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Canonical heights on varieties with morphisms. (English) Zbl 0826.14015
Let \(V\) be an algebraic variety over a number field equipped with a morphism \(\varphi : V \to V\). Suppose there is a divisor class \(\eta\) of \(V\) with the property that \(\varphi^* (\eta) = \alpha \eta\) for some \(\alpha \in \mathbb{R}\) with \(\alpha > 1\). In these circumstances the authors construct a canonical height function \(\widehat h : V (\overline K) \to \mathbb{R}\) which is characterized by the fact that it is a Weil height function for the class \(\eta\) and by the property: \(\widehat h (\varphi P) = \alpha \widehat h (P)\) for all \(P \in V (\widehat K)\). This construction generalizes the well-known Néron-Tate height on abelian varieties and the recently constructed canonical height for certain \(K_3\)-surfaces by J. H. Silverman [Invent. Math. 105, No. 2, 347-373 (1991; Zbl 0754.14023)].
In the rest of the paper the authors develop a theory of their canonical height functions. Following Néron, the authors show how to decompose their height function as a sum of local height functions. They estimate the difference between their height and any given Weil height à la Demjanenko and Zimmer. They study the behaviour of the height functions in a family \({\mathcal V} \to T\) parametrized by a curve \(T\). Following Tate the authors also exhibit rapidly converging series that might be used to explicitly compute their height functions. Finally the authors relate their non-archimedean local height pairings to intersection theory.
Reviewer: R.Schoof (Roma)

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