Valuations, intersections and Belyi functions--some remarks on the construction of heights
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.
scientific article; zbMATH DE number 5173104
scientific article