Controlling Selmer groups in the  higher core rank case

Barry Mazur; Karl Rubin

doi:10.5802/jtnb.933

Controlling Selmer groups in the higher core rank case

Barry Mazur ¹ ; Karl Rubin ²

¹ Department of Mathematics Harvard University Cambridge, MA 02138, USA
² Department of Mathematics UC Irvine Irvine, CA 92697, USA

Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 145-183.

Abstract (VO)
Résumé

We define Kolyvagin systems and Stark systems attached to $p$ -adic representations in the case of arbitrary “core rank” (the core rank is a measure of the generic Selmer rank in a family of Selmer groups). Previous work dealt only with the case of core rank one, where the Kolyvagin and Stark systems are collections of cohomology classes. For general core rank, they are collections of elements of exterior powers of cohomology groups. We show under mild hypotheses that for general core rank these systems still control the size and structure of Selmer groups, and that the module of all Kolyvagin (or Stark) systems is free of rank one.

Nous définissons les systèmes de Kolyvagin et les systèmes de Stark attachés aux représentations $p$ -adiques dans le cas de “core rank” (qui est une mesure du rang de Selmer générique dans une famille de groupes de Selmer) arbitraire. Les travaux antérieurs traitaient seulement le cas de “core rank” $1$ , où les systèmes de Kolyvagin et de Stark sont des collections de classes cohomologiques. Dans le cas général, ce sont des collections d’éléments de produits extérieurs de groupes cohomologiques. Nous montrons, sous des conditions faibles, que ces systèmes contrôlent encore la taille et la structure des groupes de Selmer, et que le module des systèmes de Kolyvagin (ou de Stark) est libre de rang $1$ .

Reçu le : 2013-12-16
Accepté le : 2015-05-28
Publié le : 2017-01-02

MR Zbl

DOI : 10.5802/jtnb.933

Classification : 11G40, 11F80, 11R23, 11R34
Mots-clés : Euler systems, Kolyvagin systems, core rank, Selmer groups

Affiliations des auteurs :

Barry Mazur ¹ ; Karl Rubin ²

¹ Department of Mathematics Harvard University Cambridge, MA 02138, USA
² Department of Mathematics UC Irvine Irvine, CA 92697, USA

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Barry Mazur; Karl Rubin. Controlling Selmer groups in the  higher core rank case. Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 145-183. doi : 10.5802/jtnb.933. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.933/

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Cité par 14 documents. Sources : Crossref, zbMATH