On the balanced Voronoi formula for GL$_N$

TL;DR

This paper extends the balanced Voronoi summation formula for GL_N from the rational numbers to arbitrary number fields, enabling broader applications in spectral reciprocity laws over number fields.

Contribution

It generalizes the balanced Voronoi formula to number fields, building on prior work over  and Templier, thus expanding its applicability.

Findings

Established a balanced Voronoi formula over number fields.

Extended spectral reciprocity law results to number fields.

Built upon Ichino and Templier's Voronoi summation formula.

Abstract

S.D. Miller and F. Zhou have proved a balanced Voronoi summation formula for GL$_N$ over $\mathbb Q$, which allows one to control the dimensions of the Kloosterman sums appearing on either side of the Voronoi formula. In this note, we prove a balanced Voronoi formula over an arbitrary number field, starting with the Voronoi summation formula of A. Ichino and N. Templier over number fields, allowing one to extend recent results on spectral reciprocity laws to number fields, in special cases.