759 名前：At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular 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 H/Γ with H the upper half-plane and Γ 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 Mobius transformation may be taken to be {\displaystyle \{0,1,\infty \}}\{0,1,\infty \}. 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 1 le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball. http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball
https://en.wikipedia.org/wiki/G._V._Belyi G. V. Belyi Belyi won a prize of the Moscow Mathematical Society in 1981, and was an invited speaker at the International Congress of Mathematicians in 1986.[1] []
