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

Let f : AN AN be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions Gf,v and Gf1,v (i.e., the v-adic canonical height functions) for f and f1. 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 f1, which was independently obtained by Lee by a different method.

In memory of Professor Masaki Maruyama

canonical height, local canonical height, regular polynomial automorphism
Mathematical Subject Classification 2010
Primary: 37P30
Secondary: 11G50, 37P05, 37P20
Received: 10 April 2012
Revised: 6 August 2012
Accepted: 4 September 2012
Published: 6 September 2013
Shu Kawaguchi
Department of Mathematics
Kyoto University
Oiwake-cho Kitashirakawa, Sakyo-ku
Kyoto-shi 606-8502