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A Poincaré series on hyperbolic space (1707.07790v1)
Published 25 Jul 2017 in math.RT and math.NT
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.