# Growth of $p$-parts of ideal class groups and fine Selmer groups in $\mathbb{Z}_q$-extensions with $p\neq q$

## TL;DR

This paper investigates the behavior of $p$-parts of ideal class groups and fine Selmer groups in $bZ_q$-extensions with $p
eq q$, showing boundedness and stabilization under certain conditions in specific number field settings.

## Contribution

It provides new results on the boundedness of $p$-parts of class groups and the stabilization of fine Selmer groups in $bZ_q$-extensions, extending Iwasawa theory to these contexts.

## Key findings

- Bounded $p$-parts of class groups in anticyclotomic $bZ_q$-extensions of imaginary quadratic fields.
- Conditions for the stabilization of $p$-parts of fine Selmer groups over $bZ_q$-extensions.
- Applicability to abelian varieties with $p$-torsion points over the base field.

## Abstract

Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $\mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]\subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $\mathbb{Z}_q$-extension of $F$ to stabilize.