The classical Birch and Swinnerton-Dyer’s conjecture asserts that the order of the zero at of the -function of an elliptic curve defined over is equal to the rank of its group of rational points. This is a theorem if or , but there is no result relating and if . We will explain how Kato proves that the -adic function attached to has, at , a zero of order at least .
La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre du zéro en de la fonction d’une courbe elliptique définie sur est égal au rang du groupe de ses points rationnels. On sait démontrer cette conjecture si ou , mais on n’a aucun résultat reliant et si . Nous expliquerons comment Kato démontre que la fonction -adique attachée à a, en , un zéro d’ordre supérieur ou égal à .
Mot clés : courbe elliptique, fonction $L$ $p$-adique
Keywords: elliptic curve, $p$-adic $L$ function
@incollection{SB_2002-2003__45__251_0, author = {Colmez, Pierre}, title = {La conjecture de {Birch} et {Swinnerton-Dyer} $\mathbf {p}$-adique}, booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923}, series = {Ast\'erisque}, note = {talk:919}, pages = {251--319}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {294}, year = {2004}, mrnumber = {2111647}, zbl = {1094.11025}, language = {fr}, url = {http://www.numdam.org/item/SB_2002-2003__45__251_0/} }
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%0 Book Section %A Colmez, Pierre %T La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique %B Séminaire Bourbaki : volume 2002/2003, exposés 909-923 %A Collectif %S Astérisque %Z talk:919 %D 2004 %P 251-319 %N 294 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://www.numdam.org/item/SB_2002-2003__45__251_0/ %G fr %F SB_2002-2003__45__251_0
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://www.numdam.org/item/SB_2002-2003__45__251_0/
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