On the theorem of Belyi (Q1919153)

A famous theorem of Belyi asserts that on any smooth projective geometrically connected algebraic curve \(C\) defined over \(\overline\mathbb{Q}\) there exists a function \(f:C\to \mathbb{P}^1\) unramified outside \(\{0,1, \infty\}\). The author shows that this function \(f\) can be chosen without nontrivial automorphism, i.e. if \(\sigma\) is an automorphism of \(C\) with \(f\circ \sigma=f\) then \(\sigma=1\). As a consequence, for \(K\subset \mathbb{C}\) a finite extension of \(\mathbb{Q}\), anany \(K\)-isomorphism class of smooth projective geometrically connected algebraic curves can be characterized by a ``dessin d'enfant de Grothendieck'', i.e. an isomorphism class of finite connected topological coverings of the sphere minus three points.