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

Abstract

Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(\zeta_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(\zeta_{\ell}))[\ell] \neq 0$. 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.