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

The analytic theory of some indefinite theta series

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.

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Acknowledgements

The author thanks Kathrin Bringmann, Olav Richter, and Sander Zwegers for helpful discussions and for their remarks. He is especially grateful to one of the referees who helped with his comments to work around some stylistic glitches.

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Competing interests The author declares that they have no competing interests.

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Correspondence to Martin Westerholt-Raum.

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Westerholt-Raum, M. \({\mathrm {H}}\)-Harmonic Maaß-Jacobi forms of degree 1. Mathematical Sciences 2, 12 (2015). https://doi.org/10.1186/s40687-015-0032-y

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Keywords

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

Mathematics Subject Classification

  • Primary 11F50
  • Secondary 11F27