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Elliptic \(K3\) surfaces with geometric Mordell-Weil rank \(15\). (English) Zbl 1162.14024
M. Kuwata [Comment. Math. Univ. St. Paul. 49, 91–100 (2000; Zbl 1018.14013)] gave examples of elliptic \(K3\) surfaces over \(\mathbb{Q}\) with rank of the Mordell-Weil group \(r\) for any \(0\leq r\leq 18\), \(r\not= 15\). In the paper under review, R. Kloosterman completes the list of Kuwata by giving an explicit example for \(r=15\) (the existence of such \(K3\) surfaces over \(\mathbb{C}\) was given by D. A. Cox [Duke Math. J. 49, 677–689 (1982; Zbl 0503.14018)]).
The \(K3\) surface studied by Kloosterman has equation \[ y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1) \] and belongs to a bigger family of \(K3\) surfaces, where the generic \(K3\) in the family over \(\mathbb{C}\) has \(r=15\).

MSC:
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28 \(K3\) surfaces and Enriques surfaces
11G05 Elliptic curves over global fields
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