UC Berkeley

Since Automorphic representations for general groups are very difficult to study individually, they are often studied in families instead. The Arthur-Selberg trace formula lends itself naturally to answering questions about averages of various parameters of the local components of automorphic representations in so-called harmonic families. In their 2016 work, Shin and Templier realized that, in the special case of representations with discrete series at infinity, the trace formula simplified dramatically enough to compute statistics with good error bounds. These bounds were good enough for applications: first, an averaged Sato-Tate law analogous to Sato-Tate for families of elliptic curves and second, computations of the specific random-matrix statistics that low-lying zeros of $L$-functions in the family follow. Following Shin-Templier's idea, we solve two further problems about discrete-at-infinity families.

First, Shin-Templier's work used the invariant trace formula which disallowed families that distinguish representations with infinite component in the same $L$-packet. However, which member of this $L$-packet a representation might correspond to determines some important characteristics---whether the representation is holomorphic or quaternionic for example. Methods related to the stable trace formula can remove this restriction. The key idea is applying a certain "hyperendoscopy" formulation of stabilization used first by Ferrari, though many technical difficulties come up.

Second, while the equidistribution results achieved are interesting in their own right, they also provide a proof-of-concept that the tools developed for proving them are sufficient for studying very general questions about discrete-at-infinity families. As a further demonstration, we also use these methods to solve a very different problem of computing explicit dimensions of spaces of quaternionic forms on the exceptional group $G_2$.