Vanishing of Poincar\'e series for congruence subgroups

TL;DR

This paper investigates conditions under which Poincaré series for congruence subgroups vanish or not, providing new results that extend previous findings by Rankin and Mozzochi.

Contribution

The authors establish new criteria for the non-vanishing of Poincaré series for $ ext{Gamma}_0(N)$, improving upon earlier results.

Findings

Identified parameter ranges where Poincaré series do not vanish

Extended previous non-vanishing results by Rankin and Mozzochi

Provided methods to determine vanishing behavior for specific weights and indices

Abstract

We consider the problem of the vanishing of Poincar\'e series for congruence subgroups. Denoting by $P_{k,m,N}$ the Poincar\'e series of weight $k$ and index $m$ for the group $\Gamma_0(N)$, we show that for certain choices of parameters $k,m,N$, the Poincar\'e series does not vanish. Our methods improve on previous results of Rankin (1980) and Mozzochi (1989).