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

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 note 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.