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Points of Low Height on Elliptic Curves and Surfaces I: Elliptic Surfaces over \({\mathbb P}^1\) with Small d

  • Noam D. Elkies
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

Abstract

For each of n=1,2,3 we find the minimal height \({\hat{h}}(P)\) 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 \({\hat{h}}(P)\) 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,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the three cases.

Keywords

Elliptic Curve Integral Point Elliptic Curf Homogeneous Polynomial Elliptic Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Noam D. Elkies
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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