Local and global canonical height functions for affine space regular automorphisms

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.