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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



Families of quasimodular forms and Jacobi forms: The crank statistic for partitions

Author: Robert C. Rhoades
Journal: Proc. Amer. Math. Soc. 141 (2013), 29-39
MSC (2010): Primary 11P82, 11P84, 11P83, 11P55, 05A17
Published electronically: April 30, 2012
MathSciNet review: 2988708
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Abstract: Families of quasimodular forms arise naturally in many situations, such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this paper we give examples of such expansions that arise in the study of partition statistics.

The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two-variable generating function for the crank partition statistic is a Jacobi form. Exploiting the structure inherent in the Jacobi theta function, we construct explicit expressions for the functions of Atkin and Garvan. Furthermore, this perspective opens the door for further investigation, including a study of the moments in arithmetic progressions. We conduct a thorough study of the crank statistic restricted to a residue class modulo 2.

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Additional Information

Robert C. Rhoades
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Keywords: Jacobi forms, quasimodular forms, crank, partition statistic, partitions
Received by editor(s): June 4, 2011
Published electronically: April 30, 2012
Additional Notes: The author is supported by an NSF Mathematical Sciences Postdoctoral Fellowship.
Communicated by: Ken Ono
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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