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Section 2.1 Overview (Angus)

Subsection 2.1.1 Belyi morphisms

Let \(X\) be an algebraic curve over \(\CC\) (i.e. a compact Riemann surface) when is \(X\) defined over \(\overline \QQ\text{?}\)

Definition 2.1.2. Ramified.

(AG) A morphism \(f \colon X \to Y\) is ramified at \(x \in X\) if on local rings the induced map \(f^\# \colon \sheaf O_{Y,f(x)} \to \sheaf O_{X,x}\) descended to

\begin{equation*} \sheaf O_{Y,f(x)}/\ideal m \to \sheaf O_{X,x}/ f^\#(\ideal m) \end{equation*}

is not a finite separable field extension.

(RS) A morphism \(f \colon X \to Y\) is ramified at \(x \in X\) if there are charts around \(x\) and \(f(x)\) such that \(f(x) = x^n\text{.}\) This \(n\) is the ramification index.

Definition 2.1.3. Belyi morphisms.

A Belyi morphism is one ramified only over \(\{0,1,\infty\}\)

A clean Belyi morphism or pure Belyi morphism is a Belyi morphism where the ramification indices over 1 are all exactly 2.

If \(\alpha \colon X\to \PP^1 \CC\) is Belyi, then \(\beta = 4\alpha(1-\alpha)\) is a clean Belyi morphism.

Subsection 2.1.2 Dessin d'Enfants

Definition 2.1.5.

A dessin d'Enfant (or Grothendieck Dessin or just Dessin) is a triple \((X_0,X_1,X_2)\) where \(X_2\) is a compact Riemann surface, \(X_1\) is a graph, \(X_0 \subset X_1\) is a finite set of points, where \(X_2 \smallsetminus X_1\) is a collection of open cells. \(X_1 \smallsetminus X_0\) is a disjoint union of line segments

Definition 2.1.7. Clean dessins.

A clean dessin is a dessin with a colouring (white and black) on the vertices such that adjacent vertices do not share a colour.

Subsection 2.1.3 The Grothendieck correspondence

Given a Belyi morphism \(\beta\colon X \to \PP^1 \CC\) the graph \(\beta^{-1}(\lb 0,1\rb)\) defines a dessin.

Example 2.1.9.
\begin{equation*} \PP^1 \CC\to \PP^1\CC \end{equation*}
\begin{equation*} x\mapsto x^3 \end{equation*}
\begin{equation*} \PP^1 \CC\to \PP^1\CC \end{equation*}
\begin{equation*} x\mapsto x^3 + 1 \end{equation*}

Subsection 2.1.4 Covering spaces and Galois groups

A Belyi morphism defines a covering map.

\begin{equation*} \tilde \beta\colon \tilde X \to \PP^1 \CC\smallsetminus \{0,1,\infty\} \end{equation*}

the coverings are controlled by the profinite completion of

\begin{equation*} \pi_1 (\PP^1 \CC\smallsetminus \{0,1,\infty\}) = \ZZ * \ZZ = F_2\text{.} \end{equation*}

By Belyi's theorem every elliptic curve \(E/\overline \QQ\) admits a Belyi morphism. For each \(j \in \overline\QQ\) there exists an elliptic curve \(E_j/\overline \QQ\) with \(j\)-invariant \(j\text{.}\)

Given \(\sigma \in \absgal{\QQ}\text{,}\)

\begin{equation*} \sigma(E_j) = E(\sigma(j)) \end{equation*}

assume \(\sigma \mapsto 1\text{,}\)

\begin{equation*} E_j \cong E_{\sigma(j)}\ \forall j \end{equation*}
\begin{equation*} j = \sigma(j) \ \forall j \end{equation*}

a contradiction.

Subsection 2.1.5 Exercises

Compute the Dessins for the following Belyi morphisms

  1. \begin{equation*} \PP^1\CC\to \PP^1\CC,\,x\mapsto x^4 \end{equation*}
  2. \begin{equation*} \PP^1\CC\to \PP^1\CC,\,x\mapsto x^2(3-2x) \end{equation*}
  3. \begin{equation*} \PP^1\CC\to \PP^1\CC,\,x\mapsto \frac{1}{x(2-x)} \end{equation*}

Give an alternate proof of the fact that \(X\) admits a Belyi morphism if and only if it admits a clean Belyi morphism using dessins and the Grothendieck correspondence.

Prove that a Belyi morphism corresponding to a tree, that sends \(\infty\) to \(\infty\) is a polynomial.