Canonical heights and the arithmetic complexity of morphisms on projective space

TL;DR

This paper introduces a new measure of arithmetic distance between morphisms on projective space, relates it to height functions, and establishes finiteness results for morphisms within bounded distance over number fields.

Contribution

It defines the arithmetic distance d(F,G) between morphisms and proves comparison theorems, showing finiteness of morphisms within bounded distance over number fields.

Findings

The arithmetic distance d(F,G) is well-defined and relates to naive height functions.

For fixed G, morphisms F with d(F,G)<B form a set of bounded height.

There are finitely many such F over any given number field.

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.