Some remarks on rational periodic points (Q1574758)

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.