An infinite family of overpartition congruences mod powers of 2

TL;DR

This work addresses congruences for the overpartition function $\overline{p}(n)$ modulo powers of 2 by combining an Atkin $U_2$-driven refinement of a Garvan–Morrow identity with modular-forms techniques. The authors introduce and exploit explicit eta-quotient modular equations between Hauptmoduln $G_2$ on $\\Gamma_0(2)$ and $G_8$ on $\\Gamma_0(8)$, obtaining concrete $U_2$-action formulas and inductive bounds on $2$-adically divided coefficients. The main contribution is an infinite family of congruences: for odd primes $\ell$ and $\alpha\ge1$, $$\ell^3 \overline{p}(2^\alpha \ell^2 n) + \ell \left(\frac{-2^\alpha n}{\ell}\right) \overline{p}(2^\alpha n) + \overline{p}\left(\dfrac{2^\alpha n}{\ell^2}\right) \equiv (\ell^3+1) \overline{p}(2^\alpha n) \pmod{2^{\alpha+12}}$$ for all $n\ge0$. The results illuminate the interaction between partition-theoretic generating functions and modular-curve arithmetic, with potential deeper implications for $2$-adic phenomena in partition congruences and related modular-forms identities.

Abstract

We prove an infinite family of Hecke-like congruences for the overpartition function modulo powers of 2. Starting from a recent identity of Garvan and Morrow and iterating Atkin's $U_2$ operator, we determine lower bounds on the 2-adic valuations of the coefficients that arise at each step. Our approach yields new modular equations relating the Hauptmoduln $G_2$ on $Γ_0(2)$ and $G_8$ on $Γ_0(8)$, together with explicit $U_2$-action formulas.

Theorems & Definitions (26)

• Theorem 2.1: Newman NewmanEtaQuotient
• Theorem 2.2: Ligozat Ligozat
• Theorem 2.3: Gordon-Hughes GordonHughes
• Theorem 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof