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Arithmetic Statistics for Quaternion Algebras

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. In the case of inner forms of GL(2), that is to say groups of units of quaternion algebras, the Selberg trace formula is a powerful method to handle it. There is a way to define a suitable notion of size, the analytic conductor, allowing to truncate the universal family to a finite one amenable to arithmetic statistics methods. A counting law for the truncated universal family is established, with a power savings error term in the totally definite case and a geometrically meaningful constant. This Weyl's law is generalized to an 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 provided, leading to uncover part of the type of symmetry of quaternion algebras.Strong evidence is provided that further ground groups should be amenable as well to the same methods and conjectural counting laws are given in the case of symplectic and unitary groups of low ranks.
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Didier Lesesvre. Arithmetic Statistics for Quaternion Algebras. Commutative Algebra [math.AC]. Université Sorbonne Paris Cité, 2018. English. ⟨NNT : 2018USPCD040⟩. ⟨tel-02491158v2⟩

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