Canonical heights and the arithmetic complexity of morphisms on projective space

Details Canonical heights and the arithmetic complexity of morphisms on projective space Canonical heights and the arithmetic complexity of morphisms on projective space

2007-06-14

arxiv.org/abs/0706.2166v1

Pure and Applied Mathematics Quarterly 5 (2009), 1201--1217

Mathematics

Number Theory

submitted to the Quarterly Journal of Pure and Applied Mathematics

Scientific paper

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.

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