On (automorphic) density theorems for certain families of congruence lattices in $\mathrm{SL}_n$

Abstract: Roughly speaking automorphic density theorems show, in a suitable quantitative way, that the number of non-tempered cusp forms is small. In applications this can serve as a convenient replacement for the generalized Ramanujan Conjecture, which appears to be out of reach of current technology. In this talk we will discuss a general approach to such density theorems for $\mathrm{SL}_n$ using the Kuznetsov formula. Doing so we will highlight some important ingredients that are necessary for a successful execution of this strategy. Finally, we will look at different families of congruence lattices in order to see what can be done about the aforementioned ingredients in practice.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)