On the arithmetic of a family of degree - Two K3 surfaces

Let P denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let X be the generic element of the family of surfaces in P given byX : w(2) = x(6) + y(6) + z(6) + tx(2)y(2)z(2).The surface X is a K3 surface over the function field Q(t). In this paper, we explicitly compute the geometric Picard lattice of X, together with its Galois module structure, as well as derive more results on the arithmetic of X and other elements of the family X.