On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises

Breuil, Christophe¹ ; Conrad, Brian^2, 3 ; Diamond, Fred⁴ ; Taylor, Richard²

¹ Département de Mathématiques, CNRS, Université Paris-Sud, 91405 Orsay cedex, France
² Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
³ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
⁴ Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454

Journal of the American Mathematical Society, Tome 14 (2001) no. 4, pp. 843-939

Résumé

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

Détail

DOI : 10.1090/S0894-0347-01-00370-8

Bibliographie

[1] Berthelot, Pierre, Breen, Lawrence, Messing, William Théorie de Dieudonné cristalline. II 1982

[2] Breuil, Christophe Schémas en groupe et modules filtrés C. R. Acad. Sci. Paris Sér. I Math. 1999 93 97

[3] Carayol, Henri Sur les représentations 𝑙-adiques associées aux formes modulaires de Hilbert Ann. Sci. École Norm. Sup. (4) 1986 409 468

[4] Carayol, Henri Sur les représentations galoisiennes modulo 𝑙 attachées aux formes modulaires Duke Math. J. 1989 785 801

[5] Conrad, Brian Ramified deformation problems Duke Math. J. 1999 439 513

[6] Conrad, Brian, Diamond, Fred, Taylor, Richard Modularity of certain potentially Barsotti-Tate Galois representations J. Amer. Math. Soc. 1999 521 567

[7] De Jong, A. J. Finite locally free group schemes in characteristic 𝑝 and Dieudonné modules Invent. Math. 1993 89 137

[8] Deligne, Pierre, Serre, Jean-Pierre Formes modulaires de poids 1 Ann. Sci. École Norm. Sup. (4) 1974

[9] Diamond, Fred The refined conjecture of Serre 1995 22 37

[10] Diamond, Fred On deformation rings and Hecke rings Ann. of Math. (2) 1996 137 166

[11] Diamond, Fred, Taylor, Richard Lifting modular mod 𝑙 representations Duke Math. J. 1994 253 269

[12] Ekedahl, Torsten An effective version of Hilbert’s irreducibility theorem 1990 241 249

[13] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV Inst. Hautes Études Sci. Publ. Math. 1967 361

[14] Rosenblatt, Alfred Sur les points singuliers des équations différentielles C. R. Acad. Sci. Paris 1939 10 11

[15] Fontaine, Jean-Marc, Mazur, Barry Geometric Galois representations 1995 41 78

[16] Gã©Rardin, Paul Facteurs locaux des algèbres simples de rang 4. I 1978 37 77

[17] Langlands, Robert P. Base change for 𝐺𝐿(2) 1980

[18] Manoharmayum, J. Pairs of mod 3 and mod 5 representations arising from elliptic curves Math. Res. Lett. 1999 735 754

[19] Mazur, B. Number theory as gadfly Amer. Math. Monthly 1991 593 610

[20] Raynaud, Michel Schémas en groupes de type (𝑝,…,𝑝) Bull. Soc. Math. France 1974 241 280

[21] Ribet, Kenneth A. Galois representations attached to eigenforms with Nebentypus 1977 17 51

[22] Shepherd-Barron, N. I., Taylor, R. 𝑚𝑜𝑑2 and 𝑚𝑜𝑑5 icosahedral representations J. Amer. Math. Soc. 1997 283 298

[23] Serre, Jean-Pierre Local fields 1979

[24] Serre, Jean-Pierre Sur les représentations modulaires de degré 2 de 𝐺𝑎𝑙(\overline{𝐐}/𝐐) Duke Math. J. 1987 179 230

[25] Shimura, Goro On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields Nagoya Math. J. 1971 199 208

[26] Shimura, Goro Introduction to the arithmetic theory of automorphic functions 1971

[27] Tate, J. T. 𝑝-divisible groups 1967 158 183

[28] Taylor, Richard, Wiles, Andrew Ring-theoretic properties of certain Hecke algebras Ann. of Math. (2) 1995 553 572

[29] Tunnell, Jerrold Artin’s conjecture for representations of octahedral type Bull. Amer. Math. Soc. (N.S.) 1981 173 175

[30] Weil, Andr㩠Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen Math. Ann. 1967 149 156

[31] Weil, Andr㩠Scientific works. Collected papers. Vol. I (1926–1951) 1979

[32] Wiles, Andrew Modular elliptic curves and Fermat’s last theorem Ann. of Math. (2) 1995 443 551