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) 2, 2, -18 . We construct a primitive E₈ + E₇ 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 Jac(C) ~ Eₖ² over KC. The associated elliptic factor is a non-CM degree-$2$ Q-curve over Q(√-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 Q(ζ₈), with intrinsic inner twist χ₋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) 2, 2, -18 Rank-$3$ lattice orthogonal to the Picard lattice Pic(Y) inside H²(Y, Z) Singular Fiber Type E₈ + E₇ Configuration of reducible fibers in the elliptic fibration (Inose pencil) Jacobian Decomposition Jac(C) ~ Eₖ² Isogeny decomposition of the Jacobian variety of the genus-$2$ curve C Base Field & Arithmetic Q(√-15), 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 Q-curve Modular Weight & Level $N = 7200, k = 2$ Level conductor and weight of the associated modular form Coefficient Field Q(ζ₈) Hecke eigenvalue field generated by an $8$-th root of unity Inner Twist Character χ₋₃ = (-3/·) Quadratic Dirichlet character associated with the CM/inner twist symmetry