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Points of low height on elliptic curves and surfaces I: Elliptic surfaces over \${\mathbb P}\$ with small \$d\$. (English)
Hess, Florian (ed.) et al., Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23‒28, 2006. Proceedings. Berlin: Springer (ISBN 3-540-36075-1/pbk). Lecture Notes in Computer Science 4076, 287-301 (2006).
Summary: For each of \$n =1,2,3\$ we find the minimal height \${\widehat{h}}(P)\$ of a nontorsion point \$P\$ of an elliptic curve \$E\$ over \${\bold C} (T)\$ of discriminant degree \$d =12n\$ (equivalently, of arithmetic genus \$n\$), and exhibit all \$(E, P)\$ attaining this minimum. The minimal \${\widehat{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 naïve height zero) for each \$m =1,2,\ldots, M\$, where \$M =6,8,9\$ for \$n =1,2,3\$; this, too, is maximal in each of the three cases.