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Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions

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Abstract

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the \(\mu \)-invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

Résumé

Dans cet article, nous étudions la croissance des groupes de Selmer (fins) de courbes elliptiques dans certaines extensions de Galois infinies où le groupe de Galois G est un groupe de Lie uniforme, pro-p et p-adique. En comparant la croissance des groupes de Selmer (fins) avec celle des groupes de classes, nous démontrons qu’il est possible que l’invariant \(\mu \) du groupe de Selmer (fins) devienne arbitrairement grand dans une certaine classe d’extension de Lie nilpotentes, uniformes et pro-p. Nous étudions également la croissance des groupes de Selmer fins dans de fausses extensions de courbe de Tate.

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Acknowledgements

We thank Kumar Murty for his continued support and all members of the GANITA Lab for listening to the details. We extend our gratitude to Prof. Sujatha Ramdorai, Prof. Christian Maire, Prof. Karl Rubin, Meng Fai Lim and Jishnu Ray for answering many questions along the way. We thank Prof. Antonio Lei for his comments on an earlier draft of this article. Finally, we thank the referee for a careful reading of the article which helped improve the exposition.

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Correspondence to Debanjana Kundu.

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Kundu, D. Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions. Ann. Math. Québec 45, 347–362 (2021). https://doi.org/10.1007/s40316-020-00147-1

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