子供の絵

[Header Image Credit] 
A monodromy group of order 35838544379904000000 in
"the mystery Manin-Marcolli monoid"

Preface

数学の研究者が「子供の絵」なる日本語を用いるのか否かは判りませんが、
Dessin d'enfant の直訳です。つい最近勉強した以下の本の題に現れます：

Quick overview

The following article which contains a YouTube video by Gareth A. Jones is very handy:
https://serious-science.org/dessins-denfants-10245

Gareth A. Jones is a coauthor of the following paper:

The YouTube video in the above article is separately accessible:

It has a Japanese transcription, which I never recommend… unfortunately it is an unacceptably lousy work.

First encounter

In June 2025, I came across Grothendieck's Dessins d'Enfants.
Accurately speaking, ChatGPT inspired me to study Belyi map, which led me to the following paper:
    "Belyi maps in number theory: a survey" by John Voight
It shows a very simple slide after the fundamental Riemann-Hurwitz formula:

Then I noticed the following eye-catching phrase in this paper:
    "Esquisse d'un programme"
The title sounded a little strange without mentioning any mathematical object but rather intriguing, and It led me to "dessin d'enfant" (子供の絵)⬇️

And here is an excerpt from its English translation⬇️

Riemann Surfaces (リーマン面)

Riemann surfaces are indeed a cornerstone of modern mathematics, acting as a crucial bridge that connects diverse and seemingly unrelated fields:

• Complex Analysis 複素解析（函数論）

• Algebraic Geometry 代数幾何学（代数曲線論）

• [Algebraic] Topology [代数的]位相幾何学

• Differential Geometry 微分幾何学

• Algebraic Number Theory 代数的整数論

Therefore, there are so many textbooks written on Riemann Surfaces.
N.B. Riemann Surfaces are real 2-dimensional but complex 1-dimensional, which is why they are called surfaces and algebraic curves.

As mentioned in the beginning, I finished reading the following book:
    "Introduction to Compact Riemann Surfaces and Dessins d'Enfants"
    by Ernesto Girondo & Gabino González-Diez
Its contents are:

1. Compact Riemann surfaces and algebraic curves

2. Riemann surfaces and discrete groups

3. Belyi's Theorem

4. Dessins d'enfants

Chapter 1 & 2 are fundamental, and I thought these chapters were nothing new, but these chapters are written in a style different from that of books I studied long time ago, and quite refreshing. Not only I find these two chapters good for reviewing my knowledge but also I learned some important techniques using hyperelliptic curves (超楕円曲線) in this book.

It is especially enlightening that the famous Klein quartic, a genus 3 Riemann surface, the group of automorphisms of which is PSL₂(F₇), the second smallest simple group of order 168, is repeatedly used as an example for computation from different perspectives ✨

Belyi's Theorem

It is astoundingly simple stated as follows:

The proof by Girondo & González-Diez of this profound result is elementary:

• No heavy scheme theory, no cohomology, no moduli stacks

• Just careful analysis, covering theory, monodromy, and the geometry of compact Riemann surfaces

Reading the proof, I grinned like the Cheshire Cat while constructing a Belyi function from triangle decomposition as if solving a beautifully rigged puzzle: nothing forced, everything falling into place almost embarrassingly naturally.

Dessins d'enfants

The definition is again very simple.
According to the book by Girondo & González-Diez,

and here are excerpts from my note:

These do not evoke the image of "dessins d'enfants", but the following image definitely does:

Here M₁₂ stands for Mathieu group of order 95040 = 12·11·10·9·8, one of the 26 sporadic groups (散在単純群).

Applications

Two major applications are:

1. ABC Conjecture for polynomials (多項式版ABC予想)

2. Inverse Galois Problem (逆ガロア問題)

① ABC Conjecture for polynomials (多項式版ABC予想)

所謂 IUT (Inter-universal Teichmüller theory) で知られた ABC予想、或いは Oesterlé–Masser conjecture のことではなく、Mason–Stothers theorem のことであり、以下のように記述されます：

Belyi's Theoremを用いた証明は以下に示されています：
    "The ABC-conjecture for polynomials" (Abhishek Parab)
更に以下の論文が ABC予想に関して Belyi's Theorem に言及しています。

• ”ABC for polynomials, dessins d'enfants and uniformization - a survey” (Jürgen Wolfart)

• ”ABC implies Mordell” (Noam D. Elkies)

② Inverse Galois Problem (逆ガロア問題)

The definition of this problem is very simple:

It's an unsolved problem in mathematics:

Jean-Pierre Serre refers to Belyi's paper in “Groupes de Galois sur Q”⬇️

I am still learning this particular topic with the following book by J. P. Serre:

This is an exceptionally enlightening book without any exaggeration. While the title sounds subtle, it explains "inverse Galois problem".
尚、"Galois Theory" と題していますが、所謂ガロア理論の説明書ではなく、『逆ガロア問題』に関する重要な手法を説明しています。
🟪