Colmez, Pierre
La conjecture de Birch et Swinnerton-Dyer 𝐩-adique
Séminaire Bourbaki, Tome 45 (2002-2003) , Exposé no. 919 , p. 251-320
Zbl 1094.11025 | MR 2111647 | 1 citation dans Numdam
URL stable : http://www.numdam.org/item?id=SB_2002-2003__45__251_0

Classification:  11-02,  11F11,  11F67,  11F80,  11F85,  11G05,  11G16,  11G40,  11R33,  11R39,  11R56,  11S80,  11S99,  14F30,  14
Mots clés: courbe elliptique, fonction L p-adique
La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre r du zéro en s=1 de la fonction L d’une courbe elliptique E définie sur 𝐐 est égal au rang r du groupe de ses points rationnels. On sait démontrer cette conjecture si r =0 ou 1, mais on n’a aucun résultat reliant r et r si r 2. Nous expliquerons comment Kato démontre que la fonction L p-adique attachée à E a, en s=1, un zéro d’ordre supérieur ou égal à r.
The classical Birch and Swinnerton-Dyer’s conjecture asserts that the order r of the zero at s=1 of the L-function of an elliptic curve E defined over 𝐐 is equal to the rank r of its group of rational points. This is a theorem if r =0 or 1, but there is no result relating r and r if r 2. We will explain how Kato proves that the p-adic L function attached to E has, at s=1, a zero of order at least r.

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