Determining monodromy groups of abelian varieties

TL;DR

This paper presents a method to determine the identity components of various monodromy and motivic groups associated with abelian varieties over number fields, assuming the Mumford-Tate conjecture, using Frobenius polynomials.

Contribution

It introduces a practical probabilistic algorithm to compute these groups and their representations from Frobenius polynomials, advancing understanding of their structure.

Findings

Groups are connected and reductive, expressible via root datum.

Algorithm can recover groups up to isomorphism from Frobenius data.

Provides a way to estimate the dimension of Hodge classes probabilistically.

Abstract

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-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.