Let A be an abelian variety of dimension g defined over a number field K.  As defined by Serre, the Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) equipped with a map that sends each Frobenius element of the absolute Galois group of K at primes p of good reduction for A to a conjugacy class of ST(A) whose characteristic polynomial is determined by the zeta function of the reduction of A at p.  Under a set of axioms proposed by Serre that are known to hold for g <= 3, up to conjugacy in Usp(2g) there is a finite list of possible Sato-Tate groups that can arise for abelian varieties of dimension g over number fields.  Under the Sato-Tate conjecture (which is known for g=1 when K has degree 1 or 2), the asymptotic distribution of normalized Frobenius elements is controlled by the Haar measure of the Sato-Tate group.

In this talk I will present a complete classification of the Sato-Tate groups that can and do arise for g <= 3.

This is joint work with Francesc Fite and Kiran Kedlaya.