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

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Didier Lesesvre (Université Paris 13/ Georg-August Universität Göttingen)

What
  • SFB-Kolloquium
When Jul 05, 2018
from 03:30 pm to 04:30 pm
Where Mainz, 05-432 (Hilbertraum)
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Abstract: 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.