- Print publication year: 2011
- Online publication date: January 2012

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/CBO9780511984440.008
- pp 188-211

Introduction

Our main goal in this article is to talk about recent theorems of Taylor and his co-workers on modularity and potential modularity of Galois representations, particularly those attached to elliptic curves. However, so as to not bog down the exposition unnecessarily with technical definitions right from the off, we will build up to these results by starting our story with Wiles' breakthrough paper [Wil95], and working towards the more recent results. We will however assume some familiarity with the general area – for example we will assume the reader is familiar with the notion of an elliptic curve over a number field, and a Galois representation, and what it means for such things to be modular (when such a notion makes sense). Let us stress now that, because of this chronological approach, some theorems stated in this paper will be superseded by others (for example Theorem 1 gets superseded by Theorem 6 which gets superseded by Theorem 7), and similarly some conjectures (for example Serre's conjecture) will become theorems as the story progresses. The author hopes that this slightly non-standard style nevertheless gives the reader the feeling of seeing how the theory evolved.

We thank Toby Gee for reading through a preliminary draft of this article and making several helpful comments, and we also thank Matthew Emerton and Jan Nekovář for pointing out various other inaccuracies and ambiguities.

Semistable elliptic curves over Q are modular

The story, of course, starts with the following well-known result proved in [Wil95] and [TW95].

[BCDT01] On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic). MR1839918(2002d: 11058) , , , and ,

[BDJ10] On Serre's conjecture for mod ℓ galois representations over totally real fields, Duke Math. J. 155 (2010), no. 1, 105–161. , , and ,

[BLGGT] , , , and , Potential automorphy and change of weight, Preprint.

[BLGHT] , , , and , A family of Calabi-Yau varieties and potential automorphy ii, to appear in P.R.I.M.S.

[BM02] Multiplicités modulaires et représentations de GL2(Zp) et de Gal(Qp/Qp) en l = p, Duke Math. J. 115 (2002), no. 2, 205–310, With an appendix by Guy Henniart. MR1944572(2004i:11052) and ,

[Bre00] Groupes p-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549. MR1804530(2001k:14087) ,

[Car83] Sur les représentations l-adiques attachées aux formes modulaires de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 629–632. MR705677(85e:11039) ,

[CDT99] Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2, 521–567. MR1639612(99i:11037) , , and ,

[CHT08] Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR2470687(2010j:11082) , , and ,

[DDT97] Elliptic curves, modular forms & Fermat's last theorem ( , 1993), Int. Press, Cambridge, MA, 1997, pp. 2–140. MR1605752(99d: 11067b) , , and , Fermat's last theorem,

[Dia96] On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166. MR1405946(97d:11172) ,

[Dia97] The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391. MR1440309(98c:11047) ,

[Dic] On the modularity of certain 2-adic Galois representations, Duke Math. J. 109 (2001), no. 2, 319–382. MR1845182(2002k: 11079) ,

[Edi92] The weight in Serre's conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594. MR1176206(93h:11124) ,

[FL82] Construction de représentations padiques, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547–608 (1983). MR707328(85c:14028) and ,

[Fon77] Groupes p-divisibles sur les corps locaux, Société Mathématique de France, Paris, 1977, Astérisque, No. 47-48. MR0498610(58\#16699) ,

[Fuj] , Deformation rings and Hecke algebras in the totally real case, preprint available at http://arxiv.org/abs/math/0602606.

[Gee] , Automorphic lifts of prescribed types, to appear in Math Annalen.

[Gee06] A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), no. 5-6, 805–811. MR2280776(2007m:11065) ,

[Gee07] Companion forms over totally real fields. II, Duke Math. J. 136 (2007), no. 2, 275–284. MR2286631(2008e:11053) ,

[Gee09] Erratum—a modularity lifting theorem for weight two Hilbert modular forms [mr2280776], Math. Res. Lett. 16 (2009), no. 1, 57–58. MR2480560(2010c:11057) ,

[Jar99] Mazur's principle for totally real fields of odd degree, Compositio Math. 116 (1999), no. 1, 39–79. MR1669444(2001a:11081) ,

[JPSS81] Relèvement cubique non normal, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 12, 567–571. MR615450(82i:10035) , , and ,

[Kha07] L-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 270–299. MR2392357(2009g:11066) , Serre's modularity conjecture: a survey of the level one case,

[Kis07] L-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, With an appendix by Ofer Gabber, pp. 438–470. MR2392362(2009j:11086) , Modularity for some geometric Galois representations,

[Kis09] Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. MR2600871 ,

[KW09a] Serre's modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR2551763(2010k:11087) and ,

[KW09b] Serre's modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. MR2551764(2010k:11088) ,

[Lan80] Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton N.J., 1980. MR574808(82a:10032) , Base change for GL(2),

[Maz89] Galois groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR1012172(90k:11057) , Deforming Galois representations,

[MT] Reprsentations galoisiennes, diffrentielles de Khler et “conjectures principales”, Inst. Hautes tudes Sci. Publ. Math. 71 (1990), 65–103. MR1079644(92e:11060) and ,

[MB89] Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 161–179, 181–194. MR1005158(90i:11065) ,

[Nek] On the parity of ranks of Selmer groups. IV (with an appendix by Jean-Pierre Wintenberger), Compos. Math. 145, no. 6, 1351– 1359. MR2575086(2010j:11106) ,

[Raj01] On the levels of mod l Hilbert modular forms, J. Reine Angew. Math. 537 (2001), 33–65. MR1856257(2002i:11041) ,

[Ram93] On a variation of Mazur's deformation functor, Compositio Math. 87 (1993), no. 3, 269–286. MR1227448(94h:11054) ,

[Ray74] Schémas en groupes de type (p, …, p), Bull. Soc. Math. France 102 (1974), 241–280. MR0419467(54\#7488) ,

[Rib90] On modular representations of Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR1047143(91g:11066) ,

[Sav05] On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J. 128 (2005), no. 1, 141–197. MR2137952(2006c:11060) ,

[Sch08] Weights in Serre's conjecture for Hilbert modular forms: the ramified case, Israel J. Math. 166 (2008), 369–391. MR2430440(2009e:11090) ,

[Ser87] Sur les représentations modulaires de degré 2 de Gal(Q/Q), Duke Math. J. 54 (1987), no. 1, 179–230. MR885783(88g: 11022) ,

[Ski] , Nearly ordinary deformations of residually dihedral representations (draft), Preprint.

[Sno] , On two dimensional weight two odd representations of totally real fields, preprint available at http://arxiv.org/abs/0905.4266.

[SW99] Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. (1999), no. 89, 5–126 (2000). MR1793414(2002b:11072) and ,

[SW01] Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, 185–215. MR1928993(2004b:11073) and ,

[Tay02] Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), no. 1, 125–143. MR1954941(2004c:11082) ,

[Tay06] On the meromorphic continuation of degree two L-functions, Doc. Math. (2006), no. Extra Vol., 729–779 (electronic). MR2290604(2008c: 11154) ,

[Tun81] Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173–175. MR621884(82j: 12015) ,

[TW95] Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR1333036(96d:11072) and ,

[Wil95] Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR1333035(96d:11071) ,