Une borne pour l'action de l'inertie sauvage sur la torsion d'un module de Drinfeld. (A bound for the wild inertia action on the torsion of a Drinfeld module)
type: Item
instance of: scholarly article
title: Une borne pour l'action de l'inertie sauvage sur la torsion d'un module de Drinfeld. (A bound for the wild inertia action on the torsion of a Drinfeld module)
zbMATH Open document ID: 1019.11015
author: Francis Gardeyn
published in: Archiv der Mathematik
publication date: 2002-12-01T00:00:00Z
review text: The author gives an upper bound for the differents of Galois extensions of global function fields by torsion points of rational Drinfeld modules. Let \(K/\mathbb{F}_q(T)\) be a global function field and \(K\{\tau\}\) the skew polynomial ring with the commutation rule \(\tau c=c^q\tau\) for any \(c\in K\). Consider a rational Drinfeld module \(\phi:A=\mathbb{F}_q[T]\rightarrow K\{\tau\}\), \(a\mapsto\phi_a\), of rank \(r\). Such a module generates the Galois extension \(K[a]=K(\phi[a])\) for any \(a\in A\) where \(\phi[a]=\text{Ker}(\phi_a|K^s)\) is the finite \(A\)-module of \(a\)-torsion points. The author gives an upper bound for the different \(\mathfrak{D}(K[a]/K)\) in terms of the Galois structure of the \(a\)-torsion module \(\phi[a]\). Combining this result with an effective version of the Chebotarev density theorem, the author refines a theorem of \textit{Y. Taguchi} [Ramifications arising from Drinfeld modules, In: The arithmetic of function fields, de Gruyter, Berlin (1992; Zbl 0801.11027)] allowing for the efficient determination of whether two Drinfeld modules are isogenous.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1837409
zbMATH Keywords: Drinfeld module
zbMATH Keywords: Galois module structure
zbMATH Keywords: differents of Galois extensions
zbMATH Keywords: torsion points
zbMATH Keywords: wild inertia
zbMATH Keywords: upper bound
zbMATH Keywords: global function field
zbMATH Keywords: skew polynomial ring
zbMATH Keywords: Chebotarev density theorem
reviewed by: Igor Potemine
MaRDI profile type: Publication
full work available at URL: s00013-002-8310-5
OpenAlex ID: W2383653930
DOI: 10.1007/S00013-002-8310-5
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description: scientific article; zbMATH DE number 1837409