Weyl's law and beyond: Arithmetic statistics for quaternion algebras

Automorphic forms are central objects in modern number theory.
Despite their ubiquity, they remain mysterious and their behavior is far
from understood. Embedding them in wider families has a smoothing
effect, allowing results on average: these are the aims of arithmetic
statistics. The whole family of automorphic representations of a given
reductive group, referred to as its universal family, is of fundamental
importance.

A suitable notion of size, namely the analytic conductor, allows to
truncate the universal family to a finite one amenable to arithmetical
statistics methods. The key tool consists in recasting these questions
in spectral terms that can be handled by trace formula methods. We
present a counting law for the truncated universal family, with a power
savings error term in the totally definite case and a geometrically
meaningful constant. This Weyl's law is generalized to a Plancherel
equidistribution result with respect to an explicit measure, and leads
to answer the Sato-Tate conjectures in this case. Statistics on
low-lying zeros are also investigated, leading to uncover part of the
type of symmetry of quaternion algebras. We could mention strong
evidences towards other ground groups that seem amenable to the same
methods and counting laws are given in the case of symplectic and
unitary groups of low ranks.