Local and global canonical height functions for affine space regular automorphisms

Kawaguchi, Shu

doi:10.2140/ant.2013.7.1225

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Local and global canonical height functions for affine space regular automorphisms

Shu Kawaguchi

Vol. 7 (2013), No. 5, 1225–1252

DOI: 10.2140/ant.2013.7.1225

Abstract

Let $f : A^{N} \to A^{N}$ be a regular polynomial automorphism defined over a number field $K$ . For each place $v$ of $K$ , we construct the $v$ -adic Green functions $G_{f, v}$ and $G_{f^{- 1}, v}$ (i.e., the $v$ -adic canonical height functions) for $f$ and $f^{- 1}$ . Next we introduce for $f$ the notion of good reduction at $v$ , and using this notion, we show that the sum of $v$ -adic Green functions over all $v$ gives rise to a canonical height function for $f$ that satisfies a Northcott-type finiteness property. Using an earlier result, we recover results on arithmetic properties of $f$ -periodic points and non- $f$ -periodic points. We also obtain an estimate of growth of heights under $f$ and $f^{- 1}$ , which was independently obtained by Lee by a different method.

In memory of Professor Masaki Maruyama

Keywords

canonical height, local canonical height, regular polynomial automorphism

Mathematical Subject Classification 2010

Primary: 37P30

Secondary: 11G50, 37P05, 37P20

Milestones

Received: 10 April 2012

Revised: 6 August 2012

Accepted: 4 September 2012

Published: 6 September 2013

Authors

Shu Kawaguchi
Department of Mathematics Kyoto University Oiwake-cho Kitashirakawa, Sakyo-ku Kyoto-shi 606-8502 Japan

Vol. 7, No. 5, 2013