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Abstract

Fix two distinct primes $p$ and $\ell $. Let $A$ be an abelian variety over $\mathbb Q(\zeta _{\ell })$, the cyclotomic field of $\ell $th roots of unity. Suppose that $A(\mathbb Q (\zeta _{\ell }))[\ell ] \neq 0$. We show that there exists a number field $L$ and a $\mathbb Z_p$-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 that there are certain (non-cyclotomic) $\mathbb Z_p$-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.