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On harmonic weak Maass forms of half integral weight


Authors: Bumkyu Cho and YoungJu Choie
Journal: Proc. Amer. Math. Soc. 141 (2013), 2641-2652
MSC (2010): Primary 11F11, 11F30; Secondary 11F37, 11F50
DOI: https://doi.org/10.1090/S0002-9939-2013-11549-2
Published electronically: May 2, 2013
MathSciNet review: 3056554
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Abstract: Since Zwegers found a connection between mock theta functions and harmonic weak Maass forms, this subject has been of vast research interest. In this paper, we obtain isomorphisms among the space $ H_{k + \frac {1}{2}}^{+}(\Gamma _0(4m))$ of (scalar valued) harmonic weak Maass forms of half integral weight whose Fourier coefficients are supported on suitable progressions, the space $ H_{k + \frac {1}{2x1}, \bar {\rho }_L}$ of vector valued ones, and the space x1 $ \mathbb{\widehat {J}}_{k+1,m}^{cusp}$ of certain harmonic Maass-Jacobi forms of integral weight:

$\displaystyle H_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)) \simeq H_{k + \frac {1}{2}, \bar {\rho }_L} \simeq \mathbb{\widehat { J}}_{k+1,m}^{cusp}$

for $ k$ odd and $ m = 1$ or a prime. This is an extension of a result developed by Eichler and Zagier, which shows that

$\displaystyle M_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)) \simeq M_{k + \frac {1}{2}, \bar {\rho }_L} \simeq J_{k+1,m}.$

Here $ M_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)), M_{k + \frac {1}{2}, \bar {\rho }_L} $ and $ J_{k+1,m}$ are the Kohnen plus space of (scalar valued) modular forms of half integral weight, the space of vector valued ones, and the space of Jacobi forms of integral weight, respectively. To extend the result, another approach is necessary because the argument by Eichler and Zagier depends on the dimension formulas for the spaces of holomorphic modular forms, but the dimensions for the spaces of harmonic weak Maass forms are not finite. Our proof relies on some nontrivial properties of the Weil representation.

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Additional Information

Bumkyu Cho
Affiliation: Department of Mathematics, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul, 100-715, Republic of Korea
Email: bam@dongguk.edu

YoungJu Choie
Affiliation: Department of Mathematics, Pohang Mathematics Institute (PMI), POSTECH, Pohang, Republic of Korea
Email: yjc@postech.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2013-11549-2
Received by editor(s): January 31, 2011
Received by editor(s) in revised form: June 6, 2011, and November 10, 2011
Published electronically: May 2, 2013
Additional Notes: The first author was partially supported by the Dongguk University Research Fund of 2012 and NRF 2010-0008426
The second author was partially supported by NRF-2011-0008928, NRF-2011-0030749 and NRF- 2008-0061325
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.