Weighted averages of $\operatorname{SL}_2(\mathbb{R})$ automorphic kernel, Part I: non-oscillatory functions

TL;DR

This paper introduces a new spectral method for evaluating weighted averages of automorphic sums associated with congruence subgroups of SL2(R), improving the preservation of symmetries in number theory applications.

Contribution

It develops a novel approach applying spectral methods directly to the automorphic kernel, avoiding traditional sum decompositions, and sets the stage for new results in number theory.

Findings

Spectral methods applied directly to automorphic kernels.

Enhanced symmetry preservation in automorphic sum analysis.

Foundation for future number theoretical applications.

Abstract

We prove a theorem that evaluates weighted averages of sums parametrised by congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$. In the proof, spectral methods are applied directly to the automorphic kernel instead of going over sums of Kloosterman sums. In number theoretical applications this better preserves the specific symmetries throughout the application of spectral methods. In a separate paper we apply the main theorem to quadratic polynomials and obtain new results about their greatest prime factor and the equidistribution of their roots to prime moduli.