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Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theoryin two different settings. Let $K$ be an imaginary quadratic field of classnumber 1 such that both $p$ and $q$ split in $K$. We show that underappropriate hypotheses, the $p$-part of the ideal class groups is bounded overfinite 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 thefine Selmer groups of $A$ over finite subextensions of a$\mathbb{Z}_q$-extension of $F$ to stabilize.

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