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

Longo, Matteo

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

Soit $E/F$ une courbe elliptique modulaire définie sur un champ de nombres totalement réel $F$ et soit $\phi$ 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 $[F:\mathbb{Q}]$ est pair et $\phi$ n’est pas nouveau en aucun idéal premier.

Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi$ 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 $[F:\mathbb{Q}]$ is even and $\phi$ not new at any prime.

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