Fourth-Moment Collision K3 Surfaces, Genus-Two Splitting, and a Projective Modular Packet

We study an explicit rational-phase exponential-sum family through the geometry of its mixed fourth moment. After saturating away the ordered pairing components, the residual collision surface is birational to a double plane whose minimal resolution is a $K3$ surface. A symplectic involution produces an explicit elliptic $K3$ surface $Y$ with Picard number $19$ and integral transcendental lattice $$T(Y) \\simeq \\langle 2, 2, -18 \\rangle.$$ We construct a primitive $E_8 + E_7$ Inose-type Jacobian pencil, recover an explicit smooth genus-$2$ curve $C$ by Kumar's marked $K3$ construction, and exhibit two complementary primitive degree-$6$ maps to a common elliptic target, yielding $$\\operatorname{Jac}(C) \\sim E_k^2 \\quad \\text{over } K_C.$$ The associated elliptic factor is a non-CM degree-$2$ $\\mathbb{Q}$-curve over $\\mathbb{Q}(\\sqrt{-15})$ with sign Brauer class $(-15, 2)$. Galois-equivariant Shioda–Inose descent gives a global projective-adjoint description of the normalized transcendental representation. A minimal scalar lift has conductor $7200$ and is unconditionally modular of weight $2$. Its Hecke field is $\\mathbb{Q}(\\zeta_8)$, with intrinsic inner twist $\\chi_{-3}$, and exact characteristic-zero Hecke computation isolates a four-constituent relative twist orbit. Key Mathematical Notations Breakdown Context / Term LaTeX Representation Description Transcendental Lattice $T(Y) \\simeq \\langle 2, 2, -18 \\rangle$ Rank-$3$ lattice orthogonal to the Picard lattice $\\operatorname{Pic}(Y)$ inside $H^2(Y, \\mathbb{Z})$ Singular Fiber Type $E_8 + E_7$ Configuration of reducible fibers in the elliptic fibration (Inose pencil) Jacobian Decomposition $\\operatorname{Jac}(C) \\sim E_k^2$ Isogeny decomposition of the Jacobian variety of the genus-$2$ curve $C$ Base Field & Arithmetic $\\mathbb{Q}(\\sqrt{-15})$, $\\mathbb{Q}$-curve Arithmetic elliptic factor defined over quadratic extension with degree-$2$ isogenies to its Galois conjugate Brauer Obstruction $(-15, 2)$ Local/global quaternion algebra invariant classifying the $\\mathbb{Q}$-curve Modular Weight & Level $N = 7200, k = 2$ Level conductor and weight of the associated modular form Coefficient Field $\\mathbb{Q}(\\zeta_8)$ Hecke eigenvalue field generated by an $8$-th root of unity Inner Twist Character $\\chi_{-3} = \\left(\\frac{-3}{\\cdot}\\right)$ Quadratic Dirichlet character associated with the CM/inner twist symmetry