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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Shintani theta lifts of harmonic Maass forms
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by Claudia Alfes-Neumann and Markus Schwagenscheidt PDF
Trans. Amer. Math. Soc. 374 (2021), 2297-2339 Request permission


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:
  • Markus Schwagenscheidt
  • Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931 Cologne, Germany
  • MR Author ID: 1094068
  • Email:
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 2297-2339
  • MSC (2020): Primary 11F03, 11F12, 11F27, 11F37
  • DOI:
  • MathSciNet review: 4223017