# Local and global canonical height functions for affine space regular automorphisms

## TL;DR

This paper constructs local and global canonical height functions for affine space automorphisms over number fields, providing tools to analyze the arithmetic dynamics of these automorphisms and their points.

## Contribution

It introduces v-adic Green functions for automorphisms and their inverses, and defines a notion of good reduction to establish a global canonical height with finiteness properties.

## Key findings

- Constructed v-adic Green functions for automorphisms and their inverses.
- Defined good reduction at places to build a global canonical height.
- Provided height growth estimates under automorphisms and their inverses.

## 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 the Northcott-type finiteness property. Using previous results, 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 is independently obtained by Lee by a different method.