Non-vanishing of Hilbert Poincar\'e series

Abstract

We prove some non-vanishing results of Hilbert Poincar\'e series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincar\'e series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et. al.