La conjecture de Birch et Swinnerton-Dyer $𝐩$-adique

Colmez, Pierre

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 919, pp. 251-319.

La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_{\infty }$ 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 $r_{\infty }=0$ ou $1$, mais on n’a aucun résultat reliant $r_{\infty }$ et $r$ si $r_{\infty }\ge 2$. 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 $r_{\infty }$ 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 $r_{\infty }=0$ or $1$, but there is no result relating $r$ and $r_{\infty }$ if $r_{\infty }\ge 2$. 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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