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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:
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
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
-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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