Belyi's theorem: coverings, dessins, and fields of definition

Abstract

Belyi's theorem asserts that a smooth projective curve over $\mathbb{C}$ is defined over a number field if and only if it admits a non-constant morphism to $\mathbb{P}^1$ ramified over at most three points. This article gives a complete, self-contained account of the theorem -- both implications, with all descent machinery proved rather than quoted -- together with a study of the objects the proof produces. Beyond the exposition it contains original results, proved in full. The principal one is a structure theorem for the Belyi polynomials $P_{m,n}(z)=\frac{(m+n)^{m+n}}{m^{m}n^{n}}z^{m}(1-z)^{n}$: their dessins are double stars, they satisfy the exact identity $P_{m,n}=π_{k}\circ P_{m/k,n/k}$ with $k=\gcd(m,n)$ and $π_{k}(w)=w^{k}$, and their monodromy is the wreath product $\mathfrak{S}_{(m+n)/k}\wr\mathbb{Z}/k$, the full symmetric group precisely when $\gcd(m,n)=1$. Around the sharp lower bounds $d\geq 2g+1$ (all Belyi maps) and $d\geq 4g$ (clean maps) we study the extremal maps attaining $d=2g+1$: their monodromy lies in the alternating group, they need not be cyclic -- the smallest non-cyclic one has degree $5$, genus $2$, monodromy $A_{5}$, and is defined over $\mathbb{Q}$ -- and they satisfy the exact mass formula $\sum 1/|\mathrm{Aut}|=2(d-1)!/d(d+1)$, obtained from Boccara's cycle-factorization count -- reproved by a self-contained Frobenius computation -- and verified by complete enumeration for $d\leq7$. We also bound the degree of the rationalization step of Belyi's algorithm by $N!$ in the number $N$ of irrational branch values. The theory is illustrated by fully computed examples, and by an explicit $G_{\mathbb{Q}}$-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field $\mathbb{Q}(\sqrt[3]{2})$.