Mordell exceptional locus for subvarieties of the additive group
type: Item
instance of: scholarly article
title: Mordell exceptional locus for subvarieties of the additive group
zbMATH Open document ID: 1178.11047
author: Dragos Ghioca
published in: Mathematical Research Letters
publication date: 2008-07-21T00:00:00Z
review text: Let \(X\) be an irreducible closed subvariety of a semiabelian variety \(G\) (an extension of an abelian variety by a torus) over an algebraically closed field \(K\). The Mordell exceptional locus \(Z(X) \subset X(K)\) is defined as the union of all translates of positive-dimensional algebraic subgroups of \(G\) that lie inside \(X\). \textit{D. Abramovich} [Compos. Math. 90, No. 1, 37--52 (1994; Zbl 0814.14041)] has shown that \(Z(X)\) is Zariski closed. In the present paper the author shows the analogous statement for subvarieties of a product of Drinfeld modules of generic characteristic. More precisely, let \(K\) be an algebraically closed field containing the polynomial ring \(A = \mathbb{F}_p[t] \) and \(\phi_1, \ldots, \phi_n : A \to \mathrm{End}(\mathbb{G}_{a,K}) \) a collection of Drinfeld modules of generic characteristic. Then \(\phi_1 \times \cdots \times \phi_n\) defines an \(A\)-module structure on \( E = \mathbb{G}_{a,K}^n \). Let \(V\) be a closed subvariety of \(E\). Define \( Z(V) \subset V\) to be the union of all translates of positive-dimensional algebraic sub-\(A\)-modules of \(E\) that lie inside \(V\), then it is shown that \(Z(V)\) is closed.
reviewed by: Lenny Taelman
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 14L10
Mathematics Subject Classification ID: 14K12
zbMATH DE Number: 5302012
zbMATH Keywords: Mordell exceptional locus
zbMATH Keywords: Drinfeld modules
MaRDI profile type: Publication
DOI: 10.4310/MRL.2008.V15.N1.A4
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description: scientific article; zbMATH DE number 5302012