Belyi's Theorem

Posted on December 2, 2018

Belyi’s Theorem is an arithmetic-geometric result that has the unusual quality of being both fairly recent and provable with only the material from a first course in classical algebraic geometry.

In what follows \mathbb{A}^n,\mathbb{P}^n are over \mathbb{C} unless otherwise specified.

Theorem (Belyi, 1979). Let C\subset\mathbb{P}^2 be an irreducible non-singular projective curve defined as the zero set of F\in\overline{\mathbb{Q}}[X,Y,Z]. Then there exists a finite regular map C\to\mathbb{P}^1 ramified only over a subset of \{0,1,\infty\}.

The proof strategy takes a projection \pi of C onto \mathbb{P}^1 and then use a sequence of regular maps \{\psi_i\} to gather the ramification points of \pi into the set \{0,1,\infty\}.

Lemma 1. The projection away from the point (0:0:1) is a finite regular map \pi:C\to\mathbb{P}^1 ramified over the points \pi(P) such that F_Z(P)=0.

Let Q=(x_0:y_0)\in\mathbb{P}^1. Then \pi is ramified over Q if and only if there exists P\in\pi^{-1}(Q) such that e_P(\pi)>1. Note that      t_Q=(y_0X-x_0Y)/H(X,Y) is a local parameter for Q, where H(X,Y) is any linear form not vanishing at Q. Thus      e_P(\pi)=\text{ord}_P(\pi^*(t_Q)). Hence e_P(\pi)>1 if and only if y_0X-x_0Y is zero on the tangent line \Theta_{C,P}. The tangent line is described by      F_X(P)X+F_Y(P)Y+F_Z(P)Z=0. We care whether the tangent line contains the line y_0X-x_0Y, i.e., if the tangent line passes through (0:0:1). This is equivalent to F_Z(P)=0.

Claim 1. The map \pi is ramified only over points in \overline{Q}\cup\{\infty\}.

By Lemma 1 above we know that the ramification points of \pi are the images of points in C\cap V(F_Z). By Bezout’s Theorem      C\cap V(F_Z)\subset\mathbb{P}^2(\overline{\mathbb{Q}}) contains precisely \text{deg}F\cdot\text{deg}F_Z points when counted with multiplicity. By Bezout again, C\cap V(F_Z) contains precisely the same number of points when considered in \mathbb{P}^2(\mathbb{C}). Since \mathbb{P}^2(\overline{\mathbb{Q}})\subset\mathbb{P}^2(\mathbb{C}) all intersection points are contained in \mathbb{P}^2(\overline{\mathbb{Q}}).

Lemma 2. A regular map \psi:\mathbb{A}^1\to\mathbb{A}^1 defined by x\mapsto G(x) is ramified precisely over the images of points x_0 where G_X(x_0)=0. Hence it extends to a map \mathbb{P}^1\to\mathbb{P}^1 ramified over only over those affine ramification points and \infty.

Note that if \psi is ramified over y_0\in\mathbb{A}^1 if and only if there exists x_0\in\mathbb{A}^1 such that G(x_0)=y_0 and e_{x_0}(\psi)>1.

Observe that      e_{x_0}(\psi)=\text{ord}_{x_0}(\psi^*(Y-y_0)). We find that      \psi^*(Y-y_0)=G(X)-G(x_0)=H(X), and thus G_X=H_X. Thus if G_X(x_0)=0, then      H(x_0)=H_X(x_0)=0 and (X-x_0)^2\mid H(X). Conversely if we write H(X)=(X-x_0)^mu(X) for some unit u(X)\in\mathcal{O}_{\mathbb{A}^1,x_0} then H_X(x_0)=0 if and only if m>1.

Let \varphi:C\to\mathbb{P}^1 be a regular map ramified only over a finite set S\subset\overline{\mathbb{Q}}\cup\{\infty\}.

Claim 2. If S\subset\overline{\mathbb{Q}}\cup\{\infty\}, then there exists \psi:\mathbb{P}^1\to\mathbb{P}^1 such that \psi\circ\phi is ramified only over points in \mathbb{Q}\cup\{\infty\}.

Let n be the maximum degree over \mathbb{Q} of an element in S and p be the number of elements in S of degree n. We will proceed by double induction on n and p.

Our object is to construct a map \psi:\mathbb{P}^1\to\mathbb{P}^1 such that \psi\circ\phi is ramified only over points in S&39;\subset \overline{\mathbb{Q}}\cup\{\infty\} where the number of elements in S&39; of degree n over \mathbb{Q} is less than p and the maximum degree of elements in S&39; is less than or equal to n.

Let \alpha\in S be an element of degree n over \mathbb{Q} and let G\in\mathbb{Q}[X] be its minimal polynomial. Define \psi:\mathbb{P}^1\to\mathbb{P}^1 by x\mapsto G(x). Let \{\beta_1,\ldots,\beta_r\} be the roots of G_X. Note that since \text{deg}(G_X)<n the elements \beta_1,\ldots,\beta_r are of degree n-1 or fewer over \mathbb{Q}.

Observe that ramification points of \psi\circ\phi are      S&39;=\psi(\{\beta_1,\ldots,\beta_r\})\cup\psi(S). Moreover, applying a polynomial over \mathbb{Q} cannot increase the degree of an algebraic element over \mathbb{Q}. Thus the maximum degree over \mathbb{Q} of elements in S&39; is n and there are at most p-1 such elements since \phi(\alpha)=0.

Claim 3. If S\subset\mathbb{Q}\cup\{\infty\} then there exists a regular map \psi:\mathbb{P}^1\to\mathbb{P}^1 such that \psi\circ\varphi is ramified only over points in \{0,1,\infty\}.

If |S|\le 3 then clearly there is an automorphism \psi such that \psi(S)\subset\{0,1,\infty\}.

Suppose that |S|\ge 4. We will construct a map \psi:\mathbb{P}^1\to\mathbb{P}^1 such that \psi\circ\varphi is ramified over some set of points S&39;\subset\mathbb{Q}\cup\{\infty\} with |S&39;|<|S|.

By applying an appropriate automorphism of \mathbb{P}^1 we may assume that S=\{0,1,\infty,\lambda_1,\ldots,\lambda_r\}. Subsequently we may apply the automorphisms (X,Y)\mapsto (1-X,Y) and (X,Y)\mapsto (Y,X) as necessary to ensure that 0<\lambda_1<1; note that each maps \{0,1,\infty\}\to\{0,1,\infty\}.

Write \lambda_1=m/(m+n). Let c=\frac{(m+n)^{m+n}}{m^nn^m}. Let us define \psi:\mathbb{P}^1\to\mathbb{P}^1 by x\mapsto G(x) where g(X)=cX^m(1-X)^n. Observe that      G_X=cX^{m-1}(1-X)^{n-1}(m-(n+m)X) which has zeros only at 0,1 and \lambda_1. Thus \psi is ramified only over      \psi(S)=\{0,1,\infty,\psi(\lambda_2),\ldots,\psi_(\lambda_r)\}. Continuing inductively and composing we construct a map \psi:\mathbb{P}^1\to\mathbb{P}^1 such that \psi(S)\subset\{0,1,\infty\} and \psi is ramified only over the values in S.

Belyi’s Theorem follows from claims 1, 2, and 3.

Example. Consider the elliptic curve C defined by the affine equation      F(X,Y)=Y^2-X(X-1)(X-\sqrt{2})=0. Note the curve has one point “at infinity,” namely the point (0:1:0). Let \pi be the projection away from this point, that is, (x,y)\mapsto x on the affine patch z=1. Note that that this extends to a projection C\to\mathbb{P}^1 since C is nonsingular. This is a slightly different projection than the one in the proof above, but works nicely in this case.

Observe that \text{deg}(\pi)=2 since [\mathbb{C}(C):\mathbb{C}(X)]=2. Therefore projection \pi is ramified over \infty and the images of the points in C where F_Y(X,Y)=2Y=0. These are the points 0,1,\sqrt{2}.

The minimal polynomial of \sqrt{2} over \mathbb{Q} is X^2-2. Thus we consider the map \psi_1 defined by t\mapsto t^2-2; note this map is ramified only at 0. We compute      \psi_1(\{0,1,\infty, \sqrt{2}\})=\{0,-1,-2,\infty\}. We next apply the map \phi_2 defined by t\mapsto -1/t which achieves the dual purpose of mapping \{0,-1,\infty\} to \{0,1,\infty\} and placing \psi_2(-2)=1/2\in (0,1). Finally, we apply the Belyi map t\mapsto 4t(1-t) which sends \{0,1,\infty,1/2\} to \{0,1,\infty\}.