Rational elliptic curves are modular

Edixhoven, Bas

Séminaire Bourbaki : volume 1999/2000, exposés 865-879, Astérisque, no. 276 (2002), Exposé no. 871, 28 p.

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     title = {Rational elliptic curves are modular},
     booktitle = {S\'eminaire Bourbaki : volume 1999/2000, expos\'es 865-879},
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     pages = {161--188},
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Edixhoven, Bas. Rational elliptic curves are modular, dans Séminaire Bourbaki : volume 1999/2000, exposés 865-879, Astérisque, no. 276 (2002), Exposé no. 871, 28 p. https://www.numdam.org/item/SB_1999-2000__42__161_0/

Bibliographie
Cité par

[1] P. Berthelot - Altérations de variétés algébriques. Séminaire Bourbaki, 1995-96, exposé 815, Astérisque 241, S.M.F. (1997), 273-311. | Numdam | MR | Zbl

[2] C. Breuil - Schémas en groupes et modules filtrés. C. R. Acad. Sci. Paris 328 (1999), 93-97. | MR | Zbl

[3] C. Breuil - Groupes $p$ -divisibles, groupes finis et modules filtrés. Ann. of Math. (2) 152 (2000), no. 2, 489-549. Mathematics. | MR | Zbl

[4] C. Breuil, B. Conrad, F. Diamond, R. Taylor - On the modularity of elliptic curves over $ℚ$ : wild $3$ -adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939. | MR | Zbl

[5] C. Breuil, A. Mezard - Multiplicités modulaires et représentations de ${GL}_{2} (ℤ_{p})$ et de $Gal ({\bar{ℚ}}_{p} / {\bar{ℚ}}_{p})$ en $l = p$ . Preprint, June 2000, Orsay.

[6] C. J. Bushnell - Smooth representations of $p$ -adic groups: the role of compact open subgroups. Proceedings of the international congress of mathematicians, Vol. 1, 2 (Zürich, 1994), 770-779, Birkhaüser, Basel, 1995. | MR | Zbl

[7] K. Buzzard, M. Dickinson, N. Shepherd-Barron, R. Taylor - On icosahedral Artin representations. Duke Math. J. 109 (2001), no. 2, 283-318. | MR | Zbl

[8] K. Buzzard, R. Taylor - Companion forms and weight one forms. Ann. of Math. (2) 149 (1999), no. 3, 905-91 | MR | Zbl

[9] H. Carayol - Sur les représentations $ℓ$ -adiques associées aux formes modulaires de Hilbert. Ann. Sci. Éc. Norm. Sup. 19 (1986), 409-468. | Numdam | MR | Zbl

[10] L. Clozel - Motifs et formes automorphes. Automorphic forms, Shimura varieties, and $L$ -functions, Vol. I, edited by L. Clozel and J.S. Milne, Perspectives in Mathematics 10, Academic Press (1990), 77-159. | MR | Zbl

[11] P. Colmez, J.-M. Fontaine - Construction des représentations $p$ -adiques semi-stables. Invent. Math. 140 (2000), no. 1, 1-43. | MR | Zbl

[12] B. Conrad - Finite group schemes over bases with low ramification. Compositio Math. 119 (1999), 239-320. | MR | Zbl

[13] B. Conrad - Ramified deformation problems. Duke Math. J. 97 (1999), 439-513. | MR | Zbl

[14] B. Conrad, F. Diamond, R. Taylor - Modularity of certain potentially Barsotti-Tate Galois representations. J.A.M.S. 12 (1999), 521-567. | MR | Zbl

[15] G. Cornell, J. Silverman, G. Stevens, EDITORS - Modular Forms and Fermat's Last Theorem Springer-Verlag, 1997. | MR | Zbl

[16] H. Darmon - A proof of the full Shimura-Taniyama-Weil conjecture is announced. Notices of the A.M.S., December 1999. | MR | Zbl

[17] H. Darmon, F. Diamond, R. Taylor - Fermat's Last Theorem. Elliptic curves, modular forms and Fermat's last theorem (Hong Kong, 1993). Second edition. International Press, Cambridge, MA, 1997, 2-140. | MR | Zbl

[18] P. Deligne - Formes modulaires et représentations $ℓ$ -adiques. Séminaire Bourbaki, 1968-69, exposé 355, Lecture Notes in Mathematics 179, Springer (1969), 139-172. | EuDML | Numdam | Zbl

[19] P. Deligne - Formes modulaires et représentations de $G L (2)$ . Antwerp II, Lecture Notes in Math. 349, Springer (1973), 55-106. | MR | Zbl

[20] B. De Smit, H. Lenstra - Explicit construction of universal deformation rings. In [15], 313-326. | MR | Zbl

[21] B. De Smit, K. Rubin, R. Schoof - Criteria for complete intersections. In [15], 343-355. | Zbl

[22] F. Diamond - The refined conjecture of Serre. In Elliptic curves, modular forms and Fermat's last theorem (Hong Kong, 1993). International Press, 1995, 22-37. | MR | Zbl

[23] F. Diamond - An extension of Wiles' results. In [15], 475-498. | MR | Zbl

[24] F. Diamond - On deformation rings and Hecke rings. Ann. Math. 144 (1996), 137-166. | MR | Zbl

[25] F. Diamond - The Taylor-Wiles construction and multiplicity one. Invent. math. 128 (1997), 379-391. | MR | Zbl

[26] F. Diamond, J. Im - Modular forms and modular curves. In "Seminar on Fermat's last theorem", Canadian Mathematical Society Conference Proceedings 17, 1995 (V. Kumar Murty, editor). | MR | Zbl

[27] F. Diamond, R. Taylor - Lifting modular mod $ℓ$ representations. Duke Math. J. 74 (1994), no. 2, 253-269. | MR | Zbl

[28] M. Dickinson - On the modularity of certain $2$ -adic Galois representations. Duke Math. J. 109 (2001), no. 2, 319-382 | MR | Zbl

[29] S. J. Edixhoven - The weight in Serre's conjectures on modular forms. Invent. math. 109 (1992), 563-594. | EuDML | MR | Zbl

[30] J.-M. Fontaine - Groupes $p$ -divisibles sur les corps locaux. Astérique 47-48, S.M.F., 1977. | Numdam | MR | Zbl

[31] J.-M. Fontaine - Représentations $p$ -adiques semi-stables. In "Périodes $p$ -adiques," Astérisque 223, S.M.F., 1994, p. 113-184. | Numdam | MR | Zbl

[32] 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 | Zbl

[33] K. Fujiwara - Deformation rings and Hecke rings in the totally real case. Preprint.

[34] C. Khare - A local analysis of congruences in the $(p, p)$ -case. II. Invent. Math. 143 (2001), no. 1, 129-155. | MR | Zbl

[35] C. Khare - On isomorphisms between deformation and Hecke rings. Work in progress.

[36] R. P. Langlands - Base change for $G L (2)$ . Ann. of Math. Studies 96, Princeton University Press, Princeton, 1980. | MR | Zbl

[37] J. Manoharmayum - Pairs of mod $3$ and mod $5$ representations arising from elliptic curves. Thesis, Cambridge University, 1998. | MR

[38] L. Moret-Bailly - Groupes de Picard et problèmes de Skolem II. Ann. Sci. ENS 22 (1989), 181-194. | EuDML | Numdam | MR | Zbl

[39] R. Noot - Abelian varieties with $ℓ$ -adic Galois representation of Mumford's type J. Reine Angew. Math. 519 (2000), 155-169 | MR | Zbl

[40] J. Oesterlé - Nouvelles approches du "théorème" de Fermat. Séminaire Bourbaki, 1987-88, exposé 694, Astérisque 161-162, S.M.F. (1988), 165-186. | EuDML | Numdam | MR | Zbl

[41] J. Oesterlé - Travaux de Wiles (et Taylor,...), partie II. Séminaire Bourbaki, 1994-95, exposé 804, Astérisque 237, S.M.F. (1996), 333-355. | EuDML | Numdam | MR | Zbl

[42] R. Ramakrishna - On a variation of Mazur's deformation functor. Compositio Math. 87 (1993), 269-286. | EuDML | Numdam | MR | Zbl

[43] R. Ramakrishna - Lifting Galois representations. Invent. math. 138 (1999), 537-562. | MR | Zbl

[44] R. Ramakrishna - Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur. Preprint. | MR

[45] K. A. Ribet, - On modular representations of $Gal (\bar{ℚ} / ℚ)$ arising from modular forms. Invent. math. 100 (1990), 431-476. | EuDML | MR | Zbl

[46] K. Rubin - Modularity of mod 5 representations. In [15], 463-474. | MR | Zbl

[47] T. Saito - Modular forms and $p$ -adic Hodge theory. Invent. math. 129 (1997), 607-620. | MR | Zbl

[48] J-P. Serre - Représentations $ℓ$ -adiques. Kyoto Int. Symposium on Algebraic Number Theory, Japan Soc. for the Promotion of Science (1977), pp. 177-193, (Oeuvres, t. III, pp. 384-400). | MR | Zbl

[49] J-P. Serre - Sur les représentations modulaires de degré $2$ de $Gal (\bar{ℚ} / ℚ)$ . Duke Math. J. 54 (1987), 179-230. | MR | Zbl

[50] J-P. Serre - Travaux de Wiles (et Taylor,...), partie I. Séminaire Bourbaki, 1994-95, exposé 803, Astérisque 237, S.M.F. (1996), 319-332. | EuDML | Numdam | MR | Zbl

[51] C. Skinner, A. Wiles - Residually reducible representations and modular forms. Pub. Math. IHES. 89 (1999), 5-126. | EuDML | Numdam | MR | Zbl

[52] C. Skinner, A. Wiles - Base change and a problem of Serre. Duke Math. J. 107 (2001), no. 1, 15-25. | MR | Zbl

[53] C. Skinner, A. Wiles - Nearly ordinary deformations of irreducible residual representations. Preprint.

[54] N. Shepherd-Barron, R. Taylor - Mod $2$ and mod $5$ icosahedral representations. J. Amer. Math. Soc. 10 (1997), 281-332. | MR | Zbl

[55] J. Tate - $p$ -divisible groups. Proceedings of a conference on local fields (Driebergen, 1966), Springer, 1967, 158-183. | MR | Zbl

[56] J. Tate - Number theoretic background. Proceedings of Symposia in Pure Mathematics 33 (1979), part 2, 3-26. | MR | Zbl

[57] R. Taylor - Icosahedral Galois representations. Pacific J. Math.: Special issue: Olga Taussky-Todd: in memoriam (1997), 337-347. | MR | Zbl

[58] R. Taylor - Remarks on a conjecture of Fontaine and Mazur. Preprint, http : //www. math.harvard.edu/ | MR

[59] R. Taylor - On icosahedral Artin representations II. Preprint, http://www.math.harvard.edu/ | MR

[60] R. Taylor, A. Wiles - Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141 (1995), 553-572. | MR | Zbl

[61] J. Tunnell - Artin's conjecture for representations of octahedral type. Bull. A.M.S. 5 (1981), 173-175. | MR | Zbl

[62] M. Volkov - Les représentations $ℓ$ -adiques associées aux courbes elliptiques sur $ℚ_{p}$ . J. Reine Angew. Math. 535 (2001), 65-101. | MR | Zbl

[63] A. Weil - Exercices dyadiques. Invent. math. 27 (1974), 1-22. | EuDML | MR | Zbl

[64] A. Wiles - Modular elliptic curves and Fermat's Last Theorem. Ann. Math. 141 (1995), 443-551. | MR | Zbl