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# Elliptic $K3$ Surfaces with Geometric Mordell--Weil Rank $15$

Published:2007-06-01
Printed: Jun 2007
• Remke Kloosterman
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## Abstract

We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$.
 MSC Classifications: 14J27 - Elliptic surfaces 14J28 - $K3$ surfaces and Enriques surfaces 11G05 - Elliptic curves over global fields [See also 14H52]