ComponentsDocument loadersDropboxOn this pageDropboxDrobpox is a file hosting 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).Intructions​`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)    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 more.\n\n![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523574441249_paper-insert.png)\n\n\n\n**Add emojis** to your doc or comment by typing `**:**` ****and choosing a character. \n\n# 👍 👎 👏 ✅ ❌ ❤️ ⭐ 💡 📌\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![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523473869783_Hot_Sauce.jpg)\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![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523564536543_Clock_Melt.png)\n![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523564528339_Boom_Box_Melt.png)\n![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523564549819_Soccerball_Melt.png)\n\n![You can add captions too](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523564518899_Cacti_Melt.png)\n![What a strange, melting toaster!](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523564508553_Toaster_Melt.png)\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 out.\n\n\n![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523575138939_paper-embed.png)\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 +                                   | ⭐     |\n| ------------------------------------------------------------------------------------------------------- | ----- |\n| To delete, select rows/columns and click the trash can                                                  | ⭐ ⭐   |\n| To delete the entire table, click inside a cell, then click the dot in the top left corner of the table | ⭐ ⭐ ⭐ |\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![](https://paper-attachments.dropbox.com/s_72143DBFDAF4C9DE702BB246920BC47FE7E1FA76AC23CC699374430D94E96DD2_1523574876795_paper-invite.png)\n\n\n**Make 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 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, 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 inﬁnitum 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 Classiﬁcation. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for ﬁnancial 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 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 ﬁnal calculation of a precise upper 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 ﬁnish 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 ﬁrst time inﬁnitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit inﬁnite 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 ﬁeld, 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 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 satisﬁed 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\nDeﬁnition. An elliptic curve over Q is a nonsingular curve deﬁned 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 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) deﬁnes 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, deﬁne 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 ﬁnite set of points called cusps. The cusps are the ﬁnitely 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\nDeﬁnition. 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 deﬁnitions of modularity (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 deﬁnition given in below.\n\n§\n\n§\n\n1.5. Semistability.\n\nDeﬁnition. 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 deﬁned 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 will work with a deﬁnition of modularity which uses modular forms.\n\nDeﬁnition. If N is a positive integer, a modular form f of weight k for Γ0(N ) is C which satisﬁes 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 satisﬁes 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 ﬁxed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (ﬁnite-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\nDeﬁnition. 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 ﬁnite ﬁeld with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at inﬁnity). We now give a second deﬁnition of modularity for an elliptic curve.\n\nDeﬁnition. An elliptic curve E over Q is modular if there exists an eigenform\n\n∞ n=1 ane2πinz such that for all but ﬁnitely 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 ¯ρE,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 subﬁeld of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at inﬁnity 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¯ρE,p : GQ\n\n→\n\n(deﬁned up to isomorphism) such that\n\n(6)\n\ndet(¯ρE,p) = ¯εp\n\nand for all but ﬁnitely many primes q,\n\n#(E(Fq))\n\n(7)\n\ntrace(¯ρE,p(Frobq))\n\nq + 1\n\n(mod p).\n\n≡ (See Appendix A for the deﬁnition 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 ﬁeld 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) ¯ρE,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 ﬁnitely 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 ¯ρE,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 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 deﬁne the notion of cusp forms on § Γ1(N ). See Chapter 3 of [35] for the deﬁnitions of the Hecke operators on the space of weight-one cusp forms for Γ1(N ).\n\nTheorem 2.2 (Langlands-Tunnell). Suppose ρ : 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(ρ(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 ﬁnitely many primes q,\n\n→\n\n−\n\nP\n\n(8)\n\nbq = trace(ρ(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(ρ(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 ¯ρE,3 is irreducible. Then E is modular.\n\nProof. It suﬃces to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satisﬁed 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 deﬁned 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) ¯ρE,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 ρ = ψ ◦ det(ρ(τ )) = Using that ¯ρE,3 is irreducible, one can show that ρ 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 ρ. 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 ﬁnd an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satisﬁes (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