Elliptic <i>K</i>3 Surfaces with Geometric Mordell–Weil Rank 15

Abstract

Abstract We prove that the elliptic surface y 2 = x 3 + 2( t 8 + 14 t 4 + 1) x + 4 t 2 ( t 8 + 6 t 4 + 1) has geometric Mordell–Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic K 3-surfaces with geometric Mordell–Weil ranks 0, 1, … , 14, 16, 17, 18.