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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.

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    $\begingroup$ A very interesting question!!! (at least to my ignorant mind...) $\endgroup$ Oct 4 at 22:37
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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.

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  • $\begingroup$ @ArianJavanpeykar Thank you for the interesting answer! So seems that there is only a polynomial upper bound for the Faltings height in terms of the Belyi height, and we are not aware of a lower bound, right? $\endgroup$
    – KhashF
    Oct 5 at 22:20
  • $\begingroup$ @KhashF Yes, the "best" bound you can expect is polynomial for the Faltings height in terms of the Belyi height. This is because in the case of modular curves $X(n)$ (where the Belyi degree is about $n^2$) one can show that its Faltings height really grows polynomially (maybe even quadratically or linearly, but I don't remember). On the other hand, the bound on $h_F(X)$ provided by the theorem above is probably not best. In its proof, the "arithmetic" contributions are at most cubic in the Belyi degree, and its only the "analytic" contributions which make the bound get so big. One can.... $\endgroup$ Oct 6 at 9:00
  • $\begingroup$ ....probably do much better than $10^8 \deg_B(X)^6$, but then the question becomes "why would one"? Also, a lower bound for the Faltings height in terms of the Belyi degree does not exist. Explanation below: $\endgroup$ Oct 6 at 9:01
  • $\begingroup$ Suppose that $\deg_B(X)$ were bounded by a function in the Faltings height (e..g., $\exp(h_F(X)^2)$). Then, bounding the Faltings height would bound the Belyi degree and imply that the set of isomorphism classes of curves of height $h_F(X)$ at most a fixed constant $C$ is finite. But that's not true. $\endgroup$ Oct 6 at 9:02
  • $\begingroup$ @AriyanJavanpeykar: I am a bit confused with your last claim. Can you elaborate? I remember seeing papers (arxiv.org/pdf/alg-geom/9402013.pdf, for example) claiming to have "proved" things based on finiteness of $h_{F}(X)$, though in stable instead of semi-stable category. I thought Bost investigated the lower bound issue and had some work done in (link.springer.com/content/pdf/10.1007/BF01231533.pdf). I am not sure if bringing Belyi maps would give better estimates for his lower bound. But I guess you must have thought about this... $\endgroup$ Oct 6 at 22:02
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I think there is basically no hope of proving such a comparison (and it is probably not true, but my argument for this is less convincing).

Such a comparison theorem would have two parts:

(1) Proof that curves with a simple dessins have small Weil height.

(2) Proof that curves with small Weil height have simple dessins.

Moreover the two results proven should be comparable in the sense that the upper and lower bounds for the simplicity of the dessins in terms of the Weil height should not be too far apart.

The proof of part (2) would certainly be a strengthening of a proof that algebraic curves defined over $\mathbb Q$ have a dessins at all. So we could either try to come up with a new proof of this fact or look at the existing proof and see if it can be strengthened to prove this result.

I think the main proof known is still the original one of Belyi, which proceeds by an explicit construction.

For a proof by explicit construction to be comparable with a proof of (1), as discussed above, it should certainly have the property that the, for a curve generated from a dessins, the construction produces a new dessins not too much more complicated than the original (for whatever measure of complexity).

This is flagrantly untrue for Belyi's method, which produces a dessins (i.e. a map to $\mathbb P^1 $ ramified over $0,1, \infty$) by a iterative process of composing maps which are simple to define but can have quite high degree. So they will typically be enormously larger than the original map, in degree/number of edges or whatever other invariant you choose, unless you "feed in" knowledge of the correct map as the starting point of the algorithm (which the (1)-(2) comparability can't assume).

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If I am not mistaken, this precise issue was attacked by Dirk Smit during 1990s in a series of papers on Communications of Mathematical Physics. I am not sure why his papers are not quoted more widely in the number theory community nowadays. One reason may be he joined industry. But experts in my social circle are certainly familiar with his work.

Some relevant links are His paper on height function over moduli spaces using triangulations and His paper on obtaining "critical" string measure (for the dimension=26 case?) as well as an earlier paper on a continuous version of the height function (see the end).

I would suggest to start reading from the third paper and moving back to the first and second paper. The first paper basically answers your question and gives much more (not just how to triangulate the surface and how this is related to dessins d'enfants, but also the distribution of the height function near the boundary of $\mathcal{M}_g$). A general survey by Yuri Manin can be found via this link.

I should add some suggestions on questions like "It has been about 30 years...So what next???". To follow up on Dirk's work, I would suggest to focus on explicit examples like $g=2,3$ and do some concrete calculations. Recently a paper independently replicating a lot of Dirk's result (https://arxiv.org/abs/1902.02420) was published. I do not think the discrepancy he raised in the end of the first paper was ever addressed, though.

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