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Eichler–Shimura theory for mock modular forms

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Abstract

We use mock modular forms to compute generating functions for the critical values of modular \(L\)-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on \(M_{k}^{!}\) in terms of periods.

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Notes

  1. We note that this definition implies that weakly holomorphic modular forms are mock modular forms. Several papers require mock modular forms to correspond to those harmonic Maass forms for which \({\mathcal F }^{-}\ne 0.\)

  2. Throughout we omit the dependence on the level in the case of \(SL_2({\mathbb Z }).\)

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Acknowledgments

The authors thank the American Institute for Mathematics for their hospitality during the “Mock modular forms” conference in March 2010. The authors also thank Jan Bruinier, Michael Dewar, Marvin Knopp, Winfried Kohnen, Wissam Raji, and Don Zagier for insightful comments. The second author is grateful for the support of the Simons Foundation.

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Authors and Affiliations

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

    Kathrin Bringmann

  2. Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, HI, 96822-2273, USA

    Pavel Guerzhoy

  3. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Zachary Kent & Ken Ono

Authors
  1. Kathrin Bringmann
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  2. Pavel Guerzhoy
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  3. Zachary Kent
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  4. Ken Ono
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Corresponding author

Correspondence to Ken Ono.

Additional information

The authors thank the NSF for their support. The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation. The fourth author thanks the support of the Candler Fund at Emory University.

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Bringmann, K., Guerzhoy, P., Kent, Z. et al. Eichler–Shimura theory for mock modular forms. Math. Ann. 355, 1085–1121 (2013). https://doi.org/10.1007/s00208-012-0816-y

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  • DOI: https://doi.org/10.1007/s00208-012-0816-y

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