<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$U_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> 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 Hauptmoduln <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma _0(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$G_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma _0(8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, together with explicit <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$U_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-action formulas.</p>

An infinite family of overpartition congruences mod powers of 2

期刊论文
发表日期：

2025年12月6日

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Zhumagali Shomanov,
Frank Garvan

摘要

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 $U_2$ $U_{2}$ 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 Hauptmoduln $G_2$ $G_{2}$ on $\Gamma _0(2)$ $Γ_{0} (2)$ and $G_8$ $G_{8}$ on $\Gamma _0(8)$ $Γ_{0} (8)$ , together with explicit $U_2$ $U_{2}$ -action formulas.