Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called *dessins d'enfants* which are bipartite graphs embedded on an oriented surface so that each component of their complement is a topological disk. To be more precise, given a smooth projective curve $X$ of genus $g$ over $\overline{\Bbb{Q}}$, Belyi's theorem guarantees the existence of a branched cover $f:X(\Bbb{C})\rightarrow\Bbb{P}^1(\Bbb{C})$ unramified outside $\{0,1,\infty\}$. Then, $f^{-1}([0,1])$ is an embedded graph that can be considered as the $1$-skeleton of a CW structure on $X(\Bbb{C})$. Conversely, such a graph on an oriented compact surface of genus $g$ equips it with a complex structure and such a branched cover.


**My Question:** Can dessins be used to defined a "reasonable" (e.g. comparable to a Weil height) height function on the moduli space $\mathcal{M}\_g(\overline{\Bbb{Q}})$? For instance, is the minimum possible number of edges of a dessin (i.e. the minimum possible degree of a Belyi map on the complex curve) be used as a height function? Any reference to the literature on this question is highly appreciated.



Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
If $X$ is a (smooth projective) curve over $\overline{\mathbb{Q}}$, we define



> 
> 
> > 
> > The Belyi degree $\deg\_B(X)$ of $X$ to be the minimum degree of a Belyi map $X\to \mathbb{P}^1\_{\overline{\mathbb{Q}}}$.
> > 
> > 
> > 
> 
> 
> 


The Belyi degree is a function on $\mathcal{M}\_g(\overline{\mathbb{Q}})$ which satisfies the following Northcott-type finiteness property.


**Proposition. (Strong Northcott)** For every integer $d$, the set of $\overline{\mathbb{Q}}$-isomorphism classes of curves $X$ over $\overline{\mathbb{Q}}$ with $\deg\_B(X)\leq d$ is finite.


*Proof.* Like all finiteness statements, this one also boils down to some "general" finiteness statements. In this case, the statement (seemingly arithmetic in nature) is a consequence of a (topological) finiteness property of the fundamental group of $\mathbb{P}^1\setminus \{0,1,\infty\}$. Indeed, the proposition can be proven using the fact that the fundamental group of $\mathbb{P}^1\setminus \{0,1,\infty\}$ is finitely generated, and that a finite generated group has only finitely many finite index subgroups of index at most $d$. QED


Note that this Proposition can be used to enumerate all curves over $\overline{\mathbb{Q}}$. Simply "write" down the curves of Belyi degree at most $3$, then $4$, then $5$, etc.


Note that the Northcott property satisfied by the Belyi degree is much stronger than that of any Weil height $h$. The Northcott property for a Weil height usually requires in addition a bound on the *degree* of the point.


The Strong Northcott property implies that, given a Weil height $h$ (or any function!) on $\overline{\mathbb{Q}}$, there is a function $f(\deg\_B(-))$ such that


$$ h(X) \leq f(\deg\_B(X)).$$


Thus, any function on $\ {\mathcal{M}\_g}(\overline{\mathbb{Q}})$ is bounded by a function in the Belyi degree (simply because of the above Proposition). For example, the genus of $X$ is bounded by $\deg\_B(X)$. This follows from the Riemann-Hurwitz formula.


There are a few natural (arithmetic) invariants on $\ {\mathcal{M}\_g}(\overline{\mathbb{Q}})$ such as the Faltings height for which one can write down explicit bounds. For example:


**Theorem.** If $X$ is a curve over $\overline{\mathbb{Q}}$ with Faltings height $h\_F(X)$, then $$h\_F(X) \leq 10^8 \deg\_B(X)^6.$$


This (with many more explicit inequalities) is proven in [1]. The motivation for proving such inequalities is that they can be used to control the running time of certain algorithms computing coefficients of modular forms.


The question of actually computing the Belyi degree of a curve is an interesting one. An algorithm (which I would not recommend trying to implement) for doing so is given in [2].


[1] A. Javanpeykar. *Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by Peter Bruin.* Algebra and Number Theory, Vol. 8 (2014), No. 1, 89–140.


[2] A. Javanpeykar and J. Voight. *The Belyi degree of a curve is computable*
Contemp. Math., 2019, 722, p. 43-57.


