A new $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant of dessins d'enfants

Abstract

We study the action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the category of Belyi functions (finite, étale covers of ${\mathbb{P}}_{\overline{\mathbb{Q}}}^1\setminus \{0,1,\infty \}$). We describe a new combinatorial $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant for whose monodromy cycle types above $0$ and $\infty$ are the same. We use a version of our invariant to prove that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of Belyi functions whose monodromy cycle types above 0 and $\infty$ are the same; the proof of this result involves a version of Belyi's Theorem for odd degree morphisms. Using our invariant, we obtain that for all $k<2^{\sqrt{\frac{2}{3}}}$ and all positive integers $N$, there is an $n\le N$ such that the set of degree $n$ Belyi functions of a particular rational Nielsen class must split into at least $\Omega \left(k^{\sqrt{N}}\right)$ Galois orbits.

25 pages, 2 figures; several revisions

References in corpus (1)

• Belyi-extending maps and the Galois action on dessins d'enfants