Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture

TL;DR

This paper explicitly formulates Serre's generalization of the Sato-Tate conjecture for motives using fiber functors, extending previous work on abelian varieties to motives of weight 3, and discusses conditions for reducing the conjecture to connected components.

Contribution

It provides an explicit description of Serre's generalization of the Sato-Tate conjecture for motives via fiber functors, extending prior results from abelian varieties to motives of weight 3.

Findings

Classification of Sato-Tate groups for certain motives of weight 3

Conditions under which the conjecture reduces to the identity component

Extension of previous classifications from abelian surfaces to motives of weight 3

Abstract

We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under which verification of the Sato-Tate conjecture reduces to the identity connected component of the corresponding Sato-Tate group.