Galois groups arising from families with big orthogonal monodromy

We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle \'etale cohomology of hypersurfaces in $\mathbb{P}_{\mathbb{F}_q}^{2n+1}$ of degree at least $3$. We also consider the $L$-functions of quadratic twists of fixed degree of an elliptic curve over a function field $\mathbb{F}_q(t)$. To determine the typical Galois group in the elliptic curve setting requires using some known cases of the Birch and Swinnerton-Dyer conjecture. This extends and generalizes work of Chavdarov, Katz and Jouve.