A Poincaré series on hyperbolic space

Abstract

Let L be the unique even self-dual lattice of signature (25,1). The automorphism group Aut(L) acts on the hyperbolic space H25. We study a Poincaré series E(z,s) defined for z in H25, convergent for Re(s)>25, invariant under 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 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.