Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d

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dc.creatorElkies, Noam D.
dc.date2006-08-24
dc.date.accessioned2026-07-07T07:22:06Z
dc.date.available2026-07-07T07:22:06Z
dc.descriptionFor each of n=1,2,3 we find the minimal height h^(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 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 naive 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.
dc.description15 pages; some lines in the TeX source are commented out with "%" to meet the 15-page limit for ANTS proceedings
dc.identifierhttps://arxiv.org/abs/math/0608593
dc.identifierhttp://arxiv.org/abs/math/0608593
dc.identifierLecture Notes in Computer Science 4076 (proceedings of ANTS-7, 2004; F.Hess, S.Pauli, and M.Pohst, ed.), 287--301
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/115538
dc.subjectAlgebraic Geometry
dc.subjectNumber Theory
dc.subject11G50; 11G05
dc.titlePoints of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d
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