Pink-type results for general subgroup of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$

Abstract

We study open subgroups $G$ of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$ in terms of some associated Lie algebras without assuming that $G$ is a pro-$\ell$ group, thereby extending a theorem of Pink. The result has applications to the study of families of Galois representations.