Integral points on subvarieties of semiabelian varieties. I
type: Item
instance of: scholarly article
title: Integral points on subvarieties of semiabelian varieties. I
zbMATH Open document ID: 1011.11040
published in: Inventiones Mathematicae
publication date: 1999-07-01T00:00:00Z
review text: We cite from the author's well-written introduction: Let \(X\) be a closed subvariety of an abelian variety \(A\), and assume that both are defined over some number field \(k\). Then a conjecture of \textit{S. Lang} [Publ. Math., Inst. Hautes Étud. Sci. 6, 319-335 (1960; Zbl 0112.13402)] states that the set of rational points is as small as one might reasonably expect: Theorem 0.1. The set \(X(k)\) is contained in a finite union \(\bigcup B_i(k)\), where each \(B_i\) is a translated abelian subvariety of \(A\) contained in \(X\). In [Ann. Math. (2) 133, 549-576 (1991; Zbl 0734.14007)] \textit{G. Faltings} proved this in the special case where \(X\times_k \overline{k}\) contains no nontrivial translated abelian subvarieties of \(A\times_k \overline{k}\); the conclusion in that case simplifies to the assertion that \(X(k)\) is finite. He proved this in general in [Perspect. Math. 15, 175-182 (1994; Zbl 0823.14009)]. This proof is also described in detail in [\textit{P. Vojta}, Lect. Notes Math. 1553, 164-208 (1993; Zbl 0846.14009)]; we follow the latter exposition closely here. In this paper we generalize Theorem 0.1 to cover the corresponding statement for integral points on closed subvarieties of semiabelian varieties: Theorem 0.2. Let \(k\) be a number field, with ring of integers \(R\). Let \(S\) be a finite set of places of \(k\), containing the set of Archimedean places, and let \(R_S\) be the localization of \(R\) away from (non-Archimedean) places in \(S\). Let \(X\) be a closed subvariety of a semiabelian variety \(A\); assume both are defined over \(k\). Let \({\mathcal X}\) be a model for \(X\) over \(\operatorname {Spec}R_S\). Then the set \({\mathcal X}(R_S)\) of \(R_S\)-valued points in \({\mathcal X}\) equals a finite union \(\bigcup{\mathcal B}_i(R_S)\), where each \({\mathcal B}_i\) is a subscheme of \({\mathcal X}\) whose generic fiber \(B_i\) is a translated semiabelian subvariety of \(A\). Another paper will address similar questions for certain op…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14K12
zbMATH DE Number: 931987
author: Paul Vojta
MaRDI profile type: Publication
full work available at URL: s002220050092
OpenAlex ID: W1978331323
DOI: 10.1007/S002220050092
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description: scientific article; zbMATH DE number 931987