Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On the modularity of elliptic curves over $\mathbf{Q}$: Wild $3$-adic exercises

Author(s): Christophe Breuil; Brian Conrad; Fred Diamond; Richard Taylor
Journal: J. Amer. Math. Soc. 14 (2001), 843-939.
MSC (2000): Primary 11G05; Secondary 11F80
Posted: May 15, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We complete the proof that every elliptic curve over the rational numbers is modular.

References:

[BBM]
P. Berthelot, L. Breen, W. Messing, Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics 930, Springer-Verlag, Berlin, 1982. MR 85k:14023
[Br1]
C. Breuil, Schémas en groupes et modules filtrés, C. R. Acad. Sci. Paris 328 (1999), 93-97. MR 99m:14086

[Br2]
C. Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés, Annals of Math. 152 (2000), 489-549. CMP 2001:06

[Ca1]
H. Carayol, Sur les représentations $\ell$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Sup. 19 (1986), 409-468. MR 89c:11083

[Ca2]
H. Carayol, Sur les représentations galoisiennes modulo $\ell$ attachées aux formes modulaires, Duke Math. J. 59 (1989), 785-801. MR 91b:11058
[Co]
B. Conrad, Ramified deformation problems, Duke Math. J. 97 (1999), 439-514. MR 2000h:11055

[CDT]
B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), 521-567. MR 99i:11037

[deJ]
A.J. de Jong, Finite locally free group schemes in characteristic $p$ and Dieudonné modules, Inv. Math. 114 (1993), 89-138. MR 94j:14043

[De]
P. Deligne, Formes modulaires et représentations $\ell$-adiques, in: Lecture Notes in Math. 179, Springer-Verlag, 1971, pp. 139-172.

[DS]
P. Deligne, J.-P. Serre, Formes modulaires de poids $1$, Ann. Sci. Ec. Norm. Sup. 7 (1974), 507-530. MR 52:284
[Di1]
F. Diamond, The refined conjecture of Serre, in Elliptic Curves, Modular Forms and Fermat's Last Theorem (Hong Kong, 1993), International Press, 1995, pp. 22-37. MR 97b:11065

[Di2]
F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), 137-166. MR 97d:11172

[DT]
F. Diamond, R. Taylor, Lifting modular mod $\ell$representations, Duke Math. J. 74 (1994), 253-269. MR 95e:11052

[E]
T. Ekedahl, An effective version of Hilbert's irreducibility theorem, in ``Séminaire de Théorie des Nombres, Paris 1988-89'', Birkhäuser, 1990. MR 92f:14018
[EGA]
A. Grothendieck, Éléments de Géométrie Algébrique IV$_4$, Publ. Math. IHES 32, 1966-7. MR 39:220

[Fa]
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inv. Math. 73 (1983), 349-366. MR 85g:11026a; erratum MR 85g:11026b

[FM]
J.-M. Fontaine, B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms and Fermat's Last Theorem (Hong Kong, 1993), International Press, 1995, pp. 41-78. MR 96h:11049

[G]
P. Gérardin, Facteurs locaux des algèbres simples de rang 4. I, in Groupes Réductifs et Formes Automorphes, I (Paris, 1976-77) Univ. Paris VII, 1978, pp. 37-77. MR 84f:22023

[Kh]
C. Khare, A local analysis of congruences in the $(p,p)$case: Part II, Invent. Math. 143 (2001), no. 1, 129-155. CMP 2001:06

[Kl]
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, 1884.

[L]
R.P. Langlands, Base Change for ${{\GL}}(2)$, Annals of Math. Studies 96, Princeton Univ. Press, Princeton, 1980. MR 82a:10032

[Man]
J. Manoharmayum, Pairs of mod $3$ and mod $5$representations arising from elliptic curves, Math. Res. Lett. 6 (1999), 735-754. MR 2000m:11045

[Maz]
B. Mazur, Number theory as gadfly, Amer. Math. Monthly 98 (1991), 593-610. MR 92f:11077
[Ra]
M. Raynaud, Schémas en groupes de type $(p,p,\dots,p)$, Bull. Soc. Math. France 102 (1974), 241-280. MR 54:7488

[Ri]
K. Ribet, The $\ell$-adic representations attached to an eigenform with Nebentypus: a survey, in: Lecture Notes in Math. 601, Springer-Verlag, 1977, pp. 17-52. MR 56:11907

[SBT]
N. Shepherd-Barron, R. Taylor, Mod $2$ and mod $5$icosahedral representations, J. Amer. Math. Soc. 10 (1997), 283-298. MR 97h:11060
[Se1]
J.-P. Serre, Local Fields, Springer-Verlag, 1979. MR 82e:12016

[Se2]
J.-P. Serre, Sur les représentations modulaires de degré $2$ de ${\Gal}(\overline{\ensuremath{\mathbf{Q}} }/\ensuremath{\mathbf{Q}} )$, Duke Math. J. 54 (1987), 179-230. MR 88g:11022

[Sh1]
G. Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199-208. MR 45:5111

[Sh2]
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, Princeton, 1971. MR 47:3318

[Sh3]
G. Shimura, Response to 1996 Steele Prize, Notices of the AMS 43 (1996), 1344-1347.

[T]
J. Tate, $p$-divisible groups, in ``Proceedings of a conference on local fields (Driebergen, 1966)'', Springer, 1967, 158-183. MR 38:155

[TW]
R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553-572. MR 96d:11072

[T]
J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. 5 (1981), 173-175. MR 82j:12015

[We1]
A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1968), 149-156. MR 34:7473

[We2]
A. Weil, Scientific works. Collected papers. III (1964-1978), Springer-Verlag, 1979. MR 80k:01067c

[Wi]
A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. Math. 141 (1995), 443-551. MR 96d:11071

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11G05, 11F80

Retrieve articles in all Journals with MSC (2000): 11G05, 11F80

Additional Information:

Christophe Breuil
Affiliation: Département de Mathématiques, CNRS, Université Paris-Sud, 91405 Orsay cedex, France
Email: Christophe.BREUIL@math.u-psud.fr

Brian Conrad
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: bconrad@math.harvard.edu, bdconrad@math.lsa.umich.edu

Fred Diamond
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454
Email: fdiamond@euclid.math.brandeis.edu

Richard Taylor
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: rtaylor@math.harvard.edu

DOI: 10.1090/S0894-0347-01-00370-8
PII: S 0894-0347(01)00370-8
Keywords: Elliptic curve, Galois representation, modularity
Received by editor(s): February 28, 2000
Received by editor(s) in revised form: January 1, 2001
Posted: May 15, 2001
Additional Notes: The first author was supported by the CNRS. The second author was partially supported by a grant from the NSF. The third author was partially supported by a grant from the NSF and an AMS Centennial Fellowship, and was working at Rutgers University during much of the research. The fourth author was partially supported by a grant from the NSF and by the Miller Institute for Basic Science.
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google