178 Normat 59:3-4, 178–191 (2011)

Galois theory and coverings

Dennis Eriksson, Ulf Persson

Hausdorﬀ Institute for Mathematics

Bonn,Germany

dennis.eriksson.se@gmail.com

Department of Mathematics

Chalmers University of Technology

Göteborg, Swden

ulfp@chalmers.se

1 Introduction

In this overview we will focus on the theory of coverings of topological spaces and

their usage in algebraic geometry and number theory. Galois theory is in its essense

the theory of correspondence between symmetry groups of ﬁeld extensions and the

ﬁeld extensions, providing a link between group theory and ﬁeld theory. Coverings

of topological spaces are provided with the same type of interpretation. Here a cov-

ering of a topological space X is basically a topological space with a map Y æ X

such that Y and X "look similar" locally. The Galois theory of coverings will be

a correspondence between symmetries of such covers and the fundamental group,

the latter playing the role of the Galois group, and we recall this in the ﬁrst s ection.

This is in fact more than just an analogy, and in the case of curves we can es-

tablish a direct link between topological covers and ﬁeld extensions of C(z) going

back to Riemann. This will be the subject of the next sections.

If one considers in particular the case of coverings of the sphere with three critical

points, this somehow incredibly sets up a correspondance between algebraic curves

deﬁned over number ﬁelds and topological covers. These covers can moreover be

easily realized by simple drawings. So simple, that essentially any drawing on paper

by a child gives an example, and Alexander Grothendieck baptized them "dessins

d’enfants" (French: children’s drawings). In fact he writes

This discovery, which is technically so simple, made a very strong impression

on me, and it represents a decisive turning point in the course of my reﬂec-

tions, a shift in particular of my center of interest in mathematics, which

suddenly found itself strongly focused. I do not believe that a mathematical

fact has ever struck me quite so strongly as this one, nor had a comparable

psychological impact. This is surely because of the very familiar, non-technical

nature of the objects considered, of which any child’s drawing scrawled on a

bit of paper (at least if the drawing is made without lifting the pencil) gives

translation from Esquisse d’un programme [5], by Leila Schneps

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 179

aperfectlyexplicitexample.Tosuchadessinweﬁndassociatedsubtlearith-

metic invariants, which are completely turned topsy-turvy as soon as we add

one more stroke.

These provide a way to encode information on the Galois group of the rational

numbers in terms of combinatorial data.

All in all, these notes are meant to suggest some intriguing connections between

more or less classical topology and complex analysis to much more modern devel-

opments in algebraic and arithmetic geometry, which provide new ways to look at

the Galois group of Q.

2 Fundamental groups and topological Galois coverings

Let (X, •) be any pointed topological space. Recall that the fundamental group of

(X, •) is the group of loops starting and ending at •, up to continuous deformation.

It is denoted ﬁ

(X, •)=ﬁ

(X). The group structure is given by composition of

loops. In what follows we will only consider well-behaved topological spaces, to

avoid pathologies, namely CW -complexes or topological manifolds.

We will give a second standard characterization of the fundamental group in terms

of coverings. Recall that, given a topological space X,acoveringY of X is a

topological s pace Y , with a map f : Y æ X, such that for every point p œ X,

there is a neighborhood U

of p and a set T (with discrete topology) together with

a commutative diagram

≠1

)

T ◊ U

where the upper map is a homemorphism.

Intuitively, locally around each point p, the inverse image of f are a number of

copies of X indexed by the set T . The covering is said to be trivial if we can take

= X, and to avoid this case will assume Y is connected in what follows.

Example 1. Consider the circle S

= {z œ C, |z| =1}. The map z ‘æ z

is a

covering map of the circle with itself, with the set T being the cyclic group Z/n.

Another covering is given by f : R æ S

,f(t)=exp(2ﬁit).

A loop in X is a map “ : S

æ X or equivalenty a map “ :[0, 1] æ X such

that “(0) = “(1). A loop can in general not be lifted to a cover, but the map “ on

the interval can be lifted to “

but then typically “

(0) ”= “

(1).Ifweﬁxx = “(0)

then it turns out that y = “

(1) œ f

≠1

(x) only depends on the homotopy class of

“. Thus we get a map ﬁ

(X, x) æ f

≠1

(x) (easily seen to be surjective), if we ﬁx

y œ f

≠1

(x) we can think of ﬁ

(Y,y) as the subgroup of ﬁ

(X, x) consisting of loops

such that “

(1) = y. We can in particular conclude

Example 2. If X is simply connected, i.e. ﬁ

(X, x)=0, then any covering of X

is necessarily trivial.

180 Dennis Eriksson, Ulf Persson Normat 3-4/2011

Any topological space admits a universal covering space. This is a simply con-

nected topological space

X with a covering p :

X æ X, such that any other

connected covering f : Y æ X,thereisacoveringg :

X æ Y such that fg = p.

It will be unique in the following sense. If we ﬁx a point x œ X, and for every

connected covering f : Y æ X a point y œ Y such that f(y)=x, then for the

universal covering p :(

X,Âx) æ (X, x), and a covering f :(Y, y) æ (X, x), the map

g :(

X,Âx) æ (Y,y) is the unique one such that g(Âx)=y.

Remark 1. If Y is the universal covering of X then ﬁ

(X) is in a 1-1 correspon-

dence with any ﬁber f

≠1

(x).

Example 3. If a group G acts on X properly discontinously (i.e. for each orbit

Gp we can ﬁnd an open cover of disjoint sets U

– gp permuted by G), then the

map X æ X/G is a Galois covering.

Example 4. The universal cover of S

is R while the universal cover of C

(ho-

motopic to S

)isC the covering given by z æ e

,whilethecoveringrestrictedto

the upper half-plane H (given by Im(z) > 0) gives a map onto the punctured unit

disc (0 < |z| < 1).

Example 5. A more involved example, with special relevance to this article, is

given by a group-action on the upper half plane. The subgroup of Moebius trans-

formations with integral coeﬃcients PSL(2, Z) given by matrices

with

determinant 1 is called the modular group  and acts on the upper half-plane but

not properly because of ﬁx-points. However if we look at the normal subgroup

given by the kernel (2) of the surjection  æ PSL(2, Z

)(= S

) given by reducing

the entries modulo 2 we get a properly discontinuous action. The quotient will be

isomorphic to H/(2)=P

\{0, 1, Œ}. T he latter space is homotopic to the ﬁgure

eight, whose fundamental group is freely generated by the two obvious loops.

For any covering f : Y æ X, the group of deck transformations Aut(f) is the

group of automorphisms of Y , commuting with the map to X. Locally it means

that the various covers are permuted. In particular, deck transformations induce

automorphisms of the ﬁber f

≠1

(x) and an inclusion Aut(f) ™ Aut(f

≠1

(x)).

We say that a covering f : Y æ X is a Galois covering, if Y is connected and G =

Aut(f) acts transitively on f

≠1

(x) for some (and thus any) x œ X. Equivalently, if

Y ◊

Y = {(z, z

) œ Y ◊ Y,f (z)=f(z

)},

then the map

G ◊ Y æ Y ◊

given by (g, z) ‘æ (z, gz) is an homeomorphism.

Theorem 2.1 (Fundamental theorem of Galois theory for topological spaces). If

the covering is moreover a universal covering space p :(

X,Âx) æ (X, x), there is an

isomorphism between the fundamental group and the group of deck transformations:

ﬁ

(X, x) æ Aut(p).

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 181

Moreover, there is a correspondance between conjugacy classes of subgroups of

ﬁ

(X) and coverings of X. The normal subgroups correspond to Galois covers.

Equivalently, Galois coverings with group G correspond to surjective homomor-

phisms ﬁ

(X) æ G.

The conjugacy classes are related to the fact that once we omit the point x,

ﬁ

(X) is only determined up to inner automorphism.

Example 6. In Example 1 we have ﬁ

)=Z. There are three types of subgroup

of Z. First of all, there is the whole group, this corresponds to the trivial cover with

the identity map. Secondly, there are the groups generated by an integer n ”=0, ±1,

which corres ponds to the covers with cyclic group Z/n. Finally, there is the 0- group,

which corresponds to the universal cover R æ S

Remark 2. Let us say that a subgroup of a group G is co-ﬁnite if it has ﬁnite

index, and let us assume that the intersection of all co-ﬁnite normal-subgroups only

consists of the identity. (The trivial example being G ﬁnite). We can then deﬁne a

Hausdorﬀ topology by letting those be a basis for the open sets around the identity.

This will in general not be a com plete space, its completion will be a compact and

totally disconnected space, and referred to as the pro-ﬁnite completion of G and

denoted by

‚

G. The groups G and

‚

G will have the s ame ﬁnite quotients

,butexcept

in the trivial cas e (when both are equal) the latter will be un-countable, while the

ﬁrst will typically be countable.

Example 7. The pro-ﬁnite completion of the fundamental group of an algebraic

variety w ill be called the algebraic fundamental group, the idea being that from an

algebraic point of view one only ’sees’ the ﬁnite covers. The algebraic fundamental

group of C

will be the product of all p≠adic integers Z

Example 8. The algebraic closure Q of Q is the union of all ﬁnite extensions of

Q. As a ﬁeld extension of Q it has of course a group of automorphisms, but it is

not so easy to explicitly determine it. However if we do it locally i.e. noting that

morally any ﬁnite Galois group Gal(K/Q) ought to be the quotient of Gal(Q/K)

we can recapture such a group in a formal way, given by its ﬁnite quotients. It is

however impossible to give any example of non-trivial elements in it, apart from

complex conjugation, as such will depend on an inﬁnite number of choices. In the

case of F where F is a ﬁnite ﬁeld, we clearly have two candidates, the bona ﬁde

Galois group generated by Frobenius, and its proﬁnite completion, which we have

already encountered in Example 7 .

So far we have only been discus sing the topological picture but as our examples

show, in practice they often come equipped with additional structure. One obvious

such is a complex structure and it should be obvious that a topological c over of a

complex manifold gets canonically a complex structure, and in this case one s ays the

covering is analytical. A beautiful set of results mainly due to J.-P. Serre, usually

called the GAGA-principle

, provides a s trong link between topological, analytical

and algebraic coverings. One of the theorems surrounding the GAGA-principle

states (with some intentional imprecision) that if we are given a suitable open subset

assuming G is ﬁnitely generated

Géométrie Analyt ique et Géométrie Algèbrique, [6]

182 Dennis Eriksson, Ulf Persson Normat 3-4/2011

U of an algebraic variety X, and a ﬁnite topological covering U

æ U, then this can

be uniquely extended to a ﬁnite holomorphic and algebraic map X

æ X,where

is algebraic. In short, ﬁnite topological coverings of (suitable) open subsets U of

X correspond to ﬁnite algebraic maps X

æ X which are coverings when restricted

to U.

Example 9. Suppose , 

™ C are open subsets. An analytical covering f :

 is a holomorphic function f such that f

(z) ”=0for all z œ 

. This follows from

the inverse function theorem, which states that under this assumption, f admits a

local holomorphic inverse around z.

The map z ‘æ z

is thus a analytic covering

outside of z =0above which it is given by taking the quotient with the natural Z

action given by z ‘æ e

2ﬁi

z. Cf. Example 6.

Remark 3. From example 5 we get a proof of Picard’s theorem to the eﬀ ec t that

any entire function which omits two values (three with Œ) is a constant! Such a

function gives a map into P

\{0, 1, Œ} and can thus be lifted to a holomorphic

function into the unit disc. But any bounded entire function is constant.

3 Algebraic Curves and Riemann Surfaces

The most basic examples of varieties are the compact varieties of com plex dimen-

sion one. Since they have real dimension two, they have also been called Riemann

surfaces. In what follows, we use the term "algebraic curve" to emphasize the alge-

braic nature, and the term "Riemann surface" to emphasize the analytical structure.

The ﬁrst are deﬁned globally, while the second local point of view was ﬁrst intro-

duced by Bernhard Riemann in the second half of the 19th century, and are still a

central topic of study. Their top ological classiﬁcation is simple: they are all homeo-

morphic (in fact diﬀeomorphic) to a sphere with handles attached. The number of

those is the genus of the curves. They can however have various diﬀerent complex

structures which distinguish them, but as opposed to the higher-dimensional case

they are all projective, given as the locus cut out by homogenous polynomials in

some projective space. They come in three types discussed below.

Example 10. The simplest Riemann surface is the (unique up to bi-holomorphism)

Riemann surface on genus 0, P

, homeomorphic to the 2-sphere, S

. It is realized as

the complex line C with a single point at inﬁnity, so that P

= Cﬁ{Œ}, the so called

Riemann sphere. Any algebraic map from P

to P

is given by is given by a rational

function f

where f

and f

are polynomials. Since the fundamental group of

the sphere is trivial, these can never be topological covers, unless it is a trivial

cover, which corresponds to the degrees being one, and the rational function given

by a Moebius transformation. Note incidentally that those form the automorphism

group of the ﬁeld C(z) preserving C.

The same example works in higher dimensions, given that we instead use the condition that

the determinant of the Jacobian is non-zero det f ”=0.

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 183

Example 11. The tori are given by C/ where  is a lattice of rank two over the

reals. They get complex structures from the one on C, (which will constitute the uni-

versal cover) but those will vary depending on the . By doubly-periodic functions

they can be embedded as a cubic in P

with an equation that can be normalized

to ZY

= X

+ aZ

X + Z

b, such that 4a

+ 27b

”=0. Two such equations will

deﬁne bi-holomorphic curves iﬀ they have the same j-invariant j(a, b)=

+27b

They are typically presented via period-parallelograms, which can be normalized as

spanned by 1,· where · œ H will be the parameter. The condition of biholomorphy

then translates into parameters belonging to the same orbit of the modular group

 (cf. Example 5).

Example 12. All the rest of the curves (i.e. g Ø 2) can be described in interesting

ways as quotients of the unit-disc (which hence is their universal cover). Examples

are given by plane curves of degree d>3 by the se t of zeros of homogenous

polynomials of degree d in three variables and with non-vanishing gradients. They

will constitute compact Riemann surfaces of genus g =(d ≠ 1)(d ≠ 2)/2.

Because all compact Riemann surfaces are algebraic it follows that all essen-

tial information about them is encoded algebraically in their function ﬁelds. For a

Riemann surface C we will denote by C(C), the ﬁeld given by all meromorphic func-

tions C æ C (or equivalently all holomorphic maps C æ P

). The function ﬁeld of

is C(z) which is the simplest and plays the same role as Q in classical ﬁeld theory,

while other function ﬁelds will be ﬁnite extensions of it. Conversely to any func-

tion ﬁeld (i.e. an extension of C of transcendence degree one) corresponds a unique

Riemann surface C. Furthermore any non-trivial holomorphic map f : C

æ C

gives rise to an injection C(C) æ C(C

) by sending a function C æ P

to the

composition C

æ C æ P

(a so called pull-back). Conversely any ﬁeld extension

C(C) æ C(C

) corresponds to a holomorphic map f : C

æ C.Whilethereis

only one Q in a number ﬁeld (a ﬁnite extension of Q) one can express C(C) in

an inﬁnite number of ways as a ﬁeld extension of C(z). In particular, a compact

Riemann surface toge ther with a meromorphic function „ is the same thing as a

ﬁnite ﬁeld extension of C(z) together with a polynomial equation P œ C(z)[T ] for „.

Example 13. In Example 11, x = X/Z can be thought of as a meromorphic

function on E c lassic ally represented as the doubly periodic meromorphic function

˝(z) on C. The ﬁeld of such meromorphic functions is C(E) and can be thought

of as a quadratic extension of C(x) given by y

= x

+ ax + b (with y = Y/Z).

Remark 4. For an polynomial P œ C(z)[T ], expanding the relation P („)=0and

clearing denominators, we get a polynomial relation P („, z)=0and by considering

„ just as a variable we get a plane curve which we can compactify by homogenizing

the equation. This curve will in general be singular, but it will have a unique

desingularization C which identﬁes the ﬁeld with C(C). In Example 13 we recapture

E as the cubic Y

Z = X

+ aXZ

+ bZ

The degree of an extension C(C) æ C(C

) can also be seen as the degree of

the associated map „ which is the same as the maximal cardinality d of a ﬁbre

„

≠1

(x),xœ C

which is the cardinality of all but a most ﬁnite number of exceptions.

This follows from the fact that the ﬁbre „

≠1

(x) can be put in a 1-1 correspondence

184 Dennis Eriksson, Ulf Persson Normat 3-4/2011

with the complex solutions to the equation P =0obtained by substituting the

value x for the formal variable z. The map is said to be ramiﬁed over the points

x when #„

≠1

(x) <d. A map is unramiﬁed iﬀ it is a cover, so in particular P

being simply connected, has no unramiﬁed non-trivial covers. Any holomorphic

multi-degree map over P

has to be ramiﬁed at at least two points, those which

are only ramiﬁed at two points are easily classiﬁed as being given up to Moebius

transformations by z æ z

. Maps ramiﬁed over exactly three points will be the

ones of main interest to us.

If we remove the ramiﬁcation points we get a covering map, and in fact the Galois

theory of ﬁelds corresponds exactly to our Galois theory of covers. In particular if

C(C

) is a Galois extension of C(C) with Galois group G the corresponding map

can be thought of as exhibiting C as C

/G, with the action of G free away from the

inverse images of ramiﬁcation points. Those occur exactly as the images of points

with non-trivial stabilizers.

We summarize our observations in the following translation be tween compact

Riemann surfaces and their function ﬁelds:

Compact riemann surfaces Function ﬁelds

Compact Riemann surface C Field K/C, trdeg

(K)=1

C(z)

Holomorphic map C

æ C Field extension K ™ L

Meromorphic function C æ P

Irreducible polynomial over C(z)

Degree of holomorphic map C

æ C Degree of ﬁeld extension K ™ L

Holomorphic map C

æ C Field extension K ™ L

with Galois group G with Galois group G

Example 14. (cf example15) Setting

K(z)=

(z ≠ 1)

+ z + 1)

(z + 1)

≠ z + 1)

we get a sequence of ﬁeld extensions

C(k(z)) µ C(z

) µ C(z)

corresponding to P

æ P

and given succesfully by the equations

≠ (

k+1

k≠1

)w +1=0and z

≠ w =0(where w is the element w(z)=z

This correspondance provides a very strong link between algebra and topology,

which we illustrate with the relation with the fundamental group. Let  ™ P

be a ﬁnite set of points of cardinality r say. Then a d-sheeted cover of P

\ 

corresponds to, by Galois theory, a map from ﬁ

\ ) to the symmetric group

on d elements. The group ﬁ

\ ) is generated by clockwise loops ¸

,¸

,...,¸

around each point in , with the relationship ¸

· ... · ¸

=1.Thisisthefree

group on r ≠1 elements. Using the GAGA-principle from the previous section, this

means that any compact Riemann surface with a d-sheeted map C æ P

ramiﬁed

in r ﬁxed points corresponds to a map from {1,...,r ≠ 1}æS

such that the

group generated by the image acts transitively on {1,...,d}, giving us a wealth of

examples of algebraic curves. This characterization was already known to Riemann,

and it is usually known as Riemann’s existence theorem.

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 185

Remark 5. The Eulernumber 2 ≠ 2g of the cover C is given by

d(2 ≠ r)+dr ≠

≠ 1) = 2d ≠

≠ 1)

where r

is the ramiﬁcation index at the point c œ C. This allows us to give a

complete classiﬁcation of all the Galois coverings ramiﬁed over P

\  which are

still P

i.e. g =0, in particular if follows that the cardinality of  is at most three.

Conversely it gives a complete classiﬁcation of all the ﬁnite subgroups of the group

of Moebiustransformations PGL(2, C).

4 Dessins d’enfants

In this section we will focus on the case of coverings ramiﬁed in at most three

points. By a Moebius transformation, we can suppose the points are 0, 1 and Œ.

While all compact Riemann surfaces can be realized as ramiﬁed covers of P

,three

points adds extra restrictions. An important observation is that in this case, then

the corresponding curve can be represented by polynomials with coeﬃcients in a

number ﬁeld, i.e. a ﬁnite ﬁeld extension of Q. This is rather astonishing, as the

curves which can be deﬁned over a number ﬁeld are very special (in the same sense

the algebraic numbers are special in the set of complex numbers). From the point

of view of algebraic geometry, it is however essentially a standard usage of Weil’s

descent theory.

In the 70’s, Alexander Grothendieck wondered if it could possibly be true that the

converse is also true, i.e. if the curve is de ﬁned over a number ﬁeld, can it be real-

ized as a cover over P

ramiﬁed at three points? This seemed as a very optimistic

assertion, even though there was no counter example. To his as tonishment, during

the International Congress of Mathematic ians in Helsinki ’78, the Russian math-

ematician G. V. Belyi announced this precise statement (see [1]). Moreover, the

proof was completely elementary, and only used general properties of polynomials.

The complete argument ﬁt easily on two pages.

More precis ely, the above discussion says that any curve C, together with a holo-

morphic map — : C æ P

only ramiﬁed at three points, is necessarily deﬁned over

a number ﬁeld. This says that the ﬁeld extensions of C(z), only ramiﬁed at z,z ≠1

and 1/z (in the sense of algebraic number theory) correspond to ﬁeld extensions

Q(z) only ramiﬁed at the same places. The pair (C, —) is called a Belyi pair, and

— a Belyi function.

We will now describe a topological recipe which describes such pairs. On the sphere,

color the point 0 white, and the point 1 black, and draw the line I =[0, 1] between

them. If we are given a Belyi pair (C, —), we can then consider =—

≠1

(I).Thisis

a graph traced on C, whose nodes are the white and black inverse images of 0 and

1 respectively, and it is bipartite, that is every edge has exactly one white and one

black node. Furthermore the complement C \ splits up into disjoint open sets U

each one c ontaining exactly one star point s i.e. an e leme nt in the inverse image

—

≠1

(Œ). As those sets are the inverse images of the simply connected set P

\ I

ramiﬁed at exactly one point they will be patches biholomorphic to the open unit

186 Dennis Eriksson, Ulf Persson Normat 3-4/2011

disc, and the map analytically equivalent to z ‘æ z

”

where ”

is the ramiﬁcation

index of — at s. (Note that U

will be a polygon bordered by 2”

edges and its

image given by U

/(Z

”

)).

Example 15. The ﬁgure below illustrates the case of P

æ P

with S

imbed-

ded in SO(3, R) µ PGL(2, C) and with the Belyi function

K(z)=

(z ≠ 1)

+ z + 1)

(z + 1)

≠ z + 1)

Note that we have three slices (each delineated by two meridans) permuted by

z ‘æ e

2ﬁi

z, and each of those are rotated by the involution around its star point.

As to the dessin at the right above, the missing white node is at inﬁnity. Note also

that we have a complete bipartite graph. This example is readily generalized to

any action of dihedral groups D

We can also extend the graph by dotted lines. In fact draw an equator on the

sphere by extending I and thus also including Œ.Itwillbisectthespherein

two halves, say a gray and a white, and each of those will have inverse images

biholomorphic to the unit-disc. For each edge we can clearly ﬁnd two wings of

diﬀerent colors which are attached to it and making up a butterﬂy

with the edge

as its body. We can then think of the map to P

by ﬂapping the wings, making

them attach on their edges. And this map becomes global, by doing it for each

edge. In the picture above we see the tesselation given by butterﬂies.

Conversely, suppose we are given a bipartite graph  traced on a g-hole donut

S, such that each component of S \ is homeomorphic with a disc. Then put a star

in each of those components and connect it to all the nodes in its closure. This will

give a tesselation of triangles, which should be colored according to whether 0, 1, Œ

deﬁnes a positive or negative orientation. Each edge will be adjacent to exactly one

triangle of each color, forming a butterﬂy ready to have its wings ﬂapped. This

gives local maps to S

to have them ﬁt together we need to identify elements in

diﬀerent images. It is not a priori clear how to do that, as we have a lot of freedom

to do so, the one restraint being that points on the dotted edges go down to two

diﬀerent S

, but by having done that, getting a ﬁxed target sphere, we can endow

it with a complex structure, letting the images of the white, black and star nodes

go to 0, 1, Œ respectively. By GAGA we can induce a complex structure on this

covering from its target and then getting a Belyi pair (C, —),

to use the terminology from [7].

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 187

Example 16. That basically any drawing by a child corresponds to a "dessin

d’enfant" is a theorem, which says that any ﬁnite graph (which can always be

made bipartite by adding a color in the middle of two equally colored nodes if

necessary) can be embe dded into a g-holed donut. In fact, suppose we are given a

cyclic ordering of the edges around every node on our bipartite graph .Thenwe

can attach open discs to edges in a coherent way, to obtain a polygon which is our

surface S, which can be veriﬁed to be compact and orientable, and so homeomorphic

to a g-holed donut.

Example 17. Let us explain, in line with the Galois theory of coverings, how a

dessin corresponds to a map {0, 1}æS

, such that the image generates a group

acting transitively on {1,...,d}. Number the edges from 1,...,d. Connect edges

lying on the same orbit of the image of 0 with a white dot, and similarily for the

edges on the same orbit of the image of 1 by a black dot. This deﬁnes a cyclic

ordering around each node, and a bipartite graph, and the above example shows

how to ass ociate a surface S where it is embedded. The transitivity just means that

the graph is connected.

Let us consider dessins which come from polynomial functions P œ C(x),deﬁning

a map P : P

æ P

which is totally ramiﬁed at Œ over Œ. By normalization we

can write P (x)=kx

(x ≠1)

„(x),(where„(0),„(1) ”=0), and a multiple solution

to P (x)=⁄ is found by looking at the zeros of P

(x)=kx

n≠1

(x ≠ 1)

m≠1

((n(x ≠

1) + mx)„(x)+x(x ≠ 1)„

(x)) = kx

n≠1

(x ≠ 1)

m≠1

Â(x).Theinterestingzeroes

are of course those of Â(x)=(n (x ≠ 1) + mx)„(x)+x(x ≠ 1)„

(x).BeingaBelyi

polynomial is equivalent with P (–

)=P (–

) for any two such roots –

,–

and by

chosing k appropriately we can assume that the common value is 1.

The dessin of any such polynomial will be a tree, and we draw it in the plane

C with its natural embedding in C ﬁ{Œ} = P

, and each white or black node will

have a valence given by its multiplicity. This is usually enough in simple cases to

determine the graph, but as we will see not always so.

n m

Example 18. „(x)=1then Â(x)=(n +

m)x ≠ n and setting k =

(n+m)

n+m

(≠m)

we nor-

malize. (Note when n = m =1we simply have

a polynomial ramiﬁed at just two points Œ,

)

n m

Example 19. „(x)=(x ≠ a) then

Â(X)=(n+m+1)x

≠(a(n+m)+n≠1)x+an

and thus its discriminant =(n + m)

≠

(2n(n + m) ≠6n ≠2m)a +(n ≠ 1)

.Whena is

chosen so that =0, then automatically we

get a Belyi polynomial with a dessin on the left.

When the discriminant is non-zero, we need to

choose a such that both roots to the quadratic

gives the same value of P . This is more involved.

When done we get the one on the left.

188 Dennis Eriksson, Ulf Persson Normat 3-4/2011

Notice that if n ”= m we get two diﬀerent

choices. We can work out the case n =3,m=2

then Â(x)=6x

≠ (5a + 2)x +3a. The two so-

lutions are of the form –±

D where – =

5a+2

and D = –

≠

. The condition that they

give the same value of the polynomial P is by

Galois theory that it belongs to Q(a).Setting

’ = – +

D we can explicitly work out P (’)=

A(–)+B(–)

D and the condition that P (’) œ

Q(a) is that B(–)=0. B turns out to be a

quintic polynomial which can be worked out ex-

plicitly as 400–

≠ 800–

+688–

≠ 276–

+27–+1.

Reducing modulo 3 one sees that it is indeed irreducible, and using Maple one can

conclude that its Galois group is S

. As the Galois group acts transitvely on the

polynomials P and hence on the two two graphs, we conclude that the graphs are

invariant under A

but are switched by odd permutations.

Example 20. We can also compose a Belyi

polynomial — with any polynomial map unrami-

ﬁed outside the inverse image under — of 0, 1, Œ

in particular we can consider x

≠1).On

the left we have the case n =1

Example 21. Deﬁne the Chebyshev polyno-

mial T

by T

(cos x) = cos nx and look at the

ﬁbers of T

= a.Ifa = cos z we can choose

x =

2ﬁ

k the corresponding cos x will all be

distinct unless z = mﬁ i.e. a = ±1.Thenthe

polynomial is

is ramiﬁed above 0, 1, Œ.

Even if a polynomial is given is it hard to determine the graph, to say nothing

about producing such pairs given the combinatorial data, which can be stated as

the ﬁrst basic problem of dessins.

The Fermat cubic (associated with the lattice

Z[ﬂ]) has an action by Z

the quotient is P

giv-

ing a meromorphic map of degree three totally

ramiﬁed. The associated — satisﬁes the equation

—

= x(x ≠ 1). (Or more elegantly — = z given

the Fermat equation x

+ y

= z

ramiﬁed at

≠1, ≠ﬂ, ﬂ

). Note that we can choose in this case

a regular hexagon as fundamental domain for the lattice action. Geometrically this

corresponds to the fact that Fermat cubics are characterized by having three ﬂexed

tangents pass through a point. Projection from such a point gives the required map.

The example above can readily be generalized to any polygon with 4d +2sides

and with opposite edges identiﬁed. This will give a 2d +1 sheeted cover totally

ramiﬁed over three points and of genus d

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 189

Dessins d’enfants and their relations to covers of the sphere were already used in

work by Felix Klein in 1978/79 ([3], [4], without Belyi’s theorem). There, he called

them Linienzüge (German: plural of "line-track").

5 Galois actions on dessins

The fact that a curve is deﬁned over a number ﬁeld K allows us to deﬁne an action

of the corresponding Galois group G = Gal(K/Q) on Belyi pairs. More speciﬁcally,

a Belyi function — : C æ P

satisﬁes a polynomial P œ K(z)[ T ] and hence we can

deﬁne for any ‡ œ G the polynomial P

‡

by acting on the coeﬃcients of the rational

functions. This deﬁnes a new curve C

‡

that does not have to be isomorphic to C

(but will topologically be the s ame) and also a new function —

‡

: C

‡

æ P

which

is also only ramiﬁed at three points. However, the corresponding dessins may look

quite diﬀerent see example 19.

Example 22. Consider again Example 11, and suppose that a, b œ

Q. The proof

of B elyi’s theorem associates to the function (x, y) ‘æ x from E(a, b):y

= x

ax + b to P

a Belyi pair (E(a, b),—).ThenE(a, b)

‡

= E(‡ · a, ‡ · b), and this is

biholomorphic to E(a, b) if and only if their j-invariants are equal, i.e. if j(a, b)=

+27b

is ﬁxed by ‡.

Before describing this action in more detail, let us explain how the outer auto-

morphisms of F

gives automorphisms of the set of dessins. Thinking of a dessin

as a Belyi pair, and hence a ﬁnite index subgroup H of ﬁ

\{0, 1, Œ})=F

it corresponds to a surjection p : F

æ F

/H. Since we have not ﬁxed a base

point in the fundamental group, we only care about F

up to inner automorphism,

so an automorphism should really be an outer automorphism (outomorphism?). If

we apply such a „ œ Out(F

), the composition p„ deﬁnes another surjection with

kernel the group „(H), and so another dessin. A somehow more natural, but more

complicated group of automorphisms on dessins is given by the outomorphisms of

„

, the proﬁnite completion of F

. This group turns out to have the same ﬁnite

quotients as F

(cf. Remark 2), so its outomorphisms, which is much bigger than

Out(F

), also acts on dessins in the same type of way.

We are now ready to "describe" which automorphisms of dessins come from the

Galois group. We have already noted that ﬁnite covers of P

\{0, 1, Œ} correspond

to certain ﬁeld extensions of Q(z). Given two diﬀerent coverings corresponding to

two subgroups N

and N

of F

, they are dominated by a third covering corre-

sponding to N

ﬂ N

. This means that the corresponding two ﬁeld extensions of

Q(z) is contained in a third one, and if we take the union of all of them we obtain

aﬁeldM, which is some weak algebraic analogue of the universal covering space

of P

\{0, 1, Œ}. The Galois group Gal(M/Q(z)) is then the correspondance be-

tween Galois coverings and Galois extensions the proﬁnite completion of F

„

The sequence of ﬁeld extensions Q(z) ™ Q(z) ™ M induces by Galois theory an

isomorphism

Gal(M/Q(z))/ Gal(M/Q) = Gal(Q(z)/Q(z)).

190 Dennis Eriksson, Ulf Persson Normat 3-4/2011

Since Gal(Q(z)/Q(z)) = Gal(Q/Q) this says

Gal(M/Q(z))/

„

= Gal(Q/Q).

In general, if we have a normal subgroup H of a group G, there is a map G ‘æ

Aut(H), given by g ‘æ [ h ‘æ ghg

≠1

]. The image of H is by deﬁnition the group of

inner automorphisms of H, and we get a map G/H æ Aut(H)/ Inn(H)=Out(H).

Applying this in the case above, we obtain a map Gal(Q/Q) æ Out(

„

). Example

22 above moreover proves the following important corollary to Belyi’s theorem :

Corollary 5.1. The action of Gal(Q/Q) on the set of dessins is faithful, i.e. if

‡ œ Gal(Q/Q) acts trivially on all dessins, then ‡ is itself trivial. That is, the

constructed map

Gal(Q/Q) æ Out(

„

)

is injective.

The proof is essentially by noticing that for any non-trivial element ‡ in the

Galois group, we can pick an element ⁄ œ Q on which it acts non-trivially. Then,

in Example 22 we can always solve for j(a, b)=⁄, and the element ‡ will deﬁne a

new cubic curve which will not be isomorphic to the original one.

From this description it is not at all c lear if a given action on dessins coming from

an element in Out(

„

) come from the Galois group. In [2], Drinfel’d deﬁnes a much

smaller group

‰

GT , the Grothendieck-Teichmüller group, which still contains the

Galois group. It is not known whether they are equal or not.

The s ec ond main question on dessins is related to this Galois-action. It is not

obvious when two dessins are related by a Galois conjugation. Since the dessins are

so simple, and admit an action of the Galois group, it should be possible to extract

some type of data which distinguish whether two dessins are Galois conjugate or

not. Are there any combinatorial, topological or even algebraic invariants which

can distinguish whether two dessins are in the same Galois-orbit? There are some

rather easy invariants that necessarily must be invariant. For example, the genus of

the surface a dessin is traced on is an obvious invariant. Another simple invariant

is the number of white and black nodes, number of e dges or the number of faces. A

slightly more subtle invariant is the so called degree-sequence. This is a decreasing

sequence of numbers, stating the number of edges coming out of the white (or

black) nodes. A yet more c omplicated combinatorial invariant is the subgroup of

in Example 17. It is known that these invariants, or other more complicated

known combinatorial invariants, do not distinguish Galois-orbits.

For the interested reader further references, from which also some of the above

material is taken, can be found in [7] (for an early account of the theory), [8], [9]

and of course the original Esquisse d’un Programme in [5].

References

[1] G. Belyi, On Galois extensions of a maximal cyclotomic ﬁeld Izv. Akad. Nauk

SSSR, Ser. Mat. 43:2 (1979), 269-276 (in Russian), [English transl.: Math. USSR

Izv. 14 (1979), 247-256].

Normat 3-4/2011 Dennis Eriksson, Ulf Persson 191

[2] V. Drinfel’d On quasitriangular quasi-Hopf algebras and a group closely connected

to Gal(Q/Q), Leningrad Math. J., 2 (1991), 829–860.

[3] F. Klein, Über die Transformation siebenter Ordnung der elliptischen Funktionen

Math. Annalen 14: collected as pp. 90–135 (in Oeuvres, Tome 3).

[4] F. Klein, Ueber die Transformation elfter Ordnung der elliptischen Functionen

Mathematische Annalen 15 (3–4): 533–555, collected as pp. 140–165 (in Oeuvres,

Tome 3).

[5] A. Grothendick , Esquisse d’un Programm e manuscript, published in “Geometric

Galois Actions”, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes

242,Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283.

[6] J.-P. Serre, Géométrie algébrique et géométrie analytique, Annales de l’Institut

Fourier 6 (1956), 1–42.

[7] G.B. Shabat, V. A. Voevodsky, Drawing curves over number ﬁelds, The

Grothendieck Festschrift Volume III, Birkhäuser (1984), 199–227.

[8] F. Herrlich, G. Schmithüsen, Dessins d’enfants and origami curves in Handbook of

Teichmüller Theory, Volume II, IRMA Lectures in Mathematics and Theoretical

Physics Vol. 13, 2009.

[9] L. Schneps, Dessins d’enfants on the Riemann sphere in The Grothendieck Theory

of Dessins d’Enfant, London Math. Soc. Lecture Notes 200, Cambridge Univ.

Press, 1994