The Sato-Tate conjecture is concerned with the statistical distribution of
the number of points on the reduction 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 such abelian variety a compact
subgroup of the unitary symplectic group USp(2g), the Sato-Tate group,
whose Haar measure governs the distribution of certain arithmetic data
attached to the abelian variety. While the Sato-Tate conjecture remains
open for all g>1, I will present recent work that has culminated in a
complete classification of the Sato-Tate groups that can arise when g=2
(and proofs of the Sato-Tate conjecture in some special cases), and
highlight some of the ongoing work in dimension 3. I will also present
numerical computations that support the conjecture, along with animated
visualizations of this data.
This is joint work with Francesc Fit\'{e}, Victor Rotger, and Kiran S.
Kedlaya, and also with David Harvey.