Non-vanishing of vector-valued Poincar\'e series

TL;DR

This paper establishes a vector-valued non-vanishing criterion for Poincaré series and demonstrates how to construct vector-valued modular forms, with applications to classical and elliptic cases.

Contribution

It extends Muić's non-vanishing criterion to vector-valued Poincaré series and provides methods for constructing such forms.

Findings

Proves a vector-valued version of Muić's non-vanishing criterion.

Provides a construction method for vector-valued modular forms via Poincaré series.

Shows non-vanishing results for classical and elliptic vector-valued Poincaré series.

Abstract

We prove a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on the upper half-plane $ \mathcal H $. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincar\'e series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincar\'e series.