[Paper Review] Belyi's theorem revisited

Bernhard Köck|arXiv (Cornell University)|Aug 31, 2001

Algebraic Geometry and Number Theory9 references21 citations

TL;DR

This paper presents a new, elementary, and self-contained proof of Belyi's Theorem, establishing that a complex smooth projective curve is defined over a number field if and only if it admits a finite morphism to the projective line with at most three critical values. The key contribution is an explicit bound for the degree of the defining number field: for a morphism of degree $ d $ with $ a $ automorphisms, the field of definition has degree at most $ \frac{d}{a} M_d $ over $ \mathbb{Q} $, where $ M_d $ is the number of index-$ d $ subgroups in a free group of rank 2.

ABSTRACT

We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.

Motivation & Objective

• To provide a self-contained, elementary, and concise proof of Belyi’s Theorem, particularly focusing on the 'if' direction.
• To clarify and simplify the proof of the field of definition for curves with morphisms to $ \mathbb{P}^1 $ having at most three critical values.
• To derive an explicit upper bound for the degree of the number field over which such curves and morphisms are defined.
• To introduce and utilize the concept of the relative moduli field of a morphism, replacing the absolute moduli field used in prior approaches.

Proposed method

• Define the relative moduli field $ \mathbb{C}^{U(X,t)} $ as the fixed field of automorphisms $ \sigma \in \mathrm{Aut}(\mathbb{C}) $ preserving the morphism $ t: X \to \mathbb{P}^1_{\mathbb{C}} $ up to isomorphism.
• Prove that if critical values of $ t $ lie in $ \{0,1,\infty\} $, then the moduli field of $ t $ is a number field (Assertion (a)).
• Establish that $ X $ and $ t $ are defined over a finite extension of this moduli field (Assertion (b)), using descent techniques inspired by Grothendieck and Coombes/Harbater.
• Use the fact that there are only finitely many isomorphism classes of coverings of fixed degree and critical value set to bound the number of Galois conjugates of $ (X,t) $.
• Apply group-theoretic bounds via the number $ M_d $ of subgroups of index $ d $ in a free group of rank 2 to estimate the degree of the field of definition.
• Derive the explicit bound $ [K:\mathbb{Q}] \leq \frac{d}{a} M_d $, where $ d $ is the degree of $ t $, and $ a $ is the number of automorphisms of $ t $.

Experimental results

Research questions

• RQ1 Can Belyi’s Theorem be proven in a way that is both elementary and self-contained, avoiding advanced algebraic geometry?
• RQ2 What is the precise degree of the smallest number field over which a curve with a 3-critical-value morphism to $ \mathbb{P}^1 $ can be defined?
• RQ3 How does the relative moduli field of a morphism $ t: X \to \mathbb{P}^1 $ relate to the field of definition of $ X $ and $ t $?
• RQ4 Can the number of isomorphism classes of such morphisms be used to bound the degree of the field of definition?
• RQ5 What is the sharpest possible bound for the degree of the field of definition in terms of the morphism’s degree and automorphism group?

Key findings

• The moduli field of a morphism $ t: X \to \mathbb{P}^1_{\mathbb{C}} $ with critical values in $ \{0,1,\infty\} $ is a number field.
• The curve $ X $ and the morphism $ t $ are defined over a finite extension of this moduli field, confirming the 'if' direction of Belyi’s Theorem.
• An explicit upper bound for the degree of the field of definition is $ [K:\mathbb{Q}] \leq \frac{d}{a} M_d $, where $ d $ is the degree of $ t $, $ a $ is the number of automorphisms of $ t $, and $ M_d $ is the number of index-$ d $ subgroups in a free group of rank 2.
• The bound $ M_d $ can be replaced by a smaller number counting only isomorphism classes with the same ramification data, yielding potentially sharper estimates.
• The proof avoids advanced tools like generic points or positive characteristic techniques, relying instead on elementary group theory and Galois descent.
• The bound in Corollary (3.7) is new and explicit, providing a quantitative measure for the field of definition in terms of group-theoretic invariants of the covering.