Belyi's theorem: coverings, dessins, and fields of definition
Abstract
Belyi's theorem asserts that a smooth projective curve over ℂ is defined over a number field if and only if it admits a non-constant morphism to ℙ1 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 Pm,n(z)=((m+n)m+n)/mmnnzm(1−z)n: their dessins are double stars, they satisfy the exact identity Pm,n=πk∘ Pm/k,n/k with k=gcd(m,n) and πk(w)=wk, and their monodromy is the wreath product S(m+n)/kwrℤ/k, the full symmetric group precisely when gcd(m,n)=1. Around the sharp lower bounds d≥ 2g+1 (all Belyi maps) and d≥ 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 A5, and is defined over ℚ -- and they satisfy the exact mass formula ∑ 1/|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≤7. 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 Gℚ-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field ℚ(√[3]2).
Related
- A sandwich theorem for the spectrum of graph coverings
- Factorization patterns and fields of definition of ℓ-torsion points on the Jacobians of genus 3 hyperelliptic curves
- Fields of definition of p-torsion points of elliptic curves and their ramification
- The Iskovskih Theorem for regular surfaces over imperfect fields
- An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup
BibTeX
@misc{dinamo2026belyis,
title = {{Belyi's theorem: coverings, dessins, and fields of definition}},
author = {Djounvouna Dinamo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19053},
eprint = {2609.19053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19053},
}
RIS
TY - UNPB
TI - Belyi's theorem: coverings, dessins, and fields of definition
AU - Djounvouna Dinamo
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19053
UR - https://arxiv.org/abs/2609.19053
AB - Belyi's theorem asserts that a smooth projective curve over ℂ is defined over a number field if and only if it admits a non-constant morphism to 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)=((m+n)^m+n)/m^mn^nz^m(1−z)^n: their dessins are double stars, they satisfy the exact identity P_m,n=π_k∘ P_m/k,n/k with k=gcd(m,n) and π_k(w)=w^k, and their monodromy is the wreath product S_(m+n)/kwrℤ/k, the full symmetric group precisely when gcd(m,n)=1. Around the sharp lower bounds d≥ 2g+1 (all Belyi maps) and d≥ 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 ℚ -- and they satisfy the exact mass formula ∑ 1/|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≤7. 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_ℚ-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field ℚ(√[3]2).
ER -