Canonical heights and Drinfeld modules
type: Item
instance of: scholarly article
title: Canonical heights and Drinfeld modules
zbMATH Open document ID: 0764.11027
DOI: 10.1007/BF01934322
author: Laurent Denis
published in: Mathematische Annalen
publication date: 1993-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/164998
review text: Let \(A\) be the polynomial ring \(\mathbb{F}_ q[t]\) with quotient field \(k\), and \(\bar k\) its algebraic closure. In the paper, two principal results about Drinfeld \(A\)-modules and \(t\)-modules (= G. Anderson's generalization of Drinfeld modules) are obtained: Theorem 1: If \(\varphi\) is a \(t\)-module over \(\bar k\) of dimension \(N\), there exists a canonical height function \(\hat h\) on \((\mathbb{G}_ a)^ N(\bar k)\) with properties (1)\dots(8) similar to the Néron-Tate height on an abelian variety. Theorem 2: Let \(C\) be the Carlitz module. There exists a real constant \(\eta\) (effectively computable, depending only on \(q\)) such that for separable points \(\alpha\) of \(C\) of degree \(\delta\) the following estimate holds: \[ \hat h(\alpha)\geq \eta\delta^{-1}(\log \delta/\log\log \delta)^{-3}. \] The last theorem is similar to results of Dobrowolski and Laurent about the multiplicative group and elliptic curves with complex multiplication, respectively.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 86198
zbMATH Keywords: \(t\)-modules
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: canonical height function
zbMATH Keywords: Carlitz module
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
cites work: \(t\)-motives
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OpenAlex ID: W2064160983
description: scientific article; zbMATH DE number 86198