高次元における軌道の数論的性質について (Algebraic Number Theory and Related Topics 2013)

Abstract

This note is an overview of recent extensions to higher-dimensions of Silverman s result on finiteness of S-integral points in orbits of rational functions. These extensions show that when a self-map $phi$ on mathbb{P}^{N} and a divisor D satisfy certain geometric conditions, the orbit points whose global heights come largely from local heights for S are Zariski-non-dense. The main results are obtained under assuming a deep Diophantine conjecture of Vojta, while we outline many specific examples for which this conjecture is unnecessary. The geometric conditions change depending on whether $phi$ is a morphism or merely a rational map.