# Motivic Serre group and Sato--Tate conjecture

## TL;DR

This paper advances the understanding of the Sato--Tate conjecture by extending the algebraic framework to more general motivic categories and weights, facilitating explicit computation of Sato--Tate groups.

## Contribution

It generalizes the algebraic approach to the Sato--Tate conjecture to include broader motivic categories and weights, and refines the connection between $l$-adic representations and motives.

## Key findings

- Extended the algebraic framework to general weights and motivic categories.
- Provided new results in the odd weight case.
- Facilitated explicit computation of Sato--Tate groups.

## Abstract

This paper concerns the Algebraic Sato--Tate and Sato--Tate conjectures, based on Serre's original motivic formulation, with an eye towards explicit computations of Sato--Tate groups. We build on the algebraic framework for the Sato--Tate conjecture introduced in a previous paper, which used Deligne's motivic category for absolute Hodge cycles and was restricted to motives of odd weight. Here, we allow general weight and some other motivic categories, notably Andr\'e's motivic category of motivated cycles; moreover, some results are also new in the odd weight case. The paper consists of two parts; in the first part we work in the framework of strongly compatible families of $l$-adic representations associated with pure, rational, polarized Hodge structures, while in the second part we use the language of motives.