Estimates of automorphic forms over quaternion algebras

TL;DR

This paper uses geometric analysis and heat kernel methods to estimate automorphic cusp forms over quaternion algebras, proving an average holomorphic QUE conjecture and extending results to higher dimensions.

Contribution

It introduces new estimates for automorphic forms over quaternion algebras and generalizes previous results to higher-dimensional cases.

Findings

Proves an average version of the holomorphic QUE conjecture.

Derives quantitative estimates for Hilbert modular cusp forms.

Extends prior results to higher dimensions.

Abstract

In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.