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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
##### Keywords:
$$K3$$-surfaces; elliptic fibrations
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