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Serrening modullik gumoni
Maydon	Algebraik sonlar nazariyasi
Gumon qilingan	Jan-Per Ser
Gumon qilingan	1975
Birinchi dalil	Chandrashekhar Khare; Jan-Per Vintenberger
Birinchi dalil	2008

Serres modularity conjecture - Wikipedia

Yilda matematika, Serrening modullik gumonitomonidan kiritilgan Jan-Per Ser (1975, 1987), g'alati, kamaytirilmaydigan, ikki o'lchovli ekanligini bildiradi Galois vakili ustidan cheklangan maydon modulli shakldan kelib chiqadi. Ushbu taxminning yanada kuchli versiyasi modulli shaklning og'irligi va darajasini belgilaydi. 1-darajadagi gipoteza isbotlandi Chandrashekhar Khare 2005 yilda,^[1] va to'liq taxminning isboti Xare va Jan-Per Vintenberger 2008 yilda.^[2]

Formulyatsiya

Gumon quyidagilarga tegishli mutlaq Galois guruhi ${ displaystyle G _ { mathbb {Q}}}$ ning ratsional son maydoni ${ displaystyle mathbb {Q}}$ .

Ruxsat bering ${ displaystyle rho}$ bo'lish mutlaqo qisqartirilmaydi, ning doimiy, ikki o'lchovli tasviri ${ displaystyle G _ { mathbb {Q}}}$ cheklangan maydon ustida ${ displaystyle F = mathbb {F} _ { ell ^ {r}}}$ .

{ displaystyle rho colon G _ { mathbb {Q}} rightarrow mathrm {GL} _ {2} (F).}

Bundan tashqari, taxmin qiling ${ displaystyle rho}$ g'alati, ya'ni murakkab konjugatsiya tasviri -1 determinantiga ega.

Har qanday normallashtirilgan modulli o'ziga xos shakl

{ displaystyle f = q + a_ {2} q ^ {2} + a_ {3} q ^ {3} + cdots}

ning Daraja ${ displaystyle N = N ( rho)}$ , vazn ${ displaystyle k = k ( rho)}$ va ba'zilari Nebentype belgisi

{ displaystyle chi colon mathbb {Z} / N mathbb {Z} rightarrow F ^ {*}}

,

Shimura, Deligne va Serre-Deligne tomonidan berilgan teorema ${ displaystyle f}$ vakillik

{ displaystyle rho _ {f} ikki nuqta G _ { mathbb {Q}} rightarrow mathrm {GL} _ {2} ({ mathcal {O}}),}

qayerda ${ displaystyle { mathcal {O}}}$ ning cheklangan kengaytmasidagi butun sonlarning halqasi ${ displaystyle mathbb {Q} _ { ell}}$ . Ushbu vakillik barcha tub sonlar uchun shart bilan tavsiflanadi ${ displaystyle p}$ , koprime ga ${ displaystyle N ell}$ bizda ... bor

{ displaystyle operator nomi {Trace} ( rho _ {f} ( operator nomi {Frob} _ {p})) = a_ {p}}

va

{ displaystyle det ( rho _ {f} ( operatorname {Frob} _ {p})) = p ^ {k-1} chi (p).}

Ushbu namoyish modulini kamaytirish maksimal ideal ${ displaystyle { mathcal {O}}}$ tartibini beradi ${ displaystyle ell}$ vakillik ${ displaystyle { overline { rho _ {f}}}}$ ning ${ displaystyle G _ { mathbb {Q}}}$ .

Serrening gumoni shuni ta'kidlaydiki, har qanday vakil uchun ${ displaystyle rho}$ yuqoridagi kabi, modulli o'ziga xos shakl mavjud ${ displaystyle f}$ shu kabi

{ displaystyle { overline { rho _ {f}}} cong rho}

.

Gumon shaklining darajasi va vazni ${ displaystyle f}$ Serening maqolasida aniq gumon qilingan. Bundan tashqari, u ushbu gumondan qator natijalarni keltirib chiqaradi Fermaning so'nggi teoremasi va hozirda isbotlangan Taniyama-Vayl (yoki Taniyama-Shimura) gumoni, endi modullik teoremasi (garchi bu Fermaning so'nggi teoremasini nazarda tutsa-da, Serre buni to'g'ridan-to'g'ri o'z taxminidan tasdiqlaydi).

Optimal daraja va vazn

Serrning gumonining kuchli shakli modulli shaklning darajasi va vaznini tavsiflaydi.

Eng maqbul daraja Artin dirijyori ning kuchi bilan vakolatxonaning ${ displaystyle l}$ olib tashlandi.

Isbot

Gipotezaning 1-darajali va kichik vaznli holatlarining isboti 2004 yilda olingan Chandrashekhar Khare va Jan-Per Vintenberger,^[3] va tomonidan Luis Dieulefait,^[4] mustaqil ravishda.

2005 yilda Chandrashekhar Khare Serre gumonining 1-darajali holatini isbotladi,^[5] va 2008 yilda Jan-Per Vintenberger bilan hamkorlikda to'liq gumonning isboti.^[6]

Izohlar

^ Khare, Chandrashekhar (2006), "Serrening modullik gumoni: Birinchi darajadagi holat", Dyuk Matematik jurnali, 134 (3): 557–589, doi:10.1215 / S0012-7094-06-13434-8.
^ Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (I)", Mathematicae ixtirolari, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007 / s00222-009-0205-7 va Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (II)", Mathematicae ixtirolari, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007 / s00222-009-0206-6.
^ Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Gal (Q / Q) ning ikki o'lchovli mod p tasvirlari uchun Serrening o'zaro gumoni to'g'risida", Matematika yilnomalari, 169 (1): 229–253, doi:10.4007 / annals.2009.169.229.
^ Dieulefait, Luis (2007), "Serre gumonining 2-darajali og'irligi", Revista Matemática Iberoamericana, 23 (3): 1115–1124, arXiv:matematika / 0412099, doi:10.4171 / rmi / 525.
^ Khare, Chandrashekhar (2006), "Serrening modullik gumoni: Birinchi darajali voqea", Dyuk Matematik jurnali, 134 (3): 557–589, doi:10.1215 / S0012-7094-06-13434-8.
^ Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (I)", Mathematicae ixtirolari, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007 / s00222-009-0205-7 va Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (II)", Mathematicae ixtirolari, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007 / s00222-009-0206-6.

Adabiyotlar

Serre, Jan-Per (1975), "Valeurs propres des opérateurs de Hecke modulo l", Journées Arithmétiques de Bordo (Konf., Univ. Bordo, 1974), Asterisk, 24–25: 109–117, ISSN 0303-1179, JANOB 0382173
Serre, Jan-Per (1987), "Sur les représentations modulaires de degré 2 de Gal (Q/ Q) ", Dyuk Matematik jurnali, 54 (1): 179–230, doi:10.1215 / S0012-7094-87-05413-5, ISSN 0012-7094, JANOB 0885783
Shteyn, Uilyam A .; Ribet, Kennet A. (2001), "Serrning taxminlari bo'yicha ma'ruzalar", Konradda, Brayan; Rubin, Karl (tahrir), Arifmetik algebraik geometriya (Park City, UT, 1999), IAS / Park City Math. Ser., 9, Providence, R.I .: Amerika matematik jamiyati, 143–232 betlar, ISBN 978-0-8218-2173-2, JANOB 1860042

Tashqi havolalar

Serrening modullik gumoni Tomonidan 50 daqiqalik ma'ruza Ken Ribet 2007 yil 25 oktyabrda berilgan ( slaydlar PDF, slaydlarning boshqa versiyasi PDF)
Serrning taxminlari bo'yicha ma'ruzalar

[1] Khare, Chandrashekhar (2006), "Serrening modullik gumoni: Birinchi darajadagi holat", Dyuk Matematik jurnali, 134 (3): 557–589, doi:10.1215 / S0012-7094-06-13434-8.

[2] Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (I)", Mathematicae ixtirolari, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007 / s00222-009-0205-7 va Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (II)", Mathematicae ixtirolari, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007 / s00222-009-0206-6.

[3] Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Gal (Q / Q) ning ikki o'lchovli mod p tasvirlari uchun Serrening o'zaro gumoni to'g'risida", Matematika yilnomalari, 169 (1): 229–253, doi:10.4007 / annals.2009.169.229.

[4] Dieulefait, Luis (2007), "Serre gumonining 2-darajali og'irligi", Revista Matemática Iberoamericana, 23 (3): 1115–1124, arXiv:matematika / 0412099, doi:10.4171 / rmi / 525.

[5] Khare, Chandrashekhar (2006), "Serrening modullik gumoni: Birinchi darajali voqea", Dyuk Matematik jurnali, 134 (3): 557–589, doi:10.1215 / S0012-7094-06-13434-8.

[6] Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (I)", Mathematicae ixtirolari, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007 / s00222-009-0205-7 va Xare, Chandrashexar; Vintenberger, Jan-Per (2009), "Serrning modullik gumoni (II)", Mathematicae ixtirolari, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007 / s00222-009-0206-6.

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