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

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We determine the smallest possible regulator R(P,Q) for a rank-2subgroup ℤP⊕ℤQ of an elliptic curve E over ℂ(t) of discriminantdegree 12n for n = 1 (a rational elliptic surface) and n = 2(a K3 elliptic surface), exhibiting equations for all (E,P,Q) attainingthe minimum. The minimum R(P,Q) = 1/36 for a rationalelliptic surface was known [Oguiso and Shioda 91], buta formula for (E,P,Q) was not, nor was the fact that this is theminimum for an elliptic curve of discriminant degree 12 over afunction field of any genus. For a K3 surface, both the minimalregulator R(P,Q) = 1/100 and the explicit equations are new.We also prove that 1/100 is the minimum for an elliptic curve ofdiscriminant degree 24 over a function field of any genus. Theoptimal (E,P,Q) are uniquely characterized by having mP andm′Q integral for m ≤ M and m′ ≤ M′, where (M,M′) = (3, 3)for n = 1 and (M,M′) = (6, 3) for n = 2. In each case MM′ ismaximal. We use the connection with integral points to find explicitequations for the curves. As an application we use the K3surface to produce, in a new way, the elliptic curves E/ℚ withnontorsion points of smallest known canonical height. Theseexamples appeared previously in [Elkies 02].