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Let $A$ be an abelian variety defined over a number field $F$. We prove acontrol theorem for the fine Selmer group of the abelian variety $A$ whichessentially says that the kernel and cokernel of the natural restriction mapsin a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. Weemphasise that our result does not have any constraints on the reduction of $A$and the ramification of $F_\infty/F$. As a first consequence of the controltheorem, we show that the fine Tate-Shafarevich group over an arbitrary$\mathbb{Z}_p$-extension has trivial $\Lambda$-corank. We then derive anasymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmergroup in a $\mathbb{Z}_p$-extension. However, as the fine Mordell-Weil groupneeds not be $p$-divisible in general, the fine Tate-Shafarevich group needsnot agree with the $p$-torsion of the dual fine Selmer group, and so theasymptotic growth formula for the dual fine Selmer groups do not carry over tothe fine Tate-Shafarevich groups. Nevertheless, we do provide certainsufficient conditions, where one can obtain a precise asymptotic formula.

On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in $\mathbb{Z}_p$-extensions