Théorème de Baker et modules de Drinfeld. (Baker's theorem and Drinfeld modules)
type: Item
instance of: scholarly article
title: Théorème de Baker et modules de Drinfeld. (Baker's theorem and Drinfeld modules)
zbMATH Open document ID: 0767.11029
author: Laurent Denis
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: In Duke Math. J. 58, 559-575 (1989; Zbl 0687.12008) \textit{Jing Yu} established the following result: Let \(\mathbb{F}_ q\) be the field of \(q\)- elements, \({\mathcal C}\) a smooth projective geometrically irreducible curve over \(\mathbb{F}_ q\) and \(\infty\) a fixed close point. We set \(A\) to be the affine ring of \({\mathcal C}-\infty\). Let \(k\) be the function field of \({\mathcal C}\) with completion \(k_ \infty\) at \(\infty\). Let \(\overline{k}_ \infty\) be a fixed algebraic closure of \(k_ \infty\) with \(\overline{k}\subset k_ \infty\) being the algebraic closure of \(k\). Let \(\varphi\) be a Drinfeld module defined over \(\overline{k}\) associated to the \(A\)-lattice \(L\subset\overline{k}_ \infty\). Finally, we let \(e_ L(z)\) be the exponential function of \(\varphi\) and \(k_ L\) the field of multiplications of \(\varphi\). Theorem: Let \(\alpha_ 1,\dots,\alpha_ n\) be elements of \(\overline{k}_ \infty\) such that \(e_ L(\alpha_ i)\in\overline{k}\) for all \(i\). If \(\alpha_ 1,\dots,\alpha_ n\) are linearly independent over \(k_ L\), then any linear combination of \(\alpha_ 1,\dots,\alpha_ n\), with coefficients not all zero from the separable closure \(k_ L^ s\) of \(k_ L\), is transcendental over \(k\). Yu's result is an analog of the qualitative version of Baker's theorem. It is refined in the present paper when \(\varphi\) has complex multiplication. The author then strengthens the theorem by replacing \(k_ L^ s\) with \(\overline{k}\).
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 166069
zbMATH Keywords: transcendence
zbMATH Keywords: zero estimate for \(T\)-module
zbMATH Keywords: Drinfeld module
zbMATH Keywords: complex multiplication
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1993.1018
OpenAlex ID: W2030213857
DOI: 10.1006/JNTH.1993.1018
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description: scientific article; zbMATH DE number 166069