Talk
Joint Equidistribution of Closed Geodesics and CM Points
Talk by Dr. Ilya Khayutin
Date: 21.03.18 Time: 15.45 - 18.00 Room: ETH HG G 43
A packet of closed geodesics on a complex modular curve is a finite collection of closed geodesics obtained as an orbit of a Picard group of an order in a real quadratic field. A celebrated theorem of Duke states that packets of closed geodesics equidistribute in the limit when the absolute value of the discriminant goes to infinity. The same holds for Picard/Galois orbits of CM points. The equidistribution of Picard orbits of closed geodesics or CM points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. After introducing these objectsI will discuss the proof of a recent theorem making progress towards the conjecture.Currently, this problem does not seem to be amenable to methods of automorphic forms even conditionally on GRH. Nevertheless, assuming a splitting condition at two primes, the joining rigidity theorem of Einsiedler and Lindenstrauss applies. As a result the obstacle to proving equidistribution is the potential concentration of mass on intermediate algebraic measures. I will present a method to discard this possibility using a geometric expansion of a relative trace and a sieve argument.