Weyl's law and beyond: Arithmetic statistics for quaternion algebras
Didier Lesesvre (Université Paris 13/ GeorgAugust Universität Göttingen)
What 


When 
Jul 05, 2018 from 03:30 pm to 04:30 pm 
Where  Mainz, 05432 (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 SatoTate
conjectures in this case. Statistics on lowlying 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.