Belyi-extending maps and the Galois action on dessins d'enfants
Melanie Wood
备注：
13 pages, 7 figures; submitted for publication; revisions are that the paper now deals only with Galois invariants of dessins, and that material is slightly expanded
发表信息：
Publications of the Research Institute for Mathematical Sciences, 42 (2006), no. 3, 721--738
We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck's dessins d'enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We give an example of a new invariant differing on two dessins which have the same values for the other readily computable invariants.
This note is concerned with the disproof of the most general case of Parker's conjecture. The conjecture relates a certain group theoretic objects to the field of moduli of a Dessin d'enfant.
Riemann's Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins d'enfants with $d$ edges. In previous work, the first author and collaborators exploited the correspondence between Belyi maps and their generating systems to provide explicit equations for two infinite families of dynamical Belyi maps. We complete this picture by describing the dessins d'enfants for these two families.
Let $N$ be a positive integer. For any positive integer $L\leq N$ and any positive divisor $r$ of $N$, we enumerate the equivalence classes of dessins d'enfants with $N$ edges, $L$ faces and two vertices whose automorphism groups are cyclic of order $r$. Further, for any non-negative integer $h$, we enumerate the equivalence classes of dessins with $N$ edges, $h$ faces of degree $2$ with $h\leq N$, and two vertices, whose automorphism groups are cyclic of order $r$. Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with $N$ e…
Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected dir…
In this paper we associate to a dessin d'enfant an associative algebra, called a Brauer configuration algebra. This is an algebra given by quiver and relations induced by the monodromy of the dessin d'enfant. We show that the dimension of the Brauer configuration algebra associated to a dessin d'enfant and the dimension of the centre this algebra are invariant under the action of the absolute Galois group. We give some examples of well-known algebras and their dessins d'enfants. Finally we show that the Brauer configuration algebras of a dessin d'enfant and its dual share the same path algebra…
We study the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the category of Belyi functions (finite, étale covers of $\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1,\infty\}$). We describe a new combinatorial $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant for whose monodromy cycle types above $0$ and $\infty$ are the same. We use a version of our invariant to prove that $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of Belyi functions whose monodromy cycle types above 0 and $\infty$ are the same; the proof of this r…
We present an algorithmic way of exactly computing Belyi functions for hypermaps and triangulations in genus 0 or 1, and the associated dessins, based on a numerical iterative approach initialized from a circle packing combined with subsequent lattice reduction. The main advantage compared to previous methods is that it is applicable to much larger graphs; we use very little algebraic geometry, and aim for this paper to be as self-contained as possible.
Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-…
We apply machine-learning to the study of dessins d'enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg-Witten curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. T…
We first introduce hyperbolic analogues of Belyi maps, Shabat polynomials and Grothendieck's dessins d'enfant. In particular we introduce and study the Shabat-Blaschke products and the size of their hyperbolic dessin d'enfants in the unit disk. We then study a special class of Shabat-Blaschke products, namely the Chebyshev-Blaschke products. Inspired by the work of Ismail and Zhang (2007) on the coefficients of the Ramanujan's entire function, we will give similar arithmetic properties of the coefficients of the Chebyshev-Blaschke products and then use them to prove some Landen-type identities…
The connections amongst (1) quivers whose representation varieties are Calabi-Yau, (2) the combinatorics of bipartite graphs on Riemann surfaces, and (3) the geometry of mirror symmetry have engendered a rich subject at whose heart is the physics of gauge/string theories. We review the various parts of this intricate story in some depth, for a mathematical audience without assumption of any knowledge of physics, emphasizing a plethora of results residing at the intersection between algebraic geometry, representation theory and number theory.
We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichmüller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_…
We study general properties of the dessins d'enfants associated with the Hecke congruence subgroups $Γ_0(N)$ of the modular group $\mathrm{PSL}_2(\mathbb{R})$. The definition of the $Γ_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $Γ_0(N)\backslash\mathrm{PSL}_2(\mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $\mathbb{Z}/N\mathbb{Z}$. The natural action of $\mathrm{PSL}_2(\mathbb{R})$ on the lattices defines a dessin d'enfant s…
In this paper, a construction of an infinite dimensional associative algebra, which will be called a \emph{Surface Algebra}, is associated in a "canonical" way to a dessin d'enfant, or more generally, a cellularly embedded graph in a Riemann surface. Once the surface algebras are constructed we will see a construction of what we call here the associated \emph{Dessin Order} or more generally the \emph{Surface Order}. This provides a way of associating to every algebraic curve $X$, with function field $k(X)$ (defined over an arbitrary field $k$) the representation theory of its Surface Algebra a…
Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.
Four-dimensional CFTs dual to branes transverse to toric Calabi-Yau threefolds have been described by bipartite graphs on a torus (dimer models). We use the theory of dessins d'enfants to describe these in terms of triples of permutations which multiply to one. These permutations yield an elegant description of zig-zag paths, which have appeared in characterizing the toroidal dimers that lead to consistent SCFTs. The dessins are also related to Belyi pairs, consisting of a curve equipped with a map to P^1, branched over three points on the P^1. We construct explicit examples of Belyi pairs ass…
We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for som…
Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher…
A dessin d'enfant, or dessin, is a bicolored graph embedded into a Riemann surface. Acyclic dessins can be described analytically by pre-images of certain polynomials, called Shabat polynomials, and also algebraically by their monodromy groups, that is, the group generated by rotations of edges about black and white vertices. In this paper we investigate the Shabat polynomials and monodromy groups of planar acyclic dessins that are uniquely determined by their passports.
Brane tilings, sometimes called dimer models, are a class of bipartite graphs on a torus which encode the gauge theory data of four-dimensional SCFTs dual to D3-branes probing toric Calabi--Yau threefolds. An efficient way of encoding this information exploits the theory of dessin d'enfants, expressing the structure in terms of a permutation triple, which is in turn related to a Belyi pair, namely a holomorphic map from a torus to a P^1 with three marked points. The procedure of a-maximization, in the context of isoradial embeddings of the dimer, also associates a complex structure to the toru…
The object of this paper is the study of a class of dessins d'enfants, the so-called diameter four trees. These objects, first introduced by G. Shabat, can be considered as the simplest non trivial example of etale covers of the projective line minus three points. Their arithmetic properties are still mysterious, and their study can inspire the understanding of more general situations. Here, the main interest is devoted to the action of the absolute Galois group on these (isomorphism classes of) coverings. In particular, in many cases, we are able to distinguish Galois orbits and to describe t…
In this paper we study an action of the absolute Galois group $Γ$ on bicolored plane trees induced by the action of $Γ$ on equivalence classes of conservative polynomials which are the simplest example of postcritically finite rational functions. We establish some basic properties of this action and compare it with the similar action provided by the Grothendieck theory of "Dessins d'enfants".
We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has wo…
