Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-31T01:49:27.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Belyi’s theorem in characteristic two

Part of: Arithmetic problems. Diophantine geometry Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  18 December 2019

Yusuke Sugiyama  and
Seidai Yasuda
Yusuke Sugiyama
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka 560-0043, Japan email u.sugiyama.0811@gmail.com
Seidai Yasuda
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka 560-0043, Japan email s-yasuda@math.sci.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

Keywords

Belyi’s theoremalgebraic curvescharacteristic twotame covering

MSC classification

Primary: 11G20: Curves over finite and local fields
Secondary: 11G32: Dessins d'enfants, Belyĭ theory 14G17: Positive characteristic ground fields 14H05: Algebraic functions; function fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 325 - 339
Copyright
© The Authors 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anbar, N. and Tutdere, S., On Belyi’s theorems in positive characteristic, Preprint (2018), arXiv:1811.00773.Google Scholar
Belyi, G., Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276.Google Scholar
Fulton, W., Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542–575.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, Berlin, 1977).CrossRefGoogle Scholar
Hoshi, Y., Frobenius-projective structures on curves in positive characteristic, Preprint (2017), available online at http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1871.pdf.Google Scholar
Raynaud, M., Sections des fibres vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.CrossRefGoogle Scholar
Saïdi, M., Revêtements modérés et groupe fondamental de graphe de groupes, Compos. Math. 107 (1997), 319–338.CrossRefGoogle Scholar
Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, Berlin, 1979).CrossRefGoogle Scholar
Shafarevich, I. R., Basic algebraic geometry 1, third edition (Springer, Berlin, 2010).Google Scholar