Expanding horocycles on the modular surface and some deep open problems in analytic number theory

Abstract: The orbits of the horocycle flow on surfaces are classified: each orbit is either dense or a closed horocycle around a cusp. Expanding closed horocycles are asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. In a recent preprint with Uri Shapira and Shucheng Yu, we explored the asymptotic behavior of evenly spaced points along an expanding closed horocycle on the modular surface. In this problem, the number of sparse points is made to depend on the expansion rate, and the difficulty is that these points are no more invariant under the horocycle flow: Ratner’s theory does not apply. In this talk, I will sketch how this problem involves the theory of Diophantine approximation, and estimates towards the Ramanujan conjecture for Hecke-Maass forms. The goal is for this talk to be accessible for topologists; no prior background in analytic number theory will be assumed.

Mathematics

Audience: researchers in the topic

Series on open questions in Arithmetic, Geometry and Topology

Series comments: This seminar is part of the MSRI program on Random and Arithmetic Structures in Topology. The talks in this series are largely expository and focus on open questions. Registration is required. If you have any questions, please don't hesitate to contact the organizers.

Registration information for this week's seminars can be found here: Alan Reid: www.msri.org/seminars/25486 Benson Farb: www.msri.org/seminars/25556