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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 primenumber $\ell$, 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 areboth motivated by group-theoretic considerations that arise in the study ofGalois representations associated to abelian varieties.

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