Given a polynomial equation (or system of equations) with integer coefficients, one can study how the number of solutions modulo p varies with the prime p. For polynomials in one variable the answer is given by the Chebotarev density theorem. For polynomials in two variables (curves) the answer is trivial if the degree is 1 or 2, and when the degree is 3 the answer is given by the Sato-Tate conjecture, recently proved by Richard Taylor and his collaborators (Taylor received the multi-million dollar Breakthrough Prize in Mathematics for his work on this and related problems). Essentially all other cases remain open, but there has been significant progress in explicitly quantifying the expected answers in terms of a random matrix model that associates to each curve a compact Lie group (the Sato-Tate group) whose Haar measure is conjectured to govern the distribution of its Frobenius classes. A complete classification of Sato-Tate groups is known for curves of genus 1 and 2, and I will report on recent progress in genus 3, which includes all smooth plane curves of degree 4. I will also discuss a remarkable "average polynomial-time" algorithm that makes it feasible to explicitly compute the Sato-Tate distributions of such curves to far greater precision than was previously possible.
Tuesday, January 10th, at 4:00pm in HBH 227
Return to talks from Spring 2017