Trace formula and functional equation

TL;DR

This paper develops a novel approach to the functional equation of standard L-functions for holomorphic cusp forms, utilizing Petersson's formula and spectral reciprocity to go beyond traditional endoscopic methods.

Contribution

It introduces a 'beyond-endoscopic' framework for analyzing the functional equation, combining Petersson's formula with spectral reciprocity techniques.

Findings

New proof of the functional equation for holomorphic cusp forms' L-functions.

Application of spectral reciprocity in the context of automorphic forms.

Advancement in beyond-endoscopic methods for L-function analysis.

Abstract

We present a "beyond-endoscopic" treatment of the functional equation for the standard $L$-function of a holomorphic cusp form with level and nebentypus. We use Petersson's formula and methods from Venkatesh's thesis and "spectral reciprocity".