Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


$ 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
Published electronically: August 12, 2015
MathSciNet review: 3451222
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Kathrin Bringmann and Ken Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA 104 (2007), no. 10, 3725-3731. MR 2301875 (2009d:11073),
  • [2] Jan H. Bruinier, Borcherds products on O(2, $ l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920 (2003h:11052)
  • [3] Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45-90. MR 2097357 (2005m:11089),
  • [4] Jan H. Bruinier, Ken Ono, and Robert C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), no. 3, 673-693. MR 2430995 (2009f:11046),
  • [5] John D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293/294 (1977), 143-203. MR 0506038 (58 #21944)
  • [6] Pavel Guerzhoy, Zachary A. Kent, and Ken Ono, $ p$-adic coupling of mock modular forms and shadows, Proc. Natl. Acad. Sci. USA 107 (2010), no. 14, 6169-6174. MR 2630103 (2011h:11048),
  • [7] Dennis A. Hejhal, The Selberg trace formula for $ {\rm PSL}(2,\,{\bf R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197 (86e:11040)
  • [8] Jeffrey Hoffstein and Thomas A. Hulse, Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2.
  • [9] Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)
  • [10] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • [11] Michael H. Mertens and Ken Ono, Special values of shifted convolution Dirichlet series, accepted for publication in Mathematika.
  • [12] Toshitsune Miyake, Modular forms, Reprint of the first 1989 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda. MR 2194815 (2006g:11084)
  • [13] Douglas Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973), 133-145. MR 0337788 (49 #2557)
  • [14] Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 347-454. MR 2555930 (2010m:11060)
  • [15] Robert A. Rankin, Contributions to the theory of Ramanujan's function $ \tau (n)$ and similar arithmetical functions. I. The zeros of the function $ \sum ^\infty _{n=1}\tau (n)/n^s$ on the line $ {\mathfrak{R}}s=13/2$. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351-372. MR 0000411 (1,69d)
  • [16] Atle Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47-50 (German). MR 0002626 (2,88a)
  • [17] Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1-15. MR 0182610 (32 #93)
  • [18] Jean-Pierre Serre, Divisibilité des coefficients des formes modulaires de poids entier, C. R. Acad. Sci. Paris Sér. A 279 (1974), 679-682 (French). MR 0382172 (52 #3060)
  • [19] Jean-Pierre Serre, Formes modulaires et fonctions zêta $ p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191-268 (French). MR 0404145 (53 #7949a)
  • [20] Don Zagier, Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque 326 (2009), Exp. No. 986, vii-viii, 143-164 (2010). Séminaire Bourbaki. Vol. 2007/2008. MR 2605321 (2011h:11049)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F37, 11G40, 11G05, 11F67

Retrieve articles in all journals with MSC (2010): 11F37, 11G40, 11G05, 11F67

Additional Information

Kathrin Bringmann
Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany

Michael H. Mertens
Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany

Ken Ono
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30022

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

American Mathematical Society