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

  • FLORIAN BOUYER (a1), EDGAR COSTA (a2), DINO FESTI (a3), CHRISTOPHER NICHOLLS (a4) and MCKENZIE WEST (a5)...
Abstract

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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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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