Esquisse d'un Programme I: Dessins d'Enfants and the Faithfulness of the Galois ActionResearch Paper
## Motivation In *Esquisse d'un Programme* (1984), Alexandre Grothendieck describes a discovery that reorganised his mathematical interests: a finite oriented combinatorial map drawn on a surface — a **dessin d'enfant**, a child's drawing — determines canonically a smooth projective algebraic curve together with a map to the projective line ramified only above $0$, $1$ and $\infty$, and that curve and map are defined over the field $\overline{\mathbb{Q}}$ of algebraic numbers (Esquisse, §3, pp. 14–16 of the French text). Consequently the absolute Galois group $\Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on these purely combinatorial objects; in the spherical case, where the structural map is a rational function $f(z) = P(z)/Q(z)$, the action of $\gamma \in \Gamma$ is obtained simply by applying $\gamma$ to the coefficients of $P$ and $Q$. Grothendieck states in §2 (p. 9) that the resulting outer action of $\Gamma$ on the profinite fundamental group $\hat{\pi}_{0,3}$ of $\mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ is **faithful**, and in §3 that the theorem of Belyi, announced at the 1978 Helsinki congress, is what makes the dictionary between combinatorics and arithmetic exact. Timeline of the results this mission formalizes. Belyi (1979, *On Galois extensions of a maximal cyclotomic field*, Izv. Akad. Nauk SSSR) proved that a smooth projective curve over $\mathbb{C}$ is defined over a number field if and only if it admits a map to $\mathbb{P}^1$ unramified outside $\{0,1,\infty\}$; the "only if" half is an explicit construction with polynomials over $\mathbb{Q}$. Grothendieck (1984) drew the consequence that $\Gamma$ acts on dessins and asserted faithfulness of the action on $\hat{\pi}_{0,3}$. Lenstra, in an appendix to L. Schneps (ed.), *The Grothendieck Theory of Dessins d'Enfants* (LMS Lecture Notes 200, CUP 1994), showed that the action is already faithful on the much smaller class of **plane trees**, equivalently on **Shabat polynomials**. That tree-level statement is the goal of this mission, because it is the sharpest form of faithfulness that can be stated without first building the theory of étale fundamental groups. ## Setting Work over $\overline{\mathbb{Q}}$, realized as the algebraic closure of $\mathbb{Q}$, and write $\Gamma$ for its group of field automorphisms fixing $\mathbb{Q}$ pointwise. A nonconstant polynomial $P$ over a field $K$ is a **Belyi polynomial** (classically a *Shabat polynomial*) when every critical value of $P$ lies in $\{0,1\}$: for every $z \in K$ with $P'(z) = 0$ one has $P(z) = 0$ or $P(z) = 1$. Over an algebraically closed field of characteristic zero this says exactly that $P$, viewed as a degree-$n$ map $\mathbb{P}^1 \to \mathbb{P}^1$, is unramified outside the fibres over $0$, $1$ and $\infty$. The associated dessin is the preimage $P^{-1}([0,1])$, a plane tree with $n$ edges whose vertices are the points above $0$ and $1$, with vertex orders equal to the multiplicities of the corresponding roots of $P$ and of $P - 1$. Two Belyi polynomials define the same dessin exactly when they are **affinely equivalent**: $Q = P(aX + b)$ for some $a \neq 0$ and some $b$. The target coordinate is already rigidified by the normalisation of the critical values to $\{0,1\}$; only the source coordinate remains free. The group $\Gamma$ acts coefficientwise: $P^{\gamma}$ is the polynomial obtained from $P$ by applying $\gamma$ to each coefficient. This is exactly the action described in §3 of the Esquisse. It sends Belyi polynomials to Belyi polynomials, and it descends to an action on affine equivalence classes, i.e. on dessins. ## Formalization targets ### Goal — faithfulness of the Galois action on plane trees $$\forall\, \gamma \in \Gamma,\quad \gamma \neq 1 \ \Longrightarrow\ \exists\, P \in \overline{\mathbb{Q}}[X] \text{ a Belyi polynomial with } P \not\sim_{\mathrm{aff}} P^{\gamma}.$$ Equivalently: no nontrivial element of the absolute Galois group fixes every plane tree. This is the weakest stable form of the faithfulness assertion in the Esquisse: it fixes no degree, no genus and no tree, asserting only that some dessin is moved. ### Supporting targets - **Belyi's theorem, polynomial form.** For every finite set $S \subseteq \overline{\mathbb{Q}}$ there is a Belyi polynomial $f \in \mathbb{Q}[X]$ with $f(S) \subseteq \{0,1\}$. - **Descent to $\overline{\mathbb{Q}}$.** Every Belyi polynomial over $\mathbb{C}$ is affinely equivalent to one whose coefficients are algebraic over $\mathbb{Q}$. - **Galois equivariance and invariants.** $P^{\gamma}$ is again a Belyi polynomial of the same degree, and the multiplicity of $z$ as a root of $P - c$ equals the multiplicity of $\gamma(z)$ as a root of $P^{\gamma} - \gamma(c)$: the dessin's vertex and face orders are Galois invariants. - **Finiteness of the orbit.** The set of Galois conjugates of a fixed polynomial over $\overline{\mathbb{Q}}$ is finite — the "visibly finite number of conjugates" of §3. - **Finiteness in a fixed degree.** For each $n$ there are only finitely many monic Belyi polynomials of degree $n$ over $\overline{\mathbb{Q}}$ with vanishing subleading coefficient. - **Separation.** For every $\alpha \in \overline{\mathbb{Q}}$ there is a Belyi polynomial $P$ such that every $\gamma$ fixing the class of $P$ fixes $\alpha$. The goal follows from this by taking $\alpha$ with $\gamma(\alpha) \neq \alpha$. ## Significance The result itself. Faithfulness turns the combinatorics of finite maps into a faithful representation of $\Gamma$: every nontrivial automorphism of $\overline{\mathbb{Q}}$ is detected by a finite tree, so invariants of dessins (degree, valency lists, monodromy group, field of moduli) are in principle a complete set of tools for distinguishing Galois elements. It is also the entry point to the anabelian programme described in §3 of the Esquisse, since the same statement expresses that $\Gamma$ embeds into the outer automorphism group of $\hat{\pi}_{0,3}$. Formalizing it. Belyi's theorem and the faithfulness of the Galois action on trees are both established results. Mathlib at the environment revision of this mission contains no declaration mentioning Belyi maps or dessins d'enfants, and no étale fundamental group, so both statements have to be built from the polynomial and Galois-theoretic libraries. What this mission produces is a formal version of the combinatorial half of the dictionary, in a form that avoids scheme theory entirely: everything is phrased with polynomials over $\overline{\mathbb{Q}}$ and $\mathbb{C}$, so the development rests only on Mathlib's existing polynomial, field theory and Galois theory libraries. ## Difficulty The naive attack on the goal — exhibit one tree and one Galois element moving it — does not scale: the statement quantifies over all $\gamma \neq 1$, and $\Gamma$ has no accessible presentation. The real work is the separation statement, which demands, for an arbitrary algebraic number $\alpha$, a tree whose isomorphism class remembers $\alpha$; the construction must control both the existence of a Belyi polynomial with prescribed arithmetic and the rigidity that makes affine equivalence classes finite. Belyi's theorem in polynomial form is itself an induction on the degree of the field of definition of the critical values, and each step changes the polynomial, so bookkeeping of critical values through composition is the bulk of the formal proof. The descent statement over $\mathbb{C}$ is not a formal manipulation either: it needs the finiteness of the set of Belyi polynomials of a given degree up to affine equivalence, which is where the combinatorial classification enters. ## Formalization scope Conventions fixed in the Lean development, and not to be re-litigated by solvers: - $\overline{\mathbb{Q}}$ is `AlgebraicClosure ℚ`, and $\Gamma$ is its group of $\mathbb{Q}$-algebra automorphisms. - "Belyi polynomial" means: positive degree, and every root of the formal derivative is sent to $0$ or $1$. Critical values are required to lie *in* $\{0,1\}$, not to be exactly $\{0,1\}$; degenerate cases such as $X^n$ (one finite critical value) are therefore included. - Being a Belyi polynomial is stated over an arbitrary field but is only intended over algebraically closed fields ($\overline{\mathbb{Q}}$, $\mathbb{C}$), where quantifying over the field's own elements captures all critical points. - Dessin isomorphism is modelled as affine equivalence of the source variable only; conjugating by an affine map of the target is excluded, since the target is rigidified by $\{0,1\}$. - The Galois action is coefficientwise application of $\gamma$. Trivialization is ruled out as follows: the goal asserts the *existence* of a moved Belyi polynomial for each nontrivial $\gamma$, with the nondegeneracy `0 < deg P` built into the definition, so no constant or empty witness satisfies it, and no hypothesis of the goal is vacuous ($\gamma \neq 1$ is satisfiable). A complete development needs: critical values and their behaviour under composition of polynomials; the classification of Belyi polynomials of fixed degree up to affine equivalence; Galois descent for a finite set of polynomials stable under conjugation; and, for the descent target, the identification of the coefficients of a Belyi polynomial over $\mathbb{C}$ as algebraic numbers. All of these are reusable outside this mission. Contributions of general polynomial-ramification infrastructure are welcome, as are alternative formalizations of the same statements over a general algebraically closed field of characteristic zero. ## Selected references - A. Grothendieck, *Esquisse d'un Programme* (1984), published in L. Schneps and P. Lochak (eds.), *Geometric Galois Actions 1*, LMS Lecture Note Series 242, Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511758874 - G. V. Belyi, *On Galois extensions of a maximal cyclotomic field*, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276. English translation: Math. USSR-Izv. 14 (1980), 247–256. https://doi.org/10.1070/IM1980v014n02ABEH001096 - L. Schneps (ed.), *The Grothendieck Theory of Dessins d'Enfants*, LMS Lecture Note Series 200, Cambridge University Press, 1994. https://doi.org/10.1017/CBO9780511569302 - S. K. Lando and A. K. Zvonkin, *Graphs on Surfaces and Their Applications*, Encyclopaedia of Mathematical Sciences 141, Springer, 2004. https://doi.org/10.1007/978-3-540-38361-1