Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d

Abstract

For each of n = 1, 2, 3 we find the minimal height ˆh(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d = 12n (equivalently, of arithmetic genus n), and exhibit all (E, P) attaining this minimum. The minimal ˆ h(P) was known to equal 1/30 for n = 1 (Oguiso-Shioda) and 11/420 for n = 2 (Nishiyama), but the formulas for the general (E,P) were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree 12n over a function field of any genus. For n = 3 both the minimal height (23/840) and the explicit curves are new. These (E, P) also have the property that that mP is an integral point (a point of na¨ıve height zero) for each m = 1, 2, ..., M, where M = 6, 8, 9 for n = 1, 2, 3; this, too, is maximal in each of the three cases.