# Families of Automorphic Forms and the Trace Formula (14w5120)

## Organizers

(University of Bonn)

Sug-Woo Shin (Massachusetts Institute of Technology)

(Cornell University)

## Description

The Banff International Research Station will host the "Families of Automorphic Forms and the Trace Formula (HALF)" workshop from November 30th to December 5th, 2014.

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. The most primitive example is the Riemann zeta function. A great deal of information about prime numbers, Galois representations and automorphic forms is encoded in the mysterious and elusive analytic behavior of $L$-functions. The notion of families of automorphic representations is emerging as a central concept in the subject, despite the obstruction that there is a priori no way to organize (cuspidal) automorphic representations of a given reductive group (over number fields) into a family in the geometric sense. It is hoped that the consideration of families would enable one to understand the analytic behavior better and attack difficult problems, just as it does in geometry, where Deligne's proof of the Riemann hypothesis over finite fields is an example. Early achievements adopting this idea appear in many instances: subconvexity results, a complete solution of Hilbert's 11th problem, have been either obtained by using families or can be interpreted in terms of families. So it is of great interest to establish various properties of families of automorphic representations.

In this workshop we focus on the equidistribution of local invariants in the families of automorphic representations. There are numerous instances of such equidistribution including Weyl's law, limit multiplicity and Plancherel density with application to a Sato-Tate type equidistribution for families and confirmed the Katz-Sarnak heuristics explaining the connection between the statistics of low-lying zeros in a family of automorphic $L$-functions and random matrix models. By bringing together different groups of researchers with no past collaboration and distant locations, a synergetic effect is expected in generalizing previous results on equidistribution while building a unifying framework and finding more applications to automorphic $L$-functions.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).