Let be a modular elliptic curve defined over a totally real number field and let be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of over suitable quadratic imaginary extensions . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when is even and not new at any prime.
Soit une courbe elliptique modulaire définie sur un champ de nombres totalement réel et soit la forme propre associée. Ce papier présente un nouvelle méthode, inspirée par un récent travail de Bertolini et Darmon, pour contrôler le rang de sur des extensions convenables quadratiques imaginaires . En particulier, ce résultat peut être appliqué aux cas qui ne sont pas considérés dans le travail de Kolyvagin et Logachëv, i.e., quand est pair et n’est pas nouveau en aucun idéal premier.
Revised:
Accepted:
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
@article{AIF_2006__56_3_689_0, 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}, pages = {689--733}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {3}, year = {2006}, doi = {10.5802/aif.2197}, zbl = {1152.11028}, mrnumber = {2244227}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2197/} }
TY - JOUR TI - On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields JO - Annales de l'Institut Fourier PY - 2006 DA - 2006/// SP - 689 EP - 733 VL - 56 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2197/ UR - https://zbmath.org/?q=an%3A1152.11028 UR - https://www.ams.org/mathscinet-getitem?mr=2244227 UR - https://doi.org/10.5802/aif.2197 DO - 10.5802/aif.2197 LA - en ID - AIF_2006__56_3_689_0 ER -
%0 Journal Article %T On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields %J Annales de l'Institut Fourier %D 2006 %P 689-733 %V 56 %N 3 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2197 %R 10.5802/aif.2197 %G en %F AIF_2006__56_3_689_0
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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