A Genus-Two Pencil on a Kummer K3 Surface: Parabolic Cohomology and Frobenius

We study an explicit genus-two pencil on the Kummer K3 surface associated with the self-product of a Legendre elliptic curve. Resolving a double-sextic model and blowing up the two geometric base points produces a projective genus-two fibration. We determine the arithmetic parabolic cohomology of the rank-four first-cohomology local system. The main result identifies the resulting five-dimensional ℓ-adic Galois representation with the tensor square of the first cohomology of the underlying elliptic curve, together with an additional Tate line carrying the quadratic character associated with −1. The extra summand arises geometrically from the two Galois-conjugate base points. We explicitly analyze the six finite singular parameters, their geometric monodromy, and the degeneration at infinity. An independent finite-field character-sum calculation yields the complete degree-five Frobenius characteristic polynomial, including contributions from singular fibers. We also establish an equivariant decomposition into one-dimensional and four-dimensional eigenspaces under the natural involution of the pencil. The results provide a concrete geometric and arithmetic realization of parabolic cohomology for this specified Kummer pencil. The paper is self-contained and does not claim a global two-parameter perverse comparison or a proof of the higher-dimensional Sato–Tate conjecture.