Integral magneticity of the level-two K3 packet: CM theta lifts and a 2-isogeny trace contraction

Bönisch, Duhr, and Maggio introduced three meromorphic modular forms \(C_4,C_{6a},C_{6b}\) on \(\Gamma_0(2)\), 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 \(t=(\eta(2\tau)/\eta(\tau))^{24}\), \(H=\eta(\tau)^4/\eta(2\tau)^2\), \(J=2E_2(2\tau)-E_2(\tau)\), \(u=64t\), 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.