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A short proof of the Mock Theta Conjectures using Maass forms

Author(s): Amanda Folsom
Journal: Proc. Amer. Math. Soc. 136 (2008), 4143-4149.
MSC (2000): Primary 11F37
Posted: June 17, 2008
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Abstract | References | Similar articles | Additional information

Abstract: A celebrated work of D. Hickerson gives a proof of the Mock Theta Conjectures using Hecke-type identities discovered by G. Andrews. Here, we respond to a remark by K. Bringmann, K. Ono and R. Rhoades and provide a short proof of the Mock Theta Conjectures by realizing each side of the identities as the holomorphic projection of a harmonic weak Maass form.


References:

1.
G. Andrews, The fifth and seventh order mock theta functions, Trans. Am. Math. Soc. 293 (1986), 113-134. MR 814916 (87f:33011)

2.
G. Andrews and F. Garvan, Ramanujan's ``Lost'' Notebook IV: The Mock Theta Conjectures, Adv. Math. 73 (1989), 242-255. MR 987276 (90d:11115)

3.
A.O.L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84-106. MR 0060535 (15:685d)

4.
K. Bringmann and K. Ono, Dyson's ranks and Maass forms, Annals of Mathematics, accepted for publication (2007).

5.
K. Bringmann and K. Ono, The $ f(q)$ mock theta function conjecture and partition ranks, Inventiones Math. 165 (2006), 243-266. MR 2231957 (2007e:11127)

6.
K. Bringmann, K. Ono, and R. Rhoades, Eulerian series as modular forms, Journal of the Amer. Math. Soc., accepted for publication.

7.
K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc., accepted for publication.

8.
J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45-90. MR 2097357 (2005m:11089)

9.
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.

10.
D. Hickerson, A proof of the Mock Theta Conjectures, Inventiones Math. (3) 94 (1988), 639-660. MR 969247 (90f:11028a)

11.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, GTM No. 97, Springer-Verlag (1984, 1993). MR 1216136 (94a:11078)

12.
K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $ q$-series, Conference Board of the Mathematical Sciences 102, Amer. Math. Soc., Providence, RI (2004). MR 2020489 (2005c:11053)

13.
S. Ramanujan, Collected Papers, Cambridge Univ. Press, London/NY, 1927 (reprinted by Chelsea, NY). MR 2280843 (2008b:11002)

14.
G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440-481. MR 0332663 (48:10989)

15.
G. N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55-80. MR 1862757

16.
G.N. Watson, The mock theta functions (2), Proc. London Math. Soc. (2) 42 (1937), 274-304.

17.
S. P. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002).

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Additional Information:

Amanda Folsom
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: folsom@math.wisc.edu

DOI: 10.1090/S0002-9939-08-09434-3
PII: S 0002-9939(08)09434-3
Received by editor(s): November 5, 2007
Posted: June 17, 2008
Additional Notes: The author is grateful for a National Science Foundation Postdoctoral Fellowship and wishes to thank Ken Ono for suggesting this project. The author also thanks the referee for a very detailed and thoughtful report, including useful suggestions that have helped ease the exposition of this paper.
Communicated by: Ken Ono
Copyright of article: Copyright 2008, American Mathematical Society


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