Explicit Elliptic $K3$ Surfaces with Rank 15

Abstract

This note presents a relatively straightforward proof of the fact that, under certain congruence conditions on a, b, c is an element of Q, the group of rational points over (Q) over bar (t) on the elliptic curve given by y(2) = x(3) + t(3)(t(2) + at + b)(2)(t + c)x + t(5)(t(2) + at + b)(3) is trivial. This is used to show that a related elliptic curve yields a free abelian group of rank 15 as its group of (Q) over bar (t)-rational points.