A Poincaré series on hyperbolic space

2017/07/25 by Basak, Tathagata
#11F55 #11H56 #20F55 #32N15 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1707.07790

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.

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