Linear Relations Among Poincare Series via Harmonic Weak Maass Forms

TL;DR

This paper explores the linear relations among Poincaré series by embedding weakly holomorphic modular forms into harmonic weak Maass forms, revealing connections to Ramanujan's mock theta functions and their pseudomodularity.

Contribution

It introduces a new perspective by embedding weakly holomorphic forms into harmonic weak Maass forms to analyze Poincaré series and their relations to mock theta functions.

Findings

Linear relations among Poincaré series are characterized.

Connections between weakly holomorphic forms and mock theta functions are established.

Obstructions to the vanishing of Poincaré series are linked to pseudomodularity.

Abstract

We discuss the problem of the vanishing of Poincar\'e series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincar\'e series and the connection to Ramanujan's mock theta functions.