A remark on the component group of the Sato-Tate group

TL;DR

This paper characterizes the component group of the Sato-Tate group for any abelian variety over a number field, linking it to the connectedness of the Lefschetz group, thus advancing understanding of these algebraic structures.

Contribution

It provides a complete characterization of the component group of the Sato-Tate group in terms of the Lefschetz group's connectedness, applicable to abelian varieties of any dimension.

Findings

Component group characterized for all dimensions.

Connection established between Sato-Tate and Lefschetz groups.

Advances understanding of algebraic symmetries in abelian varieties.

Abstract

In this paper, we give a complete characterization of the component group of the Sato-Tate group of an abelian variety $A$ of arbitrary dimension, defined over a number field $K,$ in terms of the connectedness of the Lefschetz group associated to $A.$