Character sum, reciprocity, and Voronoi formula

Abstract We prove a novel four‐variable character sum identity that serves as a twisted, non‐Archimedean analog of Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we provide a short spectral proof of the Voronoi formulae for classical modular forms with character twists. A key aspect of this work is a new way of explicitly determining the root numbers, which entails delicate arithmetic considerations.