Congruences for modular forms and applications to crank functions

Abstract

In this paper, motivated by the work of Mahlburg, we find congruences for a large class of modular forms. Moreover, we generalize the generating function of the Andrews-Garvan-Dyson crank on partition and establish several new infinite families of congruences. In this framework, we showed that both the birank of an ordered pair of partitions introduced by Hammond and Lewis, and $k$-crank of $k$-colored partition introduced by Fu and Tang process the same as the partition function and crank.