Non-vanishing of Hilbert Poincar\'e series

TL;DR

This paper establishes non-vanishing results for Hilbert Poincaré series by analyzing their Fourier coefficients and extending existing methods to larger weights and levels.

Contribution

It introduces generalized orthogonality relations for Fourier coefficients of Hilbert Poincaré series and extends a method by Kowalski et al. to broader settings.

Findings

Fourier coefficients satisfy orthogonality relations at large weights and levels

Non-vanishing of Hilbert Poincaré series under certain conditions

Generalization of Kowalski et al.'s method to Hilbert 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.