La conjecture de Birch et Swinnerton-Dyer 𝐩-adique
[The p-adic Birch and Swinnerton-Dyer’s conjecture]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 919, pp. 251-319.

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.

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.

Classification: 11-02, 11F11, 11F67, 11F80, 11F85, 11G05, 11G16, 11G40, 11R33, 11R39, 11R56, 11S80, 11S99, 14F30, 14
Mot clés : courbe elliptique, fonction $L$ $p$-adique
Keywords: elliptic curve, $p$-adic $L$ function
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Colmez, Pierre. La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 919, pp. 251-319. http://archive.numdam.org/item/SB_2002-2003__45__251_0/

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