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Special $ L$-values and periods of weakly holomorphic modular forms


Authors: Kathrin Bringmann, Karl-Heinz Fricke and Zachary A. Kent
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
Published electronically: June 27, 2014
MathSciNet review: 3238419
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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.


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Additional Information

Kathrin Bringmann
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
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

DOI: https://doi.org/10.1090/S0002-9939-2014-12092-2
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
Article copyright: © Copyright 2014 American Mathematical Society