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The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension

Adithya Chakravarthy

Abstract

Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(ζ_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(ζ_{\ell}))[\ell] \neq 0$. We show that there exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative $p$-adic Lie extensions. Finally, we include several examples to illustrate this theorem.

The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension

Abstract

Fix two distinct primes and . Let be an abelian variety over , the cyclotomic field of -th roots of unity. Suppose that . We show that there exists a number field and a extension where the -primary fine Selmer group of grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) -extensions where the -part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative -adic Lie extensions. Finally, we include several examples to illustrate this theorem.
Paper Structure (10 sections, 16 theorems, 49 equations)

This paper contains 10 sections, 16 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Construction of the G-extension Kinfty
  3. The case $m=2$
  4. The case $m > 2$
  5. Construction of the $\Gamma$-extension $L_{\infty}/L$
  6. Growth of class groups in : the proof of Theorem \ref{['thm-class-groups']}
  7. Application to fine Selmer groups: the proof of Theorems \ref{['main-thm-gamma']}
  8. Review of fine Selmer Group
  9. Proof of Theorem \ref{['main-thm-gamma']}
  10. Examples

Key Result

Theorem 1

Let $K$ be a number field and let $K_{\infty}/K$ be a $\mathbf{Z}_p$ extension with layers $K_n$. Suppose that $p^{e^n}$ is the exact power of $p$ dividing the class number of $K_n$. Then there exist integers $\mu, \lambda, \nu$ such that for all sufficiently large values of $n$.

Theorems & Definitions (30)

  • Theorem : Iwasawa
  • Theorem 1.1: Iwasawa
  • Theorem 1.2: Washington
  • Theorem 1.3: Washington
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Proposition 2.1
  • ...and 20 more