Growth of Selmer groups and fine Selmer groups in uniform pro- p extensions
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Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions Debanjana Kundu1 Received: 3 May 2020 / Accepted: 31 August 2020 © Fondation Carl-Herz and Springer Nature Switzerland AG 2020
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 μ-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. Keywords Selmer groups · Fine Selmer groups · Class groups · p-rank Mathematics Subject Classification 11R23 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 μ 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.
1 Introduction Iwasawa theory began as the study of ideal class groups over infinite towers of number fields. Kenkichi Iwasawa introduced the notion of a μ-invariant to study the growth of ( p-ranks) of ideal class groups in Z p -extensions. In [13], he constructed Z p -extensions over number fields with arbitrarily large μ-invariants. This notion of a μ-invariant was later generalized to all uniform pro- p groups [12,30]. In [10], Hajir and Maire investigated uniform pro- p
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Debanjana Kundu [email protected] Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Room 6290, Toronto, ON M5S 2E4, Canada
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groups which are realisable as Galois groups of extensions of number fields with arbitrarily large μ-invariant. In the study of rational points on Abelian varieties, the Selmer group plays an important role. In [21], exploiting the intimate connection between class groups and Selmer groups, Mazur developed an analogous theory to study the growth of Selmer groups of Abelian varieties in Z p -extensions. He showed that the Selmer groups of Abelian varieties and ideal class groups have similar growth patterns in Z p -extensions. When the Abelian variety has good ordinary reduction at p, it is possible to associate a μ-invariant to the Selmer group. In [3], Coates and Sujatha showed that the fine Selmer group has even stronger finiteness properties than the classical Selmer group. They showed that the growth of fine Selmer groups mimics the growth of ideal class group in a general p-adic analytic extension containing the cyclotomic Z p -extension. In [20]
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