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Mock modular forms as -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
Posted: January 6, 2012
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the question of correcting mock modular forms in order to obtain -adic modular forms. In certain cases we show that a mock modular form is a -adic modular form. Furthermore, we prove that otherwise the unique correction of is intimately related to the shadow of .

References

1. Scott Ahlgren and Matthew Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487–502. MR 2000466 (2004e:11115), http://dx.doi.org/10.1007/s00222-003-0295-6
2. Scott Ahlgren and Matthew 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), http://dx.doi.org/10.1353/ajm.2007.0006
3. George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), no. 1, 113–134. MR 814916 (87f:33011), http://dx.doi.org/10.1090/S0002-9947-1986-0814916-2
4. George E. Andrews and Dean Hickerson, Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105. MR 1123099 (92i:11027), http://dx.doi.org/10.1016/0001-8708(91)90083-J
5. Kathrin Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3483–3500. MR 2491889 (2010g:11175), http://dx.doi.org/10.1090/S0002-9947-09-04553-X
6. Kathrin Bringmann and Ben 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. Kathrin Bringmann and Jeremy Lovejoy, Overpartitions and class numbers of binary quadratic forms, Proc. Natl. Acad. Sci. USA 106 (2009), no. 14, 5513–5516. MR 2504957 (2010e:11031), http://dx.doi.org/10.1073/pnas.0900783106
8. Kathrin Bringmann and Ken Ono, The 𝑓(𝑞) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266. MR 2231957 (2007e:11127), http://dx.doi.org/10.1007/s00222-005-0493-5
9. Kathrin Bringmann and Ken Ono, Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), no. 1, 419–449. MR 2630043 (2011e:11165), http://dx.doi.org/10.4007/annals.2010.171.419
10. Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357 (2005m:11089), http://dx.doi.org/10.1215/S0012-7094-04-12513-8
11. J. Bruinier and K. Ono, Heegner divisors, -functions, and Maass forms, Ann. of Math., accepted for publication.
12. 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), http://dx.doi.org/10.1007/s00208-008-0252-1
13. Youn-Seo Choi, Tenth order mock theta functions in Ramanujan’s lost notebook, Invent. Math. 136 (1999), no. 3, 497–569. MR 1695205 (2000f:11016), http://dx.doi.org/10.1007/s002220050318
14. Amanda Folsom and Ken Ono, Duality involving the mock theta function 𝑓(𝑞), J. Lond. Math. Soc. (2) 77 (2008), no. 2, 320–334. MR 2400394 (2009h:11076), http://dx.doi.org/10.1112/jlms/jdm119
15. P. Guerzhoy, Z. Kent, and K. Ono, -adic coupling of mock modular forms and shadows, Proc. Nat. Acad. Sci. USA, accepted for publication.
16. Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639–660. MR 969247 (90f:11028a), http://dx.doi.org/10.1007/BF01394279
17. Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin, 1988. With an introduction by George E. Andrews. MR 947735 (89j:01078)
18. Jean-Pierre Serre, Formes modulaires et fonctions zêta 𝑝-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, pp. 191–268. Lecture Notes in Math., Vol. 350 (French). MR 0404145 (53 #7949a)
19. Jean-Pierre Serre, Divisibilité de certaines fonctions arithmétiques, Enseignement Math. (2) 22 (1976), no. 3-4, 227–260. MR 0434996 (55 #7958)
20. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kan\cflex o Memorial Lectures, 1. MR 1291394 (95e:11048)
21. G. N. Watson, The final problem: an account of the mock theta functions, Ramanujan: essays and surveys, Hist. Math., vol. 22, Amer. Math. Soc., Providence, RI, 2001, pp. 325–347. MR 1862757
22. 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)
23. S. P. Zwegers, Mock 𝜃-functions and real analytic modular forms, physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR 1874536 (2003f:11061)
24. S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University (2002).
25. Sander Zwegers, The Folsom-Ono grid contains only integers, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1579–1584. MR 2470815 (2010m:11061), http://dx.doi.org/10.1090/S0002-9939-08-09684-6

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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

Pavel Guerzhoy
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822-2273
Email: pavel@math.hawaii.edu

Ben Kane
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: bkane@math.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05525-5
PII: S 0002-9947(2012)05525-5
Keywords: $p$-adic modular forms, mock theta functions, mock modular forms, harmonic weak Maass forms
Received by editor(s): March 23, 2010
Posted: 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 after 28 years from publication.

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