L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems

TL;DR

This paper introduces L-series for vector-valued weakly holomorphic modular forms, establishes their functional equations, and proves converse theorems for various classes of modular forms, advancing the understanding of their analytic properties.

Contribution

It develops a new framework for L-series of weakly holomorphic modular forms and proves converse theorems for several types of modular forms, extending classical results.

Findings

L-series defined via Laplace transforms satisfy functional equations.

Converse theorems characterize modular forms through their L-series.

Results apply to harmonic weak Maass forms, Jacobi forms, and half-integer weight modular forms.

Abstract

We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integer weight in Kohnen plus space.