Modular representations arising from self-dual $\ell$-adic representations of finite groups

Abstract

Suppose $\ell$ is a prime number, ${\mathbf{Q}}_{\ell }$ is the field of $\ell$-adic numbers, ${\mathbf{F}}_{\ell }$ is the finite field of $\ell$ elements, and $d$ is a positive integer. Suppose $G$ is a finite subgroup of a symplectic group $S{p}_{2d}({\mathbf{Q}}_{\ell })$. We prove that $G$ can be embedded in $S{p}_{2d}({\mathbf{F}}_{\ell })$ in such a way that the characteristic polynomials are preserved (mod $\ell$), as long as $\ell >3$.