Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Automorphic functions twisted with non-expanding cusp monodromies, and dynamics

Prof. Dr. Anke Pohl talk

**Date:** 11.03.19 **Time:** 14.00 - 15.00 **Room:** ETH HG G 43

Automorphic functions play an important role in several subareas of mathematics and mathematical physics. The correspondence principle between quantum und classical mechanics suggests that automorphic functions are closely related to geometric and dynamical entities of the underlying locally symmetric spaces. Despite intensive research, the extent of this relation is not yet fully understood. A seminal and very influencial result in this direction has been provided by Selberg, showing that for any hyperbolic surface X of finite area, the Selberg zeta function (a generating function for the geodesic length spectrum of X) encodes the spectral parameters of the untwisted automorphic forms for X among its zeros. Subsequently, this type of relation was generalized to automorphic forms twisted by unitary representations, to resonances and to spaces of infinite area by various researchers. Recently, a deeper relation could be established by means of transfer operator techniques. For the so-called Maass cusp forms, these methods allow us to provide a purely dynamical characterization of these automorphic functions themselves, not only of their spectral parameters. The structure of this approach indicates that an extension to automorphic forms with well-behaved, also non-unitary twists should be expected. While such a generalization seems to be a long-term goal, we could already show first steps in this direction on the level of the Selberg zeta functions. After surveying the transfer operator techniques, we discuss the current state of art regarding dynamical approaches towards automorphic forms with non-unitary twists.