Special $L$-values and periods of weakly holomorphic modular forms
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- by Kathrin Bringmann, Karl-Heinz Fricke and Zachary A. Kent PDF
- Proc. Amer. Math. Soc. 142 (2014), 3425-3439 Request permission
Abstract:
In this paper, we explore a method for associating $L$-series to weakly holomorphic modular forms and then proceed to study their $L$-values. As our main application, we prove a very curious limiting theorem which relates three “periods” of a mock modular form and its shadow to the ratio of their noncritical $L$-values. Critical $L$-values are then shown to fit nicely within the framework of period polynomials and an extended Eichler-Shimura theory recently studied by Guerzhoy, Ono, and the first and third authors.References
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Additional Information
- Kathrin Bringmann
- Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
- MR Author ID: 774752
- Email: kbringma@math.uni-koeln.de
- Karl-Heinz Fricke
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany
- Email: fricke.karl-heinz@freenet.de
- Zachary A. Kent
- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- Email: kent@mathcs.emory.edu
- Received by editor(s): November 7, 2012
- Published electronically: June 27, 2014
- Additional Notes: The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation.
- Communicated by: Ken Ono
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 3425-3439
- MSC (2010): Primary 11F03, 11F11, 11F30, 11F37, 11F67
- DOI: https://doi.org/10.1090/S0002-9939-2014-12092-2
- MathSciNet review: 3238419