Geometric properties of curves defined over number fields (Q2087686)

This work, having its origin in a lecture delivered by the first author at the Haverford College in 1991, is centred around the famous three point theorem of \textit{G. V. Belyĭ} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267--276 (1979; Zbl 0409.12012)]. The authors first prove that if a smooth complete algebraic curve \(X\) is defined over an algebraic number field, then there is a covering \(X\rightarrow{\mathbb{P}}^{1}\) ramified over a subset the three points set \(\{0, 1,\infty\}\), and remark that their method of proof leads to a short proof of the weak Hironaka resolution of singularities in characteristic zero, see [\textit{F. A. Bogomolov} and \textit{T. G. Pantev}, Math. Res. Lett. 3, No. 3, 299--307 (1996; Zbl 0869.14007)]. The authors give two proofs of the converse assertion in Belyi's theorem: a function-theoretical one and a shorter proof phrased in the language of schemes. In Part II of their work the authors ``discuss a number of ideas and problems which came up about the time that Belyi's original theorem was proved, in 1978''. Here is a list of the topics discussed in Part II: triangulations coming from unramified coverings; modular curves and curves ramified over three points; dominant classes of varieties; ramification over more than three points; faithful action of Gal \((\overline{\mathbb Q}\setminus\mathbb Q)\) on the profree group with two generators.