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 $\eta(24z)^\gamma \phi(24z)$, where $\eta(z)$ is the Dedekind's eta function and $\phi(z)$ is a holomorphic modular form in $M_\lambda({\rm SL}_2(\mathbb{Z}))$. 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 $p_r(n)$.