Vanishing of Poincaré series for congruence subgroups

Abstract

We consider the problem of the vanishing of Poincaré series for congruence subgroups. Denoting by $P_{k,m,N}$ the Poincaré 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 Poincaré series does not vanish. Our methods improve on previous results of Rankin (1980) and Mozzochi (1989).