For each of n = 1, 2, 3 we find the minimal height ĥ(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 ĥ(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 naïve 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. © Springer-Verlag Berlin Heidelberg 2006.
CITATION STYLE
Elkies, N. D. (2006). Points of low height on elliptic curves and surfaces I: Elliptic surfaces over ℙ1 with small d. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4076 LNCS, pp. 287–301). Springer Verlag. https://doi.org/10.1007/11792086_21
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