Some remarks on rational periodic points
type: Item
instance of: scholarly article
title: Some remarks on rational periodic points
zbMATH Open document ID: 0959.14006
author: Shu Kawaguchi
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: Let \(M\) be a finitely generated extension of \(\mathbb{Q}\) and \(X\) an algebraic variety defined over \(M\). Given a finite extension \(K/M\) and a dominant morphism \(f:X\to X\) defined over \(K\), the set \(X(K)_{\text{per},f}\) of periodic \(K\)-points with respect to \(f\) is the set of points \(p\in X(K)\) such that \(f^n(p)=p\) for some positive integer \(n\). The variety \(X\) is called periodically finite if \(X(K)_{\text{per},f}\) is finite for every finite extension \(K/M\) and every dominant \(K\)-morphism \(f:X\to X\) of degree \(\geq 2\). This paper is devoted to study the varieties which are periodically finite. The results over height functions proved by \textit{J. H. Silverman} [in: Arithmetic Geometry, Pap. Conf., Storrs/Conn. 1984, 151-166 (1986; Zbl 0604.14022)] allow the author to prove the following result: If \(X\) is a geometrically irreducible, normal projective variety over \(M\) whose Picard number is 1, then \(X\) is periodically finite. The last sections of the paper are devoted to apply this result to different classes of algebraic varieties. Concretely, there are proved characterizations of periodically finite abelian varieties and periodically finite smooth projective surfaces with non-negative Kodaira dimension.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 1489537
zbMATH Keywords: rational point
zbMATH Keywords: periodically finite variety
zbMATH Keywords: periodic point
zbMATH Keywords: height functions
zbMATH Keywords: periodically finite abelian varieties
reviewed by: J. M. Muñoz Porras
MaRDI profile type: Publication
DOI: 10.4310/MRL.1999.V6.N5.A3
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description: scientific article; zbMATH DE number 1489537