Non-vanishing of Poincar\'e Series on Average

TL;DR

This paper investigates conditions under which Poincaré series for congruence subgroups do not vanish, demonstrating that on average, non-vanishing occurs even for large indices relative to the weight, surpassing previous individual non-vanishing results.

Contribution

It introduces an average-based approach to establish non-vanishing of Poincaré series in ranges beyond current individual non-vanishing methods.

Findings

Almost all Poincaré series do not vanish when varying weight or index in a dyadic interval.

Non-vanishing is proven for indices significantly larger than the square of the weight.

Average analysis enables non-vanishing results in ranges previously inaccessible.

Abstract

We study when Poincar\'e series for congruence subgroups do not vanish identically. We show that almost all Poincar\'e series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval. Crucially, analyzing the problem `on average' over these weights or indices allows us to prove non-vanishing in ranges where the index $m$ is significantly larger than $k^2$ - a range in which proving non-vanishing for individual Poincar\'e series remains out of reach of current methods.