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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Shintani theta lifts of harmonic Maass forms


Authors: Claudia Alfes-Neumann and Markus Schwagenscheidt
Journal: Trans. Amer. Math. Soc. 374 (2021), 2297-2339
MSC (2020): Primary 11F03, 11F12, 11F27, 11F37
DOI: https://doi.org/10.1090/tran/8265
Published electronically: January 20, 2021
MathSciNet review: 4223017
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Abstract: We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \mathbb {Z}, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $\xi$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $\xi$-preimages of Zagier’s weight $1/2$ generating series of singular moduli and of some of Ramanujan’s mock theta functions.


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Additional Information

Claudia Alfes-Neumann
Affiliation: Mathematical Institute, Paderborn University, Warburger Str. 100, D-33098 Paderborn, Germany
MR Author ID: 899205
ORCID: 0000-0003-2056-5378
Email: alfes@math.uni-paderborn.de

Markus Schwagenscheidt
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931 Cologne, Germany
MR Author ID: 1094068
Email: mschwage@math.uni-koeln.de

Received by editor(s): May 25, 2019
Received by editor(s) in revised form: September 10, 2019
Published electronically: January 20, 2021
Additional Notes: The second author was supported by DFG grant BR-2163/4-2.
Article copyright: © Copyright 2021 American Mathematical Society