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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 $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by

\begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*}
The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$ , together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.



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



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