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Seminar: Algebra
The generalized Sato-Tate conjecture
Andrew Sutherland, MIT
Contact: David Zureick-Brown,
Venue: Mathematics and Science Center, Room W306
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.