A Poincaré series on hyperbolic space

Abstract

Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\mathrm{Aut}(L)$ acts on the hyperbolic space ${\mathcal{H}}^{25}$. We study a Poincaré series $E(z,s)$ defined for $z$ in ${\mathcal{H}}^{25}$, convergent for $\mathrm{Re}(s)>25$, invariant under $\mathrm{Aut}(L)$ 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 $\mathrm{Re}(s)>25/2$. 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.

17 pages, submitted