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

Published online by Cambridge University Press:  27 March 2018

School of Mathematics, Howard House, Queen's Avenue, BS81SD, University of Bristol. e-mail:
Department of Mathematics, Dartmouth College, 27 N. Main Street, 6188 Kemeny Hall, Hanover, NH 03755-3551, U.S.A. e-mail:
Institut für Mathematik, Johannes Gutenberg–Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany. e-mail:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG. e-mail:
Department of Mathematics, 1200 Academy Street, Kalamazoo MI 49006, U.S.A.Kalamazoo College e-mail:


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.

Research Article
Copyright © Cambridge Philosophical Society 2018 

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