Galois groups arising from families with big orthogonal monodromy

2020/01/20 by David Zywina, Zywina, David
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2001.07273

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 ℙ_\mathbbF_q²ⁿ⁺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 \mathbbF_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.

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