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Mock modular forms as $ p$-adic modular forms

Authors: Kathrin Bringmann, Pavel Guerzhoy and Ben Kane
Journal: Trans. Amer. Math. Soc. 364 (2012), 2393-2410
MSC (2010): Primary 11F33, 11F37, 11F11
Published electronically: January 6, 2012
MathSciNet review: 2888211
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Abstract: In this paper, we consider the question of correcting mock modular forms in order to obtain $ p$-adic modular forms. In certain cases we show that a mock modular form $ M^+$ is a $ p$-adic modular form. Furthermore, we prove that otherwise the unique correction of $ M^+$ is intimately related to the shadow of $ M^+$.

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  • 1. S. Ahlgren and S. Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487-502. MR 2000466 (2004e:11115)
  • 2. S. Ahlgren and S. Boylan, Central critical values of modular $ L$-functions and coefficients of half-integral weight modular forms modulo $ l$, Amer. J. Math. 129 (2007), no. 2, 429-454. MR 2306041 (2008c:11075)
  • 3. G. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), no. 1, 113-134. MR 814916 (87f:33011)
  • 4. G. Andrews and D. Hickerson, Ramanujan's ``lost'' notebook VII: The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60-105. MR 1123099 (92i:11027)
  • 5. K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc. 361 (2009), 3483-3500. MR 2491889 (2010g:11175)
  • 6. K. Bringmann and B. Kane, Inequalities for differences of Dyson's rank for all odd moduli, Math. Res. Lett. 17 (2010), no. 5, 927-942. MR 2727619 (2011i:11155)
  • 7. K. Bringmann and J. Lovejoy, Overpartitions and class numbers of binary quadratic forms, Proc. Nat. Acad. Sci. USA 106 (2009), 5513-5516. MR 2504957 (2010e:11031)
  • 8. K. Bringmann and K. Ono, The $ f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243-266. MR 2231957 (2007e:11127)
  • 9. K. Bringmann and K. Ono, Dyson's rank and Maass forms, Ann. of Math. 171 (2010), 419-449. MR 2630043 (2011e:11165)
  • 10. J. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45-90. MR 2097357 (2005m:11089)
  • 11. J. Bruinier and K. Ono, Heegner divisors, $ L$-functions, and Maass forms, Ann. of Math., accepted for publication.
  • 12. J. Bruinier, K. Ono, and R. Rhoades, Differential operators for harmonic Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), 673-693. MR 2430995 (2009f:11046)
  • 13. Y.-S. Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Invent. Math. 136 (1999), no. 3, 497-569. MR 1695205 (2000f:11016)
  • 14. A. Folsom and K. Ono, Duality involving the mock theta function $ f(q)$, J. Lond. Math. Soc. 77 (2008), 320-334. MR 2400394 (2009h:11076)
  • 15. P. Guerzhoy, Z. Kent, and K. Ono, $ p$-adic coupling of mock modular forms and shadows, Proc. Nat. Acad. Sci. USA, accepted for publication.
  • 16. D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639-660. MR 969247 (90f:11028a)
  • 17. S. Ramanujan, The lost notebook and other unpublished papers, Narosa Publishing House, New Delhi, 1987. MR 947735 (89j:01078)
  • 18. J.-P. Serre, Formes modulaires et fonctions zêta $ p$-adiques, Modular functions of one variable (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191-268. Lecture Notes in Math., vol. 350, Springer, Berlin, 1973. MR 0404145 (53:7949a)
  • 19. J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. (2) 22 (1976), no. 3-4, 227-260. MR 0434996 (55:7958)
  • 20. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11 Kan $ \hat {\text {o}}$ Memorial Lectures, 1, Princeton Univ. Press, Princeton, NJ, 1994. MR 1291394 (95e:11048)
  • 21. G. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55-80. MR 1862757
  • 22. D. Zagier, Ramanujan's mock theta functions and their applications [d'apres Zwegers and Bringmann-Ono], Séminaire Bourbaki 986 (2007). MR 2605321
  • 23. S. Zwegers, Mock $ \vartheta $-functions and real analytic modular forms, q-series with applications to combinatorics, number theory, and physics (Ed. B. C. Berndt and K. Ono), Contemp. Math. 291, Amer. Math. Soc. (2001), pages 269-277. MR 1874536 (2003f:11061)
  • 24. S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University (2002).
  • 25. S. Zwegers, The Folsom-Ono grid contains only integers, Proc. Amer. Math. Soc. 137 (2009), 1579-1584. MR 2470815 (2010m:11061)

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Additional Information

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

Pavel Guerzhoy
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822-2273

Ben Kane
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany

Keywords: $p$-adic modular forms, mock theta functions, mock modular forms, harmonic weak Maass forms
Received by editor(s): March 23, 2010
Published electronically: January 6, 2012
Additional Notes: The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and also by NSF grant DMS-0757907. The final details of the paper were completed at the AIM workshop “Mock modular forms in combinatorics and arithmetic geometry”. The authors would like to thank AIM for their support and for supplying a stimulating work environment.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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