Fourier expansion of light-cone Eisenstein series

TL;DR

This paper derives explicit Fourier coefficient formulas for Eisenstein series on hyperbolic spaces, revealing pole locations and norms, and applies these to improve rational point counting on spheres.

Contribution

It provides a new explicit formula for Fourier coefficients of Eisenstein series associated with specific arithmetic lattices, advancing understanding of their analytic properties.

Findings

Identified all poles of the Eisenstein series.

Determined supremum norms of the Eisenstein series.

Applied results to improve rational point counting on spheres.

Abstract

In this work we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic n+1-space. As a consequence we obtain results on location of all poles of these Eisenstein series as well as their supremum norms. We use this information to get new results on counting rational points on spheres.