On primitive roots for rank one Drinfeld modules
type: Item
instance of: scholarly article
title: On primitive roots for rank one Drinfeld modules
zbMATH Open document ID: 1233.11065
author: Wei-Chen Yao
author: Jing Yu
published in: Journal of Number Theory
publication date: 2010-01-08T00:00:00Z
review text: Let \(k\) be the function field of a smooth projective geometrically irreducible curve \(X\) over a finite field, \(\infty\) a closed point on \(X\) and \(A\) the ring of functions of the complement \(X-\infty\). Let \(K\) be an \(A\)-field of finite characteristic (that is, the map \(A\to K\) is not injective). Assume that \(K\) is a global function field. Let \(\phi\) be a Drinfeld module of rank one over \(K\) and \(\alpha \in K\). For all but finitely many places \(v\) of \(K\) one may reduce the pair \((\phi, \alpha)\) modulo \(v\) and ask if the reduced point \(\alpha \mod v\) generates the module of rational points of \(\phi \mod v\). If this happens, we call \(\alpha\) a primitive root modulo \(v\) (by analogy with primitive roots for the multiplicative group). In this paper, it is shown that the set of \(v\) so that \(\alpha\) is a primitive root modulo \(v\) has a density. They also give a criterion for this density to be positive, based on \textit{H. W. Lenstra} jun.'s approach [Invent. Math. 42, 201--224 (1977; Zbl 0362.12012)].
reviewed by: Lenny Taelman
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 5655001
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Artin conjecture
zbMATH Keywords: primitive roots
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.07.003
OpenAlex ID: W2023621841
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DOI: 10.1016/J.JNT.2009.07.003
description: scientific article; zbMATH DE number 5655001