# Families of automorphic forms and the trace formula

Research **Group Müller**

**June 22 - July 5, 2014 and **July 5 - 18, 2015

**Organizers:** Werner Müller, Sug Woo Shin, Nicolas Templier

One of the most fundamental goals in number theory is to understand automorphic representations of connected reductive groups over number fields and their most important invariants, namely their L-functions. This is the content of the Langlands program. The study of automorphic representations has far-reaching impact not only on number theory, but also on other areas.

An important concept is the notion of families of automorphic representations, with the hope that the consideration of families would enable one to understand the analytic behavior of L-functions better and attack difficult problems just as it does in geometry.

A fundamental tool to study automorphic representations is the Arthur-Selberg trace formula.

The program focused on the following topics:

- Refined spectral side of the trace formula
- Uniform bounds of orbital integrals
- Asymptotic properties of automorphic spectra