no. 3
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
Longo, Matteo1
1 Université Louis Pasteur IRMA 7, rue René Descartes 67084 Strasbourg (France)
Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 689-733.

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.

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
Mot clés : courbes elliptiques, conjecture de Birch et Swinnerton-Dyer, variétés de Shimura, congruences entre formes modulaires de Hilbert
Longo, Matteo 1

1 Université Louis Pasteur IRMA 7, rue René Descartes 67084 Strasbourg (France) 
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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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