\({\mathrm {H}}\)-Harmonic Maaß-Jacobi forms of degree 1

The analytic theory of some indefinite theta series
Open Access
Research

Abstract

It was shown in previous work that the one-variable \(\widehat{\mu }\)- function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature \((r\!+\!1,1)\) are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of “product type” for non-degenerate lattices of signature \((r\!+\!s,s)\). We thus obtain a clean generalization of \(\widehat{\mu }\) to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic Maaß-Jacobi forms, we obtain harmonic weak Maaß forms of higher depth in the sense of Zagier and Zwegers. In particular, we explain the modular completion of some, so-called degenerate indefinite theta series in the context of higher depth mixed mock modular forms. The structure theory for Heisenberg harmonic Maaß-Jacobi forms developed in this paper also explains a curious splitting of Zwegers’s two-variable \(\widehat{\mu }\)-function into the sum of a meromorphic Jacobi form and a one-variable Maaß-Jacobi form.

Keywords

Real-analytic Jacobi forms Generalized \(\widehat{\mu }\)-functions Mixed mock modular forms 

Mathematics Subject Classification

Primary 11F50 Secondary 11F27 

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Copyright information

© Westerholt-Raum. 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

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