Belyi's theorem
In
ramified covering of the Riemann sphere
, ramified at three points only.
This is a result of
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
non-congruence subgroups, it is not the conclusion that any such curve is a modular curve
.
Belyi functions
A Belyi function is a
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 (Klein 1879) to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).[1]
Applications
Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
References
- ^ le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball.
- MR 1757192.
- Klein, Felix (1879). "Über die Transformation elfter Ordnung der elliptischen Functionen" [On the eleventh order transformation of elliptic functions]. Mathematische Annalen (in German). 15 (3–4): 533–555. .
- Belyĭ, Gennadiĭ Vladimirovich (1980). "Galois extensions of a maximal cyclotomic field". Math. USSR Izv. 14 (2). Translated by MR 0534593.
Further reading
- Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts, vol. 79, Cambridge: Zbl 1253.30001
- Wushi Goldring (2012), "Unifying themes suggested by Belyi's Theorem", in Dorian Goldfeld; Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate (eds.), Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 181–214, ISBN 978-1-4614-1259-5