Congruences like Atkin's for generalized Frobenius partitions

Abstract

In the 1960s Atkin discovered congruences modulo primes $\ell \le 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\ne \ell$ is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes $\ell \ge 5$. Here we consider (for primes $m\ge 5$) the $m$-colored generalized Frobenius partition functions $c{\varphi }_m(n)$; these are natural level $m$ analogues of $p(n)$. For each such $m$ we prove that there are similar congruences for $c{\varphi }_m(n)(\mathrm{mod}\ell )$ for all primes $\ell$ outside of an explicit finite set depending on $m$. To prove the result we first construct, using both theoretical and computational methods, cusp forms of half-integral weight on ${\Gamma }_0(m)$ which capture the relevant values of $c{\varphi }_m(n)$ modulo~$\ell$. We then apply previous work of the authors on the Shimura lift for modular forms with the eta multiplier together with tools from the theory of modular Galois representations.

14 pages