Sato-Tate groups of abelian threefolds

TL;DR

This paper classifies the possible Sato-Tate groups of abelian threefolds over number fields, identifying 410 groups and providing explicit examples for the maximal cases, supported by computational verification.

Contribution

It extends the classification of Sato-Tate groups from abelian surfaces to threefolds, identifying 410 possible groups and verifying their realization through explicit examples and computations.

Findings

Identified 410 possible Sato-Tate groups for abelian threefolds.

Explicitly constructed examples for each of the 33 maximal groups.

Numerical evidence confirms the distribution of Euler factors matches theoretical predictions.

Abstract

Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fit\'e, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.