Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7
Sho Yoshikawa1
1 Gakushuin University, Department of Mathematics, 1-5-1, Mejiro, Toshima-ku, Tokyo, Japan
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 729-741.

Soit K un corps totalement réel qui est une extension abélienne finie de ℚ non ramifiée en 3,5 et 7. Nous prouvons que toute courbe elliptique E sur K est modulaire, en réduisant la question de modularité de E aux théorèmes de relèvement modulaire connus.

Let K be a totally real field which is a finite abelian extension over ℚ and is unramified at 3,5, and 7. We prove that any elliptic curve E over K is modular, by reducing modularity of E to known modularity lifting theorems.

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DOI : 10.5802/jtnb.1047
Classification : 11F80, 11G05, 11F41
Mots clés : elliptic curves, Hilbert modular forms, Galois representations
Sho Yoshikawa 1

1 Gakushuin University, Department of Mathematics, 1-5-1, Mejiro, Toshima-ku, Tokyo, Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Sho Yoshikawa. Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 729-741. doi : 10.5802/jtnb.1047. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1047/

[1] Patrick B. Allen Modularity of nearly ordinary 2-adic residually dihedral Galois representations, Compos. Math., Volume 150 (2014) no. 8, pp. 1235-1346 | DOI | MR | Zbl

[2] Christophe Breuil; Brian Conrad; Fred Diamond; Richard Taylor On the modularity of elliptic curves over ℚ: wild 3-adic exercises, J. Am. Math. Soc., Volume 14 (2001), pp. 843-939 | DOI | MR | Zbl

[3] Kevin Buzzard Potential modularity - a survey (2010) (https://arxiv.org/abs/1101.0097) | Zbl

[4] Nuno Freitas Recipes to Fermat-type equations of the form x r +y r =Cz p , Math. Z., Volume 279 (2015) no. 3-4, pp. 605-639 | DOI | MR | Zbl

[5] Nuno Freitas; Bao V. Le Hung; Samir Siksek Elliptic Curves over Real Quadratic Fields are Modular, Invent. Math., Volume 201 (2015) no. 1, pp. 159-206 | DOI | MR | Zbl

[6] Sudesh Kalyanswamy Remarks on automorphy of residually dihedral representations (2016) (https://arxiv.org/abs/1607.04750) | Zbl

[7] Alain Kraus Détermination du poids et du conducteur associés aux représentations des points de p-torsion d’une courbe elliptique, Dissertationes Mathematicae, 364, Instytut Matematyczny Polskiej Akademi Nauk, 1997 | MR | Zbl

[8] Bao V. Le Hung Modularity of some elliptic curves over totally real fields (2013) (https://arxiv.org/abs/1309.4134)

[9] Jan Nekovář On the parity of ranks of Selmer groups. IV, Compos. Math., Volume 145 (2009) no. 6, pp. 1351-1359 | DOI | MR | Zbl

[10] Jean-Pierre Serre; John Tate Good Reduction of Abelian Varieties, Ann. Math., Volume 88 (1968) no. 3, pp. 492-517 | DOI | MR | Zbl

[11] Joseph H. Silverman The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer, 1986, xii+400 pages | MR | Zbl

[12] Christopher Skinner Nearly ordinary deformation of residually dihedral representations (preprint)

[13] Christopher Skinner; Andrew Wiles Residually reducible representations and modular forms, Publ. Math., Inst. Hautes Étud. Sci., Volume 89 (1999), pp. 5-126 | Zbl

[14] Christopher Skinner; Andrew Wiles Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse, Math., Volume 10 (2001) no. 1, pp. 185-215 | DOI | Numdam | MR | Zbl

[15] Richard Taylor Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu, Volume 1 (2002) no. 1, pp. 125-143 | MR | Zbl

[16] Richard Taylor; Andrew Wiles Ring-theoretic properties of certain Hecke algebras, Ann. Math., Volume 141 (1995) no. 3, pp. 553-572 | DOI | MR | Zbl

[17] Jack Thorne Automorphy of some residually dihedral Galois representations, Math. Ann., Volume 364 (2016) no. 1-2, pp. 589-648 | DOI | MR | Zbl

[18] Andrew Wiles Modular elliptic curves and Fermat’s Last Theorem, Ann. Math., Volume 141 (1995) no. 3, pp. 443-551 | DOI | MR | Zbl

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