An algebraic Sato-Tate group and Sato-Tate conjecture

TL;DR

This paper explicitly constructs an algebraic Sato-Tate group for abelian varieties, linking it to the Mumford-Tate group and endomorphisms, and verifies its description for varieties up to dimension 3.

Contribution

It provides an explicit construction of the algebraic Sato-Tate group and relates it to endomorphisms and the Mumford-Tate group for low-dimensional abelian varieties.

Findings

Constructed algebraic Sato-Tate group explicitly.

Connected part relates to Mumford-Tate group.

Described Sato-Tate groups for varieties up to dimension 3.

Abstract

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to the Mumford-Tate group, but the group of components carries additional arithmetic information. We then check that in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fite, Kedlaya, Rotger, and Sutherland.