Families of abelian varieties and large Galois images

2019/10/30 by David Zywina, Zywina, David
Computer Science · Mathematics · #11F80 (Primary) 14K15 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1910.14174

openalex publication_date 2019/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Associated to an abelian variety A of dimension g over a number field K is a Galois representation ρ_A\colon Gal(K/K)→ GL_2g(ℤ). The representation ρ_A encodes the Galois action on the torsion points of A and its image is an interesting invariant of A that contains a lot of arithmetic information. We consider abelian varieties over K parametrized by the K-points of a nonempty open subvariety U⊆ ℙⁿ_K. We show that away from a set of density 0, the image of ρ_A will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results which assumed that the family of abelian varieties have "big monodromy". We also give a version for a family of abelian varieties with a more general base.

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