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.
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.
It follows that the Riemann surface in question can be taken to be the quotient (where H is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps.
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 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).