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Belyi’s theorem in characteristic two

Part of: Arithmetic algebraic geometry Curves Arithmetic problems. Diophantine 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
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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

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