Special subvarieties of Drinfeld modular varieties
type: Item
instance of: scholarly article
title: Special subvarieties of Drinfeld modular varieties
zbMATH Open document ID: 1297.11057
DOI: 10.1515/CRELLE.2011.136
author: Florian Breuer
publication date: 2012-08-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0503452
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 6063234
zbMATH Keywords: Drinfeld modular varieties
zbMATH Keywords: Andre-Oort Conjecture
MaRDI profile type: Publication
arXiv ID: math/0503452
cites work: CM points on products of Drinfeld modular curves
cites work: Drinfeld modular polynomials in higher rank
cites work: Equidistribution of special subvarieties
cites work: Special points on the product of two modular curves
cites work: Special points on products of modular curves
cites work: Subvarieties of Shimura varieties
cites work: Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie
cites work: Q4222388
cites work: Minimal terminal \(\mathbb{Q}\)-factorial models of Drinfeld coarse moduli schemes
cites work: Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of ``infinite characteristics''
cites work: Special points on products of two Shimura curves
cites work: on a result of moonen on the moduli space of principally polarised abelian varieties
cites work: A conjecture of Yves André's
published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(S\) be a Shimura variety over \(\mathbb{C}\) and \(X \subset S\) an irreducible algebraic subvariety. The classical André-Oort conjecture for complex Shimura varieties claims that \(X(\mathbb{C})\) contains a Zariski-dense set of special points of \(S\) if and only if \(X\) is a subvariety of Hodge type. The paper under review tries to find analogous result in the Drinfeld module case (characteristic \(p\)).NEWLINENEWLINELet \(K\) be a global function field with exact field of constants \(\mathbb{F}_q\) and \(A\) be the Dedekind domain of elements of \(K\), regular away from \(\infty\). For any open subgroup \(\mathcal{K}\) of \(\mathrm{GL}_r(\hat{A})\), let \(M_A^r(\mathcal{K})\) be the moduli scheme over \(\mathrm{Spec}(A)\) (of rank \(r\)) of Drinfeld \(A\)-modules with \(\mathcal{K}\)-level structures. Let \(r\), \(r'\) be integers such that \(r' \mid r\). Let \(K'\) be a purely imaginary extension of degree \(r/r'\) and \(A'\) be the integral closure of \(A\) in \(K'\). An irreducible subvariety \(X \subset M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}}\) is said to be special if its image under the canonical projection \(M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}} \to M_A^r(1)_{\mathbb{C}_{\infty}}\) is an irreducible component of \(T_g(M_{A'}^{r'}(\mathcal{K'})_{\mathbb{C}_{\infty}})\) for some \(g \in \mathrm{GL}_r(\mathbb{A}_f)\), \(r'\) and \(A'\). \(T_g\) is the Hecke correspondence. The author made the following analogous conjecture of the André-Oort conjecture for Drinfeld modular varieties: \vskip 0.1in \noindent { Conjecture.} Let \(X \subset M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}}\) be an irreducible subvariety. Then \(X(\mathbb{C}_{\infty})\) contains a Zariski-dense set of CM points if and only if \(X\) is special. \vskip 0.1in The main results of the paper are the following two special cases of the conjecture \vskip 0.1in \noindent (1) Suppose that \(X(\mathbb{C}_{\infty})\) contains a Zariski-dense subset \(\Sigma\) of CM points satisfying the foll…
reviewed by: Xiao Xiao
description: scientific article; zbMATH DE number 6063234