Diophantine approximation of \(p\)-adic elliptic logarithms
type: Item
instance of: scholarly article
title: Diophantine approximation of \(p\)-adic elliptic logarithms
zbMATH Open document ID: 0853.11055
author: Gaël Rémond
author: Florent Urfels
published in: Journal of Number Theory
publication date: 1996-06-04T00:00:00Z
review text: Les auteurs montrent comment la méthode des déterminants d'interpolation de M. Laurent permet d'obtenir des minorations totalement explicites de formes linéaires à coefficients algébriques de deux logarithmes elliptiques \(p\)-adiques. Plus précisément, soient \(E: Y^2= X^3- AX- B\) une courbe elliptique définie sur un corps de nombres \(K\), \(\beta\in K\) et \(u_1, u_2\in \mathbb{C}_p\), \(|u_1|, |u_2 |\leq p^{-1/ (p-1)}\). On suppose que l'exponentielle \(p\)-adique de \(E\) associe à \(u_1\), \(u_2\) des points \(\exp (u_1), \exp (u_2)\in E(K)\) et on pose \(\log e= {{p^{-1/ (p-1)}} \over {\max (|u_1 |; |u_2 |)}}\) et \(h_E= h(1: A:B)\) où \(h\) désigne la hauteur logarithmique absolue. Alors, si \(\beta \neq u_2/ u_1\) on a \[ \log|\beta u_1- u_2|\geq -5, 7.10^{26}. d^6. a_1 a_2. b.\max (1, h_E, \log b)^3 \log e \] où \(d=\max (1; {{[K: \mathbb{Q}]} \over {\log \varepsilon}})\), \(a_i= \max (1; \widehat {h} (\exp (u_1)))\) \((i=1, 2)\), \(\widehat {h}\) désignant la hauteur de Néron-Tate, et \(b= \max (a_1, a_2, d, h(\beta), h_E)\). Dans le cas complexe, une minoration totalement explicite a été établie par \textit{S. David} pour un nombre quelconque de logarithmes elliptiques [Mém. Soc. Math. Fr., Nouv. Sér. 62, 143 p. (1995)].
reviewed by: Patrice Philippon
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11J89
zbMATH DE Number: 874236
zbMATH Keywords: diophantine approximation
zbMATH Keywords: \(p\)-adic elliptic logarithms
zbMATH Keywords: \(p\)-adic approximation
zbMATH Keywords: logarithmic height
zbMATH Keywords: elliptic curve
zbMATH Keywords: linear forms in logarithms
MaRDI profile type: Publication
full work available at URL: jnth.1996.0040
OpenAlex ID: W2084022171
DOI: 10.1006/JNTH.1996.0040
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description: scientific article; zbMATH DE number 874236