Lower bounds of linear forms in logarithms for Drinfeld modules
type: Item
instance of: scholarly article
title: Lower bounds of linear forms in logarithms for Drinfeld modules
zbMATH Open document ID: 0922.11062
author: Vincent Bosser
published in: Journal of Number Theory
publication date: 1999-06-02T00:00:00Z
review text: Let \(k=\mathbb F_q(T)\) be the field of rational functions over the finite field \(\mathbb F_q\). Put \(k_\infty=\mathbb F_q((1/T))\) and denote by \(C\) the completion of the algebraic closure of \(k_\infty\) and by \(K \subset C\) a finite extension of \(k\) of degree \(D\). The author considers \(n\) Drinfeld modules \((G_a,\Phi_i)\), \(1\leq i\leq n\), defined over \(K\) and of positive rank, with exponentials \(e_{\Phi_i}\). He considers a linear form of logarithms: \[ \Lambda =\beta _0+\beta _1u_1+\ldots+\beta _nu_n, \] where \(\beta _0\), \dots, \(\beta _n\in K\), and \(u_1\), \dots, \(u_n\in C\) are such that \(e_{\Phi_i}(u_i)\in K\) for \(i=1\), \dots, \(n\) (in other words \(u_1\), \dots, \(u_n\) are ``logarithms''). When \(\Lambda \not=0\) he obtains a lower bound for \(\Lambda \) in terms of the degree \(D\), the heights of the \(\beta \)'s, the absolute values of \(u_1\), \dots, \(u_n\), the heights of the \(e_{\Phi_i}(u_i)\) and the heights of the Drinfeld modules \((G_a,\Phi_i)\). This result is effective and explicit, except for some computable constant. This is the analogue for Drinfeld modules of results obtained in the case of complex algebraic groups by N. Hirata and S. David. The estimate is almost the best possible which could be expected when compared to the complex case.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 1275047
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: Baker's theory
MaRDI profile type: Publication
full work available at URL: jnth.1998.2336
OpenAlex ID: W2042141124
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DOI: 10.1006/JNTH.1998.2336
description: scientific article; zbMATH DE number 1275047