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

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Original languageEnglish
Number of pages20
JournalMathematical Proceedings of the Cambridge Philosophical Society
Early online date27 Mar 2018
DateAccepted/In press - 7 Mar 2018
DateE-pub ahead of print (current) - 27 Mar 2018


Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let χ be the generic element of the family of surfaces in ℙ given by
χ: w2=x6+y6+z6+tx2y2z2.
The surface χ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of χ, together with its Galois module structure, as well as derive more results on the arithmetic of χ and other elements of the family x.

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