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La conjecture de Birch et Swinnerton-Dyer 𝐩 -adique

Pierre Colmez (2002/2003)

Séminaire Bourbaki

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...

Lower powers of elliptic units

Stefan Bettner, Reinhard Schertz (2001)

Journal de théorie des nombres de Bordeaux

In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program...

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