An algebraic Sato-Tate group and Sato-Tate conjecture

2011/09/21 by Banaszak, Grzegorz, Kedlaya, Kiran S. · 4 citations
#11G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1109.4449

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.

Cited by

• Degeneracy and Sato-Tate groups of y²=x^p2-1
• Connected monodromy fields of Jacobians with complex multiplication
• Positive density of primes of ordinary reduction for abelian varieties of simple signature
• Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

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• Sato-Tate groups of genus 2 curves
• The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture of Banaszak and Kedlaya
• Sato-Tate groups of abelian threefolds
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• Tate module tensor decompositions and the Sato-Tate conjecture for certain abelian varieties potentially of GL₂-type