On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo, Matteo

doi:10.5802/aif.2197

Longo, Matteo

Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 689-733.

Abstract
Résumé

Let $E / F$ be a modular elliptic curve defined over a totally real number field $F$ and let $φ$ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of $E$ over suitable quadratic imaginary extensions $K / F$ . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when $[F : ℚ]$ is even and $φ$ not new at any prime.

Soit $E / F$ une courbe elliptique modulaire définie sur un champ de nombres totalement réel $F$ et soit $φ$ la forme propre associée. Ce papier présente un nouvelle méthode, inspirée par un récent travail de Bertolini et Darmon, pour contrôler le rang de $E$ sur des extensions convenables quadratiques imaginaires $K / F$ . En particulier, ce résultat peut être appliqué aux cas qui ne sont pas considérés dans le travail de Kolyvagin et Logachëv, i.e., quand $[F : ℚ]$ est pair et $φ$ n’est pas nouveau en aucun idéal premier.

Received: 2005-02-28
Revised: 2005-08-03
Accepted: 2005-09-05

MR: 2244227 | Zbl: 1152.11028

DOI: 10.5802/aif.2197
Classification: 11G05, 11G18, 11G40, 11F30
Keywords: Elliptic Curves, Birch and Swinnerton-Dyer Conjecture, Shimura Varieties, Congruences between Hilbert Modular Forms

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Longo, Matteo. On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields. Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 689-733. doi : 10.5802/aif.2197. https://aif.centre-mersenne.org/articles/10.5802/aif.2197/

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