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q-series and weight 3/2 Maass forms

Published online by Cambridge University Press:  01 May 2009

Kathrin Bringmann
Affiliation:
Mathematisches Institut, Universität Köln, Weyertal 86-90, D-50931 Köln, Germany (email: kbringma@math.uni-koeln.de)
Amanda Folsom
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA (email: folsom@math.wisc.edu)
Ken Ono
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA (email: ono@math.wisc.edu)
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Abstract

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Despite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers–Fine basic hypergeometric series.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Andrews, G. E., q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, Conference Board of the Mathematical Sciences, vol. 66 (American Mathematical Society, Providence, RI, 1986).Google Scholar
[2]Andrews, G. E., Ramanujan’s Lost” notebook V: Euler’s partition identity, Adv. Math. 61 (1986), 156164.CrossRefGoogle Scholar
[3]Andrews, G. E., Mock theta functions, in Theta functions – Bowdoin 1987, Part 2 (Brunswick, ME, 1987), Proceedings of Symposia in Pure Mathematics, vol. 49, Part 2 (American Mathematical Society, Providence, RI, 1989), 283297.Google Scholar
[4]Andrews, G. E., Partitions: at the interface of q-series and modular forms, Ramanujan J. 7 (2003), 385400. (Rankin Memorial Issues.)Google Scholar
[5]Andrews, G. E., The number of smallest parts in the partitions of n, J. Reine Angew. Math., accepted for publication.Google Scholar
[6]Bringmann, K., On the explicit construction of higher deformations of partition statistics, Duke Math. J. 144 (2008), 195233.CrossRefGoogle Scholar
[7]Bringmann, K., Garvan, F. and Mahlburg, K., Partition statistics and quasiweak Maass forms, Int. Math. Res. Not. 2009 (2009), 6397.Google Scholar
[8]Bringmann, K. and Ono, K., The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243266.Google Scholar
[9]Bringmann, K. and Ono, K., Dyson’s ranks and Maass forms, Ann. of Math (2), to appear.Google Scholar
[10]Bringmann, K., Ono, K. and Rhoades, R., Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 10851104.Google Scholar
[11]Bruinier, J. H. and Funke, J., On two geometric theta lifts, Duke Math. J. 125 (2004), 4590.CrossRefGoogle Scholar
[12]Eichler, M., On the class number of imaginary quadratic fields and the sums of divisors of natural numbers, J. Indian Math. Soc. (N.S.) 19 (1955), 153180.Google Scholar
[13]Fine, N. J., Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27 (American Mathematical Society, Providence, RI, 1988).Google Scholar
[14]Folsom, A. and Ono, K., The spt-function of Andrews, Proc. Natl Acad. Sci., USA 105 (2008), 2015220156.Google Scholar
[15]Garvan, F., Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank, Int. J. Num. Theory, accepted for publication.Google Scholar
[16]Hirzebruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms with Nebentypus, Invent. Math. 36 (1976), 57113.Google Scholar
[17]Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory, Proceedings of the 2008Harvard–MIT Current Developments in Mathematics Conference (International Press, Boston), accepted for publication.CrossRefGoogle Scholar
[18]Rademacher, H., Topics in analytic number theory (Springer, Heidelberg, 1973).CrossRefGoogle Scholar
[19]Shimura, G., On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440481.CrossRefGoogle Scholar
[20]Zagier, D., Nombres de classes et formes modulaires de poids 3/2, C. R. Acad. Sci. Paris Sér. A–B 281 (1975), 883886.Google Scholar
[21]Zagier, D., Ramanujan’s mock theta functions and their applications [d’après Zwegers and BringmannOno], Séminaire Bourbaki, in press.Google Scholar
[22]Zwegers, S. P., Mock ϑ-functions and real analytic modular forms, q-series with applications to combinatorics, number theory, and physics, Contemporary Mathematics, vol. 291 eds B. C. Berndt and K. Ono (American Mathematical Society, Providence, RI, 2001), 269277.Google Scholar
[23]Zwegers, S. P., Mock theta functions, PhD thesis, Universiteit Utrecht (2002).Google Scholar