Counting and Equidistribution for Quaternion Algebras

TL;DR

This paper studies automorphic forms on quaternion algebras, proving their equidistribution and counting laws, which confirms Sato-Tate conjectures and provides explicit error bounds in certain cases.

Contribution

It establishes the equidistribution and counting laws for automorphic forms on quaternion algebras, advancing understanding in this area.

Findings

Proves equidistribution of automorphic forms in quaternion algebra setting

Confirms Sato-Tate conjectures for these forms

Provides explicit counting law with power savings error term

Abstract

We aim at studying automorphic forms of bounded analytic conductor in the division quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato-Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term in the totally definite case.