BUNTES Three Short Stories about Belyi's theorem (Ricky)

Section 2.11 Three Short Stories about Belyi's theorem (Ricky)

Theorem 2.11.1.

\(X /\CC\) a curve. Then \(X\) is defined over \(\overline \QQ\) iff there exists a Belyi map

\begin{equation*} \phi \colon X\to \PP^1 \end{equation*}

such that \(B(\phi) \subseteq \{0,1,\infty\}\text{.}\)

Main reference: Unifying Themes Suggested by Belyi's Theorem - Wushi Goldring

Subsection 2.11.1 The case of the Rising Degree

Definition 2.11.2.

The Belyi degree of \(X/\overline \QQ\) (a curve) is the minimal degree of \(\phi\colon X \to \PP^1\) a Belyi map.

Question, how does the Belyi degree of \(X/\overline \QQ\) relate to the arithmetic of \(X\text{?}\)

Definition 2.11.3.

The field of moduli of \(X/\overline\QQ\) is the intersection over all fields \(\subseteq \overline \QQ\) over which \(X\) is defined. Similarly for a morphism \(\phi \colon X \to Y\text{.}\)

Remark 2.11.4.

This is not the same as the field of definition always.

Given \(X/\overline \QQ\) with field of moduli \(K\) we say \(X\) has good (resp. semistable) reduction at \(\ideal p \subseteq \ints_K\) if there exists a model for \(X\) over \(\ints_{K_{\ideal p}}\) s.t. the special fibre is smooth (resp. semistable) reduction.

For \(p\in \ZZ\) we say \(X\) has good/semistable reduction at \(p\) if it dies for all \(\ideal p | p\text{.}\)

Theorem 2.11.5. Zapponi.

If \(X/\overline \QQ\) then the Belyi degree of \(X\) is at least the largest prime \(p \in \ZZ\) such that \(X\) has bad semistable reduction at \(p\text{.}\)

Remark 2.11.6.

The lower bound is not “sharp” because there exist \(E/K\) with good reduction everywhere, but no degree 1 maps \(\phi \colon E \to \PP^1\text{.}\)
If
\begin{equation*} E\colon y^2 = x^3 + x^2 + p \end{equation*}
then \(E\) has bad semistable reduction at \(p\) so the Belyi degree of \(E\) is \(\ge p\text{.}\)

Theorem 2.11.7. Beckmann.

Let \(\phi \colon X \to \PP^1\) be a Belyi map with field of moduli \(M\text{.}\) Let \(G\) be the Galois group of the Galois closure of \(\phi\text{.}\) Then for all \(p\) such that \(p \nmid |G|\text{,}\) \(\tilde \phi \colon \tilde X \to \PP^1\) has good reduction at \(p\) and \(p\) is unramified in \(M\text{.}\)

Proof.

Of Zapponi.

Let \(\phi\colon X \to \PP^1\) be a Belyi map of degree \(n\text{.}\) Let \(K\) be the field of moduli of \(X\text{,}\) \(M\) the field of moduli of \(\phi\) then \(M/K\) is a finite extension. Take \(G\) as above and let \(\ideal p \subseteq \ints_K\) be a place of bad semistable reduction for \(X\text{.}\) Then \(\wp| \ideal p\) for \(\wp \subseteq \ints_M\) is a place of bad semistable reduction for \(\phi\text{.}\) By Theorem 2.11.7 \(p \mid |G|\) for \(p\in \ZZ\) below \(\ideal p\) but \(G \hookrightarrow S_n\) which implies \(p|n!\) so \(p\le n\text{.}\)

Subsection 2.11.2 Finitists Dream

Recall that if \(k \) is a perfect field of characteristic \(p\) then

\begin{equation*} \phi \colon C_1 \to C_2 \end{equation*}

is said to be tamely ramified at \(P\in C_1\) if \(p\nmid e_\phi(P)\) (wildly ramified if \(p |e_\phi(P)\)).

Theorem 2.11.8. Wild \(p\)-Belyi.

For \(C\) a curve over \(k\) perfect of characteristic \(p\text{,}\) there exists a “wild Belyi map”

\begin{equation*} \phi \colon C\to \PP^1 \end{equation*}

such that \(B(\phi) = \{\infty\}\text{.}\) I.e. every curve \(/k\) is birational to an étale cover of \(\aff^1\text{.}\)

Example 2.11.9.

\begin{equation*} \mathbf G_m \to \aff^1 \end{equation*}

\begin{equation*} x \mapsto x^p + \frac 1x \end{equation*}

but the tame étale fundamental group of \(\aff^1\) is 0.

Theorem 2.11.10. Tame \(p\)-Belyi (Saidi).

Let \(p \gt 2\text{.}\) For \(C/\overline \FF_p\) there exists \(\phi\colon C \to \PP^1\) tamely ramified everywhere (i.e. possibly unramified) with

\begin{equation*} B(\phi) \subseteq \{0,1,\infty\}\text{.} \end{equation*}

Lemma 2.11.11. Fulton.

Let \(p \gt 2\) then for \(C/k\) (\(k\) algebraically closed of characteristic \(p\)) there exists \(\psi \colon C\to \PP^1\) such that

\begin{equation*} e_\psi(P) \le 2\text{.} \end{equation*}

Proof.

Of Tame \(p\)-Belyi

Take \(\psi \colon C\to \PP^1\) as in the lemma then

\begin{equation*} B(\psi) \subseteq \PP^1(\FF_{p^m}) \end{equation*}

for some \(m\text{.}\) Define

\begin{equation*} f\colon \PP^1\to \PP^1 \end{equation*}

\begin{equation*} x\mapsto x^{p^m -1}\text{.} \end{equation*}

Take \(\phi = f\circ \psi\text{.}\) So \(\pi\) is tamely ramified everywhere and \(B(\phi) \subseteq \{0,1,\infty\}\text{.}\)

Analogue of Fulton's lemma is that there exists

\begin{equation*} \tau \colon C\to \PP^1 \end{equation*}

for \(\characteristic (k) \ne 3\) such that \(e_\tau(P) = 1\) or \(3\text{.}\)

Subsection 2.11.3 In the Stacks

Observation 2.11.12.

\(\PP^1 \smallsetminus \{0,1,\infty\}\) is the moduli space of genus 0 curves with four (ordered) marked points.

\begin{equation*} (\PP^1, \alpha_1,\alpha_2,\alpha_3,\alpha_4) \mapsto \im(\alpha_4) \text{ when } \alpha_1 \mapsto 0,\alpha_2\mapsto 1, \alpha_3 \mapsto \infty\text{.} \end{equation*}

Definition 2.11.13.

Let \(\mathcal M_{g,n}\) be the moduli space of genus \(g\) curves with \(n\) (ordered) marked points (then \(\mathcal M_{g,\lb n\rb}\) is the same for unordered points). If \(n\) is large enough relative to \(g\) then \(\mathcal M_{g,n}\) will be a scheme (but the unordered version will not).

Example 2.11.14.

\begin{equation*} \mathcal M_{0,4} \simeq \PP^1 \smallsetminus \{0,1,\infty\} \end{equation*}

Question 2.11.15. Braungardt.

Is every \(X/\overline \QQ\) (smooth projective variety) birational to a finite étale cover of some \(\mathcal M_{g,\lb n \rb}\text{?}\)

Note 2.11.16.

There exists an étale map

\begin{equation*} \mathcal M_{g,n} \to \mathcal M_{g,[n]} \end{equation*}

by forgetting the ordering of the points.

So the dimension 1 case of the conjecture is Belyi's theorem, by

\begin{equation*} X\smallsetminus \phi\inv (B(\phi)) \to \PP^1 \smallsetminus \{0,1,\infty\} \simeq \mathcal M_{0,4} \to\mathcal M_{0,[4]}\text{.} \end{equation*}

In dimension 2 we have \(\mathcal M_{1,\lb 2 \rb}\) and \(\mathcal M_{0,\lb 5 \rb}\text{,}\) the only 2-d spaces of interest. We also have an étale map

\begin{equation*} \mathcal M_{1,[2]} \xrightarrow\alpha \mathcal M_{0,[5]} \end{equation*}

as follows:

\begin{equation*} \eta = (E; \{q_1,q_2\}) \in \mathcal M_{1,[2]} \end{equation*}

with

\begin{equation*} \alpha(\eta) = (\PP^1; \{r_1,r_2,r_3,r_4,r_5\}) \end{equation*}

where the \(r_i\) come from constructing a projection \(\phi\) from \(E\) to \(\PP^1\) situated perpendicularly to the line joining \(q_1,q_2\text{.}\) This then has 4 ramification points

\begin{equation*} B(\phi) = \{r_1,r_2,r_3,r_4\} \end{equation*}

and \(r_5 = \phi(q_1) = \phi(q_2)\text{.}\) So Braungardt for surfaces \((X/\overline \QQ)\text{?}\) Does there exist \(\phi \colon X \to \mathcal M_{0,\lb 5 \rb}\) which is étale?

Theorem 2.11.17. Braungardt.

For \(X/\overline \QQ\) an abelian surface \(X\) is birational to an étale cover of \(\mathcal M_{0,\lb 5 \rb}\text{.}\)

Proof.

Sketch.

For an abelian surface over \(\overline \QQ\) there exists another isogenous to it which is principally polarized. Such surfaces come in two flavours

\begin{equation*} E_1 \times E_2 \end{equation*}

or \(J(C)\) for \(C\) of genus 2.

Case 1:

Let \(\phi_i \colon E_i \to \PP^1\smallsetminus \{0,1,\infty\}\) be Belyi maps. Then we have \(\alpha \colon A \xrightarrow{\phi_1,\phi_2} \PP^1 \times \PP^1\text{.}\) Then \(\alpha\) restricts to a finite unramified cover

\begin{equation*} \alpha\inv (S) \xrightarrow \alpha S \end{equation*}

where

\begin{equation*} S = (\PP^1 \smallsetminus \{0,1,\infty\} \times \PP^1 \smallsetminus \{0,1,\infty\}) \smallsetminus \Delta\text{.} \end{equation*}

Note that \(S \simeq \mathcal M_{0,5}\) by

\begin{equation*} (a,b ) \mapsto (\PP^1; \{0,1,\infty, a,b\})\text{.} \end{equation*}

So \(A\) is birational to \(\alpha\inv (S)\) which is an étale cover of \(\mathcal M_{0,\lb 5 \rb}\text{.}\)

Case 2

If \(A = J(C)\) then use \(\phi \colon C \to \PP^1\) and a relation between \(A\) and \(\Sym^2(C)\text{.}\)