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

TL;DR

This paper extends Pink's theorem to analyze open subgroups of (_\u2113)^n using Lie algebras, with applications to Galois representations, without requiring the subgroup to be pro-ll.

Contribution

It generalizes Pink's theorem by removing the pro-ll assumption, providing new tools for studying Galois representations via Lie algebra methods.

Findings

Extended Pink's theorem to broader subgroup classes

Established Lie algebra techniques for non-pro-ll groups

Applied results to families of Galois representations

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.