The Galois representations associated to a Drinfeld module in special characteristic. III: Image of the group ring
type: Item
instance of: scholarly article
title: The Galois representations associated to a Drinfeld module in special characteristic. III: Image of the group ring
zbMATH Open document ID: 1173.11038
author: Richard Pink
author: Matthias Traulsen
published in: Journal of Number Theory
publication date: 2006-03-24T00:00:00Z
review text: This paper is the third in a series of three. For the other two see [\textit{R. Pink}, J. Number Theory 116, No. 2, 324--347 (2006; Zbl 1173.11036) and J. Number Theory 116, No. 2, 348--372 (2006; Zbl 1173.11037)]. Let \(k\) be a finite field of \(q\) elements, \(X/k\) a smooth and projective curve with exact field of constants \(k\), \(\infty\) a closed point on \(X\). Denote the function field of \(X\) by \(F\) and the ring of regular functions on \(X - \infty\) by \(A\subset F\). Let \(K\) be a finitely generated field over \(k\), of transcendence degree \(1\) and let \( K^s \) be a separable closure of \(K\). Let \(\phi\) be a Drinfeld \(A\)-module of rank \(r\) over \(K\). Let \(\lambda \subset A\) be a maximal ideal different from the characteristic of \(\phi\). Denote by \(T_\lambda(\phi)\) the \(\lambda\)-adic Tate module of \(\phi\). This is an \(A_\lambda \)-module free of rank \(r\) on which \(G := \text{Gal}(K^s/K)\) acts continuously. Assume that \( \phi \) has finite characteristic. Denote the ring of endomorphisms of \(\phi\) (over \(K\)) by \(E\). The following two theorems are shown in the present paper: Theorem A. Assume that \(E = A\). Then for almost all maximal ideals \(\lambda \) of \(A\) the \( A/\Lambda[ G ]\)-module \( \phi[\lambda] = T_\lambda( \phi ) \otimes A/\lambda\) is absolutely irreducible. Theorem B. For almost all maximal ideals \(\lambda \) of \(A\) the ring \(E\) and the image of \(A_\lambda[G]\) in \(\text{End}_{A_\lambda}(T_\lambda(\phi))\) are commutants of each other. These theorems have now be shown to hold for Drinfeld modules of arbitrary characteristic over finitely generated fields \(K\) of arbitrary transcendence degree by \textit{R. Pink} and \textit{E. Rütsche} [J. Number Theory 129, No. 4, 866--881 (2009; Zbl 1246.11121)].
reviewed by: Lenny Taelman
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 5015057
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: \(t\)-motives
zbMATH Keywords: Galois representation
zbMATH Keywords: Tate module
MaRDI profile type: Publication
full work available at URL: j.jnt.2005.04.012
OpenAlex ID: W2064432996
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DOI: 10.1016/J.JNT.2005.04.012
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description: scientific article; zbMATH DE number 5015057