Congruences of Multipartition Functions Modulo Powers of Primes

Abstract

Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{\gamma ,\lambda }$ be the space spanned by $\eta (24z{)}^{\gamma }\varphi (24z)$, where $\eta (z)$ is the Dedekind's eta function and $\varphi (z)$ is a holomorphic modular form in $M_{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))$. In this paper, we show that the generating function of $p_r(\frac{m^{k}n+r}{24})$ with respect to $n$ is congruent to a function in the space $S_{\gamma ,\lambda }$ modulo $m^k$. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of $p(n)$ modulo $5,7,11$ and Gandhi's congruences of $p_2(n)$ modulo 5 and $p_8(n)$ modulo 11. Furthermore, using the invariance property of $S_{\gamma ,\lambda }$ under the Hecke operator $T_{{\ell }^2}$, we obtain two classes of congruences pertaining to the $m^k$-adic property of $p_r(n)$.

19 pages