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Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition


Authors: Kathrin Bringmann, Martin Raum and Olav K. Richter
Journal: Trans. Amer. Math. Soc. 367 (2015), 6647-6670
MSC (2010): Primary 11F50; Secondary 11F60, 11F55, 11F37, 11F30, 11F27
DOI: https://doi.org/10.1090/S0002-9947-2015-06418-6
Published electronically: January 15, 2015
MathSciNet review: 3356950
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Abstract: Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers's PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers's $ \widehat {\mu }$-functions to establish a theta-like decomposition.


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Additional Information

Kathrin Bringmann
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Email: kbringma@math.uni-koeln.de

Martin Raum
Affiliation: Department of Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zürich, Switzerland
Address at time of publication: Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
Email: martin@raum-brothers.eu

Olav K. Richter
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: richter@unt.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06418-6
Received by editor(s): June 21, 2013
Received by editor(s) in revised form: February 18, 2014
Published electronically: January 15, 2015
Additional Notes: The first author was partially supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and by NSF grant DMS-$0757907$. The second author held a scholarship from the Max Planck Society and is supported by the ETH Zurich Postdoctoral Fellowship Program and by the Marie Curie Actions for People COFUND Program. The third author was partially supported by Simons Foundation grant $#200765$
Article copyright: © Copyright 2015 American Mathematical Society