The original Sato-Tate conjecture addresses the statistical distribution
of the number of points on the reductions modulo primes of a fixed elliptic
curve defined over the rational numbers. It predicts that this distribution
can be explained in terms of a random matrix model, using the Haar measure
on the special unitary group SU(2). Thanks to recent work by Richard Taylor
and others, this conjecture is now a theorem. \
The Sato-Tate conjecture generalizes naturally to abelian varieties of dimension g, where it associates to each abelian variety a compact subgroup (the Sato-Tate group) of the unitary symplectic group USp(2g), whose Haar measure governs the distribution of arithmetic data attached to the abelian variety. While the Sato-Tate conjecture remains open for all g greater than 1, I will present recent work that has culminated in a complete classification of the Sato-Tate groups that can and do arise when g is 2, including proofs of the Sato-Tate conjecture in some special cases. I will also present numerical evidence in support of the conjecture, along with animated visualizations of this data. Time permitting, I will discuss the status of current ongoing work in dimension 3. \
This is joint work with Francesc Fite, Victor Rotger, and Kiran Kedlaya, and also with David Harvey.