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From Fourier Analysis and Number Theory to Radon Transforms and Geometry pp 85–106Cite as

Duality and Differential Operators for Harmonic Maass Forms

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Part of the book series: Developments in Mathematics ((DEVM,volume 28))

Abstract

Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator \({D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}\) acting on a harmonic Maass form for integers k > 2 in terms of \({\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}\) acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the p-adic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms.

Mathematics Subject Classification (2000): 11F37, 11F25, 11F30

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Acknowledgements

The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and also by NSF grant DMS-0757907. The third author is supported by an NSF Postdoctoral Fellowship and was supported by the Chair in Analytic Number Theory at EPFL during part of the preparation of this chapter.

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Authors and Affiliations

  1. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Kathrin Bringmann & Ben Kane

  2. Stanford University, Serra Mall bldg. 380, Stanford, CA, 94305, USA

    Robert C. Rhoades

Authors
  1. Kathrin Bringmann
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  2. Ben Kane
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  3. Robert C. Rhoades
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Correspondence to Kathrin Bringmann .

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Editors and Affiliations

  1. , Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Hershel M. Farkas

  2. , Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, 08544, New Jersey, USA

    Robert C. Gunning

  3. , Department of Mathematics, Temple University, Wachman Hall 608, Philadelphia, 19122, Pennsylvania, USA

    Marvin I. Knopp

  4. , Department of Mathematics, University of Michigan, Church St. 530, Ann Arbor, 48109, Michigan, USA

    B. A. Taylor

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In memory of Leon Ehrenpreis

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Bringmann, K., Kane, B., Rhoades, R.C. (2013). Duality and Differential Operators for Harmonic Maass Forms. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_6

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