Galois groups arising from families with big orthogonal monodromy

We study the Galois groups of polynomials arising from a compatible family representations with big orthogonal monodromy. show that are usually as large possible given constraints imposed on them by functional equation and discriminant considerations. As an application, we consider Frobenius middle étale cohomology hypersurfaces in [Formula: see text] degree at least three. also text]-functions quadratic twists fixed elliptic curve over function field text]. To determine typical group setting requires using some known cases Birch Swinnerton-Dyer conjecture. This extends generalizes work Chavdarov, Katz Jouve.