �eld with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at in�nity). We now give a second de�nition of modularity for an elliptic | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Modular forms. In this paper we will work with a de�nition of modularity which uses modular forms.\n\nDe�nition. If N is a positive integer, a modular form f of weight k for Γ0(N ) is C which satis�es a holomorphic function f : H\n\n→\n\nf (γ(z)) = (cz + d)kf (z)\n\na b c d\n\nH,\n\n(3)\n\nΓ0(N ) and z\n\nfor every γ =\n\n∈\n\n∈\n\n(cid:1)\n\n(cid:0)\n\nand is holomorphic at the cusps (see Chapter 2 of [35]).\n\n1 1 0 1\n\nΓ0(N )), so � n=0 ane2πinz, with complex numbers an and it has a Fourier expansion f (z) = (cid:1) . We say f is a cusp form if it with n vanishes at all the cusps; in particular for a cusp form the coefficient a0 (the value at i\n\nA modular form f satis�es f (z) = f (z + 1) (apply (3) to\n\n∈\n\n(cid:0)\n\n0 because f is holomorphic at the cusp i\n\n≥\n\n�\n\nP\n\n) is zero. Call a cusp form normalized if a1 = 1.\n\n� For �xed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (�nite-dimensional) vector space of cusp forms of weight\n\nfor integers m two for Γ0(N ) (see Chapter 3 of [35]). If f (z) =\n\n≥\n\n� n=1 ane2πinz, then\n\nP danm/d2\n\n�\n\ne2πinz\n\n(4)\n\nTmf (z) =\n\nn=1 X\n\n(d,N )=1 d|(n,m)\n\n(cid:0) X\n\n(cid:1)\n\nwhere (a, b) denotes the greatest common divisor of a and b and a b means that a divides b. The Hecke algebra T (N ) is the ring generated over Z by these operators.\n\n|\n\nDe�nition. In this paper an eigenform will mean a normalized cusp form of weight two for some Γ0(N ) which is an eigenfunction for all the Hecke operators.\n\n� n=1 ane2πinz is an eigenform, then Tmf = amf for all m.\n\nBy (4), if f (z) =\n\nP\n\n6\n\nK. RUBIN AND A. SILVERBERG\n\n1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the �nite �eld with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at in�nity). We now give a second de�nition of modularity for an elliptic |
751 | second de�nition of modularity for an elliptic curve.\n\nDe�nition. An elliptic curve E over Q is modular if there exists an eigenform\n\n� n=1 ane2πinz such that for all but �nitely many primes q,\n\n#(E(Fq)).\n\n(5) P\n\naq = q + 1\n\n− 2. An overview\n\nThe flow chart shows how Fermat’s Last Theorem would follow if one knew the Semistable Modular Lifting Conjecture (Conjecture 2.1) for the primes 3 and 5. 1 we discussed the upper arrow, i.e., the implication “Semistable Taniyama- In § Fermat’s Last Theorem�. In this section we will discuss the Shimura Conjecture other implications in the flow chart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4).\n\n⇒\n\nFermat’s Last Theorem\n\n✻\n\nSemistable Taniyama-Shimura Conjecture\n\n✻\n\n(cid:0)\n\n� �\n\n(cid:0)\n\nSemistable Taniyama-Shimura for ¯�E,3 irreducible\n\nSemistable Modular Lifting for p = 5\n\n✻\n\n(cid:0) (cid:0)\n\n�\n\n�\n\nSemistable Modular Lifting for p = 3\n\nLanglands-Tunnell Theorem\n\nSemistable Modular Lifting Conjecture\n\nFermat’s Last Theorem .\n\n⇒\n\nRemark . By the Modular Lifting Conjecture we will mean the Semistable Modular Lifting Conjecture with the hypothesis of semistability removed. The arguments of this section can also be used to show that the Modular Lifting Conjecture for p = 3 and 5, together with the Langlands-Tunnell Theorem, imply the full Taniyama- Shimura Conjecture.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n7\n\n2.1. Semistable Modular Lifting. Let ¯Q denote the algebraic closure of Q in C, and let GQ be the Galois group Gal( ¯Q/Q). If p is a prime, write\n\nF× p\n\n¯εp : GQ\n\n→\n\nfor the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below, see [39]. If E is an elliptic curve over Q and F is a sub�eld of the complex numbers, there is a natural | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: second de�nition of modularity for an elliptic curve.\n\nDe�nition. An elliptic curve E over Q is modular if there exists an eigenform\n\n� n=1 ane2πinz such that for all but �nitely many primes q,\n\n#(E(Fq)).\n\n(5) P\n\naq = q + 1\n\n− 2. An overview\n\nThe flow chart shows how Fermat’s Last Theorem would follow if one knew the Semistable Modular Lifting Conjecture (Conjecture 2.1) for the primes 3 and 5. 1 we discussed the upper arrow, i.e., the implication “Semistable Taniyama- In § Fermat’s Last Theorem�. In this section we will discuss the Shimura Conjecture other implications in the flow chart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4).\n\n⇒\n\nFermat’s Last Theorem\n\n✻\n\nSemistable Taniyama-Shimura Conjecture\n\n✻\n\n(cid:0)\n\n� �\n\n(cid:0)\n\nSemistable Taniyama-Shimura for ¯�E,3 irreducible\n\nSemistable Modular Lifting for p = 5\n\n✻\n\n(cid:0) (cid:0)\n\n�\n\n�\n\nSemistable Modular Lifting for p = 3\n\nLanglands-Tunnell Theorem\n\nSemistable Modular Lifting Conjecture\n\nFermat’s Last Theorem .\n\n⇒\n\nRemark . By the Modular Lifting Conjecture we will mean the Semistable Modular Lifting Conjecture with the hypothesis of semistability removed. The arguments of this section can also be used to show that the Modular Lifting Conjecture for p = 3 and 5, together with the Langlands-Tunnell Theorem, imply the full Taniyama- Shimura Conjecture.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n7\n\n2.1. Semistable Modular Lifting. Let ¯Q denote the algebraic closure of Q in C, and let GQ be the Galois group Gal( ¯Q/Q). If p is a prime, write\n\nF× p\n\n¯εp : GQ\n\n→\n\nfor the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below, see [39]. If E is an elliptic curve over Q and F is a sub�eld of the complex numbers, there is a natural |
752 | of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at in�nity as the identity element. Denote this group E(F ). If p is a prime, write E[p] for the subgroup of points in E( ¯Q) of order dividing p. Then E[p] ∼= F2 p. The action of GQ on E[p] gives a continuous representation\n\nGL2(Fp)\n\n¯�E,p : GQ\n\n→\n\n(de�ned up to isomorphism) such that\n\n(6)\n\ndet(¯�E,p) = ¯εp\n\nand for all but �nitely many primes q,\n\n#(E(Fq))\n\n(7)\n\ntrace(¯�E,p(Frobq))\n\nq + 1\n\n(mod p).\n\n≡ (See Appendix A for the de�nition of the Frobenius elements Frobq ∈ to each prime number q.)\n\n−\n\nGQ attached\n\n� n=1 ane2πinz is an eigenform, let\n\nOf denote the ring of integers of the number �eld Q(a2, a3, . . . ). (Recall that our eigenforms are normalized so that a1 = 1.)\n\nIf f (z) =\n\nP\n\nThe following conjecture is in the spirit of a conjecture of Mazur (see Conjectures\n\n3.2 and 4.2).\n\nConjecture 2.1 (Semistable Modular Lifting Conjecture). Suppose p is an odd prime and E is a semistable elliptic curve over Q satisfying\n\n(a) ¯�E,p is irreducible, (b) there are an eigenform f (z) =\n\n� n=1 ane2πinz and a prime ideal λ of\n\nOf\n\nsuch that p\n\nλ and for all but �nitely many primes q,\n\n∈\n\nP\n\n#(E(Fq))\n\naq ≡\n\nq + 1\n\n(mod λ).\n\n−\n\nThen E is modular.\n\nThe Semistable Modular Lifting Conjecture is a priori weaker than the Semi- stable Taniyama-Shimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3, the existence of such a form follows from the theorem below of Tunnell [41] and Langlands [20]. Wiles then gets around condition (a) by a clever argument (described below) which, when ¯�E,3 is not irreducible, allows him to use p = 5 instead.\n\n8\n\nK. RUBIN AND A. SILVERBERG\n\n2.2. Langlands-Tunnell Theorem. In order to state the | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at in�nity as the identity element. Denote this group E(F ). If p is a prime, write E[p] for the subgroup of points in E( ¯Q) of order dividing p. Then E[p] ∼= F2 p. The action of GQ on E[p] gives a continuous representation\n\nGL2(Fp)\n\n¯�E,p : GQ\n\n→\n\n(de�ned up to isomorphism) such that\n\n(6)\n\ndet(¯�E,p) = ¯εp\n\nand for all but �nitely many primes q,\n\n#(E(Fq))\n\n(7)\n\ntrace(¯�E,p(Frobq))\n\nq + 1\n\n(mod p).\n\n≡ (See Appendix A for the de�nition of the Frobenius elements Frobq ∈ to each prime number q.)\n\n−\n\nGQ attached\n\n� n=1 ane2πinz is an eigenform, let\n\nOf denote the ring of integers of the number �eld Q(a2, a3, . . . ). (Recall that our eigenforms are normalized so that a1 = 1.)\n\nIf f (z) =\n\nP\n\nThe following conjecture is in the spirit of a conjecture of Mazur (see Conjectures\n\n3.2 and 4.2).\n\nConjecture 2.1 (Semistable Modular Lifting Conjecture). Suppose p is an odd prime and E is a semistable elliptic curve over Q satisfying\n\n(a) ¯�E,p is irreducible, (b) there are an eigenform f (z) =\n\n� n=1 ane2πinz and a prime ideal λ of\n\nOf\n\nsuch that p\n\nλ and for all but �nitely many primes q,\n\n∈\n\nP\n\n#(E(Fq))\n\naq ≡\n\nq + 1\n\n(mod λ).\n\n−\n\nThen E is modular.\n\nThe Semistable Modular Lifting Conjecture is a priori weaker than the Semi- stable Taniyama-Shimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3, the existence of such a form follows from the theorem below of Tunnell [41] and Langlands [20]. Wiles then gets around condition (a) by a clever argument (described below) which, when ¯�E,3 is not irreducible, allows him to use p = 5 instead.\n\n8\n\nK. RUBIN AND A. SILVERBERG\n\n2.2. Langlands-Tunnell Theorem. In order to state the |
753 | Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight-one modular forms for a subgroup of Γ0(N ). Let\n\na b c d\n\nSL2(Z) : c\n\n0 (mod N ), a\n\nd\n\n1 (mod N )\n\n.\n\nΓ1(N ) =\n\n∈\n\n≡\n\n≡\n\n≡\n\n(cid:1)\n\n(cid:9)\n\n(cid:8)(cid:0)\n\nReplacing Γ0(N ) by Γ1(N ) in 1.6, one can de�ne the notion of cusp forms on § Γ1(N ). See Chapter 3 of [35] for the de�nitions of the Hecke operators on the space of weight-one cusp forms for Γ1(N ).\n\nTheorem 2.2 (Langlands-Tunnell). Suppose � : GQ GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the sym- metric group on four elements ), τ is complex conjugation, and det(�(τ )) = 1. � n=1 bne2πinz for some Γ1(N ), which is an Then there is a weight-one cusp form eigenfunction for all the corresponding Hecke operators, such that for all but �nitely many primes q,\n\n→\n\n−\n\nP\n\n(8)\n\nbq = trace(�(Frobq)).\n\nThe theorem as stated by Langlands [20] and by Tunnell [41] produces an auto- morphic representation rather than a cusp form. Using the fact that det(�(τ )) = 1, standard techniques (see for example [12]) show that this automorphic repre-\n\n− sentation corresponds to a weight-one cusp form as in Theorem 2.2.\n\n2.3. Semistable Modular Lifting\n\nSemistable Taniyama-Shimura.\n\n⇒\n\nProposition 2.3. Suppose the Semistable Modular Lifting Conjecture is true for p = 3, E is a semistable elliptic curve, and ¯�E,3 is irreducible. Then E is modular.\n\nProof. It suffices to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satis�ed with the given curve E, for p = 3. There is a faithful representation\n\nGL2(Z[√\n\nGL2(C)\n\nψ : GL2(F3) ֒\n\n2])\n\n−\n\n⊂\n\n→\n\nGL2(F3),\n\nsuch that for every g\n\n∈ trace(ψ(g))\n\n(mod(1 + √\n\n(9)\n\ntrace(g)\n\n2))\n\n≡\n\n−\n\nand\n\n(10)\n\ndet(ψ(g))\n\ndet(g)\n\n(mod 3).\n\n≡\n\nExplicitly, ψ can be de�ned on | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight-one modular forms for a subgroup of Γ0(N ). Let\n\na b c d\n\nSL2(Z) : c\n\n0 (mod N ), a\n\nd\n\n1 (mod N )\n\n.\n\nΓ1(N ) =\n\n∈\n\n≡\n\n≡\n\n≡\n\n(cid:1)\n\n(cid:9)\n\n(cid:8)(cid:0)\n\nReplacing Γ0(N ) by Γ1(N ) in 1.6, one can de�ne the notion of cusp forms on § Γ1(N ). See Chapter 3 of [35] for the de�nitions of the Hecke operators on the space of weight-one cusp forms for Γ1(N ).\n\nTheorem 2.2 (Langlands-Tunnell). Suppose � : GQ GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the sym- metric group on four elements ), τ is complex conjugation, and det(�(τ )) = 1. � n=1 bne2πinz for some Γ1(N ), which is an Then there is a weight-one cusp form eigenfunction for all the corresponding Hecke operators, such that for all but �nitely many primes q,\n\n→\n\n−\n\nP\n\n(8)\n\nbq = trace(�(Frobq)).\n\nThe theorem as stated by Langlands [20] and by Tunnell [41] produces an auto- morphic representation rather than a cusp form. Using the fact that det(�(τ )) = 1, standard techniques (see for example [12]) show that this automorphic repre-\n\n− sentation corresponds to a weight-one cusp form as in Theorem 2.2.\n\n2.3. Semistable Modular Lifting\n\nSemistable Taniyama-Shimura.\n\n⇒\n\nProposition 2.3. Suppose the Semistable Modular Lifting Conjecture is true for p = 3, E is a semistable elliptic curve, and ¯�E,3 is irreducible. Then E is modular.\n\nProof. It suffices to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satis�ed with the given curve E, for p = 3. There is a faithful representation\n\nGL2(Z[√\n\nGL2(C)\n\nψ : GL2(F3) ֒\n\n2])\n\n−\n\n⊂\n\n→\n\nGL2(F3),\n\nsuch that for every g\n\n∈ trace(ψ(g))\n\n(mod(1 + √\n\n(9)\n\ntrace(g)\n\n2))\n\n≡\n\n−\n\nand\n\n(10)\n\ndet(ψ(g))\n\ndet(g)\n\n(mod 3).\n\n≡\n\nExplicitly, ψ can be de�ned on |
754 | 3).\n\n≡\n\nExplicitly, ψ can be de�ned on generators of GL2(F3) by\n\n√\n\n1 1 1 0\n\n1 1 1 0\n\n1 1\n\n1 1\n\n2 1 1 0\n\n.\n\nψ\n\n=\n\nand ψ\n\n=\n\n− −\n\n− −\n\n−\n\n−\n\n(cid:19)\n\n(cid:18)(cid:18)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\n(cid:18)(cid:18) ¯�E,3. If τ is complex conjugation, then it follows from (6) and (10) that 1. The image of ψ in PGL2(C) is a subgroup of PGL2(F3) ∼= S4.\n\n(cid:19)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\nLet � = ψ ◦ det(�(τ )) = Using that ¯�E,3 is irreducible, one can show that � is irreducible.\n\n−\n\n� n=1 bne2πinz be a weight-one cusp form for some Γ1(N ) obtained by applying the Langlands-Tunnell\n\nLet p be a prime of ¯Q containing 1 + √\n\n2. Let g(z) =\n\n−\n\nP\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n9\n\nTheorem (Theorem 2.2) to �. It follows from (6) and (10) that N is divisible by 3. The function\n\n0 if d 1 if d 1 if d\n\n0 (mod 3), 1 (mod 3), 2 (mod 3)\n\n�\n\n≡ ≡ ≡\n\nχ(d)e2πinz where χ(d) =\n\nE(z) = 1 + 6\n\n\uf8f1 \uf8f2\n\nn=1 X\n\nXd|n\n\n−\n\n� n=1 cne2πinz is a weight-one modular form for Γ1(3). The product g(z)E(z) = It is now is a weight-two cusp form for Γ0(N ) with cn ≡ bn possible to �nd an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satis�es (b) of the Semistable Modular Lifting Conjecture with p = 3 and with λ = p\n\n\uf8f3\n\nbn (mod p) for all n. P n=1 ane2πinz on Γ0(N ) such that an ≡ ∩ Of .\n\n�\n\nP\n\nProposition 2.4 (Wiles). Suppose the Semistable Modular Lifting Conjecture is true for p = 3 and 5, E is a semistable elliptic curve over Q, and ¯�E,3 is reducible. Then E is modular.\n\nProof. The elliptic curves over Q for which both ¯�E,3 and ¯�E,5 are reducible are all known to be modular (see Appendix B.1). Thus we can suppose ¯�E,5 is irreducible. It suffices to produce an eigenform as in (b) of the Semistable Modular Lifting Conjecture, but this time | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 3).\n\n≡\n\nExplicitly, ψ can be de�ned on generators of GL2(F3) by\n\n√\n\n1 1 1 0\n\n1 1 1 0\n\n1 1\n\n1 1\n\n2 1 1 0\n\n.\n\nψ\n\n=\n\nand ψ\n\n=\n\n− −\n\n− −\n\n−\n\n−\n\n(cid:19)\n\n(cid:18)(cid:18)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\n(cid:18)(cid:18) ¯�E,3. If τ is complex conjugation, then it follows from (6) and (10) that 1. The image of ψ in PGL2(C) is a subgroup of PGL2(F3) ∼= S4.\n\n(cid:19)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\nLet � = ψ ◦ det(�(τ )) = Using that ¯�E,3 is irreducible, one can show that � is irreducible.\n\n−\n\n� n=1 bne2πinz be a weight-one cusp form for some Γ1(N ) obtained by applying the Langlands-Tunnell\n\nLet p be a prime of ¯Q containing 1 + √\n\n2. Let g(z) =\n\n−\n\nP\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n9\n\nTheorem (Theorem 2.2) to �. It follows from (6) and (10) that N is divisible by 3. The function\n\n0 if d 1 if d 1 if d\n\n0 (mod 3), 1 (mod 3), 2 (mod 3)\n\n�\n\n≡ ≡ ≡\n\nχ(d)e2πinz where χ(d) =\n\nE(z) = 1 + 6\n\n\uf8f1 \uf8f2\n\nn=1 X\n\nXd|n\n\n−\n\n� n=1 cne2πinz is a weight-one modular form for Γ1(3). The product g(z)E(z) = It is now is a weight-two cusp form for Γ0(N ) with cn ≡ bn possible to �nd an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satis�es (b) of the Semistable Modular Lifting Conjecture with p = 3 and with λ = p\n\n\uf8f3\n\nbn (mod p) for all n. P n=1 ane2πinz on Γ0(N ) such that an ≡ ∩ Of .\n\n�\n\nP\n\nProposition 2.4 (Wiles). Suppose the Semistable Modular Lifting Conjecture is true for p = 3 and 5, E is a semistable elliptic curve over Q, and ¯�E,3 is reducible. Then E is modular.\n\nProof. The elliptic curves over Q for which both ¯�E,3 and ¯�E,5 are reducible are all known to be modular (see Appendix B.1). Thus we can suppose ¯�E,5 is irreducible. It suffices to produce an eigenform as in (b) of the Semistable Modular Lifting Conjecture, but this time |
755 | Modular Lifting Conjecture, but this time there is no analogue of the Langlands-Tunnell Theorem to help. Wiles uses the Hilbert Irreducibility Theorem, applied to a parameter space of elliptic curves, to produce another semistable elliptic curve E′ over Q satisfying\n\n(i) ¯�E′,5 is isomorphic to ¯�E,5, and (ii) ¯�E′,3 is irreducible.\n\n(In fact there will be in�nitely many such E′; see Appendix B.2.) Now by Proposi- � n=1 ane2πinz be a corresponding eigenform. tion 2.3, E′ is modular. Let f (z) = Then for all but �nitely many primes q, P\n\n#(E′(Fq)) trace(¯�E,5(Frobq))\n\naq = q + 1\n\ntrace(¯�E′,5(Frobq)) #(E(Fq)) q + 1\n\n−\n\n≡ ≡\n\n(mod 5)\n\n≡\n\n−\n\nby (7). Thus the form f satis�es hypothesis (b) of the Semistable Modular Lifting Conjecture, and we conclude that E is modular.\n\nTaken together, Propositions 2.3 and 2.4 show that the Semistable Modular Lifting Conjecture for p = 3 and 5 implies the Semistable Taniyama-Shimura Con- jecture.\n\n3. Galois representations\n\nThe next step is to translate the Semistable Modular Lifting Conjecture into a conjecture (Conjecture 3.2) about the modularity of liftings of Galois repre- sentations. Throughout this paper, if A is a topological ring, a representation GL2(A) will mean a continuous homomorphism and [�] will denote the � : GQ isomorphism class of �. If p is a prime, let\n\n→\n\nZ× p\n\nεp : GQ\n\n→\n\nbe the character giving the action of GQ on p-power roots of unity.\n\n10\n\nK. RUBIN AND A. SILVERBERG\n\n3.1. The p-adic representation attached to an elliptic curve. Suppose E is an elliptic curve over Q and p is a prime number. For every positive integer n, write E[pn] for the subgroup in E( ¯Q) of points of order dividing pn and Tp(E) for the inverse limit of the E[pn] with respect to multiplication by p. For every n, E[pn] ∼= (Z/pnZ)2, and so Tp(E) ∼= Z2 p. The action of GQ induces a representation\n\nGL2(Zp)\n\n�E,p : GQ\n\n→\n\nsuch that | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Modular Lifting Conjecture, but this time there is no analogue of the Langlands-Tunnell Theorem to help. Wiles uses the Hilbert Irreducibility Theorem, applied to a parameter space of elliptic curves, to produce another semistable elliptic curve E′ over Q satisfying\n\n(i) ¯�E′,5 is isomorphic to ¯�E,5, and (ii) ¯�E′,3 is irreducible.\n\n(In fact there will be in�nitely many such E′; see Appendix B.2.) Now by Proposi- � n=1 ane2πinz be a corresponding eigenform. tion 2.3, E′ is modular. Let f (z) = Then for all but �nitely many primes q, P\n\n#(E′(Fq)) trace(¯�E,5(Frobq))\n\naq = q + 1\n\ntrace(¯�E′,5(Frobq)) #(E(Fq)) q + 1\n\n−\n\n≡ ≡\n\n(mod 5)\n\n≡\n\n−\n\nby (7). Thus the form f satis�es hypothesis (b) of the Semistable Modular Lifting Conjecture, and we conclude that E is modular.\n\nTaken together, Propositions 2.3 and 2.4 show that the Semistable Modular Lifting Conjecture for p = 3 and 5 implies the Semistable Taniyama-Shimura Con- jecture.\n\n3. Galois representations\n\nThe next step is to translate the Semistable Modular Lifting Conjecture into a conjecture (Conjecture 3.2) about the modularity of liftings of Galois repre- sentations. Throughout this paper, if A is a topological ring, a representation GL2(A) will mean a continuous homomorphism and [�] will denote the � : GQ isomorphism class of �. If p is a prime, let\n\n→\n\nZ× p\n\nεp : GQ\n\n→\n\nbe the character giving the action of GQ on p-power roots of unity.\n\n10\n\nK. RUBIN AND A. SILVERBERG\n\n3.1. The p-adic representation attached to an elliptic curve. Suppose E is an elliptic curve over Q and p is a prime number. For every positive integer n, write E[pn] for the subgroup in E( ¯Q) of points of order dividing pn and Tp(E) for the inverse limit of the E[pn] with respect to multiplication by p. For every n, E[pn] ∼= (Z/pnZ)2, and so Tp(E) ∼= Z2 p. The action of GQ induces a representation\n\nGL2(Zp)\n\n�E,p : GQ\n\n→\n\nsuch that |
756 | : GQ\n\n→\n\nsuch that det(�E,p) = εp and for all but �nitely many primes q,\n\n#(E(Fq)).\n\n(11)\n\ntrace(�E,p(Frobq)) = q + 1\n\n−\n\nComposing �E,p with the reduction map from Zp to Fp gives ¯�E,p of\n\n2.1. §\n\n3.2. Modular representations. If f is an eigenform and λ is a prime ideal of Of at λ. Of , let\n\nOf,λ denote the completion of\n\nDe�nition. If A is a ring, a representation � : GQ if there are an eigenform f (z) = homomorphism ι :\n\nGL2(A) is called modular � n=1 ane2πinz, a ring A′ containing A, and a\n\n→\n\nA′ such that for all but �nitely many primes q,\n\nOf →\n\nP\n\ntrace(�(Frobq)) = ι(aq).\n\n� n=1 ane2πinz and a prime ideal λ of\n\nExamples. (i) Given an eigenform f (z) = Of , Eichler and Shimura (see\n\n7.6 of [35]) constructed a representation\n\n§\n\nP\n\n�f,λ : GQ\n\nGL2(\n\nOf,λ)\n\n→\n\nZ = pZ) and for all but �nitely many primes q,\n\nsuch that det(�f,λ) = εp (where λ\n\n∩\n\n(12)\n\ntrace(�f,λ(Frobq)) = aq.\n\nThus �f,λ is modular with ι taken to be the inclusion of\n\nOf in\n\nOf,λ.\n\n(ii) Suppose p is a prime and E is an elliptic curve over Q. If E is modular, then �E,p and ¯�E,p are modular by (11), (7), and (5). Conversely, if �E,p is modular, then it follows from (11) that E is modular. This proves the following.\n\nTheorem 3.1. Suppose E is an elliptic curve over Q. Then\n\nE is modular\n\n�E,p is modular for every p\n\n�E,p is modular for one p.\n\n⇔\n\n⇔\n\nRemark . In this language, the Semistable Modular Lifting Conjecture says that if p is an odd prime, E is a semistable elliptic curve over Q, and ¯�E,p is modular and irreducible, then �E,p is modular.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n11\n\n3.3. Liftings of Galois representations. Fix a prime p and a �nite �eld k of characteristic p. Recall that ¯k denotes an algebraic closure of k.\n\nGiven a map φ : A\n\nB, the induced map from GL2(A) to GL2(B) will also be\n\n→\n\ndenoted φ. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: : GQ\n\n→\n\nsuch that det(�E,p) = εp and for all but �nitely many primes q,\n\n#(E(Fq)).\n\n(11)\n\ntrace(�E,p(Frobq)) = q + 1\n\n−\n\nComposing �E,p with the reduction map from Zp to Fp gives ¯�E,p of\n\n2.1. §\n\n3.2. Modular representations. If f is an eigenform and λ is a prime ideal of Of at λ. Of , let\n\nOf,λ denote the completion of\n\nDe�nition. If A is a ring, a representation � : GQ if there are an eigenform f (z) = homomorphism ι :\n\nGL2(A) is called modular � n=1 ane2πinz, a ring A′ containing A, and a\n\n→\n\nA′ such that for all but �nitely many primes q,\n\nOf →\n\nP\n\ntrace(�(Frobq)) = ι(aq).\n\n� n=1 ane2πinz and a prime ideal λ of\n\nExamples. (i) Given an eigenform f (z) = Of , Eichler and Shimura (see\n\n7.6 of [35]) constructed a representation\n\n§\n\nP\n\n�f,λ : GQ\n\nGL2(\n\nOf,λ)\n\n→\n\nZ = pZ) and for all but �nitely many primes q,\n\nsuch that det(�f,λ) = εp (where λ\n\n∩\n\n(12)\n\ntrace(�f,λ(Frobq)) = aq.\n\nThus �f,λ is modular with ι taken to be the inclusion of\n\nOf in\n\nOf,λ.\n\n(ii) Suppose p is a prime and E is an elliptic curve over Q. If E is modular, then �E,p and ¯�E,p are modular by (11), (7), and (5). Conversely, if �E,p is modular, then it follows from (11) that E is modular. This proves the following.\n\nTheorem 3.1. Suppose E is an elliptic curve over Q. Then\n\nE is modular\n\n�E,p is modular for every p\n\n�E,p is modular for one p.\n\n⇔\n\n⇔\n\nRemark . In this language, the Semistable Modular Lifting Conjecture says that if p is an odd prime, E is a semistable elliptic curve over Q, and ¯�E,p is modular and irreducible, then �E,p is modular.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n11\n\n3.3. Liftings of Galois representations. Fix a prime p and a �nite �eld k of characteristic p. Recall that ¯k denotes an algebraic closure of k.\n\nGiven a map φ : A\n\nB, the induced map from GL2(A) to GL2(B) will also be\n\n→\n\ndenoted φ. |
757 | to GL2(B) will also be\n\n→\n\ndenoted φ. If � : GQ A′ for the composition of � with the inclusion of GL2(A) in GL2(A′).\n\nGL2(A) is a representation and A′ is a ring containing A, we write\n\n→\n\n�\n\n⊗\n\nDe�nition. If ¯� : GQ � : GQ Zp-algebra and there exists a homomorphism ι : A\n\nGL2(k) is a representation, we say that a representation GL2(A) is a lifting of ¯� (to A) if A is a complete noetherian local\n\n→\n\n→\n\n¯k such that the diagram\n\n→ GL2(A)\n\n✟✟✯\n\n[�]\n\n✟✟\n\nι � GL2(¯k)\n\n✲\n\nGQ\n\n[ ¯� ⊗ ¯k]\n\n¯k].\n\ncommutes, in the sense that [ι\n\n�] = [¯�\n\n\n\n⊗\n\nExamples. (i) If E is an elliptic curve then �E,p is a lifting of ¯�E,p.\n\n(ii) If E is an elliptic curve, p is a prime, and hypotheses (a) and (b) of Conjecture\n\n2.1 hold with an eigenform f and prime ideal λ, then �f,λ is a lifting of ¯�E,p.\n\n3.4. Deformation data. We will be interested not in all liftings of a given ¯�, but rather in those satisfying various restrictions. See Appendix A for the de�nition of GQ associated to primes q. We say that a representation � the inertia groups Iq ⊂ of GQ is unrami�ed at a prime q if �(Iq) = 1. If Σ is a set of primes, we say � is unrami�ed outside of Σ if � is unrami�ed at every q / ∈\n\nΣ.\n\nDe�nition. By deformation data we mean a pair\n\n= (Σ, t)\n\nD where Σ is a �nite set of primes and t is one of the words ordinary or flat.\n\nZ×\n\nA× be the composition of the\n\nIf A is a Zp-algebra, let εA : GQ\n\np →\n\n→\n\ncyclotomic character εp with the structure map.\n\nDe�nition. Given deformation data type- outside of Σ, and � is t at p (where t\n\nGL2(A) is if A is a complete noetherian local Zp-algebra, det(�) = εA, � is unrami�ed\n\n, a representation � : GQ\n\nD\n\n→\n\nD\n\nordinary, flat }\n\n; see Appendix C).\n\n∈ {\n\nDe�nition. A representation ¯� : GQ eigenform f and a prime ideal λ of\n\nmodular if | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: to GL2(B) will also be\n\n→\n\ndenoted φ. If � : GQ A′ for the composition of � with the inclusion of GL2(A) in GL2(A′).\n\nGL2(A) is a representation and A′ is a ring containing A, we write\n\n→\n\n�\n\n⊗\n\nDe�nition. If ¯� : GQ � : GQ Zp-algebra and there exists a homomorphism ι : A\n\nGL2(k) is a representation, we say that a representation GL2(A) is a lifting of ¯� (to A) if A is a complete noetherian local\n\n→\n\n→\n\n¯k such that the diagram\n\n→ GL2(A)\n\n✟✟✯\n\n[�]\n\n✟✟\n\nι � GL2(¯k)\n\n✲\n\nGQ\n\n[ ¯� ⊗ ¯k]\n\n¯k].\n\ncommutes, in the sense that [ι\n\n�] = [¯�\n\n\n\n⊗\n\nExamples. (i) If E is an elliptic curve then �E,p is a lifting of ¯�E,p.\n\n(ii) If E is an elliptic curve, p is a prime, and hypotheses (a) and (b) of Conjecture\n\n2.1 hold with an eigenform f and prime ideal λ, then �f,λ is a lifting of ¯�E,p.\n\n3.4. Deformation data. We will be interested not in all liftings of a given ¯�, but rather in those satisfying various restrictions. See Appendix A for the de�nition of GQ associated to primes q. We say that a representation � the inertia groups Iq ⊂ of GQ is unrami�ed at a prime q if �(Iq) = 1. If Σ is a set of primes, we say � is unrami�ed outside of Σ if � is unrami�ed at every q / ∈\n\nΣ.\n\nDe�nition. By deformation data we mean a pair\n\n= (Σ, t)\n\nD where Σ is a �nite set of primes and t is one of the words ordinary or flat.\n\nZ×\n\nA× be the composition of the\n\nIf A is a Zp-algebra, let εA : GQ\n\np →\n\n→\n\ncyclotomic character εp with the structure map.\n\nDe�nition. Given deformation data type- outside of Σ, and � is t at p (where t\n\nGL2(A) is if A is a complete noetherian local Zp-algebra, det(�) = εA, � is unrami�ed\n\n, a representation � : GQ\n\nD\n\n→\n\nD\n\nordinary, flat }\n\n; see Appendix C).\n\n∈ {\n\nDe�nition. A representation ¯� : GQ eigenform f and a prime ideal λ of\n\nmodular if |
758 | eigenform f and a prime ideal λ of\n\nmodular if there are an\n\nGL2(k) is Of such that �f,λ is a type-\n\n→\n\nD\n\nlifting of ¯�.\n\nD\n\nRemarks. (i) A representation with a type- fore if a representation is\n\nlifting must itself be type-\n\n. There-\n\nD\n\nD and modular.\n\nmodular, then it is both type-\n\nD\n\nD\n\n(ii) Conversely, if ¯� is type-\n\n, modular, and satis�es (ii) of Theorem 5.3 below, -modular, by work of Ribet and others (see [28]). This plays an important\n\nD\n\nthen ¯� is D role in Wiles’ work.\n\n12\n\nK. RUBIN AND A. SILVERBERG\n\n3.5. Mazur Conjecture.\n\nDe�nition. A representation ¯� : GQ ¯�\n\nGL2(k) is called absolutely irreducible if\n\n→\n\n¯k is irreducible.\n\n⊗\n\nThe following variant of a conjecture of Mazur (see Conjecture 18 of [23]; see\n\nalso Conjecture 4.2 below) implies the Semistable Modular Lifting Conjecture.\n\nConjecture 3.2 (Mazur). Suppose p is an odd prime, k is a �nite �eld of charac- GL2(k) is an absolutely irreducible teristic p, lifting of ¯� to the ring of integers of\n\nis deformation data, and ¯� : GQ -modular representation. Then every type-\n\nD\n\n→ D\n\nD a �nite extension of Qp is modular.\n\nRemark . Loosely speaking, Conjecture 3.2 says that if ¯� is modular, then every lifting which “looks modular� is modular.\n\nDe�nition. An elliptic curve E over Q has good (respectively, bad ) reduction at a prime q if E is nonsingular (respectively, singular) modulo q. An elliptic curve E over Q has ordinary (respectively, supersingular) reduction at q if E has good reduction at q and E[q] has (respectively, does not have) a subgroup of order q stable under the inertia group Iq.\n\nProposition 3.3. Conjecture 3.2 implies Conjecture 2.1.\n\nProof. Suppose p is an odd prime and E is a semistable elliptic curve over Q which satis�es (a) and (b) of Conjecture 2.1. We will apply Conjecture 3.2 with ¯� = ¯�E,p. Write τ for complex conjugation. Then τ 2 = 1, and | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: eigenform f and a prime ideal λ of\n\nmodular if there are an\n\nGL2(k) is Of such that �f,λ is a type-\n\n→\n\nD\n\nlifting of ¯�.\n\nD\n\nRemarks. (i) A representation with a type- fore if a representation is\n\nlifting must itself be type-\n\n. There-\n\nD\n\nD and modular.\n\nmodular, then it is both type-\n\nD\n\nD\n\n(ii) Conversely, if ¯� is type-\n\n, modular, and satis�es (ii) of Theorem 5.3 below, -modular, by work of Ribet and others (see [28]). This plays an important\n\nD\n\nthen ¯� is D role in Wiles’ work.\n\n12\n\nK. RUBIN AND A. SILVERBERG\n\n3.5. Mazur Conjecture.\n\nDe�nition. A representation ¯� : GQ ¯�\n\nGL2(k) is called absolutely irreducible if\n\n→\n\n¯k is irreducible.\n\n⊗\n\nThe following variant of a conjecture of Mazur (see Conjecture 18 of [23]; see\n\nalso Conjecture 4.2 below) implies the Semistable Modular Lifting Conjecture.\n\nConjecture 3.2 (Mazur). Suppose p is an odd prime, k is a �nite �eld of charac- GL2(k) is an absolutely irreducible teristic p, lifting of ¯� to the ring of integers of\n\nis deformation data, and ¯� : GQ -modular representation. Then every type-\n\nD\n\n→ D\n\nD a �nite extension of Qp is modular.\n\nRemark . Loosely speaking, Conjecture 3.2 says that if ¯� is modular, then every lifting which “looks modular� is modular.\n\nDe�nition. An elliptic curve E over Q has good (respectively, bad ) reduction at a prime q if E is nonsingular (respectively, singular) modulo q. An elliptic curve E over Q has ordinary (respectively, supersingular) reduction at q if E has good reduction at q and E[q] has (respectively, does not have) a subgroup of order q stable under the inertia group Iq.\n\nProposition 3.3. Conjecture 3.2 implies Conjecture 2.1.\n\nProof. Suppose p is an odd prime and E is a semistable elliptic curve over Q which satis�es (a) and (b) of Conjecture 2.1. We will apply Conjecture 3.2 with ¯� = ¯�E,p. Write τ for complex conjugation. Then τ 2 = 1, and |
759 | τ for complex conjugation. Then τ 2 = 1, and by (6), det(¯�E,p(τ )) = 1. Since ¯�E,p is irreducible and p is odd, a simple linear algebra argument now shows that ¯�E,p is absolutely irreducible.\n\n−\n\nSince E satis�es (b) of Conjecture 2.1, ¯�E,p is modular. Let\n\nΣ = t = ordinary if E has ordinary or bad reduction at p, t = flat if E has supersingular reduction at p,\n\np\n\nprimes q : E has bad reduction at q\n\n,\n\n•\n\n{\n\n} ∪ {\n\n}\n\n= (Σ, t).\n\nD\n\nUsing the semistability of E, one can show that �E,p is a type- (by combining results of several people; see [28]) that ¯�E,p is 3.2 then says �E,p is modular. By Theorem 3.1, E is modular.\n\nlifting of ¯�E,p and -modular. Conjecture\n\nD\n\nD\n\n4. Mazur’s deformation theory\n\nNext we reformulate Conjecture 3.2 as a conjecture (Conjecture 4.2) that the algebras which parametrize liftings and modular liftings of a given representation are isomorphic. It is this form of Mazur’s conjecture that Wiles attacks directly.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n13\n\n4.1. The universal deformation algebra R. Fix an odd prime p, a �nite �eld k of characteristic p, deformation data representation ¯� : GQ extension of Qp with residue �eld k.\n\n, and an absolutely irreducible type-\n\nD\n\nD is the ring of integers of a �nite\n\nGL2(k). Suppose\n\n→\n\nO\n\nDe�nition. We say � : GQ complete noetherian local commutes\n\n)-lifting of ¯� if � is type-\n\n, A is a → -algebra with residue �eld k, and the following diagram\n\nGL2(A) is a (\n\n,\n\nD\n\nO\n\nD\n\nO\n\nGL2(A)\n\n✟✟✯\n\n[�]\n\n✟✟\n\n� GL2(k)\n\n✲\n\nGQ\n\n[ ¯�]\n\nwhere the vertical map is reduction modulo the maximal ideal of A.\n\nTheorem 4.1 (Mazur-Ramakrishna). With p, k, an that for every ( φ� : R\n\nas above, there are D GL2(R) of ¯�, with the property -algebra homomorphism\n\n, ¯�, and\n\nO\n\nalgebra R and a (\n\n)-lifting �R : GQ )-lifting � of ¯� to A there is | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: τ for complex conjugation. Then τ 2 = 1, and by (6), det(¯�E,p(τ )) = 1. Since ¯�E,p is irreducible and p is odd, a simple linear algebra argument now shows that ¯�E,p is absolutely irreducible.\n\n−\n\nSince E satis�es (b) of Conjecture 2.1, ¯�E,p is modular. Let\n\nΣ = t = ordinary if E has ordinary or bad reduction at p, t = flat if E has supersingular reduction at p,\n\np\n\nprimes q : E has bad reduction at q\n\n,\n\n•\n\n{\n\n} ∪ {\n\n}\n\n= (Σ, t).\n\nD\n\nUsing the semistability of E, one can show that �E,p is a type- (by combining results of several people; see [28]) that ¯�E,p is 3.2 then says �E,p is modular. By Theorem 3.1, E is modular.\n\nlifting of ¯�E,p and -modular. Conjecture\n\nD\n\nD\n\n4. Mazur’s deformation theory\n\nNext we reformulate Conjecture 3.2 as a conjecture (Conjecture 4.2) that the algebras which parametrize liftings and modular liftings of a given representation are isomorphic. It is this form of Mazur’s conjecture that Wiles attacks directly.\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n13\n\n4.1. The universal deformation algebra R. Fix an odd prime p, a �nite �eld k of characteristic p, deformation data representation ¯� : GQ extension of Qp with residue �eld k.\n\n, and an absolutely irreducible type-\n\nD\n\nD is the ring of integers of a �nite\n\nGL2(k). Suppose\n\n→\n\nO\n\nDe�nition. We say � : GQ complete noetherian local commutes\n\n)-lifting of ¯� if � is type-\n\n, A is a → -algebra with residue �eld k, and the following diagram\n\nGL2(A) is a (\n\n,\n\nD\n\nO\n\nD\n\nO\n\nGL2(A)\n\n✟✟✯\n\n[�]\n\n✟✟\n\n� GL2(k)\n\n✲\n\nGQ\n\n[ ¯�]\n\nwhere the vertical map is reduction modulo the maximal ideal of A.\n\nTheorem 4.1 (Mazur-Ramakrishna). With p, k, an that for every ( φ� : R\n\nas above, there are D GL2(R) of ¯�, with the property -algebra homomorphism\n\n, ¯�, and\n\nO\n\nalgebra R and a (\n\n)-lifting �R : GQ )-lifting � of ¯� to A there is |
760 | �R : GQ )-lifting � of ¯� to A there is a unique\n\n,\n\nO\n\nD\n\nO\n\n→\n\n,\n\nD\n\nO\n\nO\n\nA such that the diagram\n\n→\n\n[�R]\n\n✲\n\nGQ\n\nGL2(R)\n\n�\n\n��\n\nφ� � GL2(A)\n\n[�]\n\n���\n\ncommutes.\n\nThis theorem was proved by Mazur [21] in the case when\n\nis ordinary and is flat. Theorem 4.1 determines R and �R up to\n\nD\n\nby Ramakrishna [25] when isomorphism.\n\nD\n\n4.2. The universal modular deformation algebra T. Fix an odd prime p, a , and an absolutely irreducible �nite �eld k of characteristic p, deformation data -modular, and �x an type- representation ¯� : GQ eigenform f and a prime ideal λ of lifting of ¯�. is the ring of integers of a �nite extension of Qp with Suppose in addition that residue �eld k, Of,λ ⊆ O\n\nD\n\nGL2(k). Assume ¯� is\n\nD\n\n→\n\nD\n\nOf such that �f,λ is a type-\n\nD\n\nO , and the diagram\n\nGL2(\n\nOf,λ) � GL2(k)\n\n✟✟✟✯ ✲\n\n[�f,λ] ✟\n\nGQ\n\n[ ¯�]\n\ncommutes, where the vertical map is the reduction map.\n\n)-lifting of ¯�, and Wiles constructs a generalized Hecke algebra T which has the following properties (recall that Hecke algebras T (N ) were de�ned in\n\nUnder these assumptions �f,λ ⊗ O 1.6).\n\nis a (\n\n,\n\nD\n\nO\n\n§\n\n(T1) T is a complete noetherian local\n\nalgebra with residue �eld k.\n\nO\n\n14\n\nK. RUBIN AND A. SILVERBERG\n\n(T2) There are an integer N divisible only by primes in Σ and a homomorphism by the Σ. By abuse of notation\n\nfrom the Hecke algebra T (N ) to T such that T is generated over images of the Hecke operators Tq for primes q / ∈ we write Tq also for its image in T.\n\nO\n\n(T3) There is a (\n\n,\n\n)-lifting\n\nD\n\nO\n\nGL2(T)\n\n�T : GQ\n\n→\n\nof ¯� with the property that trace(�T(Frobq)) = Tq for every prime q / ∈\n\nΣ. )-lifting of ¯� to A, then there is a unique\n\n(T4) If � is modular and is a (\n\n,\n\nD\n\nO\n\nalgebra homomorphism ψ� : T\n\nA such that the | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: �R : GQ )-lifting � of ¯� to A there is a unique\n\n,\n\nO\n\nD\n\nO\n\n→\n\n,\n\nD\n\nO\n\nO\n\nA such that the diagram\n\n→\n\n[�R]\n\n✲\n\nGQ\n\nGL2(R)\n\n�\n\n��\n\nφ� � GL2(A)\n\n[�]\n\n���\n\ncommutes.\n\nThis theorem was proved by Mazur [21] in the case when\n\nis ordinary and is flat. Theorem 4.1 determines R and �R up to\n\nD\n\nby Ramakrishna [25] when isomorphism.\n\nD\n\n4.2. The universal modular deformation algebra T. Fix an odd prime p, a , and an absolutely irreducible �nite �eld k of characteristic p, deformation data -modular, and �x an type- representation ¯� : GQ eigenform f and a prime ideal λ of lifting of ¯�. is the ring of integers of a �nite extension of Qp with Suppose in addition that residue �eld k, Of,λ ⊆ O\n\nD\n\nGL2(k). Assume ¯� is\n\nD\n\n→\n\nD\n\nOf such that �f,λ is a type-\n\nD\n\nO , and the diagram\n\nGL2(\n\nOf,λ) � GL2(k)\n\n✟✟✟✯ ✲\n\n[�f,λ] ✟\n\nGQ\n\n[ ¯�]\n\ncommutes, where the vertical map is the reduction map.\n\n)-lifting of ¯�, and Wiles constructs a generalized Hecke algebra T which has the following properties (recall that Hecke algebras T (N ) were de�ned in\n\nUnder these assumptions �f,λ ⊗ O 1.6).\n\nis a (\n\n,\n\nD\n\nO\n\n§\n\n(T1) T is a complete noetherian local\n\nalgebra with residue �eld k.\n\nO\n\n14\n\nK. RUBIN AND A. SILVERBERG\n\n(T2) There are an integer N divisible only by primes in Σ and a homomorphism by the Σ. By abuse of notation\n\nfrom the Hecke algebra T (N ) to T such that T is generated over images of the Hecke operators Tq for primes q / ∈ we write Tq also for its image in T.\n\nO\n\n(T3) There is a (\n\n,\n\n)-lifting\n\nD\n\nO\n\nGL2(T)\n\n�T : GQ\n\n→\n\nof ¯� with the property that trace(�T(Frobq)) = Tq for every prime q / ∈\n\nΣ. )-lifting of ¯� to A, then there is a unique\n\n(T4) If � is modular and is a (\n\n,\n\nD\n\nO\n\nalgebra homomorphism ψ� : T\n\nA such that the |
761 | homomorphism ψ� : T\n\nA such that the diagram\n\nO\n\n→ [� T]\n\n✲\n\nGL2(T)\n\nGQ\n\n�\n\n��\n\nψ� � GL2(A)\n\n[�]\n\n���\n\ncommutes.\n\nSince �T is a (\n\n,\n\n)-lifting of ¯�, by Theorem 4.1 there is a homomorphism\n\nD\n\nO\n\nT\n\nϕ : R\n\n→\n\n�R. By (T3), ϕ(trace(�R(Frobq))) = Tq for every\n\nsuch that �T is isomorphic to ϕ prime q / ∈\n\nΣ, so it follows from (T2) that ϕ is surjective.\n\n4.3. Mazur Conjecture, revisited. Conjecture 3.2 can be reformulated in the following way.\n\nConjecture 4.2 (Mazur). Suppose p, k, T is an isomorphism. above map ϕ : R\n\n, ¯�, and\n\nare as in\n\n4.2. Then the\n\nD\n\nO\n\n§\n\n→\n\nConjecture 4.2 was stated in [23] (Conjecture 18) for\n\nordinary, and Wiles\n\nD\n\nmodi�ed the conjecture to include the flat case.\n\nProposition 4.3. Conjecture 4.2 implies Conjecture 3.2.\n\nProof. Suppose ¯� : GQ -modular, A is D the ring of integers of a �nite extension of Qp, and � is a type- lifting of ¯� to A. to be the ring of integers of a sufficiently large �nite extension of Qp, and Taking and its residue �eld, respectively, we may assume that � is extending � and ¯� to A, with φ� a ( as in Theorem 4.1. By (T3) and Theorem 4.1, ψ(Tq) = trace(�(Frobq)) for all but 3.5 of [35], given such a homomorphism ψ (and viewing A as �nitely many q. By � n=1 ane2πinz where aq = ψ(Tq) for all but a subring of C), there is an eigenform �nitely many primes q. Thus � is modular.\n\nGL2(k) is absolutely irreducible and\n\n→\n\nD\n\nO )-lifting of ¯�. Assuming Conjecture 4.2, let ψ = φ� ◦\n\nO\n\nϕ−1 : T\n\n,\n\nD\n\nO\n\n→\n\n§\n\nP\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n15\n\n5. Wiles’ approach to the Mazur Conjecture\n\nIn this section we sketch the major ideas of Wiles’ attack on Conjecture 4.2. The �rst step (Theorem 5.2), and the key to Wiles’ proof, is to reduce Conjecture 4.2 to a bound on the order of the cotangent space at a prime | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: homomorphism ψ� : T\n\nA such that the diagram\n\nO\n\n→ [� T]\n\n✲\n\nGL2(T)\n\nGQ\n\n�\n\n��\n\nψ� � GL2(A)\n\n[�]\n\n���\n\ncommutes.\n\nSince �T is a (\n\n,\n\n)-lifting of ¯�, by Theorem 4.1 there is a homomorphism\n\nD\n\nO\n\nT\n\nϕ : R\n\n→\n\n�R. By (T3), ϕ(trace(�R(Frobq))) = Tq for every\n\nsuch that �T is isomorphic to ϕ prime q / ∈\n\nΣ, so it follows from (T2) that ϕ is surjective.\n\n4.3. Mazur Conjecture, revisited. Conjecture 3.2 can be reformulated in the following way.\n\nConjecture 4.2 (Mazur). Suppose p, k, T is an isomorphism. above map ϕ : R\n\n, ¯�, and\n\nare as in\n\n4.2. Then the\n\nD\n\nO\n\n§\n\n→\n\nConjecture 4.2 was stated in [23] (Conjecture 18) for\n\nordinary, and Wiles\n\nD\n\nmodi�ed the conjecture to include the flat case.\n\nProposition 4.3. Conjecture 4.2 implies Conjecture 3.2.\n\nProof. Suppose ¯� : GQ -modular, A is D the ring of integers of a �nite extension of Qp, and � is a type- lifting of ¯� to A. to be the ring of integers of a sufficiently large �nite extension of Qp, and Taking and its residue �eld, respectively, we may assume that � is extending � and ¯� to A, with φ� a ( as in Theorem 4.1. By (T3) and Theorem 4.1, ψ(Tq) = trace(�(Frobq)) for all but 3.5 of [35], given such a homomorphism ψ (and viewing A as �nitely many q. By � n=1 ane2πinz where aq = ψ(Tq) for all but a subring of C), there is an eigenform �nitely many primes q. Thus � is modular.\n\nGL2(k) is absolutely irreducible and\n\n→\n\nD\n\nO )-lifting of ¯�. Assuming Conjecture 4.2, let ψ = φ� ◦\n\nO\n\nϕ−1 : T\n\n,\n\nD\n\nO\n\n→\n\n§\n\nP\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n15\n\n5. Wiles’ approach to the Mazur Conjecture\n\nIn this section we sketch the major ideas of Wiles’ attack on Conjecture 4.2. The �rst step (Theorem 5.2), and the key to Wiles’ proof, is to reduce Conjecture 4.2 to a bound on the order of the cotangent space at a prime |
762 | on the order of the cotangent space at a prime of R. In 5.2 we § see that the corresponding tangent space is a Selmer group, and in 5.3 we outline a general procedure due to Kolyvagin for bounding sizes of Selmer groups. The input for Kolyvagin’s method is known as an Euler system. The most difficult 5.4), and the part described as “not yet complete� in his part of Wiles’ work ( § December announcement, is his construction of a suitable Euler system. In 5.5 we state the results announced by Wiles (Theorems 5.3 and 5.4 and Corollary 5.5) and explain why Theorem 5.3 suffices for proving the Semistable Taniyama-Shimura Conjecture. As an application of Corollary 5.5 we write down an in�nite family of modular elliptic curves. , ¯�, 5 �x p, k,\n\n§\n\n§\n\n� n=1 ane2πinz, and λ as in\n\n4.2.\n\nFor O By property (T4) there is a homomorphism\n\n, f (z) =\n\n§\n\n§\n\nD\n\nP\n\nπ : T\n\n→ O . By property (T2) and (12), π satis�es\n\nsuch that π π(Tq) = aq for all but �nitely many q.\n\n�T is isomorphic to �f,λ ⊗ O\n\n\n\n5.1. Key reduction. Wiles uses the following generalization of a theorem of Mazur, which says that T is Gorenstein.\n\nTheorem 5.1. There is a (noncanonical ) T-module isomorphism\n\n) ∼ →\n\nHomO(T,\n\nT.\n\nO\n\nLet η denote the ideal of\n\ngenerated by the image under the composition\n\nO HomO(T,\n\n) ∼ →\n\nT π\n\nO\n\n→ O\n\nHomO(T,\n\nof the element π ∈ choice of isomorphism in Theorem 5.1.\n\n). The ideal η is well de�ned independent of the\n\nO\n\nThe map π determines distinguished prime ideals of T and R,\n\nϕ) = ϕ−1(pT).\n\npT = ker(π),\n\npR = ker(π\n\n\n\nTheorem 5.2 (Wiles). If\n\n#(pR/p2\n\nR)\n\n#(\n\n/η) <\n\n, �\n\n≤\n\nO\n\nT is an isomorphism.\n\nthen ϕ : R\n\n→\n\nThe proof is entirely commutative algebra. The surjectivity of ϕ shows that /η). Thus if\n\n#(pR/p2 #(pR/p2\n\n#(pT/p2 #(\n\nT), and Wiles proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\n≥ | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: on the order of the cotangent space at a prime of R. In 5.2 we § see that the corresponding tangent space is a Selmer group, and in 5.3 we outline a general procedure due to Kolyvagin for bounding sizes of Selmer groups. The input for Kolyvagin’s method is known as an Euler system. The most difficult 5.4), and the part described as “not yet complete� in his part of Wiles’ work ( § December announcement, is his construction of a suitable Euler system. In 5.5 we state the results announced by Wiles (Theorems 5.3 and 5.4 and Corollary 5.5) and explain why Theorem 5.3 suffices for proving the Semistable Taniyama-Shimura Conjecture. As an application of Corollary 5.5 we write down an in�nite family of modular elliptic curves. , ¯�, 5 �x p, k,\n\n§\n\n§\n\n� n=1 ane2πinz, and λ as in\n\n4.2.\n\nFor O By property (T4) there is a homomorphism\n\n, f (z) =\n\n§\n\n§\n\nD\n\nP\n\nπ : T\n\n→ O . By property (T2) and (12), π satis�es\n\nsuch that π π(Tq) = aq for all but �nitely many q.\n\n�T is isomorphic to �f,λ ⊗ O\n\n\n\n5.1. Key reduction. Wiles uses the following generalization of a theorem of Mazur, which says that T is Gorenstein.\n\nTheorem 5.1. There is a (noncanonical ) T-module isomorphism\n\n) ∼ →\n\nHomO(T,\n\nT.\n\nO\n\nLet η denote the ideal of\n\ngenerated by the image under the composition\n\nO HomO(T,\n\n) ∼ →\n\nT π\n\nO\n\n→ O\n\nHomO(T,\n\nof the element π ∈ choice of isomorphism in Theorem 5.1.\n\n). The ideal η is well de�ned independent of the\n\nO\n\nThe map π determines distinguished prime ideals of T and R,\n\nϕ) = ϕ−1(pT).\n\npT = ker(π),\n\npR = ker(π\n\n\n\nTheorem 5.2 (Wiles). If\n\n#(pR/p2\n\nR)\n\n#(\n\n/η) <\n\n, �\n\n≤\n\nO\n\nT is an isomorphism.\n\nthen ϕ : R\n\n→\n\nThe proof is entirely commutative algebra. The surjectivity of ϕ shows that /η). Thus if\n\n#(pR/p2 #(pR/p2\n\n#(pT/p2 #(\n\nT), and Wiles proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\n≥ |
763 | proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\n≥ ≤\n\n≥\n\nO\n\n/η), then\n\nO\n\n#(pR/p2\n\nR) = #(pT/p2\n\n(13)\n\nT) = #(\n\n/η).\n\nO\n\nThe �rst equality in (13) shows that ϕ induces an isomorphism of tangent spaces. Wiles uses the second equality in (13) and Theorem 5.1 to deduce that T is a local\n\n16\n\nK. RUBIN AND A. SILVERBERG\n\ncomplete intersection over that\n\n(that is, there are f1, . . . , fr ∈ O\n\n[[x1, . . . , xr]] such\n\nO\n\nT ∼=\n\n[[x1, . . . , xr]]/(f1, . . . , fr)\n\nO\n\nas morphism.\n\nalgebras). Wiles then combines these two results to prove that ϕ is an iso-\n\nO\n\n5.2. Selmer groups. In general, if M is a torsion GQ-module, a Selmer group attached to M is a subgroup of the Galois cohomology group H 1(GQ, M ) deter- mined by certain “local conditions� in the following way. If q is a prime with decomposition group Dq ⊂\n\nGQ, then there is a restriction map\n\nresq : H 1(GQ, M )\n\nH 1(Dq, M ).\n\n→ Jq ⊆\n\nH 1(Dq, M ) : q prime\n\n= For a �xed collection of subgroups { the particular problem under consideration, the corresponding Selmer group is\n\ndepending on\n\nJ\n\n}\n\nres−1\n\nH 1(GQ, M ).\n\nS(M ) =\n\nq (Jq)\n\n⊆\n\nq \\ Write H i(Q, M ) for H i(GQ, M ), and H i(Qq, M ) for H i(Dq, M ).\n\nExample. The original examples of Selmer groups come from elliptic curves. Fix an elliptic curve E and a positive integer m, and take M = E[m], the subgroup of points in E( ¯Q) of order dividing m. There is a natural inclusion\n\nH 1(Q, E[m])\n\nE(Q)/mE(Q) ֒\n\n(14)\n\n→\n\nE( ¯Q) is any\n\nE(Q) to the cocycle σ\n\nobtained by sending x point satisfying my = x. Similarly, for every prime q there is a natural inclusion\n\nσ(y)\n\ny, where y\n\n∈\n\n7→\n\n−\n\n∈\n\nH 1(Qq, E[m]).\n\nE(Qq)/mE(Qq) ֒\n\n→ De�ne the Selmer group S(E[m]) in this case by taking the group Jq to be the image of E(Qq)/mE(Qq) in H 1(Qq, E[m]), for every q. This Selmer group is an important tool in studying the | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\n≥ ≤\n\n≥\n\nO\n\n/η), then\n\nO\n\n#(pR/p2\n\nR) = #(pT/p2\n\n(13)\n\nT) = #(\n\n/η).\n\nO\n\nThe �rst equality in (13) shows that ϕ induces an isomorphism of tangent spaces. Wiles uses the second equality in (13) and Theorem 5.1 to deduce that T is a local\n\n16\n\nK. RUBIN AND A. SILVERBERG\n\ncomplete intersection over that\n\n(that is, there are f1, . . . , fr ∈ O\n\n[[x1, . . . , xr]] such\n\nO\n\nT ∼=\n\n[[x1, . . . , xr]]/(f1, . . . , fr)\n\nO\n\nas morphism.\n\nalgebras). Wiles then combines these two results to prove that ϕ is an iso-\n\nO\n\n5.2. Selmer groups. In general, if M is a torsion GQ-module, a Selmer group attached to M is a subgroup of the Galois cohomology group H 1(GQ, M ) deter- mined by certain “local conditions� in the following way. If q is a prime with decomposition group Dq ⊂\n\nGQ, then there is a restriction map\n\nresq : H 1(GQ, M )\n\nH 1(Dq, M ).\n\n→ Jq ⊆\n\nH 1(Dq, M ) : q prime\n\n= For a �xed collection of subgroups { the particular problem under consideration, the corresponding Selmer group is\n\ndepending on\n\nJ\n\n}\n\nres−1\n\nH 1(GQ, M ).\n\nS(M ) =\n\nq (Jq)\n\n⊆\n\nq \\ Write H i(Q, M ) for H i(GQ, M ), and H i(Qq, M ) for H i(Dq, M ).\n\nExample. The original examples of Selmer groups come from elliptic curves. Fix an elliptic curve E and a positive integer m, and take M = E[m], the subgroup of points in E( ¯Q) of order dividing m. There is a natural inclusion\n\nH 1(Q, E[m])\n\nE(Q)/mE(Q) ֒\n\n(14)\n\n→\n\nE( ¯Q) is any\n\nE(Q) to the cocycle σ\n\nobtained by sending x point satisfying my = x. Similarly, for every prime q there is a natural inclusion\n\nσ(y)\n\ny, where y\n\n∈\n\n7→\n\n−\n\n∈\n\nH 1(Qq, E[m]).\n\nE(Qq)/mE(Qq) ֒\n\n→ De�ne the Selmer group S(E[m]) in this case by taking the group Jq to be the image of E(Qq)/mE(Qq) in H 1(Qq, E[m]), for every q. This Selmer group is an important tool in studying the |
764 | Selmer group is an important tool in studying the arithmetic of E because it contains (via (14)) E(Q)/mE(Q).\n\n5, let m denote the maximal ideal /mn) can be\n\nRetaining the notation from the beginning of\n\n§\n\nand �x a positive integer n. The tangent space HomO(pR/p2 R,\n\nof identi�ed with a Selmer group as follows. Let Vn be the matrix algebra M2(\n\nO\n\nO\n\n/mn), with GQ acting via the adjoint repre-\n\nO\n\nsentation σ(B) = �f,λ(σ)B�f,λ(σ)−1. There is a natural injection\n\ns : HomO(pR/p2 R,\n\n/mn) ֒\n\nH 1(Q, Vn)\n\nO\n\n→\n\nwhich is described in Appendix D (see also\n\n1.6 of [21]). Wiles de�nes a collection . Let SD(Vn) denote the associated Selmer\n\n§\n\nH 1(Qq, Vn) }\n\n=\n\nJq ⊆\n\ndepending on\n\nJ group. Wiles proves that s induces an isomorphism\n\n{\n\nD\n\n/mn) ∼ →\n\nHomO(pR/p2 R,\n\nSD(Vn).\n\nO\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n17\n\n5.3. Euler systems. We have now reduced the proof of Mazur’s conjecture to bounding the size of the Selmer groups SD(Vn). About �ve years ago Kolyvagin [19], building on ideas of his own and of Thaine [40], introduced a revolutionary new method for bounding the size of a Selmer group. This new machinery, which is crucial for Wiles’ proof, is what we now describe.\n\nH 1(Qq,M ) is } 5.2. Let ˆM = a system of subgroups with associated Selmer group S(M ) as in Hom(M, µm), where µm is the group of m-th roots of unity. For every prime q, the cup product gives a nondegenerate Tate pairing\n\nSuppose M is a GQ-module of odd exponent m and\n\n=\n\nJq ⊆ §\n\nJ\n\n{\n\nH 2(Qq, µm) ∼ → H 1(Q, ˆM ), then\n\nH 1(Qq, ˆM )\n\niq : H 1(Qq, M )\n\nZ/mZ\n\n,\n\nh\n\n×\n\n→\n\nH 1(Q, M ) and d\n\n(see Chapters VI and VII of [3]). If c\n\n∈\n\n∈\n\n(15)\n\nresq(c), resq(d) h\n\niq = 0.\n\nq X\n\nH 1(Q, ˆM ) be the Selmer\n\nis a �nite set of primes. Let S∗\n\nSuppose that\n\nL ⊆ H 1(Qq, ˆM ) }\n\nL group given by the local conditions\n\n∗ =\n\nJ ∗ q | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Selmer group is an important tool in studying the arithmetic of E because it contains (via (14)) E(Q)/mE(Q).\n\n5, let m denote the maximal ideal /mn) can be\n\nRetaining the notation from the beginning of\n\n§\n\nand �x a positive integer n. The tangent space HomO(pR/p2 R,\n\nof identi�ed with a Selmer group as follows. Let Vn be the matrix algebra M2(\n\nO\n\nO\n\n/mn), with GQ acting via the adjoint repre-\n\nO\n\nsentation σ(B) = �f,λ(σ)B�f,λ(σ)−1. There is a natural injection\n\ns : HomO(pR/p2 R,\n\n/mn) ֒\n\nH 1(Q, Vn)\n\nO\n\n→\n\nwhich is described in Appendix D (see also\n\n1.6 of [21]). Wiles de�nes a collection . Let SD(Vn) denote the associated Selmer\n\n§\n\nH 1(Qq, Vn) }\n\n=\n\nJq ⊆\n\ndepending on\n\nJ group. Wiles proves that s induces an isomorphism\n\n{\n\nD\n\n/mn) ∼ →\n\nHomO(pR/p2 R,\n\nSD(Vn).\n\nO\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n17\n\n5.3. Euler systems. We have now reduced the proof of Mazur’s conjecture to bounding the size of the Selmer groups SD(Vn). About �ve years ago Kolyvagin [19], building on ideas of his own and of Thaine [40], introduced a revolutionary new method for bounding the size of a Selmer group. This new machinery, which is crucial for Wiles’ proof, is what we now describe.\n\nH 1(Qq,M ) is } 5.2. Let ˆM = a system of subgroups with associated Selmer group S(M ) as in Hom(M, µm), where µm is the group of m-th roots of unity. For every prime q, the cup product gives a nondegenerate Tate pairing\n\nSuppose M is a GQ-module of odd exponent m and\n\n=\n\nJq ⊆ §\n\nJ\n\n{\n\nH 2(Qq, µm) ∼ → H 1(Q, ˆM ), then\n\nH 1(Qq, ˆM )\n\niq : H 1(Qq, M )\n\nZ/mZ\n\n,\n\nh\n\n×\n\n→\n\nH 1(Q, M ) and d\n\n(see Chapters VI and VII of [3]). If c\n\n∈\n\n∈\n\n(15)\n\nresq(c), resq(d) h\n\niq = 0.\n\nq X\n\nH 1(Q, ˆM ) be the Selmer\n\nis a �nite set of primes. Let S∗\n\nSuppose that\n\nL ⊆ H 1(Qq, ˆM ) }\n\nL group given by the local conditions\n\n∗ =\n\nJ ∗ q |
765 | given by the local conditions\n\n∗ =\n\nJ ∗ q ⊆\n\n, where\n\nJ\n\n{\n\nthe orthogonal complement of Jq under H 1(Qq, ˆM )\n\n,\n\nif q / if q\n\n, ∈ L . ∈ L\n\niq\n\nJ ∗ q =\n\nh\n\n(\n\nH 1(Q, ˆM ), de�ne\n\nIf d\n\n∈\n\nZ/mZ\n\nθd :\n\nJq →\n\nYq∈L\n\nby\n\nθd((cq)) =\n\ncq, resq(d) h\n\niq.\n\nXq∈L\n\nWrite resL : H 1(Q, M ) maps. By (15) and the de�nition of J ∗ in addition resL is injective on S(M ), then\n\nq∈L H 1(Qq, M ) for the product of the restriction ker(θd). If\n\n→\n\nS∗\n\nq , if d\n\nL, then resL(S(M ))\n\n∈\n\n⊆\n\nQ\n\n#(S(M ))\n\n#\n\nker(θd)\n\n.\n\n≤\n\n(cid:0) \\d∈S∗\n\nL\n\n(cid:1)\n\nThe difficulty is to produce enough cohomology classes in S∗\n\nL to show that the right side of the above inequality is small. Following Kolyvagin, an Euler system is S∗ L for a large (in�nite) collection of sets of a compatible collection of classes κ( )) primes is related to resℓ(κ( )). Once an Euler system is given, Kolyvagin has an inductive procedure for choosing a set\n\n) L\n\n∈\n\n. Loosely speaking, compatible means that if ℓ /\n\n, then resℓ(κ(\n\nℓ\n\nL\n\n∈ L\n\nL ∪ {\n\n}\n\nL\n\nsuch that\n\nL\n\nresL is injective on S(M ),\n\n•\n\nP⊆L ker(θκ(P)) can be computed in terms of κ( ∅\n\n).\n\nT\n\n18\n\nK. RUBIN AND A. SILVERBERG\n\nS∗\n\nS∗\n\n, then S∗\n\nL.)\n\nL, so κ(\n\n)\n\n(Note that if\n\nP ⊆\n\nP For several important Selmer groups it is possible to construct Euler systems for\n\n∈\n\nP ⊆ L\n\nwhich Kolyvagin’s procedure produces a set\n\nactually giving an equality\n\nL ker(θκ(P))\n\n#(S(M )) = #\n\n.\n\n(cid:0) \\P⊆L This is what Wiles needs to do for the Selmer group SD(Vn). There are several examples in the literature where this kind of argument is worked out in some detail. For the simplest case, where the Selmer group in question is the ideal class group ) are constructed from cyclotomic units, of a real abelian number �eld and the κ( | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: given by the local conditions\n\n∗ =\n\nJ ∗ q ⊆\n\n, where\n\nJ\n\n{\n\nthe orthogonal complement of Jq under H 1(Qq, ˆM )\n\n,\n\nif q / if q\n\n, ∈ L . ∈ L\n\niq\n\nJ ∗ q =\n\nh\n\n(\n\nH 1(Q, ˆM ), de�ne\n\nIf d\n\n∈\n\nZ/mZ\n\nθd :\n\nJq →\n\nYq∈L\n\nby\n\nθd((cq)) =\n\ncq, resq(d) h\n\niq.\n\nXq∈L\n\nWrite resL : H 1(Q, M ) maps. By (15) and the de�nition of J ∗ in addition resL is injective on S(M ), then\n\nq∈L H 1(Qq, M ) for the product of the restriction ker(θd). If\n\n→\n\nS∗\n\nq , if d\n\nL, then resL(S(M ))\n\n∈\n\n⊆\n\nQ\n\n#(S(M ))\n\n#\n\nker(θd)\n\n.\n\n≤\n\n(cid:0) \\d∈S∗\n\nL\n\n(cid:1)\n\nThe difficulty is to produce enough cohomology classes in S∗\n\nL to show that the right side of the above inequality is small. Following Kolyvagin, an Euler system is S∗ L for a large (in�nite) collection of sets of a compatible collection of classes κ( )) primes is related to resℓ(κ( )). Once an Euler system is given, Kolyvagin has an inductive procedure for choosing a set\n\n) L\n\n∈\n\n. Loosely speaking, compatible means that if ℓ /\n\n, then resℓ(κ(\n\nℓ\n\nL\n\n∈ L\n\nL ∪ {\n\n}\n\nL\n\nsuch that\n\nL\n\nresL is injective on S(M ),\n\n•\n\nP⊆L ker(θκ(P)) can be computed in terms of κ( ∅\n\n).\n\nT\n\n18\n\nK. RUBIN AND A. SILVERBERG\n\nS∗\n\nS∗\n\n, then S∗\n\nL.)\n\nL, so κ(\n\n)\n\n(Note that if\n\nP ⊆\n\nP For several important Selmer groups it is possible to construct Euler systems for\n\n∈\n\nP ⊆ L\n\nwhich Kolyvagin’s procedure produces a set\n\nactually giving an equality\n\nL ker(θκ(P))\n\n#(S(M )) = #\n\n.\n\n(cid:0) \\P⊆L This is what Wiles needs to do for the Selmer group SD(Vn). There are several examples in the literature where this kind of argument is worked out in some detail. For the simplest case, where the Selmer group in question is the ideal class group ) are constructed from cyclotomic units, of a real abelian number �eld and the κ( |
766 | of a real abelian number �eld and the κ( L see [29]. For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19], [31], [30], [13].\n\n(cid:1)\n\n5.4. Wiles’ geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD(Vn)) using Kolyvagin’s method. This is the most technically difficult part of Wiles’ proof and is the part of Wiles’ work he referred to as not yet complete in his December announcement. We give only general remarks about Wiles’ construction.\n\nThe �rst step in the construction is due to Flach [10]. He constructed classes consisting of just one prime. This allows one to bound the ) L\n\nS∗\n\nκ( exponent of SD(Vn), but not its order.\n\nL for sets\n\n∈\n\nL\n\nEvery Euler system starts with some explicit, concrete objects. Earlier examples of Euler systems come from cyclotomic or elliptic units, Gauss sums, or Heegner points on elliptic curves. Wiles (following Flach) constructs his cohomology classes from modular units, i.e., meromorphic functions on modular curves which are holo- morphic and nonzero away from the cusps. More precisely, κ( ) comes from an explicit function on the modular curve X1(L, N ), the curve obtained by taking the quotient space of the upper half plane by the action of the group\n\nL\n\na b c d\n\nSL2(Z) : c\n\n1 (mod L) } ≡ ℓ∈L ℓ and where N is the N of (T2) of\n\n0\n\n(mod LN ),\n\na\n\nd\n\n,\n\nΓ1(L, N ) =\n\n∈\n\n≡\n\n≡\n\n{ (cid:1) (cid:0) and adjoining the cusps, where L = The construction and study of the classes κ( [8], [9] and others.\n\n4.2. ) rely heavily on results of Faltings\n\n§\n\nL\n\nQ\n\n5.5. Wiles’ results. Wiles announced two main results (Theorems 5.3 and 5.4 below) in the direction of Mazur’s conjecture, under two different sets of hypotheses on the representation ¯�. Theorem 5.3 implies the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem. Wiles’ proof | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: of a real abelian number �eld and the κ( L see [29]. For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19], [31], [30], [13].\n\n(cid:1)\n\n5.4. Wiles’ geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD(Vn)) using Kolyvagin’s method. This is the most technically difficult part of Wiles’ proof and is the part of Wiles’ work he referred to as not yet complete in his December announcement. We give only general remarks about Wiles’ construction.\n\nThe �rst step in the construction is due to Flach [10]. He constructed classes consisting of just one prime. This allows one to bound the ) L\n\nS∗\n\nκ( exponent of SD(Vn), but not its order.\n\nL for sets\n\n∈\n\nL\n\nEvery Euler system starts with some explicit, concrete objects. Earlier examples of Euler systems come from cyclotomic or elliptic units, Gauss sums, or Heegner points on elliptic curves. Wiles (following Flach) constructs his cohomology classes from modular units, i.e., meromorphic functions on modular curves which are holo- morphic and nonzero away from the cusps. More precisely, κ( ) comes from an explicit function on the modular curve X1(L, N ), the curve obtained by taking the quotient space of the upper half plane by the action of the group\n\nL\n\na b c d\n\nSL2(Z) : c\n\n1 (mod L) } ≡ ℓ∈L ℓ and where N is the N of (T2) of\n\n0\n\n(mod LN ),\n\na\n\nd\n\n,\n\nΓ1(L, N ) =\n\n∈\n\n≡\n\n≡\n\n{ (cid:1) (cid:0) and adjoining the cusps, where L = The construction and study of the classes κ( [8], [9] and others.\n\n4.2. ) rely heavily on results of Faltings\n\n§\n\nL\n\nQ\n\n5.5. Wiles’ results. Wiles announced two main results (Theorems 5.3 and 5.4 below) in the direction of Mazur’s conjecture, under two different sets of hypotheses on the representation ¯�. Theorem 5.3 implies the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem. Wiles’ proof |
767 | and Fermat’s Last Theorem. Wiles’ proof of Theorem 5.3 depends on the not-yet-complete construction of an appropriate Euler system (as in 5.4), while his proof of Theorem 5.4 (though not yet fully checked) does not. For Theorem 5.4, Wiles bounds the Selmer group of 5.2 without constructing a new Euler system, by using results from the Iwasawa theory of imaginary quadratic �elds. (These results in turn rely on Kolyvagin’s method and the Euler system of elliptic units; see [31].)\n\n§\n\n§\n\nSince for ease of exposition we de�ned modularity of representations in terms of Γ0(N ) instead of Γ1(N ), the theorems stated below are weaker than those an- nounced by Wiles, but have the same applications to elliptic curves. (Note that by our de�nition of type-\n\n, if ¯� is type-\n\n, then det(¯�) = ¯εp.)\n\nD\n\nD\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n19\n\nIf ¯� is a representation of GQ on a vector space V , Sym2(¯�) denotes the repre-\n\nsentation on the symmetric square of V induced by ¯�.\n\nTheorem 5.3 (Wiles). Suppose p, k, the following additional conditions :\n\n, ¯�, and\n\nare as in\n\n4.2 and ¯� satis�es\n\nD\n\nO\n\n§\n\n(i) Sym2(¯�) is absolutely irreducible, (ii) if ¯� is rami�ed at q and q (iii) if p is 3 or 5, then for some prime q, p divides #(¯�(Iq)).\n\n= p, then the restriction of ¯� to Dq is reducible,\n\n6\n\nT is an isomorphism.\n\nThen ϕ : R\n\n→\n\nSince Theorem 5.3 does not yield the full Mazur Conjecture (Conjecture 4.2) for 2 to see which elliptic curves §\n\np = 3 and 5, we need to reexamine the arguments of E can be proved modular using Theorem 5.3 applied to ¯�E,3 and ¯�E,5.\n\nHypothesis (i) of Theorem 5.3 will be satis�ed if the image of ¯�E,p is sufficiently large in GL2(Fp) (for example, if ¯�E,p is surjective). For p = 3 and p = 5, if ¯�E,p satis�es hypothesis (iii) and is irreducible, then it satis�es hypothesis (i).\n\nIf E is semistable, p is an odd prime, and | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: and Fermat’s Last Theorem. Wiles’ proof of Theorem 5.3 depends on the not-yet-complete construction of an appropriate Euler system (as in 5.4), while his proof of Theorem 5.4 (though not yet fully checked) does not. For Theorem 5.4, Wiles bounds the Selmer group of 5.2 without constructing a new Euler system, by using results from the Iwasawa theory of imaginary quadratic �elds. (These results in turn rely on Kolyvagin’s method and the Euler system of elliptic units; see [31].)\n\n§\n\n§\n\nSince for ease of exposition we de�ned modularity of representations in terms of Γ0(N ) instead of Γ1(N ), the theorems stated below are weaker than those an- nounced by Wiles, but have the same applications to elliptic curves. (Note that by our de�nition of type-\n\n, if ¯� is type-\n\n, then det(¯�) = ¯εp.)\n\nD\n\nD\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n19\n\nIf ¯� is a representation of GQ on a vector space V , Sym2(¯�) denotes the repre-\n\nsentation on the symmetric square of V induced by ¯�.\n\nTheorem 5.3 (Wiles). Suppose p, k, the following additional conditions :\n\n, ¯�, and\n\nare as in\n\n4.2 and ¯� satis�es\n\nD\n\nO\n\n§\n\n(i) Sym2(¯�) is absolutely irreducible, (ii) if ¯� is rami�ed at q and q (iii) if p is 3 or 5, then for some prime q, p divides #(¯�(Iq)).\n\n= p, then the restriction of ¯� to Dq is reducible,\n\n6\n\nT is an isomorphism.\n\nThen ϕ : R\n\n→\n\nSince Theorem 5.3 does not yield the full Mazur Conjecture (Conjecture 4.2) for 2 to see which elliptic curves §\n\np = 3 and 5, we need to reexamine the arguments of E can be proved modular using Theorem 5.3 applied to ¯�E,3 and ¯�E,5.\n\nHypothesis (i) of Theorem 5.3 will be satis�ed if the image of ¯�E,p is sufficiently large in GL2(Fp) (for example, if ¯�E,p is surjective). For p = 3 and p = 5, if ¯�E,p satis�es hypothesis (iii) and is irreducible, then it satis�es hypothesis (i).\n\nIf E is semistable, p is an odd prime, and |
768 | E is semistable, p is an odd prime, and ¯�E,p is irreducible and modular, then (see the proof of Proposition 3.3) and ¯�E,p satis�es (ii) ¯�E,p is D 14 of Appendix C of [39]). Therefore by Propositions and (iii) (use Tate curves; see 4.3 and 3.3, Theorem 5.3 implies that the Semistable Modular Lifting Conjecture (Conjecture 2.1) holds for p = 3 and for p = 5. As shown in 2, the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem follow.\n\nmodular for some\n\nD\n\n§\n\n§\n\nTheorem 5.4 (Wiles). Suppose p, k, contains no nontrivial p-th roots of unity. Suppose also that there are an imaginary quadratic �eld F of discriminant prime to p and a character χ : Gal( ¯Q/F ) × such that T is the induced representation Indχ of GQ is a ( an isomorphism.\n\n, ¯�, and\n\nare as in\n\n4.2 and\n\nD\n\nO\n\n§\n\nO\n\n→ O\n\n)-lifting of ¯�. Then ϕ : R\n\n,\n\nD\n\nO\n\n→\n\nCorollary 5.5 (Wiles). Suppose E is an elliptic curve over Q with complex mul- tiplication by an imaginary quadratic �eld F and p is an odd prime at which E has good reduction. If E′ is an elliptic curve over Q satisfying\n\nE′ has good reduction at p and ¯�E′,p is isomorphic to ¯�E,p,\n\n•\n\nthen E′ is modular.\n\nProof of corollary. Let p be a prime of F containing p, and de�ne = the ring of integers of the completion of F at p,\n\nO • • •\n\n/p primes at which E or E′ has bad reduction\n\nk = Σ = t = ordinary if E has ordinary reduction at p, t = flat if E has supersingular reduction at p,\n\n,\n\nO {\n\nO\n\np\n\n,\n\n} ∪ {\n\n}\n\n= (Σ, t).\n\nD\n\nLet\n\nχ : Gal( ¯Q/F )\n\nAutO(E[p�]) ∼=\n\n×\n\n→\n\nO\n\nbe the character giving the action of Gal( ¯Q/F ) on E[p�] (where E[p�] is the group of points of E killed by the endomorphisms of E which lie in some power of p). It is not hard to see that �E,p ⊗ O\n\nis isomorphic to Indχ.\n\n20\n\nK. RUBIN AND A. SILVERBERG\n\nSince E has complex multiplication, it is well | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: E is semistable, p is an odd prime, and ¯�E,p is irreducible and modular, then (see the proof of Proposition 3.3) and ¯�E,p satis�es (ii) ¯�E,p is D 14 of Appendix C of [39]). Therefore by Propositions and (iii) (use Tate curves; see 4.3 and 3.3, Theorem 5.3 implies that the Semistable Modular Lifting Conjecture (Conjecture 2.1) holds for p = 3 and for p = 5. As shown in 2, the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem follow.\n\nmodular for some\n\nD\n\n§\n\n§\n\nTheorem 5.4 (Wiles). Suppose p, k, contains no nontrivial p-th roots of unity. Suppose also that there are an imaginary quadratic �eld F of discriminant prime to p and a character χ : Gal( ¯Q/F ) × such that T is the induced representation Indχ of GQ is a ( an isomorphism.\n\n, ¯�, and\n\nare as in\n\n4.2 and\n\nD\n\nO\n\n§\n\nO\n\n→ O\n\n)-lifting of ¯�. Then ϕ : R\n\n,\n\nD\n\nO\n\n→\n\nCorollary 5.5 (Wiles). Suppose E is an elliptic curve over Q with complex mul- tiplication by an imaginary quadratic �eld F and p is an odd prime at which E has good reduction. If E′ is an elliptic curve over Q satisfying\n\nE′ has good reduction at p and ¯�E′,p is isomorphic to ¯�E,p,\n\n•\n\nthen E′ is modular.\n\nProof of corollary. Let p be a prime of F containing p, and de�ne = the ring of integers of the completion of F at p,\n\nO • • •\n\n/p primes at which E or E′ has bad reduction\n\nk = Σ = t = ordinary if E has ordinary reduction at p, t = flat if E has supersingular reduction at p,\n\n,\n\nO {\n\nO\n\np\n\n,\n\n} ∪ {\n\n}\n\n= (Σ, t).\n\nD\n\nLet\n\nχ : Gal( ¯Q/F )\n\nAutO(E[p�]) ∼=\n\n×\n\n→\n\nO\n\nbe the character giving the action of Gal( ¯Q/F ) on E[p�] (where E[p�] is the group of points of E killed by the endomorphisms of E which lie in some power of p). It is not hard to see that �E,p ⊗ O\n\nis isomorphic to Indχ.\n\n20\n\nK. RUBIN AND A. SILVERBERG\n\nSince E has complex multiplication, it is well |
769 | E has complex multiplication, it is well known that E and ¯�E,p are mod- ular. Since E has good reduction at p, it can be shown that the discriminant of contains no nontrivial p-th roots of unity. One can show F is prime to p and that all of the hypotheses of Theorem 5.4 are satis�ed with ¯� = ¯�E,p ⊗ k. By our assumptions on E′, �E′,p ⊗ O )-lifting of ¯�, and we conclude (using the D same reasoning as in the proofs of Propositions 3.3 and 4.3) that �E′,p is modular and hence E′ is modular.\n\nO\n\nis a (\n\n,\n\nO\n\nRemarks. (i) The elliptic curves E′ of Corollary 5.5 are not semistable.\n\n(ii) Suppose E and p are as in Corollary 5.5 and p = 3 or 5. As in Appendix B.2 one can show that the elliptic curves E′ over Q with good reduction at p and with ¯�E′,p isomorphic to ¯�E,p give in�nitely many C-isomorphism classes.\n\nExample. Take E to be the elliptic curve de�ned by\n\ny2 = x3\n\nx2\n\n3x\n\n1.\n\n−\n\n−\n\n−\n\nThen E has complex multiplication by Q(√ De�ne polynomials\n\n2), and E has good reduction at 3.\n\n−\n\n1512t3 3, a4(t) = a6(t) = 40824t6 + 31104t5 + 8370t4 + 504t3\n\n2430t4\n\n396t2\n\n56t\n\n−\n\n−\n\n−\n\n−\n\n−\n\n148t2\n\n24t\n\n1,\n\n−\n\n−\n\n−\n\nQ let Et be the elliptic curve\n\nand for each t\n\n∈\n\ny2 = x3\n\nx2 + a4(t)x + a6(t)\n\n−\n\nQ, ¯�Et,3 is isomorphic to (note that E0 = E). It can be shown that for every t 0 or 1 (mod 3) (or more generally if t = 3a/b or t = 3a/b + 1 ¯�E,3. If t with a and b integers and b not divisible by 3), then Et has good reduction at 3, for instance because the discriminant of Et is\n\n∈\n\nZ and t\n\n∈\n\n≡\n\n29(27t2 + 10t + 1)3(27t2 + 18t + 1)3.\n\nThus for these values of t, Corollary 5.5 shows that Et is modular and so is any elliptic curve over Q isomorphic over C to Et, i.e., any elliptic curve over Q with j-invariant equal to\n\n3\n\n4(27t2 + 6t + 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: E has complex multiplication, it is well known that E and ¯�E,p are mod- ular. Since E has good reduction at p, it can be shown that the discriminant of contains no nontrivial p-th roots of unity. One can show F is prime to p and that all of the hypotheses of Theorem 5.4 are satis�ed with ¯� = ¯�E,p ⊗ k. By our assumptions on E′, �E′,p ⊗ O )-lifting of ¯�, and we conclude (using the D same reasoning as in the proofs of Propositions 3.3 and 4.3) that �E′,p is modular and hence E′ is modular.\n\nO\n\nis a (\n\n,\n\nO\n\nRemarks. (i) The elliptic curves E′ of Corollary 5.5 are not semistable.\n\n(ii) Suppose E and p are as in Corollary 5.5 and p = 3 or 5. As in Appendix B.2 one can show that the elliptic curves E′ over Q with good reduction at p and with ¯�E′,p isomorphic to ¯�E,p give in�nitely many C-isomorphism classes.\n\nExample. Take E to be the elliptic curve de�ned by\n\ny2 = x3\n\nx2\n\n3x\n\n1.\n\n−\n\n−\n\n−\n\nThen E has complex multiplication by Q(√ De�ne polynomials\n\n2), and E has good reduction at 3.\n\n−\n\n1512t3 3, a4(t) = a6(t) = 40824t6 + 31104t5 + 8370t4 + 504t3\n\n2430t4\n\n396t2\n\n56t\n\n−\n\n−\n\n−\n\n−\n\n−\n\n148t2\n\n24t\n\n1,\n\n−\n\n−\n\n−\n\nQ let Et be the elliptic curve\n\nand for each t\n\n∈\n\ny2 = x3\n\nx2 + a4(t)x + a6(t)\n\n−\n\nQ, ¯�Et,3 is isomorphic to (note that E0 = E). It can be shown that for every t 0 or 1 (mod 3) (or more generally if t = 3a/b or t = 3a/b + 1 ¯�E,3. If t with a and b integers and b not divisible by 3), then Et has good reduction at 3, for instance because the discriminant of Et is\n\n∈\n\nZ and t\n\n∈\n\n≡\n\n29(27t2 + 10t + 1)3(27t2 + 18t + 1)3.\n\nThus for these values of t, Corollary 5.5 shows that Et is modular and so is any elliptic curve over Q isomorphic over C to Et, i.e., any elliptic curve over Q with j-invariant equal to\n\n3\n\n4(27t2 + 6t + 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + |
770 | 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + 1)\n\n.\n\n(cid:18)\n\n(cid:19)\n\nThis explicitly gives in�nitely many modular elliptic curves over Q which are\n\nnonisomorphic over C.\n\n(For de�nitions of complex multiplication, discriminant, and j-invariant, see any\n\nstandard reference on elliptic curves, such as [39].)\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n21\n\nAppendix A. Galois groups and Frobenius elements\n\nWrite GQ = Gal( ¯Q/Q). If q is a prime number and\n\nis a prime ideal dividing\n\nQ\n\nq in the ring of integers of ¯Q, there is a �ltration\n\nGQ\n\nDQ ⊃\n\nIQ\n\n⊃ where the decomposition group DQ and the inertia group IQ are de�ned by\n\nDQ = IQ =\n\nσ\n\nGQ : σ ∈ Q ∈ DQ : σx\n\n=\n\n,\n\n{\n\nQ} x (mod\n\nσ\n\n) for all algebraic integers x }\n\n.\n\n≡ { There are natural identi�cations\n\nQ\n\nDQ/IQ ∼= Gal( ¯Fq/Fq),\n\nDQ ∼= Gal( ¯Qq/Qq),\n\nxq of GQ\n\nand FrobQ ∈ Gal( ¯Fq/Fq). If and\n\nDQ/IQ denotes the inverse image of the canonical generator x\n\n7→ for some σ\n\n′ is another prime ideal above q, then\n\n′ = σ\n\nQ DQ′ = σDQσ−1,\n\nQ\n\nQ\n\n∈\n\nFrobQ′ = σFrobQσ−1.\n\nIQ′ = σIQσ−1,\n\nSince we will care about these objects only up to conjugation, we will write Dq and GQ for any representative of a FrobQ. If � is a represen- Iq. We will write Frobq ∈ tation of GQ which is unrami�ed at q, then trace(�(Frobq)) and det(�(Frobq)) are well de�ned independent of any choices.\n\nAppendix B. Some details on the proof of Proposition 2.4\n\nB.1. The modular curve X0(15) can be viewed as a curve de�ned over Q in such a way that the noncusp rational points correspond to isomorphism classes (over C) E( ¯Q) is a subgroup of pairs (E′, 42), of order 15 stable under GQ. An equation for X0(15) is y2 = x(x + 32)(x the elliptic curve discussed in 1. There are eight rational points on X0(15), four of § which are cusps. There are four modular elliptic curves, corresponding | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + 1)\n\n.\n\n(cid:18)\n\n(cid:19)\n\nThis explicitly gives in�nitely many modular elliptic curves over Q which are\n\nnonisomorphic over C.\n\n(For de�nitions of complex multiplication, discriminant, and j-invariant, see any\n\nstandard reference on elliptic curves, such as [39].)\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n21\n\nAppendix A. Galois groups and Frobenius elements\n\nWrite GQ = Gal( ¯Q/Q). If q is a prime number and\n\nis a prime ideal dividing\n\nQ\n\nq in the ring of integers of ¯Q, there is a �ltration\n\nGQ\n\nDQ ⊃\n\nIQ\n\n⊃ where the decomposition group DQ and the inertia group IQ are de�ned by\n\nDQ = IQ =\n\nσ\n\nGQ : σ ∈ Q ∈ DQ : σx\n\n=\n\n,\n\n{\n\nQ} x (mod\n\nσ\n\n) for all algebraic integers x }\n\n.\n\n≡ { There are natural identi�cations\n\nQ\n\nDQ/IQ ∼= Gal( ¯Fq/Fq),\n\nDQ ∼= Gal( ¯Qq/Qq),\n\nxq of GQ\n\nand FrobQ ∈ Gal( ¯Fq/Fq). If and\n\nDQ/IQ denotes the inverse image of the canonical generator x\n\n7→ for some σ\n\n′ is another prime ideal above q, then\n\n′ = σ\n\nQ DQ′ = σDQσ−1,\n\nQ\n\nQ\n\n∈\n\nFrobQ′ = σFrobQσ−1.\n\nIQ′ = σIQσ−1,\n\nSince we will care about these objects only up to conjugation, we will write Dq and GQ for any representative of a FrobQ. If � is a represen- Iq. We will write Frobq ∈ tation of GQ which is unrami�ed at q, then trace(�(Frobq)) and det(�(Frobq)) are well de�ned independent of any choices.\n\nAppendix B. Some details on the proof of Proposition 2.4\n\nB.1. The modular curve X0(15) can be viewed as a curve de�ned over Q in such a way that the noncusp rational points correspond to isomorphism classes (over C) E( ¯Q) is a subgroup of pairs (E′, 42), of order 15 stable under GQ. An equation for X0(15) is y2 = x(x + 32)(x the elliptic curve discussed in 1. There are eight rational points on X0(15), four of § which are cusps. There are four modular elliptic curves, corresponding |
771 | are four modular elliptic curves, corresponding to a modular form for Γ0(50) (see p. 86 of [1]), which lie in the four distinct C-isomorphism classes that correspond to the noncusp rational points on X0(15).\n\n) where E′ is an elliptic curve over Q and\n\nC\n\nC ⊂\n\n−\n\nTherefore every elliptic curve over Q with a GQ-stable subgroup of order 15 is modular. Equivalently, if E is an elliptic curve over Q and both ¯�E,3 and ¯�E,5 are reducible, then E is modular.\n\nB.2. Fix a semistable elliptic curve E over Q. We will show that there are in�nitely many semistable elliptic curves E′ over Q such that\n\n(i) ¯�E′,5 is isomorphic to ¯�E,5, and (ii) ¯�E′,3 is irreducible. Let\n\n1 0 0 1\n\na b c d\n\na b c d\n\nSL2(Z) :\n\n(mod 5) }\n\n.\n\nΓ(5) =\n\n≡\n\n∈\n\n{\n\nLet X be the twist of the classical modular curve X(5) (see [35]) by the cocycle (cid:0) induced by ¯�E,5, and let S be the set of cusps of X. Then X is a curve de�ned over Q which has the following properties. The rational points on X − (E′, φ) where E′ is an elliptic curve over Q and φ : E[5] module isomorphism.\n\n(cid:1)\n\n(cid:0)\n\n(cid:1)\n\n(cid:1)\n\n(cid:0)\n\nS correspond to isomorphism classes of pairs E′[5] is a GQ-\n\n\n\n→\n\n22\n\nK. RUBIN AND A. SILVERBERG\n\nS is four copies of H/Γ(5), so each component of\n\nAs a complex manifold X X has genus zero.\n\n\n\n−\n\nLet X 0 be the component of X containing the rational point corresponding to (E, identity). Then X 0 is a curve of genus zero de�ned over Q with a rational point, so it has in�nitely many rational points. We want to show that in�nitely many of these points correspond to semistable elliptic curves E′ with ¯�E′,3 irreducible.\n\nThere is another modular curve ˆX de�ned over Q, with a �nite set ˆS of cusps,\n\nwhich has the following properties. The rational points on ˆX (E′, φ, module isomorphism, and As a complex manifold ˆX The map that forgets the subgroup X | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: are four modular elliptic curves, corresponding to a modular form for Γ0(50) (see p. 86 of [1]), which lie in the four distinct C-isomorphism classes that correspond to the noncusp rational points on X0(15).\n\n) where E′ is an elliptic curve over Q and\n\nC\n\nC ⊂\n\n−\n\nTherefore every elliptic curve over Q with a GQ-stable subgroup of order 15 is modular. Equivalently, if E is an elliptic curve over Q and both ¯�E,3 and ¯�E,5 are reducible, then E is modular.\n\nB.2. Fix a semistable elliptic curve E over Q. We will show that there are in�nitely many semistable elliptic curves E′ over Q such that\n\n(i) ¯�E′,5 is isomorphic to ¯�E,5, and (ii) ¯�E′,3 is irreducible. Let\n\n1 0 0 1\n\na b c d\n\na b c d\n\nSL2(Z) :\n\n(mod 5) }\n\n.\n\nΓ(5) =\n\n≡\n\n∈\n\n{\n\nLet X be the twist of the classical modular curve X(5) (see [35]) by the cocycle (cid:0) induced by ¯�E,5, and let S be the set of cusps of X. Then X is a curve de�ned over Q which has the following properties. The rational points on X − (E′, φ) where E′ is an elliptic curve over Q and φ : E[5] module isomorphism.\n\n(cid:1)\n\n(cid:0)\n\n(cid:1)\n\n(cid:1)\n\n(cid:0)\n\nS correspond to isomorphism classes of pairs E′[5] is a GQ-\n\n\n\n→\n\n22\n\nK. RUBIN AND A. SILVERBERG\n\nS is four copies of H/Γ(5), so each component of\n\nAs a complex manifold X X has genus zero.\n\n\n\n−\n\nLet X 0 be the component of X containing the rational point corresponding to (E, identity). Then X 0 is a curve of genus zero de�ned over Q with a rational point, so it has in�nitely many rational points. We want to show that in�nitely many of these points correspond to semistable elliptic curves E′ with ¯�E′,3 irreducible.\n\nThere is another modular curve ˆX de�ned over Q, with a �nite set ˆS of cusps,\n\nwhich has the following properties. The rational points on ˆX (E′, φ, module isomorphism, and As a complex manifold ˆX The map that forgets the subgroup X |
772 | manifold ˆX The map that forgets the subgroup X de�ned over Q and of degree [Γ(5) : Γ(5)\n\nˆS correspond to isomorphism classes of triples E′[5] is a GQ-\n\n\n\n−\n\n) where E′ is an elliptic curve over Q, φ : E[5]\n\nC\n\n→\n\nE′[3] is a GQ-stable subgroup of order 3.\n\nC ⊂ −\n\nˆS is four copies of H/(Γ(5)\n\nΓ0(3)).\n\n•\n\n∩ induces a surjective morphism θ : ˆX\n\nC\n\n→\n\nΓ0(3)] = 4.\n\n∩\n\nLet ˆX 0 be the component of ˆX which maps to X 0. The function �eld of X 0 is Q(t), and the function �eld of ˆX 0 is Q(t)[x]/f (t, x) where f (t, x) Q(t)[x] is irreducible and has degree 4 in x. If t′ Q is sufficiently close 5-adically to the value of t which corresponds to E, then the corresponding elliptic curve is semistable at Q so that f (t1, x) is 5. By the Hilbert Irreducibility Theorem we can �nd a t1 ∈ irreducible in Q[x]. It is possible to �x a prime ℓ = 5 such that f (t1, x) has no roots modulo ℓ. If t′ Q is sufficiently close ℓ-adically to t1, then f (t′, x) has no rational roots, and thus t′ corresponds to a rational point of X 0 which is not the image of a rational point of ˆX 0. Therefore there are in�nitely many elliptic curves E′ over Q which are semistable at 5 and satisfy\n\n∈\n\n∈\n\n6\n\n∈\n\n(i) E′[5] ∼= E[5] as GQ-modules, and (ii) E′[3] has no subgroup of order 3 stable under GQ.\n\nIt follows from (i) and the semistability of E that E′ is semistable at all primes = 5, and thus E′ is semistable. We therefore have in�nitely many semistable q elliptic curves E′ which satisfy the desired conditions.\n\n6\n\nAppendix C. Representation types\n\nSuppose A is a complete noetherian local Zp-algebra and � : GQ\n\nGL2(A) is a |Dp for the restriction of � to the decomposition group Dp.\n\n→\n\nrepresentation. Write � We say � is\n\nordinary at p if �\n\n|Dp is (after a change of basis, if necessary) of the form flat at p if � is not ordinary, and for every ideal | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: manifold ˆX The map that forgets the subgroup X de�ned over Q and of degree [Γ(5) : Γ(5)\n\nˆS correspond to isomorphism classes of triples E′[5] is a GQ-\n\n\n\n−\n\n) where E′ is an elliptic curve over Q, φ : E[5]\n\nC\n\n→\n\nE′[3] is a GQ-stable subgroup of order 3.\n\nC ⊂ −\n\nˆS is four copies of H/(Γ(5)\n\nΓ0(3)).\n\n•\n\n∩ induces a surjective morphism θ : ˆX\n\nC\n\n→\n\nΓ0(3)] = 4.\n\n∩\n\nLet ˆX 0 be the component of ˆX which maps to X 0. The function �eld of X 0 is Q(t), and the function �eld of ˆX 0 is Q(t)[x]/f (t, x) where f (t, x) Q(t)[x] is irreducible and has degree 4 in x. If t′ Q is sufficiently close 5-adically to the value of t which corresponds to E, then the corresponding elliptic curve is semistable at Q so that f (t1, x) is 5. By the Hilbert Irreducibility Theorem we can �nd a t1 ∈ irreducible in Q[x]. It is possible to �x a prime ℓ = 5 such that f (t1, x) has no roots modulo ℓ. If t′ Q is sufficiently close ℓ-adically to t1, then f (t′, x) has no rational roots, and thus t′ corresponds to a rational point of X 0 which is not the image of a rational point of ˆX 0. Therefore there are in�nitely many elliptic curves E′ over Q which are semistable at 5 and satisfy\n\n∈\n\n∈\n\n6\n\n∈\n\n(i) E′[5] ∼= E[5] as GQ-modules, and (ii) E′[3] has no subgroup of order 3 stable under GQ.\n\nIt follows from (i) and the semistability of E that E′ is semistable at all primes = 5, and thus E′ is semistable. We therefore have in�nitely many semistable q elliptic curves E′ which satisfy the desired conditions.\n\n6\n\nAppendix C. Representation types\n\nSuppose A is a complete noetherian local Zp-algebra and � : GQ\n\nGL2(A) is a |Dp for the restriction of � to the decomposition group Dp.\n\n→\n\nrepresentation. Write � We say � is\n\nordinary at p if �\n\n|Dp is (after a change of basis, if necessary) of the form flat at p if � is not ordinary, and for every ideal |
773 | at p if � is not ordinary, and for every ideal a of �nite index in A, the (cid:0) |Dp modulo a is the representation associated to the ¯Qp-points reduction of � of a �nite flat group scheme over Zp.\n\n\n\n∗ ∗ 0 χ\n\nwhere χ is unrami�ed and the * are functions from Dp to A;\n\n(cid:1)\n\n\n\nAppendix D. Selmer groups\n\nWith notation as in\n\n5 (see especially §\n\n5.2), de�ne\n\n§\n\n[ǫ]/(ǫ2, mn)\n\nOn =\n\nO\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n23\n\nwhere ǫ is an indeterminate. Then v\n\n1 + ǫv de�nes an isomorphism\n\n7→ On) : δ GL2(\n\n∼ ∈ → { HomO(pR/p2 R,\n\n(16)\n\n1 (mod ǫ) } /mn) there is a unique -algebra homomorphism → On whose restriction to pR is ǫα. Composing with the representation �R On. (In particular �0 )-lifting obtained when α = 0.) De�ne a one-cocycle cα on GQ\n\nδ\n\n.\n\nVn\n\n≡\n\nFor every α\n\nO\n\nO\n\n∈\n\nψα : R of Theorem 4.1 gives a ( denotes the ( by\n\n,\n\n)-lifting �α = ψα ◦\n\n�R of ¯� to\n\nD\n\nO\n\n,\n\nD\n\nO\n\ncα(g) = �α(g)�0(g)−1.\n\nH 1(Q, Vn). This de�nes a\n\nSince �α ≡ homomorphism\n\n�0 (mod ǫ), using (16) we can view cα ∈\n\ns : HomO(pR/p2 R,\n\n/mn)\n\nH 1(Q, Vn),\n\nO and it is not difficult to see that s is injective. The fact that �0 and �α are type- D gives information about the restrictions resq(cα) for various primes q, and using this H 1(Q, Vn) and veri�es that s information Wiles de�nes a Selmer group SD(Vn) is an isomorphism onto SD(Vn).\n\n→\n\n⊂\n\nReferences\n\n[1] B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math.,\n\nvol. 476, Springer-Verlag, New York, 1975, pp. 74–144.\n\n[2] J. Buhler, R. Crandall, R. Ernvall, and T. Mets¨ankyl¨a, Irregular primes and cyclotomic\n\ninvariants to four million, Math. Comp. 61 (1993), 151–153.\n\n[3] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1967. [4] P. Deligne and J.-P. Serre, | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: at p if � is not ordinary, and for every ideal a of �nite index in A, the (cid:0) |Dp modulo a is the representation associated to the ¯Qp-points reduction of � of a �nite flat group scheme over Zp.\n\n\n\n∗ ∗ 0 χ\n\nwhere χ is unrami�ed and the * are functions from Dp to A;\n\n(cid:1)\n\n\n\nAppendix D. Selmer groups\n\nWith notation as in\n\n5 (see especially §\n\n5.2), de�ne\n\n§\n\n[ǫ]/(ǫ2, mn)\n\nOn =\n\nO\n\nA REPORT ON WILES’ CAMBRIDGE LECTURES\n\n23\n\nwhere ǫ is an indeterminate. Then v\n\n1 + ǫv de�nes an isomorphism\n\n7→ On) : δ GL2(\n\n∼ ∈ → { HomO(pR/p2 R,\n\n(16)\n\n1 (mod ǫ) } /mn) there is a unique -algebra homomorphism → On whose restriction to pR is ǫα. Composing with the representation �R On. (In particular �0 )-lifting obtained when α = 0.) De�ne a one-cocycle cα on GQ\n\nδ\n\n.\n\nVn\n\n≡\n\nFor every α\n\nO\n\nO\n\n∈\n\nψα : R of Theorem 4.1 gives a ( denotes the ( by\n\n,\n\n)-lifting �α = ψα ◦\n\n�R of ¯� to\n\nD\n\nO\n\n,\n\nD\n\nO\n\ncα(g) = �α(g)�0(g)−1.\n\nH 1(Q, Vn). This de�nes a\n\nSince �α ≡ homomorphism\n\n�0 (mod ǫ), using (16) we can view cα ∈\n\ns : HomO(pR/p2 R,\n\n/mn)\n\nH 1(Q, Vn),\n\nO and it is not difficult to see that s is injective. The fact that �0 and �α are type- D gives information about the restrictions resq(cα) for various primes q, and using this H 1(Q, Vn) and veri�es that s information Wiles de�nes a Selmer group SD(Vn) is an isomorphism onto SD(Vn).\n\n→\n\n⊂\n\nReferences\n\n[1] B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math.,\n\nvol. 476, Springer-Verlag, New York, 1975, pp. 74–144.\n\n[2] J. Buhler, R. Crandall, R. Ernvall, and T. Mets¨ankyl¨a, Irregular primes and cyclotomic\n\ninvariants to four million, Math. Comp. 61 (1993), 151–153.\n\n[3] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1967. [4] P. Deligne and J.-P. Serre, |
774 | London, 1967. [4] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. ´Ecole Norm. Sup. (4) 7\n\n(1974), 507–530.\n\n[5] L. E. Dickson, History of the theory of numbers (Vol. II), Chelsea Publ. Co., New York, 1971. [6] H. M. Edwards, Fermat’s Last Theorem. A genetic introduction to algebraic number theory,\n\nSpringer-Verlag, New York, 1977.\n\n[7] M. Eichler, Quatern¨are quadratische Formen und die Riemannsche Vermutung f¨ur die Kon-\n\ngruenzzetafunktion, Arch. Math. (Basel) 5 (1954), 355–366.\n\n[8] G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255–299. [9]\n\n, Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Ge- ometry and Number Theory, Proceedings of the JAMI Inaugural Conference (J. I. Igusa, ed.), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80.\n\n[10] M. Flach, A �niteness theorem for the symmetric square of an elliptic curve, Invent. Math.\n\n109 (1992), 307–327.\n\n[11] G. Frey, Links between solutions of A − B = C and elliptic curves, Number Theory, Ulm 1987, Proceedings, Lecture Notes in Math., vol. 1380, Springer-Verlag, New York, 1989, pp. 31–62.\n\n[12] S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., vol. 83, Princeton\n\nUniv. Press, Princeton, NJ, 1975.\n\n[13] B. Gross, Kolyvagin’s work on modular elliptic curves, L-functions and Arithmetic, London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. [14] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fourth ed., Oxford\n\nUniv. Press, London, 1971.\n\n[15] Y. Hellegouarch, ´Etude des points d’ordre �ni des vari´et´es de dimension un d´e�nies sur un\n\nanneau principal, J. Reine Angew. Math. 244 (1970), 20–36.\n\n, Points d’ordre �ni des vari´et´es ab´eliennes de dimension un, Colloque de Th´eorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Bull. Soc. Math. France, M´em. 25, Soc. Math. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: London, 1967. [4] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. ´Ecole Norm. Sup. (4) 7\n\n(1974), 507–530.\n\n[5] L. E. Dickson, History of the theory of numbers (Vol. II), Chelsea Publ. Co., New York, 1971. [6] H. M. Edwards, Fermat’s Last Theorem. A genetic introduction to algebraic number theory,\n\nSpringer-Verlag, New York, 1977.\n\n[7] M. Eichler, Quatern¨are quadratische Formen und die Riemannsche Vermutung f¨ur die Kon-\n\ngruenzzetafunktion, Arch. Math. (Basel) 5 (1954), 355–366.\n\n[8] G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255–299. [9]\n\n, Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Ge- ometry and Number Theory, Proceedings of the JAMI Inaugural Conference (J. I. Igusa, ed.), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80.\n\n[10] M. Flach, A �niteness theorem for the symmetric square of an elliptic curve, Invent. Math.\n\n109 (1992), 307–327.\n\n[11] G. Frey, Links between solutions of A − B = C and elliptic curves, Number Theory, Ulm 1987, Proceedings, Lecture Notes in Math., vol. 1380, Springer-Verlag, New York, 1989, pp. 31–62.\n\n[12] S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., vol. 83, Princeton\n\nUniv. Press, Princeton, NJ, 1975.\n\n[13] B. Gross, Kolyvagin’s work on modular elliptic curves, L-functions and Arithmetic, London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. [14] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fourth ed., Oxford\n\nUniv. Press, London, 1971.\n\n[15] Y. Hellegouarch, ´Etude des points d’ordre �ni des vari´et´es de dimension un d´e�nies sur un\n\nanneau principal, J. Reine Angew. Math. 244 (1970), 20–36.\n\n, Points d’ordre �ni des vari´et´es ab´eliennes de dimension un, Colloque de Th´eorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Bull. Soc. Math. France, M´em. 25, Soc. Math. |
775 | Bull. Soc. Math. France, M´em. 25, Soc. Math. France, Paris, 1971, pp. 107–112.\n\n[16]\n\n, Points d’ordre �ni sur les courbes elliptiques, C. R. Acad. Sci. Paris S´er. A-B 273\n\n[17]\n\n(1971), A540–A543.\n\n24\n\nK. RUBIN AND A. SILVERBERG\n\n, Points d’ordre 2ph sur les courbes elliptiques, Acta. Arith. 26 (1974/75), 253–263. [18] [19] V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift (Vol. II) (P. Cartier et al.,\n\neds.), Birkh¨auser, Boston, 1990, pp. 435–483.\n\n[20] R. Langlands, Base change for GL(2), Ann. of Math. Stud., vol. 96, Princeton Univ. Press,\n\nPrinceton, NJ, 1980.\n\n[21] B. Mazur, Deforming Galois representations, Galois groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Math. Sci. Res. Inst. Publ., vol. 16, Springer-Verlag, New York, 1989, pp. 385–437.\n\n, Number theory as gadfly, Amer. Math. Monthly 98 (1991), 593–610.\n\n[22] [23] B. Mazur and J. Tilouine, Repr´esentations galoisiennes, diff´erentielles de K¨ahler et “conjec-\n\ntures principales�, Inst. Hautes ´Etudes Sci. Publ. Math. 71 (1990), 65–103.\n\n[24] J. Oesterl´e, Nouvelles approches du “th´eor`eme� de Fermat, S´eminaire Bourbaki no. 694\n\n(1987–1988), Ast´erisque 161/162 (1988) 165–186.\n\n, On a variation of Mazur ’s deformation functor, Compositio Math. 87 (1993), 269–\n\n[25]\n\n286.\n\n[26] P. Ribenboim, 13 lectures on Fermat ’s Last Theorem, Springer-Verlag, New York, 1979. [27] K. Ribet, On modular representations of Gal( ¯Q/Q) arising from modular forms, Invent.\n\nMath. 100 (1990), 431–476.\n\n, Report on mod ℓ representations of Gal( ¯Q/Q), Motives (U. Jannsen, S. Kleiman, and J-P. Serre, eds.), Proc. Sympos. Pure Math., vol. 55 (Part 2), Amer. Math. Soc., Providence, RI, 1994 (to appear).\n\n[28]\n\n[29] K. Rubin, The main conjecture. (Appendix to Cyclotomic �elds I and II, S. Lang), Graduate\n\nTexts in Math., vol. 121, Springer-Verlag, New York, 1990, pp. 397–419.\n\n, Kolyvagin’s | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Bull. Soc. Math. France, M´em. 25, Soc. Math. France, Paris, 1971, pp. 107–112.\n\n[16]\n\n, Points d’ordre �ni sur les courbes elliptiques, C. R. Acad. Sci. Paris S´er. A-B 273\n\n[17]\n\n(1971), A540–A543.\n\n24\n\nK. RUBIN AND A. SILVERBERG\n\n, Points d’ordre 2ph sur les courbes elliptiques, Acta. Arith. 26 (1974/75), 253–263. [18] [19] V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift (Vol. II) (P. Cartier et al.,\n\neds.), Birkh¨auser, Boston, 1990, pp. 435–483.\n\n[20] R. Langlands, Base change for GL(2), Ann. of Math. Stud., vol. 96, Princeton Univ. Press,\n\nPrinceton, NJ, 1980.\n\n[21] B. Mazur, Deforming Galois representations, Galois groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Math. Sci. Res. Inst. Publ., vol. 16, Springer-Verlag, New York, 1989, pp. 385–437.\n\n, Number theory as gadfly, Amer. Math. Monthly 98 (1991), 593–610.\n\n[22] [23] B. Mazur and J. Tilouine, Repr´esentations galoisiennes, diff´erentielles de K¨ahler et “conjec-\n\ntures principales�, Inst. Hautes ´Etudes Sci. Publ. Math. 71 (1990), 65–103.\n\n[24] J. Oesterl´e, Nouvelles approches du “th´eor`eme� de Fermat, S´eminaire Bourbaki no. 694\n\n(1987–1988), Ast´erisque 161/162 (1988) 165–186.\n\n, On a variation of Mazur ’s deformation functor, Compositio Math. 87 (1993), 269–\n\n[25]\n\n286.\n\n[26] P. Ribenboim, 13 lectures on Fermat ’s Last Theorem, Springer-Verlag, New York, 1979. [27] K. Ribet, On modular representations of Gal( ¯Q/Q) arising from modular forms, Invent.\n\nMath. 100 (1990), 431–476.\n\n, Report on mod ℓ representations of Gal( ¯Q/Q), Motives (U. Jannsen, S. Kleiman, and J-P. Serre, eds.), Proc. Sympos. Pure Math., vol. 55 (Part 2), Amer. Math. Soc., Providence, RI, 1994 (to appear).\n\n[28]\n\n[29] K. Rubin, The main conjecture. (Appendix to Cyclotomic �elds I and II, S. Lang), Graduate\n\nTexts in Math., vol. 121, Springer-Verlag, New York, 1990, pp. 397–419.\n\n, Kolyvagin’s |
776 | New York, 1990, pp. 397–419.\n\n, Kolyvagin’s system of Gauss sums, Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, and J. Steenbrink, eds.), Progr. Math., vol. 89, Birkh¨auser, Boston, 1991, pp. 309–324.\n\n[30]\n\n, The “main conjectures� of Iwasawa theory for imaginary quadratic �elds, Invent.\n\n[31]\n\nMath. 103 (1991), 25–68.\n\n[32] J.-P. Serre, Sur les repr´esentations modulaires de degr´e 2 de Gal( ¯Q/Q), Duke Math. J. 54\n\n(1987), 179–230.\n\n[33] G. Shimura, Correspondances modulaires et les fonctions ζ de courbes alg´ebriques, J. Math.\n\nSoc. Japan 10 (1958), 1–28.\n\n, Construction of class �elds and zeta functions of algebraic curves, Ann. of Math.\n\n[34]\n\n85 (1967), 58–159.\n\n, Introduction to the arithmetic theory of automorphic functions, Princeton Univ.\n\n[35]\n\nPress, Princeton, NJ, 1971.\n\n, On elliptic curves with complex multiplication as factors of the Jacobians of modular\n\n[36]\n\nfunction �elds, Nagoya Math. J. 43 (1971), 199–208.\n\n, On the factors of the jacobian variety of a modular function �eld, J. Math. Soc.\n\n[37]\n\nJapan 25 (1973), 523–544.\n\n, Yutaka Taniyama and his time. Very personal recollections, Bull. London Math.\n\n[38]\n\nSoc. 21 (1989), 186–196.\n\n[39] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-\n\nVerlag, New York, 1986.\n\n[40] F. Thaine, On the ideal class groups of real abelian number �elds, Ann. of Math. (2) 128\n\n(1988), 1–18.\n\n[41] J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc.\n\n(N.S.) 5 (1981), 173–175.\n\n[42] A. Weil, ¨Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math.\n\nAnn. 168 (1967), 149–156.\n\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: [email protected]\n\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: New York, 1990, pp. 397–419.\n\n, Kolyvagin’s system of Gauss sums, Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, and J. Steenbrink, eds.), Progr. Math., vol. 89, Birkh¨auser, Boston, 1991, pp. 309–324.\n\n[30]\n\n, The “main conjectures� of Iwasawa theory for imaginary quadratic �elds, Invent.\n\n[31]\n\nMath. 103 (1991), 25–68.\n\n[32] J.-P. Serre, Sur les repr´esentations modulaires de degr´e 2 de Gal( ¯Q/Q), Duke Math. J. 54\n\n(1987), 179–230.\n\n[33] G. Shimura, Correspondances modulaires et les fonctions ζ de courbes alg´ebriques, J. Math.\n\nSoc. Japan 10 (1958), 1–28.\n\n, Construction of class �elds and zeta functions of algebraic curves, Ann. of Math.\n\n[34]\n\n85 (1967), 58–159.\n\n, Introduction to the arithmetic theory of automorphic functions, Princeton Univ.\n\n[35]\n\nPress, Princeton, NJ, 1971.\n\n, On elliptic curves with complex multiplication as factors of the Jacobians of modular\n\n[36]\n\nfunction �elds, Nagoya Math. J. 43 (1971), 199–208.\n\n, On the factors of the jacobian variety of a modular function �eld, J. Math. Soc.\n\n[37]\n\nJapan 25 (1973), 523–544.\n\n, Yutaka Taniyama and his time. Very personal recollections, Bull. London Math.\n\n[38]\n\nSoc. 21 (1989), 186–196.\n\n[39] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-\n\nVerlag, New York, 1986.\n\n[40] F. Thaine, On the ideal class groups of real abelian number �elds, Ann. of Math. (2) 128\n\n(1988), 1–18.\n\n[41] J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc.\n\n(N.S.) 5 (1981), 173–175.\n\n[42] A. Weil, ¨Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math.\n\nAnn. 168 (1967), 149–156.\n\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: [email protected]\n\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: |
777 | University, Columbus, Ohio 43210 E-mail address: [email protected]' metadata={'source': '/var/folders/l1/lphj87z16c3282pjwy91wtm80000gn/T/tmpdh5kk5yb/tmp.pdf'} page_content='This is text file' metadata={'source': 'dropbox:///test.txt', 'title': 'test.txt'}PreviousDocugamiNextDuckDBPrerequisitesInstructionsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. | Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: University, Columbus, Ohio 43210 E-mail address: [email protected]' metadata={'source': '/var/folders/l1/lphj87z16c3282pjwy91wtm80000gn/T/tmpdh5kk5yb/tmp.pdf'} page_content='This is text file' metadata={'source': 'dropbox:///test.txt', 'title': 'test.txt'}PreviousDocugamiNextDuckDBPrerequisitesInstructionsCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
778 | Xorbits Pandas DataFrame | ü¶úÔ∏èüîó Langchain | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: Xorbits Pandas DataFrame | ü¶úÔ∏èüîó Langchain |
779 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersXorbits Pandas DataFrameXorbits Pandas DataFrameThis notebook goes over how to load data from | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersXorbits Pandas DataFrameXorbits Pandas DataFrameThis notebook goes over how to load data from |
780 | notebook goes over how to load data from a xorbits.pandas DataFrame.#!pip install xorbitsimport xorbits.pandas as pddf = pd.read_csv("example_data/mlb_teams_2012.csv")df.head() 0%| | 0.00/100 [00:00<?, ?it/s]<div><style scoped> .dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }</style><table border="1" class="dataframe"> <thead> <tr style="text-align: right;"> <th></th> <th>Team</th> <th>"Payroll (millions)"</th> <th>"Wins"</th> </tr> </thead> <tbody> <tr> <th>0</th> <td>Nationals</td> <td>81.34</td> <td>98</td> </tr> <tr> <th>1</th> <td>Reds</td> <td>82.20</td> <td>97</td> </tr> <tr> <th>2</th> <td>Yankees</td> <td>197.96</td> <td>95</td> </tr> <tr> <th>3</th> <td>Giants</td> <td>117.62</td> <td>94</td> </tr> <tr> <th>4</th> <td>Braves</td> <td>83.31</td> <td>94</td> </tr> </tbody></table></div>from langchain.document_loaders import XorbitsLoaderloader = XorbitsLoader(df, page_content_column="Team")loader.load() 0%| | 0.00/100 [00:00<?, ?it/s] [Document(page_content='Nationals', metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98}), Document(page_content='Reds', metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97}), Document(page_content='Yankees', metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95}), Document(page_content='Giants', metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94}), Document(page_content='Braves', metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94}), Document(page_content='Athletics', metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94}), Document(page_content='Rangers', metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93}), Document(page_content='Orioles', | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: notebook goes over how to load data from a xorbits.pandas DataFrame.#!pip install xorbitsimport xorbits.pandas as pddf = pd.read_csv("example_data/mlb_teams_2012.csv")df.head() 0%| | 0.00/100 [00:00<?, ?it/s]<div><style scoped> .dataframe tbody tr th:only-of-type { vertical-align: middle; } .dataframe tbody tr th { vertical-align: top; } .dataframe thead th { text-align: right; }</style><table border="1" class="dataframe"> <thead> <tr style="text-align: right;"> <th></th> <th>Team</th> <th>"Payroll (millions)"</th> <th>"Wins"</th> </tr> </thead> <tbody> <tr> <th>0</th> <td>Nationals</td> <td>81.34</td> <td>98</td> </tr> <tr> <th>1</th> <td>Reds</td> <td>82.20</td> <td>97</td> </tr> <tr> <th>2</th> <td>Yankees</td> <td>197.96</td> <td>95</td> </tr> <tr> <th>3</th> <td>Giants</td> <td>117.62</td> <td>94</td> </tr> <tr> <th>4</th> <td>Braves</td> <td>83.31</td> <td>94</td> </tr> </tbody></table></div>from langchain.document_loaders import XorbitsLoaderloader = XorbitsLoader(df, page_content_column="Team")loader.load() 0%| | 0.00/100 [00:00<?, ?it/s] [Document(page_content='Nationals', metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98}), Document(page_content='Reds', metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97}), Document(page_content='Yankees', metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95}), Document(page_content='Giants', metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94}), Document(page_content='Braves', metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94}), Document(page_content='Athletics', metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94}), Document(page_content='Rangers', metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93}), Document(page_content='Orioles', |
781 | 93}), Document(page_content='Orioles', metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93}), Document(page_content='Rays', metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90}), Document(page_content='Angels', metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89}), Document(page_content='Tigers', metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88}), Document(page_content='Cardinals', metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88}), Document(page_content='Dodgers', metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86}), Document(page_content='White Sox', metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85}), Document(page_content='Brewers', metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83}), Document(page_content='Phillies', metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81}), Document(page_content='Diamondbacks', metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81}), Document(page_content='Pirates', metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79}), Document(page_content='Padres', metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76}), Document(page_content='Mariners', metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75}), Document(page_content='Mets', metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74}), Document(page_content='Blue Jays', metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73}), Document(page_content='Royals', metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72}), Document(page_content='Marlins', metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69}), Document(page_content='Red Sox', metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69}), Document(page_content='Indians', metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68}), Document(page_content='Twins', metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66}), Document(page_content='Rockies', metadata={' "Payroll | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: 93}), Document(page_content='Orioles', metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93}), Document(page_content='Rays', metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90}), Document(page_content='Angels', metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89}), Document(page_content='Tigers', metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88}), Document(page_content='Cardinals', metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88}), Document(page_content='Dodgers', metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86}), Document(page_content='White Sox', metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85}), Document(page_content='Brewers', metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83}), Document(page_content='Phillies', metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81}), Document(page_content='Diamondbacks', metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81}), Document(page_content='Pirates', metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79}), Document(page_content='Padres', metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76}), Document(page_content='Mariners', metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75}), Document(page_content='Mets', metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74}), Document(page_content='Blue Jays', metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73}), Document(page_content='Royals', metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72}), Document(page_content='Marlins', metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69}), Document(page_content='Red Sox', metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69}), Document(page_content='Indians', metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68}), Document(page_content='Twins', metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66}), Document(page_content='Rockies', metadata={' "Payroll |
782 | metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64}), Document(page_content='Cubs', metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61}), Document(page_content='Astros', metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55})]# Use lazy load for larger table, which won't read the full table into memoryfor i in loader.lazy_load(): print(i) 0%| | 0.00/100 [00:00<?, ?it/s] page_content='Nationals' metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98} page_content='Reds' metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97} page_content='Yankees' metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95} page_content='Giants' metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94} page_content='Braves' metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94} page_content='Athletics' metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94} page_content='Rangers' metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93} page_content='Orioles' metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93} page_content='Rays' metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90} page_content='Angels' metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89} page_content='Tigers' metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88} page_content='Cardinals' metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88} page_content='Dodgers' metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86} page_content='White Sox' metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85} page_content='Brewers' metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83} page_content='Phillies' metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81} page_content='Diamondbacks' metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81} page_content='Pirates' metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79} page_content='Padres' metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76} | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64}), Document(page_content='Cubs', metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61}), Document(page_content='Astros', metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55})]# Use lazy load for larger table, which won't read the full table into memoryfor i in loader.lazy_load(): print(i) 0%| | 0.00/100 [00:00<?, ?it/s] page_content='Nationals' metadata={' "Payroll (millions)"': 81.34, ' "Wins"': 98} page_content='Reds' metadata={' "Payroll (millions)"': 82.2, ' "Wins"': 97} page_content='Yankees' metadata={' "Payroll (millions)"': 197.96, ' "Wins"': 95} page_content='Giants' metadata={' "Payroll (millions)"': 117.62, ' "Wins"': 94} page_content='Braves' metadata={' "Payroll (millions)"': 83.31, ' "Wins"': 94} page_content='Athletics' metadata={' "Payroll (millions)"': 55.37, ' "Wins"': 94} page_content='Rangers' metadata={' "Payroll (millions)"': 120.51, ' "Wins"': 93} page_content='Orioles' metadata={' "Payroll (millions)"': 81.43, ' "Wins"': 93} page_content='Rays' metadata={' "Payroll (millions)"': 64.17, ' "Wins"': 90} page_content='Angels' metadata={' "Payroll (millions)"': 154.49, ' "Wins"': 89} page_content='Tigers' metadata={' "Payroll (millions)"': 132.3, ' "Wins"': 88} page_content='Cardinals' metadata={' "Payroll (millions)"': 110.3, ' "Wins"': 88} page_content='Dodgers' metadata={' "Payroll (millions)"': 95.14, ' "Wins"': 86} page_content='White Sox' metadata={' "Payroll (millions)"': 96.92, ' "Wins"': 85} page_content='Brewers' metadata={' "Payroll (millions)"': 97.65, ' "Wins"': 83} page_content='Phillies' metadata={' "Payroll (millions)"': 174.54, ' "Wins"': 81} page_content='Diamondbacks' metadata={' "Payroll (millions)"': 74.28, ' "Wins"': 81} page_content='Pirates' metadata={' "Payroll (millions)"': 63.43, ' "Wins"': 79} page_content='Padres' metadata={' "Payroll (millions)"': 55.24, ' "Wins"': 76} |
783 | "Payroll (millions)"': 55.24, ' "Wins"': 76} page_content='Mariners' metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75} page_content='Mets' metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74} page_content='Blue Jays' metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73} page_content='Royals' metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72} page_content='Marlins' metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69} page_content='Red Sox' metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69} page_content='Indians' metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68} page_content='Twins' metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66} page_content='Rockies' metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64} page_content='Cubs' metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61} page_content='Astros' metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55}PreviousXMLNextYouTube audioCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This notebook goes over how to load data from a xorbits.pandas DataFrame. | This notebook goes over how to load data from a xorbits.pandas DataFrame. ->: "Payroll (millions)"': 55.24, ' "Wins"': 76} page_content='Mariners' metadata={' "Payroll (millions)"': 81.97, ' "Wins"': 75} page_content='Mets' metadata={' "Payroll (millions)"': 93.35, ' "Wins"': 74} page_content='Blue Jays' metadata={' "Payroll (millions)"': 75.48, ' "Wins"': 73} page_content='Royals' metadata={' "Payroll (millions)"': 60.91, ' "Wins"': 72} page_content='Marlins' metadata={' "Payroll (millions)"': 118.07, ' "Wins"': 69} page_content='Red Sox' metadata={' "Payroll (millions)"': 173.18, ' "Wins"': 69} page_content='Indians' metadata={' "Payroll (millions)"': 78.43, ' "Wins"': 68} page_content='Twins' metadata={' "Payroll (millions)"': 94.08, ' "Wins"': 66} page_content='Rockies' metadata={' "Payroll (millions)"': 78.06, ' "Wins"': 64} page_content='Cubs' metadata={' "Payroll (millions)"': 88.19, ' "Wins"': 61} page_content='Astros' metadata={' "Payroll (millions)"': 60.65, ' "Wins"': 55}PreviousXMLNextYouTube audioCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
784 | Tencent COS Directory | ü¶úÔ∏èüîó Langchain | This covers how to load document objects from a Tencent COS Directory. | This covers how to load document objects from a Tencent COS Directory. ->: Tencent COS Directory | ü¶úÔ∏èüîó Langchain |
785 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersTencent COS DirectoryOn this pageTencent COS DirectoryThis covers how to load document objects | This covers how to load document objects from a Tencent COS Directory. | This covers how to load document objects from a Tencent COS Directory. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersTencent COS DirectoryOn this pageTencent COS DirectoryThis covers how to load document objects |
786 | DirectoryThis covers how to load document objects from a Tencent COS Directory.#! pip install cos-python-sdk-v5from langchain.document_loaders import TencentCOSDirectoryLoaderfrom qcloud_cos import CosConfigconf = CosConfig( Region="your cos region", SecretId="your cos secret_id", SecretKey="your cos secret_key",)loader = TencentCOSDirectoryLoader(conf=conf, bucket="you_cos_bucket")loader.load()Specifying a prefix​You can also specify a prefix for more finegrained control over what files to load.loader = TencentCOSDirectoryLoader(conf=conf, bucket="you_cos_bucket", prefix="fake")loader.load()PreviousTelegramNextTencent COS FileSpecifying a prefixCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | This covers how to load document objects from a Tencent COS Directory. | This covers how to load document objects from a Tencent COS Directory. ->: DirectoryThis covers how to load document objects from a Tencent COS Directory.#! pip install cos-python-sdk-v5from langchain.document_loaders import TencentCOSDirectoryLoaderfrom qcloud_cos import CosConfigconf = CosConfig( Region="your cos region", SecretId="your cos secret_id", SecretKey="your cos secret_key",)loader = TencentCOSDirectoryLoader(conf=conf, bucket="you_cos_bucket")loader.load()Specifying a prefix​You can also specify a prefix for more finegrained control over what files to load.loader = TencentCOSDirectoryLoader(conf=conf, bucket="you_cos_bucket", prefix="fake")loader.load()PreviousTelegramNextTencent COS FileSpecifying a prefixCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
787 | iFixit | ü¶úÔ∏èüîó Langchain | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: iFixit | ü¶úÔ∏èüîó Langchain |
788 | Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersiFixitOn this pageiFixitiFixit is the largest, open repair community on the web. The site | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: Skip to main contentü¶úÔ∏èüîó LangChainDocsUse casesIntegrationsAPICommunityChat our docsLangSmithJS/TS DocsSearchCTRLKProvidersAnthropicAWSGoogleMicrosoftOpenAIMoreComponentsLLMsChat modelsDocument loadersacreomAirbyte CDKAirbyte GongAirbyte HubspotAirbyte JSONAirbyte SalesforceAirbyte ShopifyAirbyte StripeAirbyte TypeformAirbyte Zendesk SupportAirtableAlibaba Cloud MaxComputeApify DatasetArcGISArxivAssemblyAI Audio TranscriptsAsync ChromiumAsyncHtmlAWS S3 DirectoryAWS S3 FileAZLyricsAzure Blob Storage ContainerAzure Blob Storage FileAzure Document IntelligenceBibTeXBiliBiliBlackboardBlockchainBrave SearchBrowserlessChatGPT DataCollege ConfidentialConcurrent LoaderConfluenceCoNLL-UCopy PasteCSVCube Semantic LayerDatadog LogsDiffbotDiscordDocugamiDropboxDuckDBEmailEmbaasEPubEtherscanEverNoteexample_dataMicrosoft ExcelFacebook ChatFaunaFigmaGeopandasGitGitBookGitHubGoogle BigQueryGoogle Cloud Storage DirectoryGoogle Cloud Storage FileGoogle DriveGrobidGutenbergHacker NewsHuawei OBS DirectoryHuawei OBS FileHuggingFace datasetiFixitImagesImage captionsIMSDbIuguJoplinJupyter NotebookLarkSuite (FeiShu)MastodonMediaWiki DumpMerge Documents LoadermhtmlMicrosoft OneDriveMicrosoft PowerPointMicrosoft SharePointMicrosoft WordModern TreasuryMongoDBNews URLNotion DB 1/2Notion DB 2/2NucliaObsidianOpen Document Format (ODT)Open City DataOrg-modePandas DataFrameAmazon TextractPolars DataFramePsychicPubMedPySparkReadTheDocs DocumentationRecursive URLRedditRoamRocksetrspaceRSS FeedsRSTSitemapSlackSnowflakeSource CodeSpreedlyStripeSubtitleTelegramTencent COS DirectoryTencent COS FileTensorFlow Datasets2MarkdownTOMLTrelloTSVTwitterUnstructured FileURLWeatherWebBaseLoaderWhatsApp ChatWikipediaXMLXorbits Pandas DataFrameYouTube audioYouTube transcriptsDocument transformersText embedding modelsVector storesRetrieversToolsAgents and toolkitsMemoryCallbacksChat loadersComponentsDocument loadersiFixitOn this pageiFixitiFixit is the largest, open repair community on the web. The site |
789 | open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0.This loader will allow you to download the text of a repair guide, text of Q&A's and wikis from devices on iFixit using their open APIs. It's incredibly useful for context related to technical documents and answers to questions about devices in the corpus of data on iFixit.from langchain.document_loaders import IFixitLoaderloader = IFixitLoader("https://www.ifixit.com/Teardown/Banana+Teardown/811")data = loader.load()data [Document(page_content="# Banana Teardown\nIn this teardown, we open a banana to see what's inside. Yellow and delicious, but most importantly, yellow.\n\n\n###Tools Required:\n\n - Fingers\n\n - Teeth\n\n - Thumbs\n\n\n###Parts Required:\n\n - None\n\n\n## Step 1\nTake one banana from the bunch.\nDon't squeeze too hard!\n\n\n## Step 2\nHold the banana in your left hand and grip the stem between your right thumb and forefinger.\n\n\n## Step 3\nPull the stem downward until the peel splits.\n\n\n## Step 4\nInsert your thumbs into the split of the peel and pull the two sides apart.\nExpose the top of the banana. It may be slightly squished from pulling on the stem, but this will not affect the flavor.\n\n\n## Step 5\nPull open the peel, starting from your original split, and opening it along the length of the banana.\n\n\n## Step 6\nRemove fruit from peel.\n\n\n## Step 7\nEat and enjoy!\nThis is where you'll need your teeth.\nDo not choke on banana!\n", lookup_str='', metadata={'source': 'https://www.ifixit.com/Teardown/Banana+Teardown/811', 'title': 'Banana Teardown'}, lookup_index=0)]loader = IFixitLoader( "https://www.ifixit.com/Answers/View/318583/My+iPhone+6+is+typing+and+opening+apps+by+itself")data = loader.load()data [Document(page_content='# My iPhone 6 is typing and opening apps by itself\nmy iphone 6 is typing and opening apps by itself. How do i fix this. I | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0.This loader will allow you to download the text of a repair guide, text of Q&A's and wikis from devices on iFixit using their open APIs. It's incredibly useful for context related to technical documents and answers to questions about devices in the corpus of data on iFixit.from langchain.document_loaders import IFixitLoaderloader = IFixitLoader("https://www.ifixit.com/Teardown/Banana+Teardown/811")data = loader.load()data [Document(page_content="# Banana Teardown\nIn this teardown, we open a banana to see what's inside. Yellow and delicious, but most importantly, yellow.\n\n\n###Tools Required:\n\n - Fingers\n\n - Teeth\n\n - Thumbs\n\n\n###Parts Required:\n\n - None\n\n\n## Step 1\nTake one banana from the bunch.\nDon't squeeze too hard!\n\n\n## Step 2\nHold the banana in your left hand and grip the stem between your right thumb and forefinger.\n\n\n## Step 3\nPull the stem downward until the peel splits.\n\n\n## Step 4\nInsert your thumbs into the split of the peel and pull the two sides apart.\nExpose the top of the banana. It may be slightly squished from pulling on the stem, but this will not affect the flavor.\n\n\n## Step 5\nPull open the peel, starting from your original split, and opening it along the length of the banana.\n\n\n## Step 6\nRemove fruit from peel.\n\n\n## Step 7\nEat and enjoy!\nThis is where you'll need your teeth.\nDo not choke on banana!\n", lookup_str='', metadata={'source': 'https://www.ifixit.com/Teardown/Banana+Teardown/811', 'title': 'Banana Teardown'}, lookup_index=0)]loader = IFixitLoader( "https://www.ifixit.com/Answers/View/318583/My+iPhone+6+is+typing+and+opening+apps+by+itself")data = loader.load()data [Document(page_content='# My iPhone 6 is typing and opening apps by itself\nmy iphone 6 is typing and opening apps by itself. How do i fix this. I |
790 | and opening apps by itself. How do i fix this. I just bought it last week.\nI restored as manufactures cleaned up the screen\nthe problem continues\n\n## 27 Answers\n\nFilter by: \n\nMost Helpful\nNewest\nOldest\n\n### Accepted Answer\nHi,\nWhere did you buy it? If you bought it from Apple or from an official retailer like Carphone warehouse etc. Then you\'ll have a year warranty and can get it replaced free.\nIf you bought it second hand, from a third part repair shop or online, then it may still have warranty, unless it is refurbished and has been repaired elsewhere.\nIf this is the case, it may be the screen that needs replacing to solve your issue.\nEither way, wherever you got it, it\'s best to return it and get a refund or a replacement device. :-)\n\n\n\n### Most Helpful Answer\nI had the same issues, screen freezing, opening apps by itself, selecting the screens and typing on it\'s own. I first suspected aliens and then ghosts and then hackers.\niPhone 6 is weak physically and tend to bend on pressure. And my phone had no case or cover.\nI took the phone to apple stores and they said sensors need to be replaced and possibly screen replacement as well. My phone is just 17 months old.\nHere is what I did two days ago and since then it is working like a charm..\nHold the phone in portrait (as if watching a movie). Twist it very very gently. do it few times.Rest the phone for 10 mins (put it on a flat surface). You can now notice those self typing things gone and screen getting stabilized.\nThen, reset the hardware (hold the power and home button till the screen goes off and comes back with apple logo). release the buttons when you see this.\nThen, connect to your laptop and log in to iTunes and reset your phone completely. (please take a back-up first).\nAnd your phone should be good to use again.\nWhat really happened here for me is that the sensors might have stuck to the screen and with mild twisting, they got disengaged/released.\nI posted this in Apple | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: and opening apps by itself. How do i fix this. I just bought it last week.\nI restored as manufactures cleaned up the screen\nthe problem continues\n\n## 27 Answers\n\nFilter by: \n\nMost Helpful\nNewest\nOldest\n\n### Accepted Answer\nHi,\nWhere did you buy it? If you bought it from Apple or from an official retailer like Carphone warehouse etc. Then you\'ll have a year warranty and can get it replaced free.\nIf you bought it second hand, from a third part repair shop or online, then it may still have warranty, unless it is refurbished and has been repaired elsewhere.\nIf this is the case, it may be the screen that needs replacing to solve your issue.\nEither way, wherever you got it, it\'s best to return it and get a refund or a replacement device. :-)\n\n\n\n### Most Helpful Answer\nI had the same issues, screen freezing, opening apps by itself, selecting the screens and typing on it\'s own. I first suspected aliens and then ghosts and then hackers.\niPhone 6 is weak physically and tend to bend on pressure. And my phone had no case or cover.\nI took the phone to apple stores and they said sensors need to be replaced and possibly screen replacement as well. My phone is just 17 months old.\nHere is what I did two days ago and since then it is working like a charm..\nHold the phone in portrait (as if watching a movie). Twist it very very gently. do it few times.Rest the phone for 10 mins (put it on a flat surface). You can now notice those self typing things gone and screen getting stabilized.\nThen, reset the hardware (hold the power and home button till the screen goes off and comes back with apple logo). release the buttons when you see this.\nThen, connect to your laptop and log in to iTunes and reset your phone completely. (please take a back-up first).\nAnd your phone should be good to use again.\nWhat really happened here for me is that the sensors might have stuck to the screen and with mild twisting, they got disengaged/released.\nI posted this in Apple |
791 | got disengaged/released.\nI posted this in Apple Community and the moderators deleted it, for the best reasons known to them.\nInstead of throwing away your phone (or selling cheaply), try this and you could be saving your phone.\nLet me know how it goes.\n\n\n\n### Other Answer\nIt was the charging cord! I bought a gas station braided cord and it was the culprit. Once I plugged my OEM cord into the phone the GHOSTS went away.\n\n\n\n### Other Answer\nI\'ve same issue that I just get resolved. I first tried to restore it from iCloud back, however it was not a software issue or any virus issue, so after restore same problem continues. Then I get my phone to local area iphone repairing lab, and they detected that it is an LCD issue. LCD get out of order without any reason (It was neither hit or nor slipped, but LCD get out of order all and sudden, while using it) it started opening things at random. I get LCD replaced with new one, that cost me $80.00 in total ($70.00 LCD charges + $10.00 as labor charges to fix it). iPhone is back to perfect mode now. It was iphone 6s. Thanks.\n\n\n\n### Other Answer\nI was having the same issue with my 6 plus, I took it to a repair shop, they opened the phone, disconnected the three ribbons the screen has, blew up and cleaned the connectors and connected the screen again and it solved the issue… it’s hardware, not software.\n\n\n\n### Other Answer\nHey.\nJust had this problem now. As it turns out, you just need to plug in your phone. I use a case and when I took it off I noticed that there was a lot of dust and dirt around the areas that the case didn\'t cover. I shined a light in my ports and noticed they were filled with dust. Tomorrow I plan on using pressurized air to clean it out and the problem should be solved. If you plug in your phone and unplug it and it stops the issue, I recommend cleaning your phone thoroughly.\n\n\n\n### Other Answer\nI simply changed the power supply and problem was gone. The block that plugs | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: got disengaged/released.\nI posted this in Apple Community and the moderators deleted it, for the best reasons known to them.\nInstead of throwing away your phone (or selling cheaply), try this and you could be saving your phone.\nLet me know how it goes.\n\n\n\n### Other Answer\nIt was the charging cord! I bought a gas station braided cord and it was the culprit. Once I plugged my OEM cord into the phone the GHOSTS went away.\n\n\n\n### Other Answer\nI\'ve same issue that I just get resolved. I first tried to restore it from iCloud back, however it was not a software issue or any virus issue, so after restore same problem continues. Then I get my phone to local area iphone repairing lab, and they detected that it is an LCD issue. LCD get out of order without any reason (It was neither hit or nor slipped, but LCD get out of order all and sudden, while using it) it started opening things at random. I get LCD replaced with new one, that cost me $80.00 in total ($70.00 LCD charges + $10.00 as labor charges to fix it). iPhone is back to perfect mode now. It was iphone 6s. Thanks.\n\n\n\n### Other Answer\nI was having the same issue with my 6 plus, I took it to a repair shop, they opened the phone, disconnected the three ribbons the screen has, blew up and cleaned the connectors and connected the screen again and it solved the issue… it’s hardware, not software.\n\n\n\n### Other Answer\nHey.\nJust had this problem now. As it turns out, you just need to plug in your phone. I use a case and when I took it off I noticed that there was a lot of dust and dirt around the areas that the case didn\'t cover. I shined a light in my ports and noticed they were filled with dust. Tomorrow I plan on using pressurized air to clean it out and the problem should be solved. If you plug in your phone and unplug it and it stops the issue, I recommend cleaning your phone thoroughly.\n\n\n\n### Other Answer\nI simply changed the power supply and problem was gone. The block that plugs |
792 | supply and problem was gone. The block that plugs in the wall not the sub cord. The cord was fine but not the block.\n\n\n\n### Other Answer\nSomeone ask! I purchased my iPhone 6s Plus for 1000 from at&t. Before I touched it, I purchased a otter defender case. I read where at&t said touch desease was due to dropping! Bullshit!! I am 56 I have never dropped it!! Looks brand new! Never dropped or abused any way! I have my original charger. I am going to clean it and try everyone’s advice. It really sucks! I had 40,000,000 on my heart of Vegas slots! I play every day. I would be spinning and my fingers were no where max buttons and it would light up and switch to max. It did it 3 times before I caught it light up by its self. It sucks. Hope I can fix it!!!!\n\n\n\n### Other Answer\nNo answer, but same problem with iPhone 6 plus--random, self-generated jumping amongst apps and typing on its own--plus freezing regularly (aha--maybe that\'s what the "plus" in "6 plus" refers to?). An Apple Genius recommended upgrading to iOS 11.3.1 from 11.2.2, to see if that fixed the trouble. If it didn\'t, Apple will sell me a new phone for $168! Of couese the OS upgrade didn\'t fix the problem. Thanks for helping me figure out that it\'s most likely a hardware problem--which the "genius" probably knows too.\nI\'m getting ready to go Android.\n\n\n\n### Other Answer\nI experienced similar ghost touches. Two weeks ago, I changed my iPhone 6 Plus shell (I had forced the phone into it because it’s pretty tight), and also put a new glass screen protector (the edges of the protector don’t stick to the screen, weird, so I brushed pressure on the edges at times to see if they may smooth out one day miraculously). I’m not sure if I accidentally bend the phone when I installed the shell, or, if I got a defective glass protector that messes up the touch sensor. Well, yesterday was the worse day, keeps dropping calls and ghost pressing keys for me when I was on a | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: supply and problem was gone. The block that plugs in the wall not the sub cord. The cord was fine but not the block.\n\n\n\n### Other Answer\nSomeone ask! I purchased my iPhone 6s Plus for 1000 from at&t. Before I touched it, I purchased a otter defender case. I read where at&t said touch desease was due to dropping! Bullshit!! I am 56 I have never dropped it!! Looks brand new! Never dropped or abused any way! I have my original charger. I am going to clean it and try everyone’s advice. It really sucks! I had 40,000,000 on my heart of Vegas slots! I play every day. I would be spinning and my fingers were no where max buttons and it would light up and switch to max. It did it 3 times before I caught it light up by its self. It sucks. Hope I can fix it!!!!\n\n\n\n### Other Answer\nNo answer, but same problem with iPhone 6 plus--random, self-generated jumping amongst apps and typing on its own--plus freezing regularly (aha--maybe that\'s what the "plus" in "6 plus" refers to?). An Apple Genius recommended upgrading to iOS 11.3.1 from 11.2.2, to see if that fixed the trouble. If it didn\'t, Apple will sell me a new phone for $168! Of couese the OS upgrade didn\'t fix the problem. Thanks for helping me figure out that it\'s most likely a hardware problem--which the "genius" probably knows too.\nI\'m getting ready to go Android.\n\n\n\n### Other Answer\nI experienced similar ghost touches. Two weeks ago, I changed my iPhone 6 Plus shell (I had forced the phone into it because it’s pretty tight), and also put a new glass screen protector (the edges of the protector don’t stick to the screen, weird, so I brushed pressure on the edges at times to see if they may smooth out one day miraculously). I’m not sure if I accidentally bend the phone when I installed the shell, or, if I got a defective glass protector that messes up the touch sensor. Well, yesterday was the worse day, keeps dropping calls and ghost pressing keys for me when I was on a |
793 | and ghost pressing keys for me when I was on a call. I got fed up, so I removed the screen protector, and so far problems have not reoccurred yet. I’m crossing my fingers that problems indeed solved.\n\n\n\n### Other Answer\nthank you so much for this post! i was struggling doing the reset because i cannot type userids and passwords correctly because the iphone 6 plus i have kept on typing letters incorrectly. I have been doing it for a day until i come across this article. Very helpful! God bless you!!\n\n\n\n### Other Answer\nI just turned it off, and turned it back on.\n\n\n\n### Other Answer\nMy problem has not gone away completely but its better now i changed my charger and turned off prediction ....,,,now it rarely happens\n\n\n\n### Other Answer\nI tried all of the above. I then turned off my home cleaned it with isopropyl alcohol 90%. Then I baked it in my oven on warm for an hour and a half over foil. Took it out and set it cool completely on the glass top stove. Then I turned on and it worked.\n\n\n\n### Other Answer\nI think at& t should man up and fix your phone for free! You pay a lot for a Apple they should back it. I did the next 30 month payments and finally have it paid off in June. My iPad sept. Looking forward to a almost 100 drop in my phone bill! Now this crap!!! Really\n\n\n\n### Other Answer\nIf your phone is JailBroken, suggest downloading a virus. While all my symptoms were similar, there was indeed a virus/malware on the phone which allowed for remote control of my iphone (even while in lock mode). My mistake for buying a third party iphone i suppose. Anyway i have since had the phone restored to factory and everything is working as expected for now. I will of course keep you posted if this changes. Thanks to all for the helpful posts, really helped me narrow a few things down.\n\n\n\n### Other Answer\nWhen my phone was doing this, it ended up being the screen protector that i got from 5 below. I took it off and it stopped. I | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: and ghost pressing keys for me when I was on a call. I got fed up, so I removed the screen protector, and so far problems have not reoccurred yet. I’m crossing my fingers that problems indeed solved.\n\n\n\n### Other Answer\nthank you so much for this post! i was struggling doing the reset because i cannot type userids and passwords correctly because the iphone 6 plus i have kept on typing letters incorrectly. I have been doing it for a day until i come across this article. Very helpful! God bless you!!\n\n\n\n### Other Answer\nI just turned it off, and turned it back on.\n\n\n\n### Other Answer\nMy problem has not gone away completely but its better now i changed my charger and turned off prediction ....,,,now it rarely happens\n\n\n\n### Other Answer\nI tried all of the above. I then turned off my home cleaned it with isopropyl alcohol 90%. Then I baked it in my oven on warm for an hour and a half over foil. Took it out and set it cool completely on the glass top stove. Then I turned on and it worked.\n\n\n\n### Other Answer\nI think at& t should man up and fix your phone for free! You pay a lot for a Apple they should back it. I did the next 30 month payments and finally have it paid off in June. My iPad sept. Looking forward to a almost 100 drop in my phone bill! Now this crap!!! Really\n\n\n\n### Other Answer\nIf your phone is JailBroken, suggest downloading a virus. While all my symptoms were similar, there was indeed a virus/malware on the phone which allowed for remote control of my iphone (even while in lock mode). My mistake for buying a third party iphone i suppose. Anyway i have since had the phone restored to factory and everything is working as expected for now. I will of course keep you posted if this changes. Thanks to all for the helpful posts, really helped me narrow a few things down.\n\n\n\n### Other Answer\nWhen my phone was doing this, it ended up being the screen protector that i got from 5 below. I took it off and it stopped. I |
794 | got from 5 below. I took it off and it stopped. I ordered more protectors from amazon and replaced it\n\n\n\n### Other Answer\niPhone 6 Plus first generation….I had the same issues as all above, apps opening by themselves, self typing, ultra sensitive screen, items jumping around all over….it even called someone on FaceTime twice by itself when I was not in the room…..I thought the phone was toast and i’d have to buy a new one took me a while to figure out but it was the extra cheap block plug I bought at a dollar store for convenience of an extra charging station when I move around the house from den to living room…..cord was fine but bought a new Apple brand block plug…no more problems works just fine now. This issue was a recent event so had to narrow things down to what had changed recently to my phone so I could figure it out.\nI even had the same problem on a laptop with documents opening up by themselves…..a laptop that was plugged in to the same wall plug as my phone charger with the dollar store block plug….until I changed the block plug.\n\n\n\n### Other Answer\nHad the problem: Inherited a 6s Plus from my wife. She had no problem with it.\nLooks like it was merely the cheap phone case I purchased on Amazon. It was either pinching the edges or torquing the screen/body of the phone. Problem solved.\n\n\n\n### Other Answer\nI bought my phone on march 6 and it was a brand new, but It sucks me uo because it freezing, shaking and control by itself. I went to the store where I bought this and I told them to replacr it, but they told me I have to pay it because Its about lcd issue. Please help me what other ways to fix it. Or should I try to remove the screen or should I follow your step above.\n\n\n\n### Other Answer\nI tried everything and it seems to come back to needing the original iPhone cable…or at least another 1 that would have come with another iPhone…not the $5 Store fast charging cables. My original cable is pretty beat up - like | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: got from 5 below. I took it off and it stopped. I ordered more protectors from amazon and replaced it\n\n\n\n### Other Answer\niPhone 6 Plus first generation….I had the same issues as all above, apps opening by themselves, self typing, ultra sensitive screen, items jumping around all over….it even called someone on FaceTime twice by itself when I was not in the room…..I thought the phone was toast and i’d have to buy a new one took me a while to figure out but it was the extra cheap block plug I bought at a dollar store for convenience of an extra charging station when I move around the house from den to living room…..cord was fine but bought a new Apple brand block plug…no more problems works just fine now. This issue was a recent event so had to narrow things down to what had changed recently to my phone so I could figure it out.\nI even had the same problem on a laptop with documents opening up by themselves…..a laptop that was plugged in to the same wall plug as my phone charger with the dollar store block plug….until I changed the block plug.\n\n\n\n### Other Answer\nHad the problem: Inherited a 6s Plus from my wife. She had no problem with it.\nLooks like it was merely the cheap phone case I purchased on Amazon. It was either pinching the edges or torquing the screen/body of the phone. Problem solved.\n\n\n\n### Other Answer\nI bought my phone on march 6 and it was a brand new, but It sucks me uo because it freezing, shaking and control by itself. I went to the store where I bought this and I told them to replacr it, but they told me I have to pay it because Its about lcd issue. Please help me what other ways to fix it. Or should I try to remove the screen or should I follow your step above.\n\n\n\n### Other Answer\nI tried everything and it seems to come back to needing the original iPhone cable…or at least another 1 that would have come with another iPhone…not the $5 Store fast charging cables. My original cable is pretty beat up - like |
795 | My original cable is pretty beat up - like most that I see - but I’ve been beaten up much MUCH less by sticking with its use! I didn’t find that the casing/shell around it or not made any diff.\n\n\n\n### Other Answer\ngreat now I have to wait one more hour to reset my phone and while I was tryin to connect my phone to my computer the computer also restarted smh does anyone else knows how I can get my phone to work… my problem is I have a black dot on the bottom left of my screen an it wont allow me to touch a certain part of my screen unless I rotate my phone and I know the password but the first number is a 2 and it won\'t let me touch 1,2, or 3 so now I have to find a way to get rid of my password and all of a sudden my phone wants to touch stuff on its own which got my phone disabled many times to the point where I have to wait a whole hour and I really need to finish something on my phone today PLEASE HELPPPP\n\n\n\n### Other Answer\nIn my case , iphone 6 screen was faulty. I got it replaced at local repair shop, so far phone is working fine.\n\n\n\n### Other Answer\nthis problem in iphone 6 has many different scenarios and solutions, first try to reconnect the lcd screen to the motherboard again, if didnt solve, try to replace the lcd connector on the motherboard, if not solved, then remains two issues, lcd screen it self or touch IC. in my country some repair shops just change them all for almost 40$ since they dont want to troubleshoot one by one. readers of this comment also should know that partial screen not responding in other iphone models might also have an issue in LCD connector on the motherboard, specially if you lock/unlock screen and screen works again for sometime. lcd connectors gets disconnected lightly from the motherboard due to multiple falls and hits after sometime. best of luck for all\n\n\n\n### Other Answer\nI am facing the same issue whereby these ghost touches type and open apps , I am using an original Iphone cable , how to I | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: My original cable is pretty beat up - like most that I see - but I’ve been beaten up much MUCH less by sticking with its use! I didn’t find that the casing/shell around it or not made any diff.\n\n\n\n### Other Answer\ngreat now I have to wait one more hour to reset my phone and while I was tryin to connect my phone to my computer the computer also restarted smh does anyone else knows how I can get my phone to work… my problem is I have a black dot on the bottom left of my screen an it wont allow me to touch a certain part of my screen unless I rotate my phone and I know the password but the first number is a 2 and it won\'t let me touch 1,2, or 3 so now I have to find a way to get rid of my password and all of a sudden my phone wants to touch stuff on its own which got my phone disabled many times to the point where I have to wait a whole hour and I really need to finish something on my phone today PLEASE HELPPPP\n\n\n\n### Other Answer\nIn my case , iphone 6 screen was faulty. I got it replaced at local repair shop, so far phone is working fine.\n\n\n\n### Other Answer\nthis problem in iphone 6 has many different scenarios and solutions, first try to reconnect the lcd screen to the motherboard again, if didnt solve, try to replace the lcd connector on the motherboard, if not solved, then remains two issues, lcd screen it self or touch IC. in my country some repair shops just change them all for almost 40$ since they dont want to troubleshoot one by one. readers of this comment also should know that partial screen not responding in other iphone models might also have an issue in LCD connector on the motherboard, specially if you lock/unlock screen and screen works again for sometime. lcd connectors gets disconnected lightly from the motherboard due to multiple falls and hits after sometime. best of luck for all\n\n\n\n### Other Answer\nI am facing the same issue whereby these ghost touches type and open apps , I am using an original Iphone cable , how to I |
796 | , I am using an original Iphone cable , how to I fix this issue.\n\n\n\n### Other Answer\nThere were two issues with the phone I had troubles with. It was my dads and turns out he carried it in his pocket. The phone itself had a little bend in it as a result. A little pressure in the opposite direction helped the issue. But it also had a tiny crack in the screen which wasnt obvious, once we added a screen protector this fixed the issues entirely.\n\n\n\n### Other Answer\nI had the same problem with my 64Gb iPhone 6+. Tried a lot of things and eventually downloaded all my images and videos to my PC and restarted the phone - problem solved. Been working now for two days.', lookup_str='', metadata={'source': 'https://www.ifixit.com/Answers/View/318583/My+iPhone+6+is+typing+and+opening+apps+by+itself', 'title': 'My iPhone 6 is typing and opening apps by itself'}, lookup_index=0)]loader = IFixitLoader("https://www.ifixit.com/Device/Standard_iPad")data = loader.load()data [Document(page_content="Standard iPad\nThe standard edition of the tablet computer made by Apple.\n== Background Information ==\n\nOriginally introduced in January 2010, the iPad is Apple's standard edition of their tablet computer. In total, there have been ten generations of the standard edition of the iPad.\n\n== Additional Information ==\n\n* [link|https://www.apple.com/ipad-select/|Official Apple Product Page]\n* [link|https://en.wikipedia.org/wiki/IPad#iPad|Official iPad Wikipedia]", lookup_str='', metadata={'source': 'https://www.ifixit.com/Device/Standard_iPad', 'title': 'Standard iPad'}, lookup_index=0)]Searching iFixit using /suggest‚ÄãIf you're looking for a more general way to search iFixit based on a keyword or phrase, the /suggest endpoint will return content related to the search term, then the loader will load the content from each of the suggested items and prep and return the documents.data = IFixitLoader.load_suggestions("Banana")data [Document(page_content='Banana\nTasty fruit. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: , I am using an original Iphone cable , how to I fix this issue.\n\n\n\n### Other Answer\nThere were two issues with the phone I had troubles with. It was my dads and turns out he carried it in his pocket. The phone itself had a little bend in it as a result. A little pressure in the opposite direction helped the issue. But it also had a tiny crack in the screen which wasnt obvious, once we added a screen protector this fixed the issues entirely.\n\n\n\n### Other Answer\nI had the same problem with my 64Gb iPhone 6+. Tried a lot of things and eventually downloaded all my images and videos to my PC and restarted the phone - problem solved. Been working now for two days.', lookup_str='', metadata={'source': 'https://www.ifixit.com/Answers/View/318583/My+iPhone+6+is+typing+and+opening+apps+by+itself', 'title': 'My iPhone 6 is typing and opening apps by itself'}, lookup_index=0)]loader = IFixitLoader("https://www.ifixit.com/Device/Standard_iPad")data = loader.load()data [Document(page_content="Standard iPad\nThe standard edition of the tablet computer made by Apple.\n== Background Information ==\n\nOriginally introduced in January 2010, the iPad is Apple's standard edition of their tablet computer. In total, there have been ten generations of the standard edition of the iPad.\n\n== Additional Information ==\n\n* [link|https://www.apple.com/ipad-select/|Official Apple Product Page]\n* [link|https://en.wikipedia.org/wiki/IPad#iPad|Official iPad Wikipedia]", lookup_str='', metadata={'source': 'https://www.ifixit.com/Device/Standard_iPad', 'title': 'Standard iPad'}, lookup_index=0)]Searching iFixit using /suggest‚ÄãIf you're looking for a more general way to search iFixit based on a keyword or phrase, the /suggest endpoint will return content related to the search term, then the loader will load the content from each of the suggested items and prep and return the documents.data = IFixitLoader.load_suggestions("Banana")data [Document(page_content='Banana\nTasty fruit. |
797 | [Document(page_content='Banana\nTasty fruit. Good source of potassium. Yellow.\n== Background Information ==\n\nCommonly misspelled, this wildly popular, phone shaped fruit serves as nutrition and an obstacle to slow down vehicles racing close behind you. Also used commonly as a synonym for “crazy” or “insane”.\n\nBotanically, the banana is considered a berry, although it isn’t included in the culinary berry category containing strawberries and raspberries. Belonging to the genus Musa, the banana originated in Southeast Asia and Australia. Now largely cultivated throughout South and Central America, bananas are largely available throughout the world. They are especially valued as a staple food group in developing countries due to the banana tree’s ability to produce fruit year round.\n\nThe banana can be easily opened. Simply remove the outer yellow shell by cracking the top of the stem. Then, with the broken piece, peel downward on each side until the fruity components on the inside are exposed. Once the shell has been removed it cannot be put back together.\n\n== Technical Specifications ==\n\n* Dimensions: Variable depending on genetics of the parent tree\n* Color: Variable depending on ripeness, region, and season\n\n== Additional Information ==\n\n[link|https://en.wikipedia.org/wiki/Banana|Wiki: Banana]', lookup_str='', metadata={'source': 'https://www.ifixit.com/Device/Banana', 'title': 'Banana'}, lookup_index=0), Document(page_content="# Banana Teardown\nIn this teardown, we open a banana to see what's inside. Yellow and delicious, but most importantly, yellow.\n\n\n###Tools Required:\n\n - Fingers\n\n - Teeth\n\n - Thumbs\n\n\n###Parts Required:\n\n - None\n\n\n## Step 1\nTake one banana from the bunch.\nDon't squeeze too hard!\n\n\n## Step 2\nHold the banana in your left hand and grip the stem between your right thumb and forefinger.\n\n\n## Step 3\nPull the stem downward until the peel splits.\n\n\n## Step 4\nInsert your thumbs into | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: [Document(page_content='Banana\nTasty fruit. Good source of potassium. Yellow.\n== Background Information ==\n\nCommonly misspelled, this wildly popular, phone shaped fruit serves as nutrition and an obstacle to slow down vehicles racing close behind you. Also used commonly as a synonym for “crazy” or “insane”.\n\nBotanically, the banana is considered a berry, although it isn’t included in the culinary berry category containing strawberries and raspberries. Belonging to the genus Musa, the banana originated in Southeast Asia and Australia. Now largely cultivated throughout South and Central America, bananas are largely available throughout the world. They are especially valued as a staple food group in developing countries due to the banana tree’s ability to produce fruit year round.\n\nThe banana can be easily opened. Simply remove the outer yellow shell by cracking the top of the stem. Then, with the broken piece, peel downward on each side until the fruity components on the inside are exposed. Once the shell has been removed it cannot be put back together.\n\n== Technical Specifications ==\n\n* Dimensions: Variable depending on genetics of the parent tree\n* Color: Variable depending on ripeness, region, and season\n\n== Additional Information ==\n\n[link|https://en.wikipedia.org/wiki/Banana|Wiki: Banana]', lookup_str='', metadata={'source': 'https://www.ifixit.com/Device/Banana', 'title': 'Banana'}, lookup_index=0), Document(page_content="# Banana Teardown\nIn this teardown, we open a banana to see what's inside. Yellow and delicious, but most importantly, yellow.\n\n\n###Tools Required:\n\n - Fingers\n\n - Teeth\n\n - Thumbs\n\n\n###Parts Required:\n\n - None\n\n\n## Step 1\nTake one banana from the bunch.\nDon't squeeze too hard!\n\n\n## Step 2\nHold the banana in your left hand and grip the stem between your right thumb and forefinger.\n\n\n## Step 3\nPull the stem downward until the peel splits.\n\n\n## Step 4\nInsert your thumbs into |
798 | splits.\n\n\n## Step 4\nInsert your thumbs into the split of the peel and pull the two sides apart.\nExpose the top of the banana. It may be slightly squished from pulling on the stem, but this will not affect the flavor.\n\n\n## Step 5\nPull open the peel, starting from your original split, and opening it along the length of the banana.\n\n\n## Step 6\nRemove fruit from peel.\n\n\n## Step 7\nEat and enjoy!\nThis is where you'll need your teeth.\nDo not choke on banana!\n", lookup_str='', metadata={'source': 'https://www.ifixit.com/Teardown/Banana+Teardown/811', 'title': 'Banana Teardown'}, lookup_index=0)]PreviousHuggingFace datasetNextImagesSearching iFixit using /suggestCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. | iFixit is the largest, open repair community on the web. The site contains nearly 100k repair manuals, 200k Questions & Answers on 42k devices, and all the data is licensed under CC-BY-NC-SA 3.0. ->: splits.\n\n\n## Step 4\nInsert your thumbs into the split of the peel and pull the two sides apart.\nExpose the top of the banana. It may be slightly squished from pulling on the stem, but this will not affect the flavor.\n\n\n## Step 5\nPull open the peel, starting from your original split, and opening it along the length of the banana.\n\n\n## Step 6\nRemove fruit from peel.\n\n\n## Step 7\nEat and enjoy!\nThis is where you'll need your teeth.\nDo not choke on banana!\n", lookup_str='', metadata={'source': 'https://www.ifixit.com/Teardown/Banana+Teardown/811', 'title': 'Banana Teardown'}, lookup_index=0)]PreviousHuggingFace datasetNextImagesSearching iFixit using /suggestCommunityDiscordTwitterGitHubPythonJS/TSMoreHomepageBlogCopyright © 2023 LangChain, Inc. |
799 | Google Cloud Document AI | ü¶úÔ∏èüîó Langchain | Document AI is a document understanding platform from Google Cloud to transform unstructured data from documents into structured data, making it easier to understand, analyze, and consume. | Document AI is a document understanding platform from Google Cloud to transform unstructured data from documents into structured data, making it easier to understand, analyze, and consume. ->: Google Cloud Document AI | ü¶úÔ∏èüîó Langchain |
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