Zur Arithmetik von Drinfeld-Moduln
type: Item
instance of: scholarly article
title: Zur Arithmetik von Drinfeld-Moduln
zbMATH Open document ID: 0536.14028
DOI: 10.1007/BF01455309
author: Ernst-Ulrich Gekeler
published in: Mathematische Annalen
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163698
review text: The theory of modular curves over global function fields has been developed by \textit{V. G. Drinfel'd} in his paper ''Elliptic modules'' [Math. USSR, Sb. 23 (1974), 561-592 (1976); translation from Mat. Sb. Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]. In the paper under review, Drinfeld's elliptic modules are referred as Drinfeld modules, and endomorphism rings of Drinfeld modules are extensively investigated. First, Drinfeld modules are defined: Algebraic as well as analytic descriptions are given. Let K be the global function field over a finite field \({\mathbb{F}}_ q\) of characteristic p, \(\infty\) a fixed place, A the ring of functions integral outside \(\infty\), \(K_{\infty}\) the completion of K with respect to the absolute value \(||_{\infty}=||\), \(\bar K_{\infty}\) the algebraic closure of \(K_{\infty}\) and \(C=\hat{\bar K}_{\infty}\) the completion of \(\bar K_{\infty}\). Let L be a field with A-algebra structure \(\gamma:A\to L.\) Let \({\mathbb{G}}_ a\) denote the additive group scheme, and let \(End_ L({\mathbb{G}}_ a)=:L\{\tau \}\) denote the ring of non- commutative polynomials over L in one variable \(\tau\) with the commutation relation \(f^ p\tau =\tau f\) (\(f\in L).\) Algebraic description: A ''Drinfeld module over L'' of rank \(r\in {\mathbb{N}}\) is a ring homomorphism \(\phi:A\to End_ L({\mathbb{G}}_ a)\) with the following properties: (a) for all \(a\in A\), \(\phi(a)=\phi_ a\) is an additive polynomial with linear coefficient \(\gamma(a)\), and (b) for all \(a\in A\), \(\| a\| =:\deg \phi_ a=| a|^ r.\) A ''homomorphism'' of Drinfeld modules \(\phi\to \psi\) is an element \(u\in End_ L({\mathbb{G}}_ a)\) such that for all \(a\in A\), \(u{\mathbb{O}}\phi_ a=\psi_ a{\mathbb{O}}u\); and u is called an ''isomorphism'' if \(u\in L^*\). \(\phi\) has the following properties: Let \(End(\phi)=End_ L(\phi)=\{u\in End_ L({\mathbb{G}}_ a)| \quad u{\mathbb{O}}\phi =\phi {\mathbb{O}}u\}.\) Then En\(d(\phi)\) is a projective A-module of rank …
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3851273
zbMATH Keywords: endomorphism ring
zbMATH Keywords: stable reduction
zbMATH Keywords: good reduction
zbMATH Keywords: Tate parametrization
zbMATH Keywords: supersingular Drinfeld modules
zbMATH Keywords: Hasse invariant
reviewed by: Noriko Yui
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3851273