Galois groups arising from families with big orthogonal monodromy

Abstract

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 étale cohomology of hypersurfaces in [Formula: see text] of degree at least three. We also consider the [Formula: see text]-functions of quadratic twists of fixed degree of an elliptic curve over a function field [Formula: see text]. 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.