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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
##### MathOverflow Questions:
Find a formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1$$
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##### References:
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