Let be a regular polynomial automorphism defined over a number field . For each place of , we construct the -adic Green functions and (i.e., the -adic canonical height functions) for and . Next we introduce for the notion of good reduction at , and using this notion, we show that the sum of -adic Green functions over all gives rise to a canonical height function for that satisfies a Northcott-type finiteness property. Using an earlier result, we recover results on arithmetic properties of -periodic points and non--periodic points. We also obtain an estimate of growth of heights under and , which was independently obtained by Lee by a different method.
Shu Kawaguchi. "Local and global canonical height functions for affine space regular automorphisms." Algebra Number Theory 7 (5) 1225 - 1252, 2013. https://doi.org/10.2140/ant.2013.7.1225