Big symplectic or orthogonal monodromy modulo $\ell$

Abstract

Let $k$ be a field not of characteristic two, and let $\Lambda$ be a set consisting of almost all rational primes invertible in $k$. Suppose that we have a variety $X/k$ and strictly compatible system $\{{\mathcal M}_\ell\to X:\ell\in\Lambda\}$ of constructible $\mathbb{F}_\ell$-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of ${\mathcal M}_\ell$ is a subgroup of a corresponding isometry group $\Gamma_\ell$ over $\mathbb{F}_\ell$, and we say that it has big monodromy if it contains the derived subgroup ${\mathcal D}\Gamma_\ell$. We prove a theorem that gives sufficient conditions for ${\mathcal M}_\ell$ to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary $\ell$ and the system. We also show how it leads to new results for the inverse Galois problem