Character sum, reciprocity and Voronoi formula

TL;DR

This paper introduces a new four-variable character sum identity as a non-archimedean analogue of Weber's integrals and uses it to provide a novel spectral proof of the Voronoi formula for modular forms.

Contribution

It presents a novel character sum identity and applies it to give a new spectral proof of the Voronoi formula, linking character sums and modular form analysis.

Findings

Established a new four-variable character sum identity.

Provided a spectral proof of the Voronoi formula.

Linked character sums with classical modular form analysis.

Abstract

We prove a novel four-variable character sum identity which serves as a twisted, non-archimedean counterpart to Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we present a new, spectral proof of the Voronoi formula for classical modular forms.