Belyi's theorem#Belyi functions
{{short description|Connects non-singular algebraic curves with compact Riemann surfaces}}
In mathematics, Belyi's theorem on algebraic curves 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.
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 dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data.
Quotients of the upper half-plane
It follows that the Riemann surface in question can be taken to be the quotient
:H/Γ
(where H is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.
Belyi functions
A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Möbius transformation may be taken to be . Belyi functions may be described combinatorially by dessins d'enfants.
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 article {{Harv|Klein|1879}} to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).{{citation
| last = le Bruyn | first = Lieven
| title = Klein's dessins d'enfant and the buckyball
| year = 2008
| url = http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball}}.
Applications
Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
References
{{reflist}}
{{refbegin}}
- {{cite book|last=Serre|first=Jean-Pierre|authorlink=Jean-Pierre Serre|title=Lectures on the Mordell-Weil theorem|volume=15|series=Aspects of Mathematics|edition=Third|year=1997|doi=10.1007/978-3-663-10632-6|mr=1757192|isbn=3-528-28968-6|others=Translated from the French by Martin Brown from notes by Michel Waldschmidt|publisher=Friedr. Vieweg & Sohn, Braunschweig}}
- {{cite journal|last=Klein|first=Felix|journal=Mathematische Annalen|volume=15|issue=3–4|pages=533–555|year=1879|title=Über die Transformation elfter Ordnung der elliptischen Functionen|trans-title=On the eleventh order transformation of elliptic functions|language=German|doi=10.1007/BF02086276|url=https://zenodo.org/record/1642598}}
- {{cite journal|last=Belyĭ|first=Gennadiĭ Vladimirovich|title=Galois extensions of a maximal cyclotomic field|year=1980|journal=Math. USSR Izv.|issue=2|volume=14|pages=247–256|mr=0534593|doi=10.1070/IM1980v014n02ABEH001096|translator=Neal Koblitz|translator-link=Neal Koblitz}}
{{refend}}
Further reading
- {{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 }}
- {{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 }}
{{Algebraic curves navbox}}