Formes linéaires de logarithmes de points algébriques sur une courbe elliptique. (Linear forms of logarithms of algebraic points on an elliptic curve)
type: Item
instance of: scholarly article
title: Formes linéaires de logarithmes de points algébriques sur une courbe elliptique. (Linear forms of logarithms of algebraic points on an elliptic curve)
zbMATH Open document ID: 0957.11030
author: Mohammed Ably
publication date: 2000-03-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75413
review text: In connection with Siegel's theorem on the finiteness of integral points on a curve of genus \(\geq 1\), \textit{S. Lang} raised the following question [Conjecture 2 of Diophantine approximations on toruses. Am. J. Math. 86, 521-533 (1964; Zbl 0142.29601); = Collected Papers, Vol. I, 1952-1970, Springer (2000; Zbl 0955.01032), p. 313-325]. Given an abelian variety \(A\) of dimension \(1\) over a number field \(K\) and a non-constant rational function \(\phi\) on \(A\) defined over \(K\), there exists a positive constant \(C\) such that the height of points \(P\) in \(A(K)\) which satisfy \(|\phi(P)|\leq H(P)^{-C}\) is bounded. Here, \(H(P)\) is the absolute height of \(P\). Lang also speculated a best possible value for \(C\), but we shall only consider the mere existence of \(C\). Later he extended his conjecture to abelian varieties of arbitrary dimension [p. 673 of Transcendental numbers and diophantine approximations. Bull. Am. Math. Soc. 77, 635-677 (1971; Zbl 0218.10053); = Collected Papers, Vol. II, 1971-1977, Springer (2000; Zbl 0955.01033), p. 1-43; p. 783 of Higher dimensional diophantine problems. Bull. Am. Math. Soc. 80, 779-787 (1974; Zbl 0298.14014); = Collected Papers, Vol. II, 1971-1977, Springer (2000), p. 102-110] Non-trivial effective lower bounds for \(|\phi(P)|\) have been obtained by \textit{D. W. Masser} [Elliptic functions and transcendence. Lecture Notes Math. 437, Springer-Verlag, Berlin-New York (1975; Zbl 0312.10023)] for elliptic curves with complex multiplication, and [Linear forms in algebraic points of Abelian functions. I--III. Math. Proc. Camb. Philos. Soc. 77, 499-513 (1975; Zbl 0306.14018); ibid. 79, 55-70 (1976; Zbl 0318.14011); Proc. Lond. Math. Soc. (3) 33, 549-564 (1976; Zbl 0334.14019)], \textit{S. Lang} [Diophantine approximation on abelian varieties with complex multiplication. Adv. Math. 17, 281-336 (1975; Zbl 0306.14019); = Collected Papers, Vol. II, 1971-1977, Springer (2000), p. 113-168] and also \textit{J. Coates} and …
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11J89
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 1409062
zbMATH Keywords: linear form of logarithms
zbMATH Keywords: elliptic curve
zbMATH Keywords: elliptic curves with complex multiplication
zbMATH Keywords: Lang's conjecture
MaRDI profile type: Publication
OpenAlex ID: W2319119075
cites work: Lagrange polynomials over the integers of an imaginary quadratic field
cites work: Linear forms in the logarithms of algebraic numbers
cites work: Logarithmic forms and group varieties.
cites work: Diophantine approximation on abelian varieties with complex multiplication
cites work: Minorations de formes linéaires de logarithmes elliptiques
cites work: Q4303003
cites work: Q4143433
cites work: Elliptic functions and transcendence
cites work: Polynomial interpolation in several complex variables
cites work: Lemmes de zéros dans les groupes algébriques commutatifs
cites work: Q3033850
cites work: Linear forms in logarithms on commutative algebraic groups
cites work: Q3864596
cites work: Q4168679
cites work: Q3331283
DOI: 10.5802/AIF.1745
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1409062