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| In [[mathematics]], '''Belyi's theorem''' on [[algebraic curve]]s states that any [[non-singular]] algebraic curve ''C'', defined by [[algebraic number]] coefficients, represents a [[compact Riemann surface]] which is a [[ramified covering]] of the [[Riemann sphere]], ramified at three points only.
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| This is a result of [[G. V. Belyi]] from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of [[Dessin d'enfant|dessins d'enfant]], which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.
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| == Quotients of the upper half-plane ==
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| It follows that the Riemann surface in question can be taken to be
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| :''H''/Γ
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| with ''H'' the [[upper half-plane]] and Γ of [[finite index]] in the [[modular group]], compactified by [[cusp (singularity)|cusps]]. Since the modular group has [[non-congruence subgroup]]s, it is ''not'' the conclusion that any such curve is a [[modular curve]].
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| == Belyi functions ==
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| A '''Belyi function''' is a [[holomorphic map]] from a compact Riemann surface ''S'' to the [[complex projective line]] '''P'''<sup>1</sup>('''C''') ramified only over three points, which after a [[Möbius transformation]] may be taken to be <math> \{0, 1, \infty\} </math>. Belyi functions may be described combinatorially by [[Dessin_d'enfant|dessins d'enfants]].
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| Belyi functions and dessins d'enfants – but not Belyi's theorem – date at least to the work of [[Felix Klein]]; he used them in his {{Harv|Klein|1879}} to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).<ref>{{citation
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| | last = le Bruyn | first = Lieven
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| | title = Klein’s dessins d’enfant and the buckyball
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| | year = 2008
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| | url = http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball.html}}.</ref>
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| == Applications ==
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| Belyi's theorem is an [[existence theorem]] for Belyi functions, and has subsequently been much used in the [[inverse Galois problem]].
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *[[J.-P. Serre|Serre, J.-P.]] (1989), ''Lectures on the Mordell-Weil Theorem'', p. 71
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| *{{cite doi|10.1007/BF02086276}}
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| *{{cite doi|10.1070/IM1980v014n02ABEH001096}}
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| {{refend}}
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| ==Further reading==
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| * {{citation | last1=Girondo | first1=Ernesto | last2=González-Diez | first2=Gabino | title=Introduction to compact Riemann surfaces and dessins d'enfants | series=London Mathematical Society Student Texts | volume=79 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-74022-7 | zbl=1253.30001 }}
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| * {{citation | editor1=Dorian Goldfeld | editor2=Jay Jorgenson | editor3=Peter Jones | editor4=Dinakar Ramakrishnan | editor5=Kenneth A. Ribet | editor6=John Tate | title=Number Theory, Analysis and Geometry. In Memory of Serge Lang | publisher=Springer | year=2012 | isbn=978-1-4614-1259-5 | author=Wushi Goldring | chapter=Unifying themes suggested by Belyi's Theorem | pages=181–214 }}
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| [[Category:Algebraic curves]]
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| [[Category:Theorems in algebraic geometry]]
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