Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions

Abstract: In this article, we develop vertical control theorems for fine Selmer groups associated with Galois representations and their deformations in $p$-adic families. We apply them to provide new evidence for Conjectures A and B of Coates--Sujatha and to obtain pseudonullity of the fine Selmer groups for cyclotomic characters, elliptic curves, classical cuspidal newforms, and Hida families of $p$-ordinary cuspidal newforms.