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

Subong Lim, Wissam Raji

TL;DR

This paper develops a unified Laplace-transform framework for L-series attached to vector-valued modular objects, defining $L_f(\varphi)$ and its Mellin-family $L(s,f,\varphi)$ to capture a broad class of $L$-series for vector-valued weakly holomorphic modular forms, harmonic weak Maass forms, Jacobi forms, and Kohnen plus space forms. It proves analytic continuation and functional equations for these $L$-series and establishes converse theorems that characterize the corresponding modular objects from the analytic properties of their $L$-series, using tools such as Mellin transforms, theta decompositions, and Weil/metaplectic representations. The results cover both holomorphic and non-holomorphic components, relate the $igl(\xi_{2-k} \bigr)$-maps to cusp forms, and extend to Jacobi and half-integral weight plus-space settings, providing a cohesive approach to functoriality-type conclusions via Laplace-constructed $L$-series.

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.

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

TL;DR

This paper develops a unified Laplace-transform framework for L-series attached to vector-valued modular objects, defining and its Mellin-family to capture a broad class of -series for vector-valued weakly holomorphic modular forms, harmonic weak Maass forms, Jacobi forms, and Kohnen plus space forms. It proves analytic continuation and functional equations for these -series and establishes converse theorems that characterize the corresponding modular objects from the analytic properties of their -series, using tools such as Mellin transforms, theta decompositions, and Weil/metaplectic representations. The results cover both holomorphic and non-holomorphic components, relate the -maps to cusp forms, and extend to Jacobi and half-integral weight plus-space settings, providing a cohesive approach to functoriality-type conclusions via Laplace-constructed -series.

Abstract

We introduce the -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.
Paper Structure (5 sections, 11 theorems, 138 equations)

This paper contains 5 sections, 11 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. $L$-series of vector-valued weakly holomorphic modular forms
  3. $L$-series of vector-valued harmonic weak Maass forms
  4. $L$-series of harmonic Maass Jacobi forms
  5. $L$-series of harmonic weak Maass forms in the Kohnen plus space

Key Result

Theorem 2.2

Let $f$ be a vector-valued weakly holomorphic modular form in $M^!_{k,\chi,\rho}$.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 14 more