Belyi's theorem revisited (Q1875920)

In 1979, \textit{G.V. Belyi} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267--276 (1979; Zbl 0409.12012)] proved the ``Three point theorem'': a complex smooth projective curve \(X\) is defined over a number field, if and only if there exists a non-constant morphism \(t: X \to \mathbb P_{\mathbb C}^1\) with at most 3 critical values. The paper under review gives a self-contained and elementary proof of this theorem, making use of the moduli field of a covering. Furthermore, the author gives an explicit bound for \([K:\mathbb Q]\), depending on the degree of the morphism \(t\), such that \(X\) and \(t\) are defined over \(K\).