modularity sidhaantham

modularity sidhaantham (munbu taniama-shimura anumaanam, taniama-weyl anumaanam allengil deerghavrithaakrithiyilulla vakrangalkkulla modularity anumaanam ennu vilikkappedunnu) yukthisahamaya samkhyakalude feeldile deerghavrithaakrithiyilulla vakrangal modular roopangalumaayi bandhappettirikkunnave prasthaavikkunnu. andrew wiles semistable eliptik karvukalkkulla modularity sidhaantham theliyichu, ithu fermaattinte avasana sidhaanthathe soochippikkaan paryapthamaayirunnu. pinneedu, vailsinte muun vidyaarthikalaaya bryan konraad, fred diamond, richard tailor ennivarude prabandhangalude oru parambara, christoph bruyilumaayi oru samyuktha prabandhatthil avasaanikkukayum 2001 ll poornna modularity sidhaantham theliyikkunnathinayi vailsinte saankethikavidyakal vipuleekarikkukayum cheythu.
prasthavana.
soothravaakyathinu mukalilulla eathu deerghavrithaakrithiyilulla vakravum classical modular karvu formula ninnu poornnasankhya gunakangalulla oru yukthisahamaya bhoopadatthiloode nedaamennu sidhaantham prasthaavikkunnu. eee mappingine level formulayude modular paaraameettaraiseshan ennu vilikkunnu. atharam oru paaraameettaraiseshan kandethaan kazhiyunna ettavum cheria poornnasankhya formula _ 3 aanenkil (modularity sidhaantham thanne ippol "conductor" ennu vilikkunna oru samkhyayaayi ariyappedunnu), appol oru prathyeka tharam modular roopamaaya bharam, level formula _ 3 ennivayaal srishtikkappedunna oru maappinginte adisthaanathil paaraamettaraiseshane nirvachikkaam, ithu poornnasankhya formula _ 7-vipuleekaranamulla oru saadhaarana puthiya roopamaanu, aavashyamenkil oru aisojeni pinthudaruka.
bandhapetta prasthavanakal.
modularity sidhaantham adutha bandhamullah oru vishakalana prasthavana soochippikkunnu
formula _ 1 nuu mukalilulla oro deerghavrithaakrithiyilulla vakrathilum namukk anubandhamaaya oru ell-seerees attachucheyyaam. formula _ 9 seerees saadhaaranayaayi ezhuthappedunna oru diriclet sreniyaanu.
gunakangalude formula _ 11 nte ulppaadana pravartthanam appol
naam pakarakkaaranaanenkil
sankeernamaaya variable formula _ 15-nte oru function formula _ 14-nte forrier vipuleekaranam nammal ezhuthiyittundennu nammal kaanunnu, athinaal formula _ 7 seereesinte gunakangal formula _ 17-nte forrier gunakangalaayi kanakkaakkappedunnu. eee reethiyil labhikkuna function, shradheyamaayi, bharam randinteyum level formula _ 3-nteyum oru cusp roopamaanu, koodaathe oru eejanform (ellah heck oopparettarmaarudeyum oru eejanvektar) koodiyaanu; ithu modularity sidhaanthathil ninnu pinthudarunna haash-weyl anumaanamaan.
rund bharamullah chila modular roopangal, oru deerghavrithaakrithiyilulla vakrathinulla holomorphic differentialukalumaayi poruthappedunnu. modular karvinte jacobian (aisojeni vare) bharam 2 nte heck ijanfomukalkku anusrithamaayi irreducible abelian enangalude ulppannamaayi ezhuthaam. 1-dimensional gadakangal deerghavrithaakrithiyilulla vakrangalaanu (uyarnna dimensional gadakangalum undakam, athinaal ellah heck aijanfomukalum yukthisahamaya deerghavrithaakruthiyumaayi poruthappedunnilla). anubandha cusp roopam kandethunnathiloodeyum athil ninnu oru vakram nirmmikkunnathiloodeyum labhikkuna vakram yadhaartha vakravumaayi samaroopamaanu (pakshe pothuve, athinu samaroopamalla).
charithram.
1955ll tokyoyileyum nicoyileyum beejaganita samkhyaa sidhaanthathekkurichulla anthaaraashtra simposiyathil yoothaaka taniama anumaanathinte praadhamika (alpam thettaaya) pathipp prasthaavichu. goro shimura thaniyamayum 1957 vare athinte kaadinyam mechappeduthunnathinaayi pravarthichu. andrey weyl eee anumaanam veendum kandethukayum 1967 ll deerghavrithaakrithiyilulla vakrarekhayude chila valanja soothravaakyam _ 9 seereesinulla (anumanikkappetta) pravartthana samavakyangalil ninnu ithu pinthudarumennu kaanikkukayum cheythu; ithu anumaanam sariyaanennu karuthaamennathinte aadyathe gurutharamaaya thelivaayirunnu. deerghavrithaakrithiyilulla vakrathinte chaalakan anubandha modular roopathinte nilavaaramaayirikkanamennum weyl kaanichu. taniama-shimura-weyl anumaanam langlands prograaminte bhaagamaayi maari.
1986ll fermaattinte avasana sidhaanthathe soochippikkunnuvennu gerhard free nirddeshichappol eee anumaanam ganyamaaya thaalparyam aakarshichu. fermaattinte avasana sidhaanthathinu eathenkilum ethir udaaharanam kuranjathu oru noonn-modular deerghavrithaakrithiyilulla vakrathinte nilanilppine soochippikkumennu kaanikkan sramichukondaanu addeham ithu cheythath. 1987ll jeane-piyari sere freeyude yadhaartha kruthiyil oru kaanathaaya link (ippol epsiloon conjecture allengil ribett sidhaantham ennariyappedunnu) thiricharinjappol eee vaadam poortthiyaayi, thudarnnu rund varshathinu sesham kenn ribett epsiloon konjekcharinte thelivu poortthiyaakki.
gauravamaaya shradha nediyathinusheshavum, taniama-shimura-weyl anumaanam samakaalika ganithashaasthranjar theliyikkaan asaadhaaranamaamvidham budhimuttullatho allengil thelivu polum labhyamallathatho aayi kandu. udaaharanathinu, vailsinte piechu.dii. yadhaarthathil theliyikkaan asaadhyamaanennu" thonniyennu supervisor joonn coats prasthaavikkunnu, kenn ribett svayam "[athu] poornnamaayum axus cheyyaan kazhiyaathathaanennu vishwasicha bahubhooripaksham aalukalil oralaayi" kanakkakki.
1995-ll andrew wiles, richard taylorude sahaayathode, fermattinte avasana sidhaantham theliyikkaan upayogichirunna ellah semistable deerghavrithaakrithiyilulla vakrangalkkumulla taniama-shimura-weyl anumaanam theliyichu, koodaathe muzhuvan taniama-shimura-weyl anumaanavum oduvil vajram, konraad, diamond & tailor, bruyil, konraad, diamond & tailor enniva theliyikkukayum cheythu; vailsinte pravartthanatthe adisthaanamaakki, 1999-ll poornna phalam theliyikkunnathuvare avaa sheshikkunna kesukalil vardhicha thothil pinvalikkukayum cheythu. 
poornnamaayum theliyikkappettukazhinja, eee anumaanam modularity sidhaantham ennariyappettu.
fermaattinte avasana sidhaanthathinu samanamaaya samkhyaa sidhaanthathile niravadhi sidhaanthangal modularity sidhaanthathil ninnu pinthudarunnu. udaaharanathina oru cube rund coprime formulakalude koottaayi ezhuthaan kazhiyilla, formula _ 20th powerukal, formula _ 21. (cases formula _ 22 ithinakam ularinu ariyamaayirunnu.)
saamaanyavalkkaranangal.
roobert langlands kaaranam kooduthal pothuvaaya anumaanangalude oru prathyeka kesaanu modularity sidhaantham. langlands prograam oru samkhya feeldile oro deerghavrithaakrithiyilulla vakravum pole ganitha beejaganita jyaamithiyude kooduthal pothuvaaya vasthukkalumayi oru automorphic phoom allengil automorphic praathinidhyam (oru modular roopathinte anuyojyamaaya saamaanyavalkkaranam) gadippikkaan shramikkunnu. eee vipuleekritha anumaanangalude mikka kesukalum ithuvare theliyikkappettittilla. ennirunnaalum, yadhaartha quadratic feeldukalil nirvachichirikkunna deerghavrithaakrithiyilulla valavukal modulaaraanennu freetaas, leeii haang & sixec enniva theliyichu.
udaaharanam.
udaaharanathinu, discriminant (conductor) 37 ulla deerghavrithaakrithiyilulla vakra soothravaakyam _ 23, roopavumaayi bandhappettirikkunnu.
37 nuu thulyamallaatha abhaajya samkhyakalkku, gunakangalekkurichulla savisheshatha parisodhikkaan kazhiyum. athinaal, "l" = 3 nuu, modulo 3:,,,,,, enna samavaakyathin 6 parihaarangalund.
parihaarangalund madangivarunna eee anumaanam 1999 aayappozhekkum andrew vailsinte ashayangal upayogichu poornnamaayum theliyikkapettu, avar 1994 ll deerghavrithaakrithiyilulla valayangalude oru valia kudumbatthinu ithu theliyichu.
anumaanathinu niravadhi soothravaakyangalundu. avaa thulyamaanennu kaanikkunnath irupathaam noottaandinte randaam pakuthiyil samkhyaa sidhaanthathinte oru pradhana velluviliyaayirunnu. x" 0 ("n") muthal "e" vareyulla modular karvu muthal q nuu mukalil sthiramallaatha oru yukthisahamaya bhoopadam nirvachichittundennu paranjukondu conductor "n" nte deerghavrithaakrithiyilulla karvinte modularity prakadippikkaanum kazhiyum. prathyekichum, "iii" yude pooyintukal modular functionukal vazhi paaraameettar cheyyaan kazhiyum.
udaaharanathinu, vakra soothravaakyathinte oru modular paaraameettaraiseshan _ 25 nalkunnu
ivide, mukalil paranjathupole, "q" = exp (2π "iz"). x" ("z"), "y" ("z") ennee functionukal bhaara 0, level 37 ennivayude modular aanu; mattoru vidhathil paranjaal avaa meromorphic aanu, mukalile pakuthi-playin IM ("z")> 0 ll nirvachichirikkunnu, thripthippeduthunnu.
athupole "y" ("z") nuu, "ad"-"bc" = 1,37 ennivayulla "a, b, c, d" ennee ellah poornnasamkhyakalkkum|"sii.
mattoru formulation oru vashath deerghavrithaakrithiyilulla valavukalumaayum maruvashathu modular roopangalumaayum gadippichirikkunna galois praathinidhyangalude thaarathamyathe aasrayichirikkunnu. randaamathe formulation anumaanathinte thelivaayi upayogichittundu. roopangalude nilavaravumaayi (vakrathinte chaalakavumayulla bandham) edapedunnathu prathyekichum sookshumamaanu.
eee anumaanatthinte ettavum athisayakaramaaya prayogam fermattinte avasana sidhaanthathinte (eflt) thelivaanu. oru priminu "p" ≤5, fermet samavaakyam ennu karuthuka.
poojyamallaatha poornnasamkhyakalulla oru parihaaramund, athinaal flattinu oru ethir-udaaharanamaanu. appol aadyam shradhichathupole, deerghavrithaakrithiyilulla vakram
vivechanam
modular aakaan kazhiyilla. athinaal, deerghavrithaakrithiyilulla eee kudumbatthile (hellegowark-freeyude vakrangal ennu vilikkunna) taniama-shimura-weyl anumaanathinte thelivu flattine soochippikkunnu. gerhard free (1985) enna ashayathe adisthaanamaakkiyulla eee rund prasthaavanakalum thammilulla bandhatthinte thelivu budhimuttullathum saankethikavumaanu. 1987ll kennathu ribettaanu ithu sthaapichathu.
athum kaanuka.
sereyude modularity anumaanam

