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

Fix two distinct odd primes $p$ and $q$. We study “$p\mathrm{\ne }q$” 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 $A_{\mathrm{/}F}$ 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.