Minimal Regulators for Rank-2 Subgroups of Rational and K3 Elliptic Surfaces

We determine the smallest possible regulator R(P, Q) for a rank-2 subgroup ZP circle plus ZQ of an elliptic curve E over C(t) of discriminant degree 12n for n = 1 (a rational elliptic surface) and n = 2 (a K3 elliptic surface), exhibiting equations for all (E, P, Q) attaining the minimum. The minimum R(P, Q) = 1/36 for a rational elliptic surface was known [Oguiso and Shioda 911, but a formula for (E, P, Q) was not, nor was the fact that this is the minimum for an elliptic curve of discriminant degree 12 over a function field of any genus. For a K3 surface, both the minimal regulator R(P, Q) = 1/100 and the explicit equations are new. We also prove that 1/100 is the minimum for an elliptic curve of discriminant degree 24 over a function field of any genus. The optimal (E, P, Q) are uniquely characterized by having mP and m'Q integral form <= M and m' <= M', where (M, M') = (3, 3) for n = 1 and (M, M') = (6, 3) for n = 2. In each case MM' is maximal. We use the connection with integral points to find explicit equations for the curves. As an application we use the K3 surface to produce, in a new way, the elliptic curves E/Q with nontorsion points of smallest known canonical height. These examples appeared previously in [Elkies 02].