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L-series of harmonic Maass forms and a summation formula for harmonic lifts

Nikolaos Diamantis, Min Lee, Wissam Raji, Larry Rolen

Abstract

We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.

L-series of harmonic Maass forms and a summation formula for harmonic lifts

Abstract

We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.

Paper Structure

This paper contains 8 sections, 11 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Context and previous work
  3. Harmonic Maass forms
  4. $L$-series associated to harmonic Maass forms
  5. The converse theorem
  6. Alternative converse theorem for integral weight.
  7. Example of using the converse theorem
  8. A summation formula for harmonic lifts

Key Result

Theorem 1.1

Let $(a(n))_{n \ge -n_0}$ be a sequence of complex numbers such that $a(n)=O(e^{C \sqrt{n}})$ as $n \to \infty$, for some $C>0.$ For each $z \in \mathbb{H}$, set Suppose that the function $L_f(\varphi)$ defined, for each compactly supported smooth $\varphi: \mathbb{R}_{+} \to \mathbb{C}$ , by e:Lseriesmap_def satisfies where $\check{\varphi}$ is given by Then $f$ is a weakly holomorphic cusp fo

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Theorem 3.2: BF
  • Remark 4.1
  • Definition 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • ...and 15 more