AI 中文总结
研究二级K3包中三个亚纯模形式\(C_4\)、\(C_{6a}\)、\(C_{6b}\)的积分磁性。通过超几何变换、形式等同及相关公式，证明其分母为1的性质及奇素数可除性，处理2的情况，得出二级K3包具有全局分母为1的磁性这一主要贡献。
AI 中文摘要
博尼施、杜尔和马焦在\(\Gamma_0(2)\)上引入了三个亚纯模形式\(C_4\)、\(C_{6a}\)、\(C_{6b}\)，源于一个超几何K3族，并猜想它们的深度分别为1、2、2。我们证明了更强的分母为1的结论，即\(\frac{c_4(n)}n\)、\(\frac{c_{6a}(n)}{n^2}\)、\(\frac{c_{6b}(n)}{n^2}\in\mathbb Z\)（\(n\geq1\)）。通过超几何主模变换将权4的情况简化为逐项二项式可除性。对于权6，将两种形式与判别式为\(-8\)和\(-4\)的规范二级CM形式等同。通过明确的一对权为\(-3/2\)的向量值弱全纯形式及相关公式得到完整的奇素数可除性，2的情况单独处理。证明了无穷族收缩\(U_2\bigl(\mathcal Tt\,\mathbb Z_2[[u]]\bigr) \subseteq 2^5\mathcal Tt\,\mathbb Z_2[[u]]\)，从而得出二级K3包具有全局分母为1的磁性。
英文摘要
Bönisch, Duhr, and Maggio introduced three meromorphic modular forms \(C_4,C_{6a},C_{6b}\) on \(Γ_0(2)\), arising from a hypergeometric K3 family, and conjectured that they are magnetic of depths \(1,2,2\). Writing \[ C_4=\sum_{n\ge1}c_4(n)q^n,\qquad C_{6a}=\sum_{n\ge1}c_{6a}(n)q^n,\qquad C_{6b}=\sum_{n\ge1}c_{6b}(n)q^n, \] we prove the stronger denominator-one statements \[ \frac{c_4(n)}n,\qquad \frac{c_{6a}(n)}{n^2},\qquad \frac{c_{6b}(n)}{n^2}\in\mathbb Z \qquad(n\ge1). \] The weight-four case is reduced to a termwise binomial divisibility by a hypergeometric change of Hauptmodul. For weight six we identify the two forms with canonical level-two CM forms of discriminants \(-8\) and \(-4\): \[ f_{3,-8,0,1,1}=-64C_{6a},\qquad f_{3,-4,2,1,1}=32C_{6b}. \] An explicit pair of vector-valued weakly holomorphic forms of weight \(-3/2\) then gives the full odd-prime divisibility through the higher-level theta-lift coefficient formula of Löbrich--Schwagenscheidt. The prime \(2\) is treated independently. If \(t=(η(2τ)/η(τ))^{24}\), \(H=η(τ)^4/η(2τ)^2\), \(J=2E_2(2τ)-E_2(τ)\), \(u=64t\), and \(\mathcal T=H^4J\), we prove the infinite-family contraction \[ U_2\bigl(\mathcal Tt\,\mathbb Z_2[[u]]\bigr) \subseteq 2^5\mathcal Tt\,\mathbb Z_2[[u]]. \] Consequently \(v_2(c_{6\bullet}(2^rm))\ge5r\), which is stronger than the slope \(2r\) required for double magneticity. Thus the complete level-two K3 packet is magnetic with global denominator one.
Comments15 pages, no figures. Ancillary exact-verification script and transcript included
