For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. Under the generalized Sato-Tate conjecture, this is equal to the distribution of characteristic polynomials of random matrices in a closed subgroup ST(A) of USp(4). The Sato-Tate group ST(A) may be obtained from the Galois action on any Tate module of A, and must satisfy a certain set of constraints (the Sato-Tate axioms). Up to conjugacy, we find that there are exactly 55 subgroups of USp(4) that satisfy these axioms. By analyzing the possible Galois module structures on the R-algebra generated by the endomorphisms of A (the Galois type), we are able to establish a matching with Sato-Tate groups that shows that at most 52 of the 55 possible Sato-Tate groups can actually arise for some A and k, of which at most 34 can occur when k = Q. After a large-scale numerical search, we can now exhibit explicit examples, as Jacobians of hyperelliptic curves, that realize all 52 of these Sato-Tate groups. In each case we find statistical evidence that strongly supports the Sato-Tate conjecture.

I will give an overview of these results, starting with an introduction to the classical Sato-Tate conjecture in genus 1. My presentation will include graphic animations (in both 2-d and 3-d) of several examples. Time permitting, I will also discuss a recent breakthrough by David Harvey that may greatly facilitate extensions of this work to genus 3 (and has many other applications).

This is joint work with Francesc Fite, Victor Rotger, and Kiran Kedlaya.