Number theory seminar in memory of F. Momose

Sato-Tate groups of abelian varieties.


Kiran S. Kedlaya


We propose a definition of the algebraic Sato-Tate group and the Sato-Tate group associated to an abelian variety, in terms of which one may properly formulate the equidistribution conjecture that generalizes the classic Sato-Tate conjecture for elliptic curves. The connected parts of these groups are determined by the Mumford-Tate group, but the component groups encode additional arithmetic information. We also enumerate some group-theoretic properties implied by the definition of these groups (the ”Sato-Tate axioms”), which can be used to classify Sato-Tate groups of abelian varieties of dimension at most 3. This includes joint work with Banaszak and also joint work with Fité, Rotger, and Sutherland.


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