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Abstract
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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.
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In memory of Professor Masaki
Maruyama
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Keywords
canonical height, local canonical height, regular
polynomial automorphism
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Mathematical Subject Classification 2010
Primary: 37P30
Secondary: 11G50, 37P05, 37P20
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Milestones
Received: 10 April 2012
Revised: 6 August 2012
Accepted: 4 September 2012
Published: 6 September 2013
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