Title: The Sato-Tate conjecture for abelian varieties
Speaker: Andrew Sutherland
Speaker Info: MIT
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.Date: Monday, January 6, 2014
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 and Jeff Achter.