IUMJ

Title: An algebraic Sato-Tate group and Sato-Tate conjecture

Authors: Grzegorz Banaszak and Kiran Kedlaya

Issue: Volume 64 (2015), Issue 1, 245-274

Abstract:

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated with 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 Fit\'e, Kedlaya, Rotger, and Sutherland.