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

Abstract

We determine the smallest possible regulator $R(P,Q)$ for a rank-2 subgroup $\mathbb{Z}P\oplus\mathbb{Z}Q$ of an elliptic curve $E$ over $\mathbb{C}(t)$ 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, 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 for $m\leq M$ and $m'\leq M'$, 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 $E/\mathbb{Q}$ with nontorsion points of smallest known canonical height. These examples appeared previously in Noam D. Elkies. “Nontorsion Points of Low Height on Elliptic Curves over $mathbb{Q}$.”.