An infinite family of overpartition congruences mod powers of 2

TL;DR

This paper establishes an infinite family of congruences for the overpartition function modulo powers of 2, using modular equations, identities, and operator iteration to analyze 2-adic valuations.

Contribution

It introduces new modular equations and explicit formulas for the $U_2$ operator, extending overpartition congruences to all powers of 2.

Findings

Proves infinite overpartition congruences mod powers of 2.

Derives new modular equations relating Hauptmoduln G_2 and G_8.

Provides explicit formulas for the action of the $U_2$ operator.

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 $\Gamma_0(2)$ and $G_8$ on $\Gamma_0(8)$, together with explicit $U_2$-action formulas.