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

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

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Title: Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d
Author: Elkies, Noam
Citation: Elkies, Noam D. 2006. Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d. Lecture Notes in Computer Science 4076: 287-301.
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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.
Published Version: http://dx.doi.org/10.1007/11792086_21
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794827

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  • FAS Scholarly Articles [7106]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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