Diophantine geometry of the torsion of a Drinfeld module
type: Item
instance of: scholarly article
title: Diophantine geometry of the torsion of a Drinfeld module
zbMATH Open document ID: 1055.11037
published in: Journal of Number Theory
publication date: 2003-04-02T00:00:00Z
review text: It has been conjectured by Denis that certain qualitative Diophantine results valid for subgroups of semi-Abelian varieties might be extended to Drinfeld modules. In particular, according to Denis, this should be the case of the Manin-Mumford conjecture saying that an irreducible subvariety of a semi-Abelian variety containing a Zariski dense set of torsion points is a translate of a semialgebraic group, unless replacing \` \` semi-Abelian variety'' by \` \` power of the additive group considered as an \({\mathbb F}_p[t]\)-module via a Drinfeld module''; actually Denis permits finite extensions of \({\mathbb F}_p [t]\) but requires that the Drinfeld module have generic characteristic. The paper under review proves a version of this conjecture. Let us sketch its setting. For \(q\) a power of some prime, a smooth absolutely irreducible projective curve \(C\) over \({\mathbb F}_q\) is fixed, as well as a closed point \(\infty\), and the ring \({\mathbb A}\) of regular functions on \(C - \{ \infty \}\) is considered. Let \(G_a\) be the additive group scheme \(\text{Spec\,} {\mathbb Z}[X]\). An \({\mathbb A}\)-field is an \({\mathbb F}_q\)-morphism \(\iota\) from \({\mathbb A}\) to a field \(K\), and a Drinfeld module (over \(\iota\)) is an \({\mathbb F}_q\)-algebra homomorphism \(\varphi\) from \({\mathbb A}\) to \(\text{End}_K \, G_a\) such that \(\varphi\) and \(\iota\) have the same differential and \(\varphi({\mathbb A})\) is not contained in \(K\) (as canonically embedded into \(\text{End}_K \, G_a\)). The characteristic of \(\varphi\) is the kernel of \(\iota\), and \(\varphi\) is said to have generic characteristic if this ideal is \((0)\). For \(N\) a positive integer, \(K^N\) can be regarded as an \({\mathbb A}\)-module via \(\varphi\). The \(\varphi\)-torsion group of \(\varphi_{\text{tor}} (K^N)\) of \(K^N\) consists of the elements \(x \in K^N\) for which \(\varphi(b)(x) = 0\) for some non-zero \(b \in {\mathbb A}\). As said, the main theorem of the paper is…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 1891424
zbMATH Keywords: Manin-Mumford conjecture
zbMATH Keywords: Drinfeld module
zbMATH Keywords: difference field
author: T. J. Scanlon
reviewed by: Carlo Toffalori
MaRDI profile type: Publication
OpenAlex ID: W1994588630
cites work: Sous–variétés algébriques de variétés semi–abéliennes sur un corps fini
cites work: Model theory of difference fields
cites work: MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS
cites work: The Manin-Mumford conjecture and the model theory of difference fields
cites work: Q4714153
cites work: p-adic distance from torsion points of semi-abelian varieties
cites work: Difference algebraic subgroups of commutative algebraic groups over finite fields
DOI: 10.1006/JNTH.2001.2751
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description: scientific article; zbMATH DE number 1891424