Edgar Assing (Uni Bonn) | On Sarnak's (spherical) density hypothesis for principal congruence subgroups

Startdatum: 20.10.2022 - 14:15
Enddatum: 14.10.2022 - 15:45
Adresse: MZH 4140
Preis: 0€

The (generalized) Ramanujan conjecture for ${\rm GL}_n$ can be interpreted as a strong form of the spectral gap. It is one of the great open problems in the theory of automorphic forms. Sarnak’s density Hypothesis, which is a quantitative statement about the rareness of failures of the Ramanujan conjecture, can serve as a convenient replacement for Ramanujan’s conjecture in many applications. In this talk we will explain Sarnak’s density hypothesis in the classical setting of Maaß forms for congruence subgroups of ${\rm SL}_2(\mathbb{Z})$. We will then move on to higher rank presenting recent results for principal congruence subgroups of ${\rm GL}_n$. Finally, if time permits we will look at some applications.