Sato-Tate groups of abelian threefolds: a preview of the classification

TL;DR

This paper previews the classification of Sato-Tate groups for abelian threefolds over number fields, identifying 410 possible conjugacy classes and establishing key bounds for their realization.

Contribution

It provides the first comprehensive classification of Sato-Tate groups for abelian threefolds, including explicit realization of 33 maximal groups.

Findings

410 possible conjugacy classes identified

33 groups realized as maximal cases

Key bounds for classification established

Abstract

We announce the classification of Sato-Tate groups of abelian threefolds over number fields; there are 410 possible conjugacy classes of closed subgroups of USp(6) that occur. We summarize the key points of the "upper bound" aspect of the classification, and give a more rigorous treatment of the "lower bound" by realizing 33 groups that appear in the classification as maximal cases with respect to inclusions of finite index. Further details will be provided in a subsequent paper.