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Fine Selmer groups and ideal class groups

Sören Kleine, Katharina Müller

Abstract

Let $K_\infty/K$ be a uniform $p$-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $K_\infty$ contains sufficiently many $p$-power torsion points of $A$, then we can compare the ranks and the Iwasawa $μ$-invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $\mathbb{Z}_p$-extensions), we can also prove relations between suitable generalisations of the Iwasawa $λ$-invariant of the two types of Iwasawa modules. In the literature, different kinds of Iwasawa $λ$-invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $\mathbb{Z}_p$-extensions.

Fine Selmer groups and ideal class groups

Abstract

Let be a uniform -adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If contains sufficiently many -power torsion points of , then we can compare the ranks and the Iwasawa -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple -extensions), we can also prove relations between suitable generalisations of the Iwasawa -invariant of the two types of Iwasawa modules. In the literature, different kinds of Iwasawa -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple -extensions.

Paper Structure

This paper contains 19 sections, 53 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. General notation
  4. Uniform extensions
  5. Fine Selmer groups
  6. Abelian varieties with complex multiplication
  7. Asymptotic class number formulas
  8. $\mathds{Z}_p$-extensions
  9. Multiple $\mathds{Z}_p$-extensions
  10. Exact formulas in the non-commutative setting
  11. Estimates for quotients of ideal class groups
  12. Auxiliary results
  13. Ranks of groups
  14. Iwasawa modules
  15. Class numbers
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $A$ be an abelian variety defined over the number field $K$, and let $K_\infty/K$ be a normal uniform pro-$p$ extension with Galois group $G$. We choose a finite set $\Sigma$ of primes of $K$ which consists of the primes above $p$, the primes where $A$ has bad reduction and the primes of $K$ whi for every $v\le k$.

Theorems & Definitions (111)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2: Serre-Tate
  • proof
  • ...and 101 more