A Poincar\'e series on hyperbolic space

TL;DR

This paper investigates a Poincaré series on hyperbolic 25-space associated with an even self-dual lattice, analyzing its Fourier expansion, meromorphic continuation, and the implications for automorphic forms and Kloosterman sums.

Contribution

It provides the first detailed Fourier expansion and meromorphic continuation of a Poincaré series on hyperbolic 25-space related to the lattice L.

Findings

Fourier expansion computed at a Leech cusp

Meromorphic continuation of the series to Re(s) > 12.5

Meromorphic continuation of Fourier coefficients via Kloosterman sum zeta functions

Abstract

Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\operatorname{Aut}(L)$ acts on the hyperbolic space $\mathcal{H}^{25}$. We study a Poincar\'e series $E(z,s)$ defined for $z$ in $\mathcal{H}^{25}$, convergent for $\operatorname{Re}(s) > 25$, invariant under $\operatorname{Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a "Leech cusp" and prove that it can be meromorphically continued to $\operatorname{Re}(s) > 25/2$. Analytic continuation of Kloosterman sum zeta functions imply that the individual Fourier coefficients of $E(z,s)$ have meromorphic continuation to the whole $s$-plane.