Big symplectic or orthogonal monodromy modulo `

Let k be a field not of characteristic two and Λ be a set consisting of almost all rational primes invertible in k. Suppose we have a varietyX/k and strictly compatible system {M` → X : ` ∈ Λ} of constructible F`-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M` is a subgroup of a corresponding isometry group Γ` over F`, and we say it has big monodromy if it contains the derived subgroup DΓ`. We prove a theorem which gives sufficient conditions for M` 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 ` and the system. We also show how it leads to new results for the inverse Galois problem.