#### Vol. 7, No. 5, 2013

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Local and global canonical height functions for affine space regular automorphisms

### Shu Kawaguchi

Vol. 7 (2013), No. 5, 1225–1252
##### Abstract

Let $f:{\mathbb{A}}^{N}\to {\mathbb{A}}^{N}$ 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