Partition values and central critical values of certain modular functions
Author:
John J. Webb
Journal:
Proc. Amer. Math. Soc. 138 (2010), 12631272
MSC (2010):
Primary 11F33; Secondary 11F11
Published electronically:
December 8, 2009
MathSciNet review:
2578520
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Abstract: Let denote the number of partitions of a positive number , let and let be the least nonnegative residue of modulo . In this paper we prove congruences modulo between and ratios of central critical values of functions associated to twists of certain integer weight newforms. In 1999, Guo and Ono proved analogous results for .
 [Ahl00]
Scott
Ahlgren, Distribution of the partition function modulo composite
integers 𝑀, Math. Ann. 318 (2000),
no. 4, 795–803. MR 1802511
(2001j:11099), 10.1007/s002080000142
 [AB03]
Scott
Ahlgren and Matthew
Boylan, Arithmetic properties of the partition function,
Invent. Math. 153 (2003), no. 3, 487–502. MR 2000466
(2004e:11115), 10.1007/s0022200302956
 [AO01]
Scott
Ahlgren and Ken
Ono, Congruence properties for the partition function, Proc.
Natl. Acad. Sci. USA 98 (2001), no. 23,
12882–12884 (electronic). MR 1862931
(2002k:11187), 10.1073/pnas.191488598
 [And98]
George
E. Andrews, The theory of partitions, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976
original. MR
1634067 (99c:11126)
 [AL78]
A.
O. L. Atkin and Wen
Ch’ing Winnie Li, Twists of newforms and pseudoeigenvalues
of 𝑊operators, Invent. Math. 48 (1978),
no. 3, 221–243. MR 508986
(80a:10040), 10.1007/BF01390245
 [Gar07]
F.G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank. Int. J. Number Theory, to appear. arXiv:0710.5793
 [GO99]
Li
Guo and Ken
Ono, The partition function and the arithmetic of certain modular
𝐿functions, Internat. Math. Res. Notices 21
(1999), 1179–1197. MR 1728677
(2000m:11102), 10.1155/S1073792899000641
 [Iwa97]
Henryk
Iwaniec, Topics in classical automorphic forms, Graduate
Studies in Mathematics, vol. 17, American Mathematical Society,
Providence, RI, 1997. MR 1474964
(98e:11051)
 [Kno93]
Marvin
I. Knopp, Modular functions in analytic number theory, Markham
Publishing Co., Chicago, Ill., 1970. MR 0265287
(42 #198)
 [Ono00]
Ken
Ono, Distribution of the partition function modulo 𝑚,
Ann. of Math. (2) 151 (2000), no. 1, 293–307.
MR
1745012 (2000k:11115), 10.2307/121118
 [Ono04]
Ken
Ono, The web of modularity: arithmetic of the coefficients of
modular forms and 𝑞series, CBMS Regional Conference Series in
Mathematics, vol. 102, Published for the Conference Board of the
Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 2004. MR 2020489
(2005c:11053)
 [Ono08]
K. Ono, Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory. HarvardMIT Current Developments in Mathematics 2008, International Press, (preliminary version).
 [Shi73]
Goro
Shimura, On modular forms of half integral weight, Ann. of
Math. (2) 97 (1973), 440–481. MR 0332663
(48 #10989)
 [Wal81]
J.L.
Waldspurger, Sur les coefficients de Fourier des formes modulaires
de poids demientier, J. Math. Pures Appl. (9) 60
(1981), no. 4, 375–484 (French). MR 646366
(83h:10061)
 [Ahl00]
 S. Ahlgren, Distribution of the partition function modulo composite integers . Math. Ann. 318 (2000), no. 4, 795803. MR 1802511 (2001j:11099)
 [AB03]
 S. Ahlgren and M. Boylan, Arithmetic properties of the partition function. Invent. Math. 153 (2003), no. 3, 487502. MR 2000466 (2004e:11115)
 [AO01]
 S. Ahlgren and K. Ono, Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 1288212884. MR 1862931 (2002k:11187)
 [And98]
 G. E. Andrews, The Theory of Partitions. Cambridge University Press, Cambridge, 1998. MR 1634067 (99c:11126)
 [AL78]
 A.O.L. Atkin and W.C.W. Li, Twists of newforms and pseudoeigenvalues of operators. Invent. Math. 48 (1978), no. 3, 221243. MR 508986 (80a:10040)
 [Gar07]
 F.G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank. Int. J. Number Theory, to appear. arXiv:0710.5793
 [GO99]
 L. Guo and K. Ono, The partition function and the arithmetic of certain modular functions. Internat. Math. Res. Notices 1999, no. 21, 11791197. MR 1728677 (2000m:11102)
 [Iwa97]
 H. Iwaniec, Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)
 [Kno93]
 M. Knopp, Modular Functions in Analytic Number Theory. Chelsea Publishing Company, New York, 1993. MR 0265287 (42:198)
 [Ono00]
 K. Ono, Distribution of the partition function modulo . Ann. of Math. (2) 151 (2000), no. 1, 293307. MR 1745012 (2000k:11115)
 [Ono04]
 K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Series. CBMS Regional Conference Series in Mathematics, 102. American Mathematical Society, Providence, RI, 2004. MR 2020489 (2005c:11053)
 [Ono08]
 K. Ono, Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory. HarvardMIT Current Developments in Mathematics 2008, International Press, (preliminary version).
 [Shi73]
 G. Shimura, On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440481. MR 0332663 (48:10989)
 [Wal81]
 J.L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demientier. J. Math. Pures Appl. (9) 60 (1981), no. 4, 375484. MR 646366 (83h:10061)
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Additional Information
John J. Webb
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Caro lina 29208
Email:
webbjj3@mailbox.sc.edu
DOI:
http://dx.doi.org/10.1090/S0002993909101880
Received by editor(s):
August 28, 2009
Published electronically:
December 8, 2009
Communicated by:
Ken Ono
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
