Non-vanishing of Taylor coefficients and Poincar\'e series

TL;DR

This paper develops recursive formulas for Taylor coefficients of cusp forms like Ramanujan's Delta function, demonstrating their non-vanishing at specific points and proposing related conjectures.

Contribution

It introduces recursive formulas for Taylor coefficients at points in the upper half-plane, enabling non-vanishing results at CM points and for Poincaré series.

Findings

All Taylor coefficients of Delta at small discriminant CM points are non-zero.

Certain Poincaré series have non-vanishing Taylor coefficients.

At generic points, all Taylor coefficients are non-zero.

Abstract

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincar\'e series. At a "generic" point all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.