Congruences like Atkin's for generalized Frobenius partitions

TL;DR

The paper extends Atkin-type congruences to the $m$-colored generalized Frobenius partition function $c\phi_m(n)$ for primes $m\ge 5$ by constructing cusp forms of half-integral weight on $\Gamma_0(m)$ that capture $c\phi_m(n)$ modulo a prime $\ell\ge 5$. It combines these modular forms with the Shimura lift for eta-multiplier forms and results on modular-Galois representations to produce congruences modulo $\ell$ for all but an explicit finite set of pairs $(\ell,m)$. The authors provide both theoretical constructions and computational verifications, culminating in density results for primes $p$ that yield infinite families of congruences for $c\phi_m(n)$ in Atkin-like progressions. This work broadens the scope of congruence phenomena from the classical partition function to higher-level colored Frobenius partitions via a unified modular-analytic and Galois-representations framework, with explicit finite exceptional cases addressed case-by-case.

Abstract

In the 1960s Atkin discovered congruences modulo primes $\ell\leq 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\neq \ell$ is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes $\ell\geq 5$. Here we consider (for primes $m\geq 5$) the $m$-colored generalized Frobenius partition functions $cφ_m(n)$; these are natural level $m$ analogues of $p(n)$. For each such $m$ we prove that there are similar congruences for $cφ_m(n)\pmod \ell$ 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 $Γ_0(m)$ which capture the relevant values of $cφ_m(n)$ 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.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof : Proof of Theorem \ref{['thm:Flconstruct']} when $\ell\overline{\ell}>m^2$
• Lemma 3.3
• proof