Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 729-741.

Let K be a totally real field which is a finite abelian extension over 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.

Soit K un corps totalement réel qui est une extension abélienne finie de 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.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1047
Classification: 11F80, 11G05, 11F41
Keywords: elliptic curves, Hilbert modular forms, Galois representations
Yoshikawa, Sho 1

1 Gakushuin University, Department of Mathematics, 1-5-1, Mejiro, Toshima-ku, Tokyo, Japan
@article{JTNB_2018__30_3_729_0,
     author = {Yoshikawa, Sho},
     title = {Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {729--741},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1047},
     mrnumber = {3938624},
     zbl = {1441.11135},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1047/}
}
TY  - JOUR
AU  - Yoshikawa, Sho
TI  - Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 729
EP  - 741
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1047/
DO  - 10.5802/jtnb.1047
LA  - en
ID  - JTNB_2018__30_3_729_0
ER  - 
%0 Journal Article
%A Yoshikawa, Sho
%T Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 729-741
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1047/
%R 10.5802/jtnb.1047
%G en
%F JTNB_2018__30_3_729_0
Yoshikawa, Sho. Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 729-741. doi : 10.5802/jtnb.1047. http://www.numdam.org/articles/10.5802/jtnb.1047/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: