On the non-vanishing of Hilbert Poincaré series

Abstract

We prove that if $\nu$ has small norm with respect to the level and the weight, the $\nu$-th Hilbert Poincaré series does not vanish identically. We also prove Selberg's identity on Kloosterman sums in the case of number fields, which implies certain vanishing and non-vanishing relations of Hilbert Poincaré series when the narrow class number is $1$. Finally, we pass to the adelic setting and interpret the problem via Hecke operators.