Determining monodromy groups of abelian varieties

2020/09/16 by David Zywina, Zywina, David
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2009.07441

openalex publication_date 2020/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, ℓ-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.

Related

• An algebraic Sato-Tate group and Sato-Tate conjecture
• A Survey of the Hodge Conjecture for Abelian Varieties
• On the Mumford-Tate conjecture for abelian varieties with reduction conditions
• Galois representations, Mumford-Tate groups and good reduction of abelian varieties
• The p-adic monodromy group of abelian varieties over global function fields of characteristic p