The weight two and opposite sign cases for the Fourier relative trace formulas

TL;DR

This paper presents an adelic relative trace formula proof for the Petersson/Bruggeman-Kuznetsov formulas in specific weight and sign cases, providing refined formulas under certain hypotheses.

Contribution

It introduces a new adelic proof approach for PBK formulas in weight two and opposite sign cases, with refined results based on geometric and spectral assumptions.

Findings

Provides a new adelic proof for PBK formulas in specific cases

Derives refined formulas under geometric and spectral hypotheses

Extends understanding of trace formulas in automorphic forms

Abstract

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for $\kappa=2$ and the non-holomorphic case for $m_1m_2<0$. Given two sets of hypothesis on the non archimedean test function $f$, called the geometric and spectral assumptions, this approach allows us to obtain refined PBK formulas.