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Valuations, intersections and Belyi functions–some remarks on the construction of heights. (Valuations, intersections et fonctions de Belyi–quelques remarques sur la construction des hauteurs.) (French) Zbl 1142.14014

This note discusses several approaches to defining the height of a smooth projective curve \(X\) defined over a number field \(K\). The definitions discussed include the Néron-Tate height \(\widehat h(X)\) of \(X\) viewed as a subvariety of its Jacobian; the self-intersection of the relative dualizing sheaf with admissible metric; the least upper bound on the set of \(\epsilon>0\) for which the set \(\{P\in X(\overline K):\widehat h(\phi_{D_0}(P))\leq\epsilon\}\) is finite (from the Bogomolov conjecture); and the minimal degree of a “Belyĭ map” \(X\to\mathbb P^1\) unramified outside of \(0\), \(1\), and \(\infty\).
Comparison theorems are stated, and references are given for the proofs.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H25 Arithmetic ground fields for curves
14H30 Coverings of curves, fundamental group
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