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arXiv · 0706.2166

Canonical heights and the arithmetic complexity of morphisms on projective space

Shu Kawaguchi·Joseph H. Silverman
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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.

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Shu Kawaguchi, Joseph H. Silverman. 2007-06-14. Canonical heights and the arithmetic complexity of morphisms on projective space. https://arxiv.org/abs/0706.2166

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