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$ p$-adic properties of modular shifted convolution Dirichlet series


Authors: Kathrin Bringmann, Michael H. Mertens and Ken Ono
Journal: Proc. Amer. Math. Soc. 144 (2016), 1439-1451
MSC (2010): Primary 11F37, 11G40, 11G05, 11F67
DOI: https://doi.org/10.1090/proc/12809
Published electronically: August 12, 2015
MathSciNet review: 3451222
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Abstract: Hoffstein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms $ f_1$ and $ f_2$. The second two authors investigated certain special values of symmetrized sums of such functions, numbers which are generally expected to be mysterious transcendental numbers. They proved that the generating functions of these values in the $ h$-aspect are linear combinations of mixed mock modular forms and quasimodular forms. Here we examine the special cases when $ f_1=f_2$ where, in addition, there is a prime $ p$ for which $ p^2$ divides the level. We prove that the mixed mock modular form is a linear combination of at most two weight 2 weakly holomorphic $ p$-adic modular forms.


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Additional Information

Kathrin Bringmann
Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Email: kbringma@math.uni-koeln.de

Michael H. Mertens
Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Email: mmertens@math.uni-koeln.de

Ken Ono
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30022
Email: ono@mathcs.emory.edu

DOI: https://doi.org/10.1090/proc/12809
Received by editor(s): September 2, 2014
Received by editor(s) in revised form: April 15, 2015
Published electronically: August 12, 2015
Additional Notes: The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. The second author thanks the DFG-Graduiertenkolleg 1269 ‘Global Structures in Geometry and Analysis’ for the financial support of his research. The third author thanks the National Science Foundation and the Asa Griggs Candler Fund for their generous support.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society