Local and global canonical height functions for affine space regular automorphisms

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

2009-09-19

arxiv.org/abs/0909.3573v1

Mathematics

Algebraic Geometry

23 pages

Scientific paper

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.

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