On the distribution of closed geodesics on modular curves (Asbjørn Nordentoft, Université Sorbonne Paris Nord)Séminaire « Arithmétique »
M3 Salle de séminaire
The study of closed geodesics on Riemann surfaces is a classical topic. In the arithmetic case (i.e. modular curves) the closed geodesics can be parametrized by indefinite quadratic forms and as such has an associated discriminant. We will explain a celebrated result of Duke which shows that as the discriminant tends to infinity, the geodesics equidistribute. Secondly we will explain an analogue for the distribution of closed geodesics in the homology of the modular curves. If time permits we will explain the proofs and the connections to L-functions.