Geometric properties of curves defined over number fields
type: Item
instance of: scholarly article
title: Geometric properties of curves defined over number fields
zbMATH Open document ID: 1506.14026
published in: European Journal of Mathematics
publication date: 2022-10-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14D10
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 7604890
zbMATH Keywords: algebraic curve
zbMATH Keywords: number field
zbMATH Keywords: triangulation
author: Fedor Bogomolov
author: Dale H. Husemoller
reviewed by: B. Z. Moroz
MaRDI profile type: Publication
full work available at URL: s40879-021-00497-2
OpenAlex ID: W3197637000
cites work: ON GALOIS EXTENSIONS OF A MAXIMAL CYCLOTOMIC FIELD
cites work: Torsion of elliptic curves and unlikely intersections
cites work: Weak Hironaka theorem
cites work: On Contraction of Algebraic Points
cites work: Couniformization of curves over number fields
cites work: Dominant classes of projective varieties
cites work: Q3977509
cites work: Q3351551
cites work: Q4006891
cites work: Q4364755
cites work: Q3211418
cites work: The red book of varieties and schemes. Includes the Michigan lectures (1974) on ``Curves and their Jacobians''.
cites work: Q4303932
cites work: Q4346719
cites work: Q4325465
cites work: Q3972767
DOI: 10.1007/S40879-021-00497-2
Recommended article: Belyi's theorem revisited
Recommended article: Three point covers with bad reduction
Recommended article: Unifying themes suggested by Belyi’s Theorem
Recommended article: Stable reduction of three point covers
Recommended article: Computation and rationality of Belyi functions in genus zero
Recommended article: On explicit descent of marked curves and maps
Recommended article: Another proof of the three points theorem
Recommended article: Q3815410
Recommended article: On triple curves through a rational triple point of a surface
Recommended article: Q3488290
description: scientific article; zbMATH DE number 7604890