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, p. 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:] 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:] est pair et φ n’est pas nouveau en aucun idéal premier.

DOI : https://doi.org/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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     author = {Longo, Matteo},
     title = {On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {3},
     year = {2006},
     pages = {689-733},
     doi = {10.5802/aif.2197},
     mrnumber = {2244227},
     zbl = {1152.11028},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_3_689_0}
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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. http://www.numdam.org/item/AIF_2006__56_3_689_0/

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