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Canonical heights on projective space. (English) Zbl 0895.14006
Let \(K\) be a number field, and let \(\phi\:\mathbb P^N\to\mathbb P^n\) be a morphism of degree \(d>1\) defined over \(K\). By the theory of canonical heights on varieties relative to an endomorphism as developed by G. S. Call and J. H. Silverman [Compos. Math. 89, No. 2, 163-205 (1993; Zbl 0826.14015)], there is a unique Weil height \(\widehat h\) for \(\mathcal O(1)\) on \(\mathbb P^N\) such that \(\widehat h(\phi(P))=dh(P)\) for all \(P\in\mathbb P^N(\overline{K})\). The present paper shows that this canonical height \(\widehat h(P)\) can be expressed as a sum (over places of \(K(P)\)) of canonical local heights. For non-archimedean places of good reduction for \(\phi\), the local height is shown to be given by a simple formula; for the finitely many remaining places, some series and sequence formulas are given. Additional results are given in the case where there is a hyperplane \(W\) such that \(\phi^{*}(W)=(\deg\phi)W\), and also in the special case of \(\mathbb P^1\). The paper concludes by using the local canonical heights on \(\mathbb P^1\) to define a “\(v\)-adic filled Julia set” of \(\phi\), and to show that a point \(P\in K\) is pre-periodic for \(\phi\) if and only if it lies in the \(v\)-adic filled Julia set for all \(v\).
Reviewer: P.Vojta (Berkeley)

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G25 Global ground fields in algebraic geometry
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