Congruences of the partition function

Abstract

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$. Here the integer $k$ depends on the Hecke eigenvalues of a certain invariant subspace of $S_{m/2-1}(\Gamma_0(576),\chi_{12})$ and can be explicitly computed. More generally, we will show that for each integer $i>0$ there exists an integer $k$ such that for every non-negative integers $j\ge i$ with a properly chosen $B$ the congruence $$ p(m^j\ell^kn+B)\equiv 0\mod m^i $$ holds for all integers $n$ not divisible by $\ell$.