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

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.

12 pages