The Prime Geodesic theorem in the 3-dimensional hyperbolic space

Arithmétique

Lieu:

Salle Kampé de Fériet

Orateur:

Dimitrios Chatzakos

Affiliation:

Université de Lille

Dates:

Jeudi, 30 Novembre, 2017 - 11:00 - 12:00

Résumé:

The Prime Geodesic Theorem studies the asymptotic behaviour of lengths of primitive closed geodesics on hyperbolic manifolds.

For 2-dimensional manifolds this problem was first studied by Huber and Selberg. It turns out that the lengths of these geodesics

obey an asymptotic distribution analogous to the Prime Number Theorem, and the error term has been extensively studied by use

of the Selberg and the Kuznetsov trace formulas.

In this talk, we discuss the Prime Geodesic Theorem on 3-dimensional hyperbolic manifolds. For the Picard manifold, we improve on

the classical pointwise bound of Sarnak, using the Kuznetsov formula combined with a recent large sieve inequality of Watt. Further,

for a 3-manifold of finite area, we study the second moment of the error term using the Selberg trace formula.

This is a joint work in progress with Giacomo Cherubini and Niko Laaksonen.