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On a completed generating function of locally harmonic Maass forms

Part of: Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 March 2014

Kathrin Bringmann ,
Ben Kane  and
Sander Zwegers
Kathrin Bringmann
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany email kbringma@math.uni-koeln.de
Ben Kane
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany email bkane@math.uni-koeln.de
Sander Zwegers
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany email szwegers@math.uni-koeln.de
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Abstract

While investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the function $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.

Keywords

locally harmonic Maass formsindefinite theta functionsholomorphic projectionmock modular formsmodular forms

MSC classification

Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F25: Hecke-Petersson operators, differential operators (one variable) 11F37: Forms of half-integer weight; nonholomorphic modular forms 11E16: General binary quadratic forms 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 5 , May 2014 , pp. 749 - 762
Copyright
© The Author(s) 2014 

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