# Control Theorems for Fine Selmer Groups

Inspired by the work of Iwasawa on growth of class groups in $\mathbb{Z}_p$-extensions, Mazur developed an analogous theory to study the growth of Selmer groups of Abelian varieties in $\mathbb{Z}_p$-extensions. He proved what is nowadays called a ``control theorem'', which we describe briefly here. Let $A$ be an Abelian variety defined over a number field $F$ with potential good ordinary reduction at all primes above $p$, and let $\Linf$ be a $\mathbb{Z}_p$-extension of $F$. For every intermediate subsextension $F^\prime$ of $\mathcal{L}/F$, we have natural maps \[s_{\mathcal{L}/F^\prime}: Sel(A/F^\prime) \longrightarrow Sel(A/\mathcal{L})^{\Gal(\mathcal{L}/F^\prime)}\] on the Selmer groups, which are induced by the restriction maps on cohomology. Mazur's Control Theorem therefore asserts that the kernel and cokernel of the maps $s_{\mathcal{L}/F^\prime}$ are finite and bounded independently of $F^\prime$. This control theorem has subsequently been generalized to more general $p$-adic Lie extensions by Greenberg.

We will consider variants of the control theorem for a certain subgroup of the $p$-primary Selmer group, called the *fine Selmer group*, which in recent years has been studied extensively. The said fine Selmer group is obtained by imposing stronger conditions at primes above $p$. We prove various Control Theorems for fine Selmer groups of elliptic curves in a general $p$-adic Lie extension, where the reduction type of the elliptic curve at primes above $p$ may not be potentially good ordinary. We will discuss certain estimates on the $\mathbb{Z}p$-coranks of the kernel and cokernel of the restriction maps \[r_{\mathcal{L}/F^\prime}: R(E/F^\prime) \longrightarrow R(E/\mathcal{L})^{\Gal(\mathcal{L}/F^\prime)}\] for a $p$-adic Lie extension $\mathcal{L}/F$. We will specialize to three cases of $p$-adic Lie extensions where we can show that the kernel and cokernel of the restriction map are finite, and (under appropriate assumptions) give growth estimates for their orders. This is joint work with Meng Fai Lim.