Diophantine problems over \(t\)-modules
type: Item
instance of: scholarly article
title: Diophantine problems over \(t\)-modules
zbMATH Open document ID: 0842.11026
author: Laurent Denis
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/232828
full work available at URL: jtnb7-1.html
review text: Let \({\mathbf A}= \mathbb{F}_q [T]\) and let \(C\) be the Carlitz module defined over \({\mathbf A}\). Let \(C^{\otimes n}\) be the \(n\)-th tensor power of \(C\). It is by now well known that \(C\) plays the role of the multiplicative group \(\mathbb{G}_m\) for \({\mathbf A}\) and that its division values generate analogs of classical cyclotomic fields. Thus it makes sense to inquire if one can define a ``Fermat equation'' associated to \(C\); in fact, this is easily done by noting that \(x^n- y^n= y^n (({x \over y})^n -1)\) and that the roots of \(u^n- 1\) \((u:= x/y)\) are the \(n\) division points of \(\mathbb{G}_m\). The fields generated by the division values of \(C\) are actually geometric extensions of \({\mathbf k}\) (i.e., they contain no constant field extensions). Thus one refers to these fields as the geometric cyclotomic extensions, and the above Fermat equation as the geometric Fermat equation. The constant field extensions, which after all are obtained from roots of unity, are called the arithmetic cyclotomic extensions, and the classical Fermat equation, viewed over \({\mathbf A}\), is referred to as the arithmetic Fermat equation. In a striking change from the classical theory, it turns out that the geometric Fermat equations are amenable to a geometric attrack [see \textit{L. Denis}, Trans. Am. Math. Soc. 343, 713-726 (1994; Zbl 0811.11044)]. As one associates to \(C\) a Fermat equation, so too one can attach ``Fermat families'' to \(C^{\otimes n}\) (see \textit{L. Denis}, loc. cit.). In the paper being reviewed the author establishes that, under certain very general hypotheses, one has finitely many relatively prime solutions to these families. This is established through results of the author on heights. Various other results of interest are presented, such as estimates relating the canonical height and the Weil height. Finally, a number of open problems in the diophantine geometry of Drinfeld modules are stated.
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 858017
zbMATH Keywords: Fermat families
zbMATH Keywords: geometric Fermat equations
zbMATH Keywords: Carlitz module
zbMATH Keywords: canonical height
zbMATH Keywords: Weil height
zbMATH Keywords: open problems
reviewed by: David Goss
MaRDI profile type: Publication
OpenAlex ID: W2323385177
cites work: \(t\)-motives
cites work: Tensor powers of the Carlitz module and zeta values
cites work: Canonical heights and Drinfeld modules
cites work: Le Theoreme de Fermat-Goss
cites work: Q3134968
cites work: Q4854042
cites work: ELLIPTIC MODULES
cites work: On a Fermat equation arising in the arithmetic theory of functions fields
cites work: Harmonic analysis and the flow of a Drinfeld module
cites work: Explicit Class Field Theory for Rational Function Fields
cites work: Q4010746
cites work: Q3309989
cites work: Transcendence and special zeta values in characteristic p
cites work: On the difference of the Weil height and the Neron-Tate height
DOI: 10.5802/JTNB.133
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description: scientific article; zbMATH DE number 858017