Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Modularity of certain potentially Barsotti-Tate Galois representations

Authors: Brian Conrad, Fred Diamond and Richard Taylor
Journal: J. Amer. Math. Soc. 12 (1999), 521-567
MSC (1991): Primary 11F80; Secondary 11G18
MathSciNet review: 1639612
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that certain potentially semistable lifts of modular mod$l$ representations are themselves modular. As a result we show that any elliptic curve over the rational numbers with conductor not divisible by 27 is modular.

References [Enhancements On Off] (What's this?)

  • 1. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Spring-Verlag, 1987. MR 91i:14034
  • 2. K. Brown, Cohomology of Groups, Springer-Verlag, 1982. MR 83k:20002
  • 3. 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
  • 4. H. Carayol, Sur les représentations galoisiennes modulo $\ell$ attachées aux formes modulaires, Duke Math. J. 59 (1989), 785-801. MR 91b:11058
  • 5. B. Conrad, Finite group schemes over bases with low ramification, to appear in Compositio Mathematica.
  • 6. B. Conrad, Ramified deformation problems, to appear in Duke Math. Journal.
  • 7. J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, 1992. MR 93m:11053
  • 8. C. Curtis, I. Reiner, Methods of Representation Theory, Volume 1, Wiley & Sons, New York, 1981. MR 82i:20001
  • 9. H. Darmon, F. Diamond, R. Taylor, Fermat's Last Theorem, in Current Developments in Mathematics, 1995, International Press, 1996, pp. 1-154. CMP 98:02
  • 10. B. de Smit, H. Lenstra, Explicit construction of universal deformation rings, in Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, 1997, pp. 313-326.
  • 11. 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
  • 12. F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), 137-166. MR 97d:11172
  • 13. F. Diamond, The Taylor-Wiles construction and multiplicity one, Inv. Math. 128 (1997), 379-391. MR 98c:11047
  • 14. F. Diamond, R. Taylor, Non-optimal levels for mod $\ell$modular representations of ${\operatorname{Gal}}(\overline{\mathbf Q}/\mathbf{Q})$, Inv. Math. 115 (1994), 435-462. MR 95c:11060
  • 15. F. Diamond, R. Taylor, Lifting modular mod $\ell$representations, Duke Math. J. 74 (1994), 253-269. MR 95e:11052
  • 16. B. Edixhoven, The weight in Serre's conjectures on modular forms, Inv. Math. 109 (1992), 563-594. MR 93h:11124
  • 17. N. Elkies, Elliptic and modular curves over finite fields, and related computational issues, to appear in Computational Perspectives on Number Theory (J. Teitelbaum, ed.).
  • 18. J.-M. Fontaine, Groupes $p$-divisibles sur les corps locaux, Astérisque 47-48, Société mathématique de France, Paris, 1977. MR 58:16699
  • 19. J.-M. Fontaine, Le corps des périodes $p$-adiques, in Périodes $p$-adiques, Astérisque 223, 59-111. MR 95k:11086
  • 20. J.-M. Fontaine, Représentations $p$-adiques semi-stables, in Périodes $p$-adiques, Astérisque 223, 113-184. MR 95g:14024
  • 21. J.-M. Fontaine, Sur certains types de représentations $p$-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti-Tate, Annals of Mathematics, 115 (1982), 529-577. MR 84d:14010
  • 22. 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
  • 23. K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.
  • 24. 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
  • 25. A. Grothendieck, Groupes de Monodromie en Géométrie Algébrique (SGA 7), Lecture Notes in Math. 288, Springer-Verlag, 1972. MR 50:7134
  • 26. A. Grothendieck, Schémas en Groupes (SGA3), Lecture Notes in Math. 151, Springer-Verlag, 1970. MR 43:223a
  • 27. Y. Ihara, On modular curves over finite fields, Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, 1975, pp. 161-202. MR 53:2956
  • 28. N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton Univ. Press, 1985. MR 86i:11024
  • 29. N. Katz, W. Messing, Some consequences of the Riemann Hypothesis for varieties over finite fields, Inv. Math. 23 (1974), 73-77. MR 48:11117
  • 30. B. Mazur, Deforming Galois representations, in Galois Groups over $\ensuremath{\mathbf{Q}} $ (MSRI, 1987) Springer-Verlag, 1989, pp. 385-437. MR 90k:11057
  • 31. D. Mumford, Abelian Varieties, Oxford University Press, 1970. MR 44:219
  • 32. R. Ramakrishna, On a variation of Mazur's deformation functor, Comp. Math. 87 (1993), 269-286. MR 94h:11054
  • 33. M. Raynaud, Schémas en groupes de type $(p,p,\dots,p)$, Bull. Soc. Math. France 102 (1974), 241-280. MR 54:7488
  • 34. T. Saito, Modular forms and $p$-adic Hodge theory, Inv. Math. 129 (1997), 607-620. MR 98g:11060
  • 35. J.-P. Serre, Le problème des groupes de congruence pour ${SL}_2$, Ann. Math. 92 (1970), 489-527. MR 42:7671
  • 36. J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Inv. Math. 15 (1972), 259-331. MR 52:8126
  • 37. J.-P. Serre, Cohomologie Galoisienne, Springer-Verlag, 1973. MR 53:8030
  • 38. J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977. MR 56:8675
  • 39. J.-P. Serre, Trees, Springer-Verlag, 1980. MR 82c:20083
  • 40. J.-P. Serre, Sur les représentations modulaires de degré $2$ de ${\operatorname{Gal}}(\overline{\mathbf Q}/\ensuremath{\mathbf{Q}} )$, Duke Math. J. 54 (1987), 179-230. MR 88g:11022
  • 41. J.-P. Serre, J. Tate, Good reduction of abelian varieties, Annals of Mathematics 88 (1968), 492-517. MR 38:4488
  • 42. G. Shimura, Algebraic number fields and symplectic discontinuous groups, Ann. Math. 86 (1967), 503-592. MR 36:5100
  • 43. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton Univ. Press, 1971. MR 47:3318
  • 44. J. Tate, $p$-divisible groups, in Proceedings of a Conference on Local Fields (Dreibergen, 1966), Springer, 1967, pp. 158-183. MR 38:155
  • 45. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553-572. MR 96d:11072
  • 46. 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 (1991): 11F80, 11G18

Retrieve articles in all journals with MSC (1991): 11F80, 11G18

Additional Information

Brian Conrad
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Fred Diamond
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854

Richard Taylor

Keywords: Hecke algebra, Galois representation, modular curves.
Received by editor(s): April 1, 1998
Received by editor(s) in revised form: September 1, 1998
Additional Notes: The first author was supported by an N.S.F. Postdoctoral Fellowship, and would like to thank the Institute for Advanced Study for its hospitality. The second author was at M.I.T. during part of the research, and for another part was visiting Université de Paris-Sud supported by the C.N.R.S. The third author was supported by a grant from the N.S.F. All of the authors are grateful to Centre Émile Borel at the Institut Henri Poincaré for its hospitality at the $p$-adic semester.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society