On some characteristic polynomials attached to finite Drinfeld modules
type: Item
instance of: scholarly article
title: On some characteristic polynomials attached to finite Drinfeld modules
zbMATH Open document ID: 0880.11047
DOI: 10.1007/BF02677478
published in: Manuscripta Mathematica
publication date: 1997-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/156306
review text: Let \(K\) be a global function field over \(\mathbb{F}_q\) and \(A\) the Dedekind subring of elements regular away from a fixed place \(\infty\) of \(K\). The paper deals with questions about Drinfeld \(A\)-modules \(\varphi\) defined over a finite field \(L\) provided with a structure as an \(A\)-algebra. After recalling and discussing some results about the endomorphism ring \(\text{End} (\varphi)\) and the ``de Rham cohomology'' \(H^*_{\text{DR}} (\varphi,L)\) of \(\varphi\), the ``crystalline cohomology'' \(H^*_{\text{crys}} (\varphi,L)\) is introduced, which is a free module of \(\text{rank} r= \text{rank} (\varphi)\) over the relative Witt ring \(W(L)\) of \(L\) with respect to \(A\). It is then shown (theorem 3.2) that for each \(u\in \text{End} (\varphi)\), the characteristic polynomial of \(u\) with respect to \(K\) agrees with the characteristic polynomial of \(H^*_{\text{crys}} (u,L)\) over \(W(L)\). For purely inseparable \(u\) (e.g., a power of the Frobenius), some more detailed assertions are proved (Theorems 4.2, 4.4, 4.6).
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 1038298
zbMATH Keywords: finite Drinfeld modules
zbMATH Keywords: endomorphism rings
zbMATH Keywords: Drinfeld \(A\)-modules
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: characteristic polynomial
author: Bruno Anglès
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
cites work: On some subrings of Ore polynomials connected with finite Drinfeld modules
cites work: Q3338360
cites work: ELLIPTIC MODULES
cites work: On finite Drinfeld modules
cites work: Q4061092
description: scientific article; zbMATH DE number 1038298