Canonical heights and the arithmetic complexity of morphisms on projective space
Number Theory
2011-05-30 v1 Dynamical Systems
Abstract
Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to the morphisms F and G, respectively. We prove comparison theorems relating d(F,G) to more naive height functions and show that for a fixed G, the set of F satisfying d(F,G) < B is a set of bounded height. In particular, there are only finitely many such F defined over any given number field.
Keywords
Cite
@article{arxiv.0706.2166,
title = {Canonical heights and the arithmetic complexity of morphisms on projective space},
author = {Shu Kawaguchi and Joseph H. Silverman},
journal= {arXiv preprint arXiv:0706.2166},
year = {2011}
}