-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION

Abstract

AbstractA partition of a positive integer n is called $\ell $ -regular if none of its parts is divisible by $\ell $ . Denote by $b_{\ell }(n)$ the number of $\ell $ -regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$ .