Lifting Subgroups of Symplectic Groups over $\mathbb{Z} / \ell \mathbb{Z}$

TL;DR

This paper proves that certain subgroups of symplectic groups over $\\mathbb{Z}_\ell$ that project onto the finite symplectic group are actually the entire group, with implications for Galois representations of abelian varieties.

Contribution

It provides a self-contained proof that subgroups surjecting onto finite symplectic groups are equal to the full group, advancing understanding of symplectic group structures over local rings.

Findings

Subgroups surjecting onto finite symplectic groups are the entire group.

The proof is self-contained and group-theoretic.

Results are motivated by Galois representations of abelian varieties.

Abstract

For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we give a self-contained proof that any closed subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$ which surjects onto $\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ must in fact equal all of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.