DESSINS D'ENFANTS A N D GENUS ZERO ACTIONS by LINH TON D U O N G B.A. York University, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard s I T H E UNIVERSITY O F BRITISH C O L U M B I A September 1999 © Linh Ton Duong, 1999 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract A Dessin D'Enfant is a cellular map on a Riemann surface ramified over {0, l , o o } . We will describe Grothendieck's correspondence between the set of isomorphism classes of dessins and the set of ismorphism classes of algebraic curves defined over Q . We also describe the action of the Galois group of the algebraic numbers, Gal(Q/Q), on the dessins. Finally, we also talk about groups that admit genus zero actions on Riemann surfaces and investigate the nature of Belyi functions on these Riemann surfaces. ii Table of Contents Abstract ii Table of Contents iii Chapter 1. Introduction 1 Chapter 2. Riemann Surfaces 2.1 Definition of a Riemann Surface 2.2 Mappings on Riemann Surfaces 2.3 Riemann Surface of Many-Valued Meromorphic Functions 2.4 The Riemann Surface of an Algebraic Function 2.5 The Genus of a Compact Riemann Surface 2.6 Fundamental Group of a Space S 2.7 Covering Spaces of Riemann Surfaces 4 4 6 7 12 13 14 15 Chapter 3. Automorphisms of the Upper Half-Plane H 3.1 Uniformisation Theorem and Automorphism Groups 3.2 Conjugacy Classes of PSL{2,R) 3.3 Fuchsian Groups 3.4 Fundamental Regions 3.5 The Quotient-Space of H / r 19 19 20 21 24 27 Chapters Dessins D'Enfants 4.1 Definition of a Dessin 4.1.1 Historical Background 4.2 Belyi's Theorem 4.3 Grothendieck's Correspondence 4.3.1 Definitions and Results 4.3.2 Algebraic Curves defined over Q 4.3.3 Grothendieck's Correspondence 4.3.4 Ramified Coverings of S\{0, l,oo} 4.4 The Action of Gal (Q/Q) on dessins 4.4.1 Dessins D'Enfants Obtained from Belyi pairs (X,0) 4.4.2 Belyi Pairs Represented by Permutation Groups — 4.4.3 Plane Trees and Shabat Polynomials 29 29 30 31 33 33 36 37 39 41 43 45 48 iii iv Table of Contents 4.4.4 Chapter 5. Examples 51 Genus Zero Actions on Riemann Surfaces and Dessins D'Enfants 55 5.1 Introduction 55 5.2 Genus Zero Actions on Riemann Surfaces 56 5.2.1 Background 57 5.2.2 Correspondence between Genus Zero Actions and Short Exact Squences of Groups 58 5.3 Actions on Riemann Surfaces of Genus 0 and 1 5.4 The Zero Actions of ZM Groups G 4 ( - l ) on Riemann Surface . . 60 59 5.5 Genus Zero Action of G i(-l) P i and Dessin D'Enfants Pt Bibliography 61 .69 Index 71 iv Chapter 1 Introduction In an unpublished manuscript, titled I'Esquisse d'un Programme, Alexander Grothendieck [Gro85] introduced a new way of approaching the problem of describing the structure of the absolute Galois group Gal(Q/Q). Here, <Q> denotes the set of algebraic numbers (that is a 6 Q a is a root of some non-zero polynomial f(x) whose coefficients are elements of Q). The absolute Galois group Gal(Q/Q) is the group of automorphisms of Q which fix Q. During his attempt at solving this problem, Grothendieck became interested in studying the action of Gal(Q/Q) on the set of dessins d'enfant. "A dessin d'enfant" (meaning- a child's drawing) is a cellular map on a compact orientable topological surface. The map can be roughly visualized as a finite set of points on the surface, connected by afiniteset of edge, in such a way that the edges and vertices form a connected set (with particular properties) which cuts the surface into open cells. Figure 1.1: Plane tree drawn on a sphere Figure 1.1 is an example of a dessin d'enfant that is a plane tree drawn on a 1 Chapter 1. Introduction 2 sphere. The action of Gal(Q/Q) depends on the correspondence between isomorphism classes of dessins and equivalence (isomorphism) classes of algebraic curves defined over Q . It is a well known fact that a Riemann surface X is compact if and only if it corresponds to algebraic curve (that is we can choose a polynomial p G C[x, y] so that the curve f(x, y) = 0 is a plane model representing this class). Thus, it turns out that the correspondence between isomorphism classes of dessins and equivalence classes of algebraic curves defined over Q is a direct consequence of Belyi's Theorem which states that X is defined over Q if and only if there is a function P : X - » P (C) ramified at 1 most over {0,1, oo} (f3 is called a Belyi function). It was Grothendieck who observed this correspondence via Belyi's Theorem. Thus, we call the correspondence mentioned above, Grothendieck's correspondence. This paper will give a rough sketch of Belyi's Theorem and describe Grothendieck's correspondence. We shall also describe the action of the G a lois group of algebraic numbers, Gal(Q/Q) on the dessins. A l l this is given in Chapter Four. The purpose of Chapter Two is to provide all the background material needed for Chapter Four. Chapter Two introduces the notion of Riemann surfaces, compact Riemann surfaces associated to algebraic curves and covering spaces of Riemann surfaces. This paper will also discuss the work of Kallel, Sjerve, and Song [KS], [KSS] on genus zero actions on Riemann surfaces. A group G admits a genus zero action on a compact, connected Riemann surface S (that is G x S —>• S) if for every non-trivial subgroup H of G the orbit space S/H has genus zero (that is S/H = P ^ C ) ) . Kallel and Sjerve looked at the following problem: Which groups G admit a genus zero action? The solution of this is discussed in Chapter 5. It turns out that there is correspondence between genus zero actions of groups G on Riemann surfaces S and short exact sequences of groups: i-)-n->rAG->i, where T is a Fuchsian group and II is the fundamental group of S. Thus, Kallel and Sjerve solved the problem by looking at Fuchsian groups T and short exact sequences of groups (stated above) and by using the Riemann-Hurwitz formula, which gives the relationship between the genus g of S and the n u m ber of fixed points of the action. Chapter Three provides all the background Chapter 1. Introduction 3 material for Chapter 5. Chapter Three introduces Fuchsian groups. The Uniformisation Theorem states that there are only three simply connected Riemann surfaces: the Riemann sphere S = P (C), the complex plane C, and the upper half complex plane M. — {z € C \ Im(z) > 0}. Let U denote the universal covering space (that is a simply connected Riemann surface) of a connected Riemann surface S of genus g. It turns out that 1 ' P ^ C ) if = 0 C if g = 1 ^H if g > 1 ff W= S = U/G for some subgroup G of AutU acting discontinuously on U. In most cases, it turns out that the universal covering space of S is HI and Fuchsian groups are subgroups of AutW = PSL(2, R) which act properly discontinuously on EL Thus, Chapter Three is devoted to Fuchsian groups. One of the groups that admit a genus zero action on a connected Riemann surface S of genus g is the Zassenhaus metacyclic group, G i(—1), where p is an odd prime. It turns out that genus g = p — 1 and the Riemann surface Pt 2p S is given by an equation of the form y = — ej) where ej G C and e/s 2 j=i are distinct. The remainder of Chapter 5 looks at Belyi functions on S. It turns out that there exists a Belyi function 3 : S -> IP (C) if and only if ej e Q for all j = 1 , 2 p . We prove this fact in Chapter Five and we use the approach of Shabat and Zvonkin to investigate the nature of Belyi functions on Riemann surface 5. I would like to take this opportunity to thank my advisor, Dr. Denis Sjervefor his help and guidance. 1 Chapter 2 Riemann Surfaces The purpose of this chapter is to give an overview of Riemann Surfaces needed for later chapters. The idea of Riemann surfaces first appeared in Riemann's doctoral dissertation in 1851. H e introduced Riemann surfaces as topological aids to help us understand many-valued functions. It wasn't u n til the early 1900's that it became apparent that Riemann surfaces were considered as important mathematical objects to be studied. The definition of abstract Riemann surface first appeared in H . Weyl's book [Wey55], The C o n cept of a Riemann Surface, in 1913. The contents of this chapter are taken from Qon87] and [Mas67]. 2.1 Definition of a Riemann Surface Definition 2.1 A surface S is a Hausdorfftopological space such that every point s in S has an open neighbourhood U homeomorphic to an open subset ofC(or equivalent^ R ). 2 Definition 2.2 Any surface S is covered by a family of open sets Ui such that for each Ui there is a homeomorphism $i : Ui ->• Wi where Wi is an open set ofCA set of such pairs A = {(Ui, <&j)} is called an atlas for S. Ifs £ Ui, we call (Ui, $j) a chart at s and z% = $i(s) a local coordinate for S. Definition 2.3 Let (Ui, <&j) and (Uj, <&i) be charts at s e S and Ui n Uj ^ 0. Then the function $j o^j : 1 $j(Ui f\Uj) ->• D Uj) is called acoordinate transition (see Figure 21). An atlas A on S is called analytic if all its coordinate transition functions are analytic. function 4 Chapter 2. Riemann Surfaces 5 c Figure 2.1: co-ordinate transition function In order to give the definition of analytic or meromorphic function on S, without having to specify a particular atlas, we have the following definition of compatibility of analytic atlases. Definition 2.4 Let A = {(Ui, and B = {(Vj, <&j)} be analytic atlases. A and B are said to be compatible if whenever (Ui, $;) G A and (Vj,^>j) G B satisfy Ui nVj^Q then o T ^([/j n Vj) ->• *<(C/i D Vj) is analytic. 1 : The compatibility of atlases is an equivalence relation and an equivalence class of atlases is called a complex structure on S. A surface with a complex structure is called a Riemann surface. In each case it is sufficient to specify one atlas on S, since this can be taken as a representative for its equivalence class, thus, defining a complex structure on S. Example 2.5 Let S — S = C U {oo} (E is often referred to as the Riemann sphere). Let the topology on S be the following: Consider the 2-sphere:S = {(xi,X2,x ) 2 3 take TV = (0,0,1) (north pole of S ) then 2 G K | x\+x\+x\ 3 TT : § \ 2 = ljinl^.lfwe {N} ->• C is the stereographic projection from N. We extend this map to TT : S - » S by defining n(N) = oo. 2 We use 7r to define a topology on E by defining the open sets of E to be the images under TT of open sets of § (in its usual topology as a subspace of R ) . 2 3 Thus, E is a topological space and has the same topological properties as the sphere S . 2 Chapter 2. Riemann Surfaces 6 There is an atlas A on S consisting of two charts (Ui, i = 1,2: $!=id:Ui^C Z^T z and U = 2 $ =J U Z\{0}, 2 : 2 -> C 1 2 oo !->• 0 The atlas .4 = {(t/i, $j) | i = 1,2} is an analytic atlas, giving a complex structure on E . Thus, E is a Riemann surface. 2.2 Mappings on Riemann Surfaces Let S be a Riemann surface. Definition 2.6 A function f : 5 ->• C is analytic if/or AZZ c/wrfs (U,§) on S, the function f o ->• C is analytic on $(U). Let (f/, <&) be a chart on S and let (V, S with UnV be a chart in a compatible atlas on = ®. Then $ o tf- is analytic and / o ty' = (/ o 1 1 o ($ o tf- ) 1 is analytic. Thus, the definition of an analytic function / : S -> C depends only on the complex structure on 5 . Example 2.7 The Riemann sphere E has two charts {(Ui, <&i), (t/2, $2)} as in Example 2.5. Given / : E -> C then / o S r ; $,({7) 1 C , for i = 1,2, have the form: / o ^ - 1 : C ^ C * ^ /(*) and / o$ " 1 2 Thus, / is analytic 2 = 0. 4==> /(z) and f(^) : C -> C are analytic on C and /(j) is analytic at Chapter 2. Riemann Surf aces 7 Figure 2.2: holomorphic function f : S\ -> Sz Definition 2.8 Let Si, S2 be Riemann surfaces. Then a continuous function f : Si -> 52 is defined to be holomorphic if, whenever (Ui,$i), (U2, $2) are charts on S\ and S2 respectively with Ui n / ( L 7 ) 0 then <f> o / o $1(^1 f~ (U )) -> C is analytic. See Figure 2.2. _ 1 : n 2 2 l 2 Definition 2.9 Let S2CI2.A continuous function f : S i ->• S2 w/ricfe is holomorphic is called a meromorphic function. Example 2.10 The meromorphic functions / : E —> E are the rational functions. See Qon87] for details. 2.3 Riemann Surface of Many-Valued Meromorphic Functions Instead of constructing individual Riemann surface for each many-valued meromorphic function, we construct a single abstract Riemann surface S which is so large that it has among its subspaces, the Riemann surface of every many-valued meromorphic function. One of the difficulties is defining precisely the concept of 'single-valued branch of a many-valued function w = f(z) at a point z G E ' . It is not enough to just consider those (z, w) G E x E such that w = f(z) because 2 branches may agree at z but not near z. Example 2.11 The many-valued function f(z) = y/l + y/z has four analytic branches on E \ {0,1,00}. There are two values for rj = <Jz and each of these values determines two values for f(z) = y ! + rj. A t z = 0, two of the four 7 Chapter 2. Riemann Surf aces 8 branches have the same value and the other two branches have the same value. Before we construct the Riemann surface <S, let us state the following definition and lemma. Definition 2.12 A function element is the pair (D, f) where D is a region such that f : D -> Eisa single-valued meromorphic function on D. Lemma 2.13 Let (D, f) and (D, g) be function elements defined on the same region D. If f = g on some non-empty open subset U ofD then f = g on D. See Qon87] for the proof of Lemma 2.13. Let a £ E , Two function elements (D, f) and (E, g) are said to be equivalent at a £ U C D n £ if and only if / = g on 17. The equivalence class of (D, / ) is called the germ of / at a and is denoted by [/]„. The set M of all germs [f] , for all a G E is called the sheaf of germs of meromorphic functions. If m = [/] £ M then / is meromorphic on some neighbourhood D containing a. Then we write m = [D, f] . The following definition defines a topology on M. a a a Definition 2.14 We define the D-neighbourhood D(m) ofm in M to be D(m) — {[f]b '• b G D}. A subset A C M is defined to be open iffor each m G A there is some D-neighbourhood D(m) C A. Now, we define the map, ip : M -> E "1 = [f]a a Note the following facts: (1) ip is continuous (2) is a Hausdorff space (3) {D(m),%l)D,m) is a chart at m if ij)(m) ^ oo (4) (D(m), J o ipD,m • [f]b I) is a chart at m if ip(m) = oo. (5) the charts in (3) and (4) form an atlas for M, which makes M an abstract Riemann surface. (6) ip : M —> E is meromorphic. Chapter 2. Riemann Surfaces 9 in. 1 t — >/w ,->v-/J. C2D Figure 2.3: Given m = [f] € M, the value of /(a) is independent of the particular choice of / because of Lemma 2.13. So, we are able to define a map, a cp: A 4 - » E m ^ f(a). <p: M -> E is meromorphic. See Figure 2.3. Now, we briefly describe the notion of meromorphic continuation along a path 7. Definition 2.15 A path 7 in S from a to b is a continuous function 7 : I — > • S, where I is rTze closed unit interval [0,1] = {s e R | 0 < s < 1}. If a = 7(0) and h = 7(1) we say rTwrt 7 is a pathfroma to b. We say that 7 is a cZosed path if a — b. We say that 7 is simple ifj(s) = j(s') implies s — s' or s — 0 and s' = 1. By abuse of terminology, we refer 7(1) as the path 7. We are only interested in connected components Q of M. The set ty is an abstract Riemann surface. The connected component containing germ m = [f] is called the unbranched Riemann surface ofm and is denoted by Q{m). The condition that Q is a connected component of M is equivalent to the requirement that, (a) the map xp : M -> E is onto and (b) for every path 7 : I -> E , and every [/] € Q with rp{[f] ) — 7(0) = a, there exists a unique path 7' : I ->• (/ with 7'(0) = [/] and 7 = ip o 7' (see Figure 2.4). a a a a Chapter 2. Riemann Surf aces 10 Figure 2.4: The function 7' : I ->• M is called the meromorphic continuation of[f] along 7. See [Jon87] for details. If A(z, w) is a single-valued function of two variables z and w, then the unbranched Riemann surface, MA of the equation A(z, w) = 0 is defined to the largest open subset of M on which A(ip(m), <f){m)) = 0. Thus MA = {m = [D, / ] I z and io = f(z) satisfy A(z, w) = 0 for all z G D}. a 0 Example 2.16 For example, A(z, w) = w — 2w + 1 — z, corresponds to the many-valued function w = f(z) = + \fz which has four analytic branches (see Example 2.11). The Riemann surface MA can be identified with the unbranched Riemann surface Q(m) because MA is the single component of M which can be obtained by the meromorphic continuation of one germ (or one of the 4 branches). 4 2 Now, we need to adjoin branch-points to M (see [Jon87]) for details. Let us just give the definition of a branch-point. Definition 2.17 Let S' and S be two surfaces. Let p : S' ->• S be a continuous onto mapping with the property that each s G S has an open neighbourhood U homeomorphic to the open disc D = {z G C | \z\ < 1} such that each connected component V ofp~ (U) is mapped homeomorphically by p onto U. The map p is called a covering map. l Chapter 2. Riemann Surfaces 11 Define the map ir for n E Z such n > 1 by: n 7r n :D— > •D z z' V p a U Figure 2.5: If for each s G S has an open neighbourhood U C 5 and a homeomorphism a : £7 —» D (D as defined in Definition 2.17) such that for each connected component V of there is a homeomorphism 8 : V ->• D such that a o p = 7r o 8 for some integer n > 1 (see Figure 2.5). n Definition 2.18 Ifn > I for some V as above then s' G V n p (s) isflbranchpoint of order n — l.We also say p is ramified over s and the ramification order over s is n. We also say that s is a critical value of p. _1 We let S be the union of M. and all the branch-points which can be adjoined as described in [Jon87]. We extend tp and <f> to functions from S to S. We have the following facts about S: (a) M is embedded as an open subspace. (b) The set S \ M of branch-points is discrete. (c) <S is an abstract Riemann surface. (d) The extensions of ip and 4> are meromorphic. Each branch-point [f] is attached to a unique connected component Q (m) of M, where m G D([f] ). The connected component of S, denoted S(m), is called the branched Riemann surface ofm. The branched Riemann surface <S^ of A(z, w) = 0 is the largest open subset of <S on which A(ij), 4>) — 0. We have that MA = Mil SA, SO, SA consists of the unbranched Riemann surface MA together with any attached branch-points. c c Chapter 2. Riemann Surf aces 12 2.4 The Riemann Surface of an Algebraic Function Definition 2.19 We say that w is an algebraic function of z (could be manyvalued) if the relationship between w and z has the form A(z, w) = 0 for some polynomial A(z, w). Example 2.20 The algebraic function A(z, w) — w - 2w + l-z to the many-valued function w = y/l + y/z. 4 2 corresponds By coUecting powers of w, A(z, w) can be written as: A(z,w) = a (z)w + ai(z)w ~ n n 1 0 + Va (z), n where di(z) is a polynomial in z and a (z) ^ 0. We say that n is the degree of A. For each fixed z € S, the equation A(z, w) = 0 is a polynomial equation with n distinct roots w. Values of z for which there are n distinct w roots are called regular points for A. The values of z satisfying one or more of the following conditions: 0 (1) z = oo; (2) <*,(*) =0; (3) A(z, w) = 0 has a repeated root w. form the set CA of critical points. The set CA is finite and the branch points of the branched Riemann surface SA all lie above critical points. Example 2.21 Consider A(z, w) — w — 2w +1 — z which corresponds to the many-valued function w = f(z) = \ / l + y/z. The critical points are 0,1, oo. 4 2 See [Jon87] for the proofs of the following four theorems. Theorem 2.22 If A(z, w) is an irreducible polynomial the Riemann surface, SA is connected. Theorem 2.23 IfA(z, w) is a polynomial then the branched Riemann surface, SA is compact. Theorem 2.24 Any compact abstract Riemann surface can be identified with the Riemann surface SA of some algebraic function A(z, w) — 0. Theorem 2.25 Every Riemann surface is orientable. Chapter 2. Riemann Surfaces 13 2.5 The Genus of a Compact Riemann Surface We start by stating the following theorem. Theorem 2.26 Each compact, connected, orientable surface is homeomorphic to a surface S formed by attaching g handles to a sphere, for some unique integer g > 0. g See [Jon87] for proof of Theorem 2.26 By Theorem 2.22, if A(z, w) is irreducible then the Riemann surface SA is homeomorphic to the surface S . We call g the genus of the surface. g Definition 2.27 The Euler characteristic of a compact, connected surface S, denoted by x(5), is defined to be x(S) = x{M) = V-E + F, where M is a polygonal subdivision of S with V vertices, E edges, F faces. Theorem 2.28 The Euler characteristic of a compact, connected, orientable surface S of genus g is given by x(S ) = 2 — 2g. 9 9 See [Jon87] for proof of Theorem 2.28. The following result is known as the Riemann-Hurwitz formula. Theorem 2.29 If S is the Riemann Surface SA of an irreducible algebraic equation A(z,w) = 0 of degree n in w, and if the branch-points have orders ri\, n ,..., n then the genus g of S is given by 2 r See [Jon87] for proof of Theorem 2.29. Example 2.30 Consider the algebraic function A(z,w) = w - 2w + 1 - z. There are 2 branch-points of order 1 at z = 0,1 branch-point of order 1 at z = 1 and 1 branch-point of order 3 at z — oo. Thus, g = l - 4 + ^(2 + l + 3)=0. So, SA is homeomorphic to a sphere. A 2 r Chapter 2. Riemann Surfaces 14 2.6 Fundamental Group of a Space S Given a topological space S, and a point a G S, we can define the fundamental group of S, denoted by ni (S, a). This group is a topological invariant of 5. Let 70,71 be two pathsfromatob in S. A homotopyfrom 70 to 71 in S is a continuous function f : I ->• 5 swc/t i/wi: Definition 2.31 2 / ( s , 0 ) = 70(a), /(a, 1 ) = /(0,*)=<*, /(M) foralls,t G I= =& [0,1]. Therefore, by Definition 2.31 for each £ G I, we have a path 7 from a to b t in 5, given by 7 (s) = / ( s , i), as t increases from 0 to 1, and j is continuously 4 t deformed within S from 70 to 71, while keeping the endpoints fixed. If there exists a homotopy from 70 to 71, we say that 70 and 71 are homotopic and we write 70 ~ 71. This is an equivalence relation and equivalence classes are called homotopy classes. If 7, 5 are paths in S with 7(1) = 5(0), then the product jS of 7 and S is: (lS)(s) = { 7(2a) *(2*-l) The inverse path of 7 is 7 - 1 0<s<\ i < 2 _ S < l _ (s) = 7(1 — s) The homotopy classes [7] of closed paths 7 from a given point a G 5 to itself form the fundamental group 7ri(5, a). The product [7][<5] of 2 classes is the class [7<5]. The identity element of the group 7Ti(5, a) is the class containing the constant path j (s) a = a for all s G I. The inverse of [7] is [ 7 ] . -1 Note that the group operations defined above are independent of the choice of representatives of the homotopy classes. The operation defined above satisfies the group axioms. Chapter 2. Riemann Surf aces 15 If S is a path-connected space then there is a path a from any point a G S to any point b € S and the map: 7ri(5,o)->7ri(5,6) [7] i->- [or 7a] 1 is an isomorphism. Thus, as an abstract group, 7Ti (5, a) is independent of choice of a G S, so we denote this group by ni (S). Definition 2.32 A topological space S is simply connected if it is path-connected and 7r(5,a) is the trivial group (one element group) for some a G S and hence for every a G S. Example 2.33 Let S = E \ {0,1,00}. Let a, 3, 7 be closed loops around 0, 1, 00 respectively. Then the fundamental group of S, ni(S) — {[a], [8], [7] | [a][3}[j] = 1). See [Mas67] for details. 2.7 Covering Spaces of Riemann Surfaces Definition 2.34 A covering space of a surface S is a pair (S', p) where S' is a space and p : S' —> S is a covering map (see Definition 2.17). s G 5 then p~ (s) is called the fiber above s and the cardinality of the set p~ (s), denoted | p ( s ) | is the number of connected components of p~ (U), where U is as defined in Definition 2.17. \p~ (s) \ is locally constant on 5. But, If 1 1 l _ 1 1 if S is connected then is constant and is called the number of sheets of the covering. Definition 2.35 Let {S[,pi), (S ,P2) be covering spaces ofS. A homomorphism is a continuous map ip : S[ -> S' such that the following diagram (Figure 2.6) is commutative. 2 2 Definition 2.36 A homomorphism ip of(S[,pi) into (S' ,P2) is called an isomorphism if there exists a homomorphism ip of (S' ,P2) into (S[,pi) such that both compositions ip<p and <pip are identity maps. 2 2 Chapter 2. Riemann Surf aces 16 S Figure 2.6: Two covering spaces are said to be isomorphic if there exists an isomor- automorphism is an isomorphism of a covering space onto itself. Automorphisms of covering spaces are usually called covering transformations. Note: A homomorphism of a covering is an isomorphism if and only if it is a homeomorphism in the usual sense. A covering transformation of (S',p) is a homeomorphism <? / : S' -> S' such thatpy? = p. Covering transfor- phism of one into the other. A n mations (automorphisms) form a group under composition. We denote the group of covering transformations of the covering space (S',p) by AutS'. Example 2.37 Let S' be the unbranched Riemann surface of f(z) — (refer to Qon87] for details) and S = £ \ {0, oo}. Define p = ip-.S'->S lf]a i-> a [e f] for k — 0 , 1 , k — 1, where e = exp(2m/k); these form a cyclic group of order k. So, the covering transformations are [/] a i-» h a Let Gbea group ofhomeomorphisms of a topological space X onto itself. G acts discontinuously on X if every x e X has a neighbourhood V such that V n g(V) = 0/or all non-identity element g e G. Definition 2.38 We state the following theorem which generalizes this idea. If S' is connected, then the group of G of covering transformations ofp : S' ->• S acts discontinuously on S'; in particular, a non-identity element ofG can have no fixed-points on S'. Theorem 2.39 Chapter 2. Riemann Surfaces 17 The following definition is actually a result (see [Mas67]) but we state it as a definition. Definition 2.40 We say that (S',p) is a regular covering space of S if, for each s G S, the group G of covering transformations acts transitively on the fiber p~ (s). l The following theorem shows that regular covering spaces are essential. Theorem 2.41 lf(S',p) is a regular covering surface of S, and ifG is the group of covering transformations, then there is a homeomorphism q:S^ S'/G q(s) •->• [s'] , G where s G Sands' G p~ (s). l See [Jon87] for proof of Theorem 2.41. Theorem 2.42 Every connected surface S has a covering space {S',p) such that S' is simply connected. See [Mas67] for proof of Theorem 2.42. We call (S',p),a universal covering space for S because it can be a covering space of any covering space of S. Theorem 2.43 If S is a connected Riemann surface then the universal covering surface {S',p) is a regular covering surface of S. See Qon87] for proof of Theorem 2.43. Proposition 2.44 If(S',p) is a universal covering space ofSthen AutS' = n\{S). See [Mas67] for proof of Proposition 2.44. Let p : ITI(S) —>• AutS' be the isomorphism. Let H be some subgroup of TTI (S). If S is a connected Riemann surface then by Theorem 2.42, S has a universal covering space (S',p) and by Theorem 2.43 and Theorem 2.41, there is a homeomorphism q : S ->• S'/p(H) where p(H) is a subgroup of AutS'. By the Uniformisation Theorem (stated in the chapter Chapter 2. Riemann Surf aces 18 3, section 1) S' = £ , C, or EL Then, using q we can give S'/p(H) a complex structure. So q : S —> S'/p{H) is a conformal equivalence (see chapter 3, section 1 for definition). Thus, S = S'/p(H) for some subgroup H of ni(S). We should also note the following fact: Conjugacy classes of subgroups of 7ri(5) determine a covering surface of S up to isomorphism. That is, there is a Galois correspondence { Conjugacy classes of 1 subgroups of 7Ti (5) J See [Mas67] for details. J Covering surface of S, |^ up to isomorphism Chapter 3 Automorphisms of the Upper Half-Plane EI In this chapter we will introduce the group PSL(2, M) and its discrete subgroups. As we will see, these two groups are very important in Riemann surface theory. Again, the purpose of this chapter is intended to provide preliminary concepts for later chapters. The contents of this chapter are taken from [Jon87]. 3.1 Uniformisation Theorem and Automorphism Groups Let Si, S2 be two Riemann surfaces. Definition 3.1 If f : Si ->• S and f~ : S -» Si are holomorphic homeomorphisms and the local coordinates are transformed conformally by f and f~ then f is a conformal equivalence or conformal homeomorphism. We say that Si and S are conformally equivalent and we write Si = S . l 2 2 l 2 2 The following theorem is called the Uniformisation Theorem because it allows us to parameterize compact Riemann surfaces by single-valued functions. Theorem 3.2 Every simply connected Riemann surface is conformally equivalent to just one of: (i) the Riemann sphere S; (ii) the complex plane C; (iii) the upper half plane M = {z e C | Im(z) > 0}. 19 20 Chapter 3. Automorphisms of the Upper Half-Plane H See [Bea84] for proof of Theorem 3.2. A conformal homeomorphism / : S ->• S is called an automorphism of S. The set of Aut S of automorphisms of 5 is a group under composition. Theorem 3.3 (i) Aut E PSL(2,C); (H)AutC {z^az + b\a,b&C,a^ (iii) AutM 0}; PSL(2,R). See [Jon87] for proof of Theorem 3.3. We should note that the groups of automorphisms in Theorem 3.3 are made of linear fractional or Mobius transformations (see chapter 2 of Qon87] for more details on these transformations). The Uniformisation Theorem (Theorem 3.2) tells us that there are 3 simply connected Riemann surfaces, so we have the following result which is important to us. If Sis a connected Riemann surface not conformally equivalent to the sphere E, the plane C, the punctured plane C \ {0}, or a torus then S has universal covering space S' = M (upper half-plane) and S is conformally equivalent to M/G for some subgroup G ofPSL(2, R) acting discontinuously on EL Theorem 3.4 See Qon87] for proof of Theorem 3.4. So, Theorem 3.4 tells us that in most cases, except four cases, the universal covering space is the upper half-plane EL Thus, AutM — PSL(2, R) is very important for us. So, in the remainder of this chapter we investigate the action of subgroups of AutM = PSL{2, R) on EL 3.2 Conjugacy Classes of PSL(2, R) PSL(2, R) = | T ( Z ) = : a, b, c, d G M, ad — be = 1 We classify the elements in PSL(2, R) in the following way: Parabolic Elements Parabolic elements are elements that have one fixed point in R U {oo}. These Chapter 3. Automorphisms of the Upper Half-Plane H 21 elements have trace 2 (i.e. \a + d\ = 2). There are two conjugacy classes of parabolic elements in PSL(2, R). Hyperbolic Elements Hyperbolic elements are elements that have two fixed-point in R U {oo}. These elements have trace greater than 2 (i.e. \a + d\ > 2). Every hyperbolic element of PSL(2, R) is conjugate to a unique element of the form: U\(z) — \z where A > 1. Elliptic Elements Elliptic elements are elements that have a pair of complex conjugate fixedpoints. These elements have trace less than 2 (i.e. \a+d\ < 2). Elliptic elements are conjugate to elements W € PSL(2, E) where ^4^ = ^ ( 1 ^ ) , 0< <2n. W(z) +i z + i' All elements in PSL(2, R) of finite order are elliptic. 9 K 3.3 Fuchsian Groups PSL(2, R) is a group and a topological space because it can be identified with the set {(a,b,c,d) E M ! \ad — be = 1} modulo the involution (a,b,c,d) i-» (—a, — b, — c, — d). 4 Definition 3.5 A Fuchsian group is a discrete subgroup ofPSL(2, R). (1) The hyperbolic element z i-> Xz, (A > 1) generates a Fuchsian group which consists of hyperbolic elements and the identity. (2) The Modular group Example 3.6 PSL(2,Z) = \T{Z) = °?L±}- ; a,b,c,d € ad — be = l\ is a Fuchsian group. Fuchsian groups do not have discontinuous behaviour (see Definition 2.38) because elliptic elements fix points, so these points do not have a neighbourhood V as in Definition 2.38. But, Fuchsian groups are discontinuous in a slightly weaker way. Chapter 3. Automorphisms of the Upper Half-Plane H 22 Definition 3.7 Let Gbea group ofhomeomorphisms of a topological space Y. Then G acts properly discontinuously on Y if each point y eY has a neighbourhood V such that ifg(V) D V ^ $for g^G, then g(y) = y. Theorem 3.8 Let Tbea subgroup ofPSL(2, R). Then F is a Fuchsian group if and only ifT acts properly discontinuous on EL See Qon87] for proof of Theorem 3.8. Corollary 3.9 Let T be a subgroup of PSL(2,R). Then T is a Fuchsian group if and only if for all z e HL the T-orbit of z, is a discrete subset o/HL See Qon87] for proof of Corollary 3.9. Now, we describe how to construct triangle groups. Definition 3.10 The unique path of shortest hyperbolic length between any two given points in EL is called a hyperbolic line segments or H-line segments. Note the following: (1) The H-line segments in M are arcs of semi-circles with center on the real axis or segments of Euclidean lines perpendicular to the real axis. (2) The elements of PSL(2,R) map H-line segments to H-line segments. Definition 3.11 Let Q be an H-line. Then H-reflection in Q is an H-isometry of HL other than the identity, which fixes every point ofQ. Theorem 3.12 An H-reflection in Q is the restriction of a Euclidean inversion in Q to the upper half-plane. See Qon87] for the proof of Theorem 3.12. Let r be an H-triangle with vertices v\, V2, v$, and angles ir/l, ir/m, at these vertices, respectively and sides Mi, M , 2 ir/n M 3 , at opposite sides of the vertices (I, m, n are positive integers). See Figure 3.1. Let B 4 be the H-reflection in the H-line containing Mj for i = 1,2,3. Note that every H-refection is an anti-conformal homeomorphism (it preserves angles but reverses orientation) of H and has the form: az + b z !->• —; -, a, 0, c, a € K, aa — be = —1. cz + a Chapter 3. Automorphisms of the Upper Half-Plane H 23 Mi Figure 3.2: Let T' = (Ri,R , R ). But Rt $ PSL(2, R), so, V is not a Fuchsian group. If r = V n PSL(2, R), it will turn out that T is a Fuchsian group. 2 3 we consider The image of T under Ri is the hyperbolic triangle -Ri(r) with sides Mi, Ri(M ), Ri{M ). Here R R Ri is the H-reflection in Ri{M ) because it fixes R\(m ) pointwise and R\R R~[ transforms the hyperbolic triangle RI(T) to RiRrf^iR^r)) = RIR (T). See Figure 3.2. l i?i(Mi) = 2 3 x 2 2 l 2 2 2 Continuing this, we see that the hyperbolic triangles surrounding the vertex v are 3 T, R^T), RIR {T), 2 {RiR^^R^r). RIR RI(T), 2 Note that RiR 2 is a product of 2 H-reflections fixing v , so it can be thought of as a hyperbolic 3 rotation about v through the angle 27r/n and (RiR ) n 3 2 = 1. Chapter 3. Automorphisms of the Upper Half-Plane H 24 The set {T(r) : T G T'} forms a tessellation of H (meaning that no two T'-images of r overlap and every point of M belongs to some T'-image of r.) If we let q be a point in r then the T'-images of q are the corresponding points of the other triangles of the tessellation and hence form a discrete set. Therefore, T-orbit of q is a discrete set, Vg G EL So, by Corollary 3.9, T is a Fuchsian group. Fuchsian groups constructed as above are called triangle groups. Every triangle in the tessellation of EI is of the form T(r) where T can be written as a finite product of the R 4 , i = 1,2,3. A n d we have the relations: R\ = R\ = R\ = (R-[R ) = (R R ) n 2 1 2 3 = (RiR3) m — 1. A n y other relations in the group can be deduced from these relations. Thus, in group theory, we can say that r = (A, B : A = B = (AB) = 1) where A = R R , B = R Rz n l m X 2 2 is a presentation of Y. Note: since we are working in the hyperbolic plane, 1/Z + 1/m + l / n < 1. 3.4 Fundamental Regions Let T be a Fuchsian group. Definition 3.13 We say F is a fundamental region for Y if F is a closed set such that: (i) U T e r T(F) = EL (ii) int(F) n T(int(F)) = $for allTeY\ {1}, where int(F) is the interior ofF. Example 3.14 (1) Let Y be a triangle group obtained from a hyperbolic triangle r (in Section 3.3). Then T\JR\ (T) is easily seen to be a fundamental region for r. (2) The fundamental region for the modular group PSL(2, R) is F = {z e H : \z\ > 1 and \Re(z)\ < 1/2}. See [Jon87] for details. See Figure 3.3. Chapter 3. Automorphisms of the Upper Half-Plane H 25 F Figure 3.3: z = i=$& and z = x 2 1 ± ^- We define the Dirichlet region for a Fuchsian group T. Suppose p G EI is not fixed by any element of F \ {1}. Then the Dirichlet region for F centered at p is: D (F) = {z e M : h(z,p) < h(T(z),p) for all T € F} , P where h is the hyperbolic distance between two points in EL Note that if p is not fixed by any element of F \ {1}, then D (F) is a connected fundamental p region for F. For Fuchsian groups, Dirichlet regions are bounded by H-lines and possibly by sections of the real axis. If two such H-lines intersect in M then their point of intersection is called a vertex of the Dirichlet region. Vertices of Dirichlet regions are isolated and therefore the boundary of the Dirichlet region consists of a union of (may be infinitely many) H lines and maybe sections of the real axis. Example 3.15 The Dirichlet region of the modular group F centered at p = hi, where k > 1 is D {F) = {z E M: \z\ > 1 and \Re(z)\ < 1/2}. ki Definition 3.16 Let F be a Dirichlet region for a Fuchsian group F and let u, v be vertices ofF. u and v are called congruent if there exists T e F satisfying T(u) = v. Congruence is an equivalence relation on the vertices ofF and the equivalence classes Chapter 3. Automorphisms of the Upper Half-Plane H 26 are called cycles. If'u is fixed by an elliptic element R then v is fixed by the elliptic element TRT~ . If one vertex of a cycle is fixed by an elliptic element then so are all vertices of that cycle. Such a cycle is called an elliptic cycle and the vertices of that cycle are called elliptic vertices. l Theorem 3.17 There is a one-to-one correspondence between the elliptic cycles ofF and the conjugacy classes of non-trivial maximalfinitecyclic subgroup ofT. See [Jon87] for the proof of Theorem 3.17. Definition 3.18 The orders of the maximal finite subgroups ofT are called the periods of T. Each period is repeated as many times as there are conjugacy classes of maximalfinitesubgroups ofT of that order. Example 3.19 A triangle group obtained from a hyperbolic triangle with angles 7r/Z, ir/m, ir/n (as described in Section 3.3) has periods Z, m, n. The Dirichlet region F is a hyperbolically convex region which is bounded by a union of H-lines. The intersection of F with one of these H-lines is either a single point or a segment of an H-line. These segments are called sides of F. Definition 3.20 Let rbea side of a Dirichlet region. IfT e T \ {1} and T(r) is also a side of F then r and T(r) are called congruent sides. Example 3.21 Recall the fundamental region for the modular group (Example 3.15). The two vertical sides of the region are congruent (because one of the sides are mapped to the other side by the transformation z H » Z + 1). The side on the unit circle between z\ = (—1 + iy/3)/2 and z — (1 + iy/3)/2 is 2 mapped to itself by the elliptic element z t-> — 1 /z of order 2. Theorem 3.22 Let {Ti} be the subset ofT consisting of those elements which pair the sides of somefixedDirichlet region F. Then {Ti} is a set of generators for T. Example 3.23 By Theorem 3.22, the modular group P S L ( 2 , Z) is generated z z + 1 and z — l/z. by Chapter 3. Automorphisms of the Upper Half-Plane H 27 3.5 The Quotient-Space of H/T Let [Z]T denote the T-orbit of z and the natural projection map n: m ->• H / r z [z] r Theorem 3.24 H / r is a connected Riemann surface and U : W -> W/T is a holomorphic map. Therefore, quotient-spaces of Fuchsian groups (H/r) are Riemann surfaces. Case 1: Let S be a connected Riemann surface not homeomorphic to E , C, and torus. By Theorem 3.4, the universal covering space of S is H and S = H / A (conformally equivalent), where S is a properly discontinuous subgroup of PSL(2, R) acting without fixed points (i.e. it contains no elliptic elements). Let A be a properly discontinuous group of automorphisms of H acting without fixed points (i.e. no elliptic elements). Then by Theorem 3.8, A is a Fuchsian group containing no elliptic elements. Case 2: Let A be a Fuchsian group containing element elements. Then every Riemann surface can be represented as the quotient-space H / $ A , for some Fuchsian group A . The benefit of using Fuchsian groups without elliptic elements to represent Riemann surfaces is seen by the following theorem. Theorem 3.25 Let A i , A 2 , be Fuchsian groups without elliptic elements. Then H / A i and H / A are conformally equivalent if and only if there exists T e PSL(2, R) such that T A i T " = A . 2 1 2 Note: If we consider two triangle groups with different periods then these two groups are not isomorphic and so, not conjugate in PSL(2, R). But the quotient-space H / A , where A is a triangle group, is the Riemann sphere E . Theorem 3.26 If F is a Dirichlet region for T then F/T is homeomorphic to H / r . Chapter 3. Automorphisms of the Upper Half-Plane H 28 We now state a result about Fuchsian groups whose quotient-spaces are compact. Theorem 3.27 Let F be a Dirichlet region for a Fuchsian group Y. Then U/Y is compact if and only if F is compact subset ofM. If T is a Fuchsian group such that M/Y is compact, then by Theorem 3.27, a Dirichlet region F for Y is compact. Thus, it has a finite number of sides (see Qon87] for details). A s a result, F has finitely many vertices and hence only finitely many elliptic cycles. By Theorem 3.17, Y has a finite number mi,...,m . (g | m i , ...,m ). of periods If r W/Y has genus g, then we say that Y has signature r The presentation of a group with signature (g | m i , m ) is r (A ,B ,A ,B ,...,A ,B ,X ,...,X 1 1 2 2 g g 1 \ X™> = • • • = X?* r = X ...X A B A-{ B^...A B A- Bl l r l l See Lehner [Leh64] for details. 1 g g g 1 = 1). Chapter 4 Dessins D'Enfants In this chapter, we will discuss what a "dessin d'enfant" is and describe the bijection between the set of isomorphism classes of dessins and the set of isomorphism classes of algebraic curves defined over Q . We shall also describe the action of Galois group of the algebraic numbers, Gal ( Q / Q ) , on the dessins. We will define the monodromy group M and the cartographic group C of a dessin. It will turn out that these groups are invariant under the action of Gal ( Q / Q ) on the dessin. The contents of this chapter is taken from [Sch94] and [JS97]. 4.1 Definition of a Dessin A dessin d'enfant (meaning-a child's drawing) is a cellular map on a compact orientable topological surface. The map can be thought of as a finite number of points (vertices) on a surface, with a finite number of edges connecting these vertices such that: Definition 4.1 (1) set of vertices and edges is connected (2) this set cuts the surface into open cells (3) a bipartite structure can be put on the map (we can mark the vertices with • and o such that direct neighbours of any vertex are all of opposite mark) Examples of dessins (all are drawn on Riemann surfaces of genus 0) are given in Figure 4.1. 29 Chapter 4. 30 Dessins D'Enfants Ife} CM (C} Figure 4.1: Examples of dessins drawn on Riemann surfaces of genus 0 There are 3 types of dessins. The first type is called "pre-clean". The pre- clean dessins are the ones whose o vertices have valencies (number of edges corning out of the vertex) less than or equal to 2 (see Figure 4.1 (b)). The second type is called "clean". The clean dessins are the ones whose o vertices have valency exactly equal to 2 (see Figure 4.1 (c)). The third type is called the general dessin (see Figure 4.1 (a)). 4.1.1 Historical Background In an unpublished manuscript, titled I'Esquisse d'un Programme, Alexander Grothendieck introduced a new way of approaching the problem of describing the structure of the absolute Galois group Gal (Q/Q). During his attempt at solving this problem, Grothendieck became interested in studying the action of Gal (Q/Q) on the set of dessins. Let us define a Belyi function. Definition 4.2 A meromorphic function 3 : X ->• S which is unbranched outside the set {0,1, oo} is called a Belyi function. We call 3 a pre-clean Belyi function if all the ramification orders over 1 are less than or equal to 2. We call 3 a clean Belyi function if all ramification orders over 1 are all exactly equal to 2. By Theorem 2.24, we know that a Riemann surface X is compact if and only if it corresponds to an equivalence class of algebraic curves. We can pick some polynomial A G C[z, w] such that the curve A(z, w) = 0 is a plane model representing the equivalence class. Belyi's theorem states that X is defined over Q (for us, this means that we can pick A G Q[z, w]) if and only if there is a Belyi function 3 : X -> E . Chapter 4. Dessins D'Enfants 31 So, to any Belyi pair (X, 8) (ie: the pair (X, 8), where X is an algebraic curve defined over Q and 8 is a Belyi function) we can identify a dessin by taking a simple dessin V\ on the Riemann sphere £ , and use 8 to lift T>\ to a dessin V on X. For example , if we take £>i to be the bipartite map with 2 vertices at 0 and 1 joined by a single edge [0,1] and take V = /3 ([0,1]). _1 A s a convention, we consider «'s and o's to symbolize the pre-images of 0 and 1 respectively (and mark the pre-image of oo by * such that exactly one • falls somewhere into each open cell). Under this identification, the pre-clean dessins correspond to pre-clean Belyi functions and the clean dessins correspond to clean Belyi functions. Grothendieck used Belyi's theorem to show that to any dessin, one can associate a covering 8 : X —> £ , where 8 is a Belyi function and X is a Riemann surface. A rigorous combinatorial proof of this was given in a 1975 article by two Grothendieck's students, J. Malgoire and C . Voisin [VM77]. D. Singerman and G . Jones QS78] also proved it around the same time. In VEsquisse d'un Programme, Grothendieck posed the following question: C a n every algebraic curve defined over Q be obtained by a dessin? The answer was yes via a corollary to Belyi's theorem which states that an algebraic curve defined over C is defined over Q if and only if there exists a clean Belyi function 8 : X -)• E . Therefore, we have a bijection between abstract dessins and isomorphism classes of Belyi pairs (X, 8). This correspondence will be explained in detail in later sections of this chapter. 4.2 Belyi's Theorem In this section we shall give a rough outline of the proof of Belyi's theorem and the corollary. The proofs can be found in [Ser89] and [Sch94] and [Bel79]. These two results are essential to the description of the Grothendieck's correspondence. The following result is called Belyi's Theorem (1979). Theorem 4.3 Let X be an algebraic curve defined over C. Then X is defined over Q if and only if there exists a Belyi function f : X -> F C. 1 Sketch of Proof. We only consider the one implication. (=$•) X is defined over Q . It is a fact that every curve is a ramified Chapter 4. Dessins D'Enfants (branched) covering of F C 1 32 (i.e. there exists / : X ->• P Q . We take / to be X defined over Q . Let S be the finite set of critical values of / (i.e. the set of images of the points of X where the differential is zero) and 5 c Q U {oo}. Lemma 4.4 If S c Q U {oo} is afiniteset then there is a covering 4>: ¥ C ->• P * C defined over Q and ramified over 0,1, oo such that <f>(S) C {0,1, oo}. l Take -K = 4> o / . Then by Lemma 4.4, <j> is a Belyi function and we are finished. We prove Lemma 4.4 for the special case when 5 C Q U {oo}. Let <f> : P C -> P C be a non-constant function. Let C^ be the set of critical values of 1 X <j), let = cf>(S) U Off, and let the cardinality of S be equal to n (i.e. \S\ = n). (a) If n < 3, then there exists an automorphism 4> of P C (namely a Mobius 1 transformation) defined over Q such that qb(S) C {0,1, oo}. (b) If n > 4, then we shall construct (f> : P C -> P C defined over Q such X that X C Q U {oo} and IS^I < n — 1. Then by induction on n, our Lemma 4.4 is proved for this case. Construction ofcf): Suppose 0,1, oo, a e S and a e Q and a £ {0,1, oo}. We try to look for <f> in the form of <j>(z) = z (z—l) where r, s axe rational integers, r ^ 0,s / 0,r + s ^ 0. Then (f)(0), (f)(oo), <p(l) 6 {0,oo}. The ramification of <> \ r s is given by dd> r , s = -dz H -dz = 0 4> z z—1 Choose r, s E Z such that a — z = Then r r+s C {0, oo, ^(a)} U <fr(S'), where S' = S\ {0,1,oo,a}. Hence, |0(S")| < n - 4. See [Sch94], [Ser89] or [Bel79] for the general case. • Corollary 4.5 An algebraic curve defined over C is defined over Q if and only if there exists a clean Belyi function 3 : X ->• P C . 1 Proof. If a : X ->• ¥ C is Belyi function then, 3 — 4a(1 - a) is a clean Belyi l function. • The following two examples are taken from [JS97]. Example 4.6 Let X = S and let 3(x) = x n for some integer n > 1. This covering is an n-sheeted covering which is ramified over 0, oo. So, 3 is a Belyi function. Chapter 4. Dessins D'Enfants Example 4.7 33 A n elliptic curve is a Riemann surface X of genus 1. It can be written in the following form: A), where A E C \ {0,1}. y = x(x -l)(x2 This form is called the Legendre normal form. Let X\ be this surface. By Belyi's Theorem 4.3, X\ is defined over Q <=> A E Q . When this is true, we can find a Belyi function 3 on X\. Consider the projection map TT : X -> £ x (Note: set of critical values of 7r is CV = { 0 , 1 , oo, A}). We obtain 3 by composing 7r with appropriate rational functions r : E ->• E . For example, let us take A = 1 + \ / 2 , which is an algebraic number with minimal polynomial p(x) = x 2 — 2x — 1 over Q . The map a : x takes CTT into { 1 , 2 , oo, 0}. N o w consider the function 7 : x i-> —: — — which 4(a; — l j maps the set { 1 , 2 , 0 0 , 0 } into { 0 0 , 1 , 0 0 , 0 } respectively. Composing these three functions, we get a Belyi function 3 = 7 o a o 7r: —>• E , of degree 8 on X ^ . 4.3 Grothendieck's Correspondence In this section, we will describe the Grothendieck's correspondence. But before we do that we state the precise definition of "dessins d'enfants", other key concepts, and some results needed for the existence of the correspondence. 4.3.1 Definitions and Results The definitions in this section are taken from [Sch94]. Definition 4.8 A Grothendieck dessin is a triple X c X C X where X is the topological model of a compact connected Riemann surface, XQ is afiniteset of points, Xi \ X is afinitedisjoint union of segments and X \ Xi is afinitedisjoint union of open cells, such that a bipartite structure can be put on the set of vertices XQ; namely the vertices can be marked with two distinct marks in such a way that the direct neighbours of any given vertex are all of the opposite mark. 0 0 x 2 2 2 Chapter 4. Dessins D'Enfants 34 Definition 4.9 Two dessins D = X C X c X and D' = X' c X[ C X' are isomorphic if there exists a homeomorphism from X into X inducing a homeomorphism from X into X[ and one from X into X' . We sometimes use the terminology abstract dessins for an isomorphism class of dessins. This indicates that the structure of the dessin is determined, but it is not associated with any particular embedding into a complex topological surface. 0 x 2 0 2 x 0 2 2 Q By the Corollary 4.5, we need only consider the clean dessins. Grothendieck also did this in VEsquisse d'un Programme. So now, we have the following def- inition. Definition 4.10 A pre-clean Grothendieck dessin is a triple X c X c X where X is the topological model of a compact connected Riemann surface, X is a finite set of points, X\ \ XQ is a finite disjoint union of segments and X \ X\ is a finite disjoint union of open cells. 0 x 2 2 0 2 In definition 4.10, all vertices in XQ are considered to have the same "mark" and somewhere on each edge is a vertex of the "opposite mark" (this can be anywhere on the edge, even on the end but it cannot be any of vertices in XQ). This gives us a natural bipartite structure on the dessin. Definition 4.11 A marking on a pre-clean dessin is afixedchoice of one point on each component of X \ Xo, and one point in each open cell of X \X\. We will always use the notation «for a point in X , ofor a point in X\ \ X and -kfor a point mX \X . x 2 0 2 0 x Definition 4.12 Let Dbea pre-clean dessin with afixedmarking. Then the flag set !F{D) of D is the set of triangles whose three vertices are marked •, o, and • in such a way that • is in the closure of the segment containing o, and that segment is in the closure of the open cell containing •. The oriented flag set T (D) is the set of flags the order of whose vertices is * - • - o when read counterclockwise. + Definition 4.13 The cartographical group Ci is given by three generators UQ, G\ and (72 together with the relations a\—a\ = a -=\ and [OQOI) = 1. The oriented cartographical group C is the subgroup of index 2 of Ci is given by all even words 0/C2. A generating set is given by po = o\o§, pi — GQU2 and pi — oio\, with the 2 2 2 relations p\ — 1 and P0P1P2 = 1- Chapter 4. Dessins D'Enfants 35 Figure 4.3: Now, we describe the action of the C on the flag set F(D). If F e T(D) is 2 a flag given in Figure 4.2 then OQ(F), cri(F) and 02(F) are given in Figure 4.3. We can denote an oriented flag F by a vertex • and an edge —> as above + because the • point and o edge of an oriented flag determines the position of the open cell • . If F + and p2(F ) + is an oriented flag given Figure 4.4 then p (F ), + 0 p\ (F ), + are given in Figure 4.5. This completely describes the action of C^ on the F (D). We consider C + 2 acting on the left, so that in the example of po = o\o§, 00 acts first. We usually consider only the set F (D) + of positively oriented flags. The set F (D) + has the following properties: (a) For each edge of a dessin, there are exactly two flags in F + (D). (b) Considered as a set together with the action of a particular group C^, F + (D) is independent of of the marking on TJ. It depends only on the abstract dessin(isomorphism class of dessin D). Note: If we consider only the set T (D), not all markings on D are required because, as stated earlier, each flag in F (D) can be represented by one vertex + + • and one edge —>• coming out of it. Chapter 4. Dessins D'Enfants 36 Figure 4.5: Lemma 4.14 Let D be a pre-clean dessin, F G F (D) be a fixed flag and BF,D = {a G C \a(F) — F}. Then BF,D is a subgroup of finite index in C£. Let F' G F (D) be another flag and B i be the stabilizing subgroup of F'. Then Bp>,D is conjugate to BF,D in C^. Moreover Bp,D depends only on the abstract dessin. + + + F i D See [Sch94] for proof of Lemma 4.14. Theorem 4.15 There is a bijection between the isomorphism classes of clean dessins and the conjugacy classes of subgroups ofC^ offiniteindex. See [Sch94] for proof of Theorem 4.15. Note: The forward implication (==>) of Theorem 4.15 is given by Lemma 4.14. Theorem 4.15 is due to Malgoire-Voisin [VM77]. Similar results were given by Jones and Singerman [JS78]. 4.3.2 Algebraic Curves defined over Q Let X be an algebraic curve defined over Q . Let 7Ti(S \ {0,1, oo}) denote the fundamental group of S \ {0, l , o o } , generated by closed loops 70, 71, and Chapter 4. 37 Dessins D'Enfants 7oo around 0, 1, and oo respectively with the relation 7071700 = 1- See Example 2.33. Lemma 4.16 There is a bijection between the conjugacy classes of subgroups offinite index o / 7 r i ( £ \ {0,1,00}) and isomorphism classes offinite coverings XofT, ramified only over 0,1,00. See chapter 2, section 2.7 for details of the correspondence. It should be noted that the ramification indices over 0 , 1 , 0 0 are given by the lengths of the orbits in the quotient space 7 r i ( E \ { 0 , l , o o } ) / i f under the action of 7 0 , 7 1 , 7<x> respectively, where i f is a subgroup of -K\ ( £ \ {0,1,00}) of finite index. Let ei )7i/ ...,efc i)7i denote the orbit lengths for i = 0 , 1 , 0 0 . The degree d of the covering is given by the index [71-1 (E \ {0,1,00}) : H] and the genus g of X is given by Hurwitz's formula (see Theorem 2.29 in chapter 2, section 2.5): k- 2g-2 = -2d + J2 I>;.7<-1)ie{0,l,oo} j=l Let ir'i = 7ri (E \ {0,1,00})/(•y ). Then the following result is a consequence 2 of Lemma 4.16. Corollary 4.17 There is a bijection between the conjugacy classes of subgroups of finite index of ir[ and the isomorphism classes of coverings of E ramified only over 0,1, and 00, such that all the ramification orders over 1 is at most 2. 4.3.3 Grothendieck's Correspondence We can now state the main result of this section, which is a consequence of Lemma 4.16 and Theorem 4.15 (Note: groups TT[ and C^ are canonically isomorphic). Theorem 4.18 There is a bijection between the set of abstract clean dessins and the set of isomorphism classes of clean Belyi pairs. We describe the Grothendieck's correspondence using the following topological construction. Given a clean Belyi pair (X, 3), let X 2 be the topological Chapter 4. Dessins D'Enfants model of X. Let X 38 = 8~ (0) and X = /3 ([0,1]) where [0,1] is the segment _1 l 0 ± of the real line on E . Then B~ (oo) gives a point in each open cell. The condil tion that 8 is a clean Belyi function is equivalent to the condition that every edge of the dessin has a vertex on each end. Figure 4.6: Now, given a dessin, we can associate a curve to the dessin. We may interpret the association in the following manner. The flags (considered as triangles with vertices • , o, and *) tessellate the topological surface X 2 with lozenges made of pairs of adjacent flags, one of which is positively oriented and the other one is negatively oriented. The common side of the two flags is of * - • type (See Figure 4.6). Identifying corresponding sides of the lozenge gives something homeomorphic to the sphere. We identify all these lozenges with E by the following association: • 0 O h-> 1 •k \—*r OO This identification gives us a morphism 8 : X -> E , ramified only over 2 0,1, oo with ramification orders corresponding to the valencies in the tessellation. We put a Riemann surface structure on X 2 by requiring 8 to be holo- morphic. We should note that the Belyi function associated to a given abstract dessin is not well denned because the dessin corresponds to an isomorphism class of Belyi pairs (X, 8), where 8 is defined only u p to automorphisms of the Riemann surface X. For example, if X is of genus 0 then 8 is defined u p to Chapter 4. Dessins D'Enfants 39 automorphisms of the Riemann sphere, PSL(2, C). 4.3.4 Ramified Coverings of E \ {0,1, oo} In this section we describe the Grothendieck's correspondence by considering X as a finite covering of E ramified only over {0, l , o o } such that all the ramification orders over 1 are at most 2. Let a; € E \ {0,1, oo} and let p~ {x) = {x\,X2,Xd} ing X. l be the fiber over x where d is the degree of the cover- Let 7 0 , 7 i and 7oo be loops going clockwise once around 0,1, oo respectively with initial and terminal point x. Then 7 0 , 71 and 700 induce permutations 0-0,0-1,0-00 of the set p~ (x) such that o- 010"oo = 1 (see [Mas67] for details l 0 of the action). Note the following facts: 1. ex 1 is of order 2. 2. Let M = (<7o,0-i I cr = 1)- M is a subgroup of the symmetric group Sd, 2 called the monodromy group of the branched coverings : X —> E . 3. If the covering X is connected then (0-0,01 | cr? = 1) is transitive on the set p~ {x). l 4. Conversely, given (cro, 01 I ° i = 1)/ there exists a connected covering of X of E ramified over 0,1,00 corresponding to it and all ramification orders over 1 are at most 2. 5. If all the ramification orders over 1 are exactly equal to 2, then d must be even and o\ must be a product of d/2 disjoint transpositions. Now, given an X as described above, or equivalently, given an even integer d and two permutations cro, <7i € Sd, where o\ is the product of d/2 disjoint transpositions and the subgroup ( 0 0 , 0 1 | of = 1) is transitive, we will describe how to draw the pre-clean dessin associated to X. Set O Q O = (oooi) . -1 The genus g of X can be calculated from the decomposition of the cr; into disjoint cycles and using the Hurwitz's formula: 2g-2 = d - n - n - n ^ , (*) 0 1 where rij is the number of disjoint cycles occurring in o-j. We begin to draw the dessin by writing a o o as a product of I disjoint cycles si, S 2 , s i . For 1 < j < I, let kj be the length of Sj and write Sj = Chapter 4. Dessins D'Enfants {hj,i2,j,—,ikj,j)- 40 For each SJ, 1 < j < I draw a kj-gon. Orient the edges of every /%-gon by going around it in a counter clockwise direction. Going around the edges of each /%-gon in order (you can start at any edge). We then label the edges with transpositions c r i ( i i j ) ) , j , cri(i . j ) ) . Each of fc these transpositions actually occurs in the disjoint cycle decomposition of o\. Now, we glue the I polygons in the following manner: identify the sides labeled by the same transposition, in the same direction. Clearly every edge is identified with exactly one other edge, so we get a compact topological surface S with no boundary and a natural dessin drawn on it by the identified edges of the polygons. There is a natural morphism 3 from this surface S to E (see [Sch94] for details). The morphism 3 marks the dessin and identifies the lozenges with E . By our construction, the covering 3 : S -> E has the same ramification properties as X, and so, S is isomorphic to X. We end this section with the following example which is taken from [Ton87]. Example 4.19 Consider X = {(z,w) | A{z,w) = w - 2w + 1 - z). The al4 2 gebraic curve X is defined over Q (see Example 2.16 and Example 2.20). We know by Belyi's theorem that there exists a Belyi function 3 on X. Consider the projection map: (z,w) ' y z. This is a 4-sheeted covering ramified only over 0 , 1 , oo. Thus, 3 is a Belyi function on X. Let 70,71,7oo be the loops around 0 , 1 , 0 0 . These loops induce the permutation CXQ, cri, Coo respectively, on the s e t p ( : r ) where x 6 E \ { 0 , 1 , 0 0 } . _1 We can order the set {ooOiY = 1 CT^CTQ 1 = p~ (x) so that a l 0 = (12)(34),ai = (1)(24)(3), (24)(34)(12) = (1234). The subgroup (CT ,CTI | O\ 0 = 1) = of the symmetric group 5 4 is the monodromy group of the branched covering 3 : X ->• E . Note that the permutations 00, ^1,^00 describe the branching pattern of 3 above 0 , 1 , 0 0 . The genus g of X can be calculated from the Gi's. Recall the following formula: 2g — 2 = d — no — ni — n ^ . In this example, d — 4, no = 2, iii = 3 and = 1. Substituting theses values into our for- mula, we get that g = 0. Thus, X is homeomorphic to the Riemann sphere E . We will draw the dessin associated to the Belyi pair (X, 3) in Section 4.4.1. Chapter 4. Dessins D'Enfants 4.4 The Action of Gal(Q/Q) The Galois group Gal(Q/Q) 41 on dessins is the automorphism group of the field Q of al- I'Esquisse d'un programme, Grothendieck stated that the gebraic numbers. In natural action of Gal(Q/Q) on Belyi pairs {X,/3) induces an action on these dessins. The action has the following properties: 1) The action of Gal ( Q / Q ) on the dessins is faithful. By faithful, we mean that each non-identity element a € GaZ(Q/Q) sends some dessin V to a nonisomorphic dessin V . a In fact, Lenstra and Schneps [Sch94] proved that the action of Gal(Q/Q>) on the set of plane trees (maps on £ with a single face) is faithful. 2) The action of GaZ ( Q / Q ) on the dessins preserves genus g, number of vertices, edges, faces, valencies of the vertices and other numerical parameters. 3) The action of GaZ ( Q / Q ) preserves the groups of orientation-preserving automorphisms of the dessins. 2) and 3) are widely-known, but explicit proofs are hard to come by. Thus, it makes it difficult to determine whether two dessins are conjugate under GaZ ( Q / Q ) . It is necessary but not sufficient that the invariants in 2) and 3) should be equal. We define the monodromy group of a dessin to be the monodromy group of the branched covering 3 : X —>• E. Definition 4.20 See Section 4.3.4 for an explanation of the monodromy group of a covering 3 : X —>• E . We have another version of Definition 4.9. Two dessins V and V are isomorphic if and only if their monodromy generators oi and o\ are simultaneously conjugate. In other words, there exists 8 e Sd satisfying o\ = of for i = 0,1, oo. Definition 4.21 We state the major result of this section. This result is due to G . Jones and M . Streit [JS97]. Two dessins V and V which are conjugate under Gal(Q/Q) have monodromy groups M and M respectively, which are are conjugate in Sd (i.e. which are isomorphic as permutation groups). Theorem 4.22 a a Chapter 4. Dessins D'Enfants 42 Note that two dessins can be conjugate under Gal(Q/Q) but can be non- isomorphic because the generators of their monodromy groups need not be simultaneously conjugate. Also, note that the converse of Theorem 4.22 is not true. Non-conjugate dessins can have conjugate monodromy groups. To demonstrate this fact, we use a finer invariant: cartographic group C. The cartographic group C is a transitive subgroup of S^d- Theorem 4.22 shows that conjugate dessins have conjugate cartographic groups. So, conjugacy of cartographic groups implies conjugacy of monodromy groups, but the converse is false. Examples of dessins with conjugate monodromy groups but non-conjugate cartographic groups (which means that the dessins are nonconjugate) are given in [JS97]. N o w we describe how Gal(Q/Q) acts on Belyi pairs (X, 8). Before we do that let us state the what it means for two Belyi pairs to be equivalent. Definition 4.23 Two Belyi pairs (X, 3) and (X', 3') are equivalent if there is an isomorphism p : X -> X' such that 3' o p = 3. Definition 4.23 is saying that two Belyi pairs (X, 3) and (X ,3') are equiv1 alent 4=s> covering spaces (X, 3) and (X',8 ) of £ are isomorphic (see Sec1 tion 2.7 for details) We know that X and 8 are defined over Q . Thus, each automorphism a G Gal ( Q / Q ) acts naturally on the defining coefficients of X and 8 to give a Belyi pair (X , 8 ). So now, we have the following induced a a action: Let B be the set of equivalence classes, a G Gal(Q/Q) Go/(Q/Q) and [(X, 3)} G B. xB^B a[(X,8)]^[(X°,n]. Now, we can state some examples of the action defined above. The following two examples are taken from [JS97]. Example 4.24 In Example 4.6, X and 3 are defined over Q , therefore, the equivalence class [(X, 3)} is fixed by Gal (Q/Q). Example 4.25 In Example 4.7, our algebraic curve X\ = X ^ 1+y is defined over Q(v 2). The Galois group Gal ( Q ( V 2 ) / Q ) is of order two and generated / by the automorphism a : \/2 \-> —y/2. Note a extends to an element i n Chapter 4. Dessins D'Enfants 43 Gal (Q/Q). a sends X\ to the conjugate elliptic curve X £ = X^^ and fixes Belyi function 8 (see Example 4.7) since 8 is defined over Q. It is a fact that in general, two elliptic curves X = X are isomorphic (as Riemann surfaces) T they have the same J-invariant, J(X ) = 4(1 - r + r ) / 2 7 r ( l - T ) . In 2 3 2 2 T this example, J(X ^) l+ = (19 - 3 ^ / 2 7 and J(X _ ) 1 = (19 + 3 ^ / 2 7 . y/2 Thus, they are non-isomorphic and so, the two Belyi pairs are inequivalent. 4.4.1 Dessins D'Enfants Obtained from Belyi pairs (X, 3) A s mentioned from earlier sections of Chapter 4, there is a correspondence between equivalence classes of Belyi pairs [(X, 0)} and abstract dessins. There are several ways of obtaining a dessin from a Belyi pair (X, 3). The general idea is to take some simple combinatorial structure on £ , such as a triangulation, a map or a hypermap and use 3 to lift it to X, to obtain a structure on X which covers the original structure on £ . These structures on X are known as dessins d'enfants. In this section we will describe how to obtain dessins d'enfants by using triangulations and bipartite maps. We do this because in the next section we will demonstrate how the action of the absolute Galois group Gal(Q, Q) on the Belyi pairs (as described in section 4.4) induces actions of Gal(Q, Q) on the dessins associated to the Belyi pairs. Triangulation Let 71 be a triangulation of E formed by selecting 3 vertices 0, l,oo and 3 edges along the line-segments in M joining the 3 vertices. We have 2 triangular faces, corresponding to the upper and lower half planes of C. Let 8 : X —»• E be a Belyi function of degree d, then / ? ( 7 i ) (Note: 3~ (T\) is the dessin l -1 associated to (X, 8)) is a triangulation T of X). Since, 8 is only ramified over {0,1, oo}, each of the two triangular faces of 71 lifts to d triangular faces on X and each of the 3 edges of lifts to d edges on X. Thus, triangulation T has 2d faces and 3d edges. Now, assume that over a vertex v = 0,1 or oo of 7 i , the d sheets come together in cycles of lengths Z V ) i,l , v n v such that d = l i + • • • + l n - Each vertex v has two edges coming out of it (i.e. has Vt v> v valency 2), so, B~ (v) consists of n vertices of valencies 2 l l v that Euler characteristic ofX,x V j i,2l v > n v . Recall = 2-2g = V-E + F where V is the number of vertices and E is the number of edges and F is the number of faces of Chapter 4. Dessins D'Enfants 44 triangulation T of X and g is the genus of X. In our case, V — no + n\ + n ^ and E = 3d and F = 2d. Thus, we have, 2~2g = x = n + ni+n -d. (**) 0 oo Formula (**) above is exactly the formula (*) in Section 4.3.4. We have a 2colouring of the faces of T : label the faces that cover the upper half-plane of C by + and label the faces that cover the lower half-plane by —, as a result, each edge separates faces with different labels. We can also label the vertices of T by • , o, * corresponding to the vertex v = 0,1, oo of Ti they cover respectively, so that each edge of T joins vertices with different labels. We also have that the 3 vertices on each face of T have different labels and the face is called positive or negative as its orientation (induced by 8 from the orientation of £ ) corresponds to the cyclic order of (• o •) or (* o •) of these labels. (Note: this is all consistent with the definitions stated in Section 4.3.1.) The following example is taken from [JS97]. Example 4.26 Recall Example 4.6, where X = E and 3(x) = x . The triann gulation T of X corresponding to this Belyi pair (X, 3) with two vertices of valency 2n at 0 and oo (because over 0 and oo, the sheets come together in a single cycle of length n) and n vertices of valency 2 at the n-th roots j = 0 , n e ^, 2m n — 1, of 1 (there is no branching over 1, thus, h j = 1 for each j). There are 3n edges, one joining each e ^ 2ni n to 0 and one joining each e ^ 2m n to oo, and n edges joining 0 to oo. Bipartite M a p s In this situation, we delete the vertex oo and its two incident edges in triangulation Ti of E as described above. We get a map B\ consisting of a graph G\ embedded in E . The graph G\ has one edge / = [ 0 , l j c R joining vertices 0 and 1 and it also has a single face E \ I. The Belyi function 8 lifts B\ to a map B = B~ (B\) l on X. The map B can be obtained from T by deleting the ver- tices labeled oo and their incident edges. The graph G = 3~ (I) (embedded X in X) is bipartite. In other words, the vertices of G can be coloured black • (if they cover 0) or white o (if they cover 1). Each edge of G connects vertices of different colours. There are no black vertices and n\ white vertices, of valencies ^o,i, •••) lo,n 0 ni i,ni ( ) 7 1 1 respectively. We have d (degree of 8) edges Chapter 4. Dessins D'Enfants 45 in G (each of which cover I). B has n o o faces with 2 ^ 1 , 2 1 ^ ^ sides. A s above, the genus g and characteristic of X is given by: 2 - 2g = x = n + ni + rioo - d. 0 T can be obtained from B by adding the vertex oo in each face of B and this is joined by non-intersecting edges to the vertices of B incident with that face. The triangulation T obtained in this manner is called stellation of B. Example 4.27 Recall the algebraic curve X given by the algebraic function A(z, w) = w — 2w + 1 — z} in Example 4.19, where 0 was the projection 4 2 map (z, w) i 7 z. Recall also that X is homeomorphic to the Riemann sphere S . Now, we are going to obtain the dessin associated to the Belyi pair (X, 0) by using 0 to lift I = [0,1] to X to obtain a bipartite map B. The embedded bipartite graph G = /3 ([0,1]) in X = £ looks like the following: _1 0 Figure 4.7: This is an example of a plane tree (an embedding of a tree G in C ) . We get plane trees when X — £ and 0 is a polynomial. A s stated before, the resulting structures on X obtained from Belyi functions 0 are called dessins d'enfants. The action of Gal ( Q , Q ) on Belyi pairs (X, 0) induces an action of Gal ( Q , Q ) on the various dessins associated to these pairs. We will give examples of this action in later sections. 4.4.2 Belyi Pairs Represented by Permutation Groups Before we give examples of the action of Gal(Q, Q ) on dessins, we describe how to represent Belyi pairs by permutations. A Belyi pair (X, 8) can be rep- Chapter 4. Dessins D'Enfants 46 resented by its monodromy group M or its cartographic group C. In this section we will only consider permutation representations of bipartite maps B. (Note: we can also use permutations to represent maps which are not bipartite. See [Jon97] and [JS97].) Monodromy Group Let (X, 0) be a Belyi pair and let B = 0~ (Bi) l be the bipartite map on X (see Section 4.4.1). The d sheets of the covering 0 can be associated with the set E of edges of the graph G = 0~ ([O,1]). The positive orientation of E lifts (by l using 0) to an orientation of the surface X. This results in a cyclic ordering of the edges around the black and white vertices of graph G. Each edge e € E joins a unique black vertex in the set 0~ (O) and a unique white vertex in 1 the set 0~ (1). The cyclic orderings around the black and white vertices of G X form the disjoint cycles of permutations cro and o\ respectively, of the set E. In other words, at each black vertex, the orientation of X induces the cyclic permutation of the incident edges because as stated above, each edge meets a unique black vertex. The local rotations of the edges about the black vertices are the disjoint cycles of a permutation cro of the set E of edges of B. Similarly, the local rotations around the white vertices gives us a permutation o\ of E. Permutations o\ and <JQ tell us how the sheets are permuted by seeing how the edges of G are permuted by rotations around their incident black and white vertices. Thus, the cycle lengths of cro and o\ are valencies of the black and white vertices respectively. Note that we get the partitions: l \ + • • • + Vy h,n of d associated with the critical values 0 and 1. Let M = (oo,o\) be the v subgroup of Sd (set of permutations of the set E) generated by UQ and o\. The graph G is connected, so the group M is transitive. The group M is called the monodromy group of the Belyi pair ( X , 0) because M is the monodromy group of the branched covering 0 : X ->• £ (See Definition 4.20). Let (X, 0) and (X ,0') be Belyi pairs with bipartite maps B and B' and monodromy groups M = (er , a\) and M' = (a' , a[) in Sd- Then the following are equivalent: a) the Belyi pairs (X, 0) and (X ', 0') are equivalent; b) the bipartite maps B and B' are isomorphic; c) the pairs (ao, ai) and (cr , a[) are conjugate in Sd (i.e. 3 a £ Sd such that cT~ GiO — o\for i = 0,1). Lemma 4.28 1 0 1 0 l 0 Chapter 4. Dessins D'Enfants 47 Condition c) tells us that M and M' are conjugate in Sd which implies that they are isomorphic. Note: the permutation o-Qo = (cro^i) -1 € M describes the permutation of sheets induced by a rotation around oo, each cycle of length d corresponding to a 2d-gonal face of B. We can also obtain a permutation representation of Belyi pairs by using the triangulation T (see [JS97] for details). Cartographic G r o u p We have another permutation group associated with a Belyi pair (X, (3) called the cartographic group C. This method of permutation representation was developed by Malgoire and Voisin [VM77] and by Jones and Singerman QS78]. The cartographic group can be defined for any oriented map (it may or may not be bipartite) M (see [JS97] and Qon97] for details). Thus, the cartographic group is more general than the monodromy group. We will only look at the cartographic group for a bipartite map B. Suppose the bipartite graph G has d edges then the set £2 has 2d darts (or directed edges) of B. We define C = (770, r/i). C is subgroup of the symmetric group S d (C consists of permu2 tations of the set £2). 770 uses the orientation of X to rotate the darts around the vertex to which they point, and 771 is the involution which reverses the direction of each dart (772 = (770771) -1 rotates the darts around faces). Graph G is connected, so, C is transitive on the set £2. (X, (3), we obtain a bipartite map B = (3~ (B{). B has a cartographic group C = (770,771) < S d (which is defined to be the carl Given a Belyi pair 2 tographic group of (X, (3)). Theorem 4.22 tells us that (C, £2) like the monodromy group (M, E) is invariant under Gal(Q/Q). Let us state a few defini- tions before we continue. Let G be a transitive permutation group on the finite set A. A block is a non-empty subset D of A such that for all 0 £ G either o(D) = D or o(D) n D = 0 (note: a(D) = {o(d) | d £ D}). Definition 4.29 A (transitive) group G acting on a set A is said to be primitive if the only blocks in A are the trivial ones: the sets of size 1 and A itself. Definition 4.30 The monodromy group M of (X, (3) may be primitive, but the cartographic group C is always imprimitive. The cartographic group C = (^o^i) preserves a non-trivial equivalence relation on the set of darts £2: 2 Chapter 4. Dessins D'Enfants 48 darts of B are equivalent if they point to vertices of the same colour. The two equivalence classes fio and Q\ consists of d black darts (darts pointing to • vertex) and white darts (darts pointing to o vertex). 770 preservesfio>^1 and 771 transposes them. Thus, C acts imprimitively on the set Q,, with two blocks fio and Q,\. The action of C on set {fio, ^1} induces a permutation representation, 7T{j7 ,ni}/ of C. The permutation representation 7!"{n ,ni} is 0 0 and and the kernel of the n n { 0tCll} a homomorphism is H = {rjo,^ ), [rfc = ^riom)-Thus, 1 1 H is a normal subgroup of C of index 2. One can verify that 770 and rf^ permute the 1 elements in SIQ, while <TO and o\ permute their underlying edges. 770 and 77^ act on fii as o\ and OQ respectively act on E. So, H acts as M on blocks Qo, ^1 and the two actions differ only by a transposition of generators. Thus, H can be embedded as a subgroup o f M x M and in a similar manner, the cartographic group C can be embedded in the wreath product M \ S 2 (semi-direct product o f M x M and the form ((mi,m2),cr) where S2 (mi,m2) = ( M x M) >\ S2 (vi))- The elements in M I S2 are of E M x M, o € S2. The operation of — this group is defined as follows: Let ip be a homomorphism defined as: ip :S2^Aut(M x M) Then ((mi,m ),cr)((m' ,m ),cr') = ( ( m m ) ( 7 - (m'^ra'^aa') 2 1 2 2 1 ) =((mi, m )(/?(o-)(m' , m ) , 00') 2 1 2 =((mi, m )cr"" ( i, ^ ' 2 ) ^ °" ')1 m <7 2 We identify (12) with 771, so 771 acts on M x M by conjugation transposing the two factors. M I S is a group of order | M | | 5 | . So, we have 2\M\ < \C\ < 2 2 2 2\M\ . 2 4.4.3 Plane Trees and Shabat Polynomials A s mentioned before, Schneps [Sch94] proved that the action of G a / ( Q , Q) on plane trees is faithful. Recall that plane trees are maps on S with a single face. Plane trees are the simplest class of bipartite maps. We will focus Chapter 4. Dessins D'Enfants 49 our attention on plane trees for the remainder of this paper because it offers the following advantages: a) Plane trees are easy to draw, b) Belyi functions involved are polynomials, so the computations involved are a little easier, and c) X — £ , so the bipartite graphs G and G\ are embedded in the same Riemann surface. The automorphism group of the Riemann sphere E , Aut(E) = PSL{2, C) = {T : z — i> \ \a,b,c,d E C,ad - be = 1}. cz + d a Z + The meromorphic functions on E are the rational functions P(z)/Q(z), where P and Q are polynomials with complex coefficients. The simplest rational functions are polynomials which can be thought as meromorphic functions with a unique pole, located at oo. Let Pbea polynomial which has oo as a critical value (ifdeg(P) > I) and it will be called a Belyi polynomial if it is a Belyi function on E (i.e. if its finite critical values lie in {0,1}). Definition 4.31 We have a more general definition. Definition 4.32 A polynomial P is called a Shabat polynomial or a generalized if it has at most two critical values in C. Chebyshev polynomial Given a Shabat polynomial P, we can always find a, b E C(a ^ 0) such that aP+b is a Belyi polynomial (of the same degree as P)(i.e. the finite critical values of aP + b lie in the set {0,1}) and the only other critical value in E is oo (if deg(P) > 1). Therefore, Shabat polynomials and Belyi polynomials are essentially the same. Thus, P : E ->• E can be considered as a Belyi function. We now describe how to construct a bipartite map B = Bp from a Shabat polynomial P. Let the set of critical values of P be { c o , c i } , let I be the E u - c\ in C, and let G = P~ {l) C C. The graph G is embedded in E with d = deg(P) edges mapped homeomorphically onto I l clidean line-segment from co to by P. The vertices of G are members of the sets P _ 1 (c ) and P~ {c\). We mark l 0 Chapter 4. 50 Dessins D'Enfants the elements of P _ 1 ( c ) by • and the elements of P ( c i ) by o. G is bipartite - 1 0 because each edge in G, joins vertex • to vertex o. Let B = B P be the bipartite map formed from this embedding of G in £ . The covering P : E -> E has a unique pole at oo. Thus, the sheets of the covering P : E —> E all come together in a cycle of length d at oo, so, B has a single face which is simplyconnected. The complement G of the face is connected and is a tree. We call G a plane tree associate with P because G is embedded in C. The bipartite map B = Bp has no • vertices and n\ o vertices of valencies k,i,--,lo,n 0 a n d / i , ! , ...,h,ni respectively, where Z ,H 0 Mo,n = 0 h,i + ---h,m = d. To make our notation a little easier, let n =p,m= 0 q, l 0tj = ~fj and = Sj. So, jj and Sj are the orders of the zeros of P and P — 1 respectively such that 7H We have that 1- 7p = <*i H \-S = d. q and 3 = 0 and substituting these values into formula (**) in Section 4.4.1, gives us no + n\ = d + 1, which in turn gives us, p + q = d+l.(***) S denote the partitions of d, that is, 7 = 7 1 , j and S = Si,S . We will call the ordered pair (7; S) the type of Bp. Partitions 7 and S contain a Let 7 and p q total of d+1 parts. Every finite tree is bipartite and as a result has a type which is unique up to transposition of 7 and S. Conversely, given a pair of partitions 7 and S of d with d+1 parts, we get a type of some tree with d edges. In general, given a pair of partitions, we may get several trees and each tree may have several plane embeddings, but the number of non-isomorphic plane trees of a given type must be a finite number. Two bicoloured plane trees are called isomorphic if there is an orientation-preserving self-homeomorphism ofC taking one to the other such that the vertex-colours are preserved. Definition 4.33 A s described in Section 4.4.2, we can represent a Belyi function 3 by a permutation group (co, o~\) < Sd (also known as the monodromy group of the Chapter 4. Dessins D'Enfants covering 3 : X 51 S . We can do the same thing for a Shabat polynomial P . The requirement that the bipartite map B have a single face is equivalent to the P = (CTOOI) condition that -1 is a permutation consisting of a single d-cycle and also, the requirement that the bipartite map be planar is equivalent to the permutations cro and o\ having a total of d + 1 cycles. Isomorphism classes of bicoloured plane trees are in one-to-one correspondence with conjugacy classes of subgroups (<ro, o\) in Sd- (Note: (CTI, <T ) is the monodromy group of 0 the branched covering P : £ ->• £ and (oi, cr ) acts transitively on the set of d 0 edges. If the Shabat polynomial P is defined over Q then a e Gal(Q, Q) sends P into a Shabat polynomial P°', which has an associated plane tree B . p tion of Gal(Q, Q) preserves the type (7,6) and the monodromy group The ac(01, co) but it does not generally preserve the isomorphism class of a bicoloured plane tree. 4.4.4 Examples We end Chapter 4 with examples of Belyi pairs ( £ , P ) where P is a Shabat polynomial and their permutation groups M and C. These examples are taken from [JS97]. Example 4.34 Let P be a Shabat polynomial of degree d and let M be the monodromy group of (S, P). The polynomial P has a unique pole at 00, so 0"oo = (0001 ) - 1 G M is cycle of length d. So, we have (coo) — Cd < M, where Cd is a cyclic group of order d. Let us consider the situation: M = ( O =• C . d Recall from the previous Section 4.4.3 ber of elements in the set P - 1 P (0) -1 that we define p and q to be the num- (we label the elements in this set by •) and ( l ) (we label the elements in this set by o) respectively. So cro must con- sists of p cycles of length d/p and o\ must consists of q cycles of length d/q. Recall the formula (***) in the previous Section 4.4.3, p+ q= d+l. Chapter 4. Dessins D'Enfants 52 So, p and q can only take on the following values, Case 1: p = 1 and q — dor Case 2: p = d and q — 1. Case 1: <7i must consist of q (oWi) - 1 = erg" = > 1 d cycles of length 1, = ^ CTI = 1 (identity). So, = cro = CT^ . This tells us that the d-sheets of the covering 1 P : S - » S come together in a cycle of length d over 0. The plane tree we obtain from this: Figure 4.8: This plane tree corresponds to the Shabat polynomial P(z) = z. d Case 2: The plane tree we get for this case, p = d and q = 1 is: Figure 4.9: This plane tree corresponds to the Shabat polynomial P(z) = 1 — z. d Now, let us consider the cartographic groups of these trees. Let C be the Chapter 4. Dessins D'Enfants 53 cartographic group. C < S d and C is transitive on the set of darts Cl = 2 {1,2,..., 2d}. Recall from Section 4.4.2, C = (r/ , r/i) where, 770 rotates the darts 0 around the vertex to which they point (using the orientation of X — £ and rji is the involution which reverses the direction of the darts. So, 770 consists of a single cycle of length d and 771 is a product of d transpositions. rjo and T]Q = rj^rjorii are disjoint d-cycles. So, 770 and 77Q commute ==>• H = (770,^o ) - ft Cd. From Section 4.4.2, we know H is a normal Thus 1 x 1 subgroup of C (of index 2) and 771 acts by conjugation on H, transposing the two factors. Thus, we have C is the wreath product Cd I £ 2 of order Example 4.35 2d . 2 Let P be a Shabat polynomial of degree d and let M be the monodromy group of P. Consider the next simplest case \M\ = 2d. Again, (coo) — Cd < M. Since \M : {ooo) \ = 2, (ooo) is normal subgroup in M. Recall the formula (***) in the previous Section 4.4.3, p + q = d + l. This formula tells us that at least one of GO and o\ must have a fixed-point. Assume that GQ has a fixed-point (we can do this by transposing colours of our vertices if required). We label the edges of our bipartite graph G with the elements of Z<j = { 0 , 1 , d — 1} in such a way that O-QQ acts on as a translation j H> j + 1 and UQ : 0 H-> 0. Since M is transitive on the set of edges E = "Ld, there is only one equivalence class. So, the number of elements in the equivalence class containing 0 is \M : MQ\ = d, where Mo = {m G G I m-0 = 0}. Thus, the stabilizer of edge 0 has order |Mo| = 2. Since GQ G MQ, {GQ) GO — Mo. So, GQ is an involution. Since acts on (CTOO) (O-QO) < ] G, GQ normalizes (GOO). SO, = Z^ by conjugation and therefore GO can be identified as an automorphism of additive group Z^: GQ • I'd ->• Id j ^ for some involution u eUd example, if u = -l(d uj, (Ud is the multiplicative group of units in Z^). For > 2) = > cr : j ^ 0 -j- So, M = D d (D d is the dihedral group of order 2d). Claim: p = (d + 2)/2 and q = d/2 when d is even. p = (d + l)/2 and q = [d + l)/2 when d is odd. Chapter 4. 54 Dessins D'Enfants Verify claim: o\ : j i-> -j + 1, |oi| = 2 and |<7 | 0 = 2, thus o\ is a product of g transpositions and UQ is a product of p transpositions. If d is even then o\ has no fixed-point = > q = d/2. We substitute this into p + g = d + 1 and get p- (d + 2)/2. If d is o d d , then o\ has one fixed-point ==> g = (d + l)/2 and so,p= (d+l)/2. • The corresponding plane tree is the path of d edges: • — e — • • • -•—o &—e- ... - e — • — o Figure 4.10: O n the left is the case d is odd and on the right is the case d is even. The plane tree corresponds to the Shabat polynomial (T<2 + l ) / 2 (Ta(z) cos(n c o s - 1 = z) is the d-th degree Chebyshev polynomial with finite critical val- ues all equal to ± 1 . See [JS97] for details.) The cartographic group C is the dihedral group D d of order 4d. 2 Jones and Streit [JS97] also give an example of plane trees of type (7; 8) = (3,2,1,1; 3,2,1,1). There are 9 plane trees of this type. They all have the same monodromy groups, but they are not conjugates of each other. In fact, under the action of Gal ( Q / Q ) , the 9 trees form 2 orbits (orbits of length 3 and length 6). See QS97] for details. This example demonstrates that monodromy groups will not always distinguish non-conjugate trees of the same type. This example also shows that the converse of Theorem 4.22 is false. In this case, it turns out that the cartographic groups distinguish the orbits (see [JS97] for details). Thus, the cartographic groups are a finer invariant than monodromy groups. Chapter 5 Genus Zero Actions on Riemann Surfaces and Dessins D'Enfants 5.1 Introduction In this chapter, we consider a finite group G acting on a Riemann surface S (i.e. G x S —> S). S is compact, connected, orientable and without boundary. We look at the following problem. Which finite groups G admit an action on some compact, connected Riemann surface S so that if H is any non-trivial subgroup of G then the orbit space S/H has genus zero (i.e. S/H — £ ) ? If G admits such an action on S, we say that G has genus 0. We also say that the action has genus zero. In this chapter, we will only summarize how to determine the groups G admitting zero actions on Riemann surface S of genus 0 and genus 1. This is taken from [KS]. The solution to the above problem is due to Sadok Kallel and Denis Sjerve [KS] and will be stated in this chapter. Kallel and Sjerve [KS] observed that the Zassenhaus metacyclic groups Gp^(-l), where p is an odd prime, admit a genus zero action on a Riemann surface S of genus g = p—1. (Note: G 4 (—1) is the metacyclic group presented by A = 1, B — 1, BAB~ = A ' where p is an odd prime.) It turns out that Pt p 4 l 1 2p S is given by an equation of the form y = Y\_( ~ j) where ej G C and e/s 2 x e are distinct. In the remainder of chapter 5, we will look at the Belyi functions on Rie- 55 Chapter 5. Genus Zero Actions 56 mann surface S. Recall Belyi's Theorem 4.3 which states that an algebraic curve X is defined over Q there exists a Belyi function / : X ->• S . Thus, there exists a Belyi function on S if 5 is defined over Q . It turns out that S is defined over Q ej e Q . We will prove this fact in this chapter. 5.2 Genus Zero Actions on Riemann Surfaces Let 5 be a Riemann surface. We assume throughout this chapter that S is compact, connected, orientable and without boundary. Let G be a finite group. We consider G acting topologically on Riemann surface S of genus g. We introduce the following definition, which is given in [KS]. Definition 5.1 A group G is said to have genus zero if there exists a Riemann surface S, an action ofGonS satisfying the following condition: If H is any non-trivial subgroup ofG then the orbit surface S/H has genus zero, that is 1 ^ H < G => S/H = F (C), X or equivalently S/Zp = F (C) for all Z < G, where p is a prime number dividing \G\. 1 p We also say that the action has genus zero. Kallel and Sjerve [KS] set out to solve the following problem: Which finite groups G admit a genus zero action on some compact, connected Riemann surface 5 ? The answer to the problem is stated in the following theorem. The groups having genus zero are the cyclic groups, the generalized quaternion groups Q{2 ), the polyhedral groups and the Zassenhaus metacyclic groups (abbreviated ZM groups) G A(—1), where p is an odd prime. Theorem 5.2 n P Chapter 5. Genus Zero Actions Let us define Q(2 ) and G n P ) 4 57 (-1). The generalized quaternion group is the group with the presen- Definition 5.3 tation Q(2 ) = (A,B\ n A~ 2n = 1,B x 2 = A ~\BAB~ 2n = x A- ). 1 Definition 5.4 Let G „ ( r ) denote the group presented as follows: m ) generators: A, B; relations: A = 1, B = 1, BAB' = A ; conditions: GCD((r - l)n, m) = 1 and r = l(mod m). m n 1 r n Kallel and Sjerve [KS] proved Theorem 5.2 by looking at Fuchsian groups T and a short exact sequence l->n-»r-»G-»l which is given by the genus zero action of G on S and the Riemann-Hurwitz formula which gives the relationship between the genus g of S and the number of fixed points of the action. Here, II Y (0 | n\, ri2,..., = -K\ (S) (fundamental group of S) and Y = n ) for some choice of rij. We will give details of the exact short r sequence and the Riemann-Hurwitz formula in later sections. But, before we do that we give some background material. 5.2.1 Background We assume in this chapter that S is a connected Riemann surface of genus g, so Theorem 2.42 Tell us that S has a universal covering space U (i.e U is simply connected Riemann surface). The Uniformisation Theorem 3.2, tells us that U is conformally equivalent to one of the following: 1) E = C U {oo} = F C 1 (Riemannsphere) 2) C (complex plane) 3) H (upper half plane). Thus, we use the notation U to denote any of these three simply connected Riemann surfaces. If g = 0 then U = P (C). If g = 1 then U = C. If g > 0 then 1 U = M Thus, every connected Riemann surface is conformally equivalent to U/Yl, where II C Aut(U) is a torsion free Fuchsian group (see Definition 3.5). We assume S is compact, so II is a Fuchsian group such that U/II is compact Chapter 5. Genus Zero Actions 58 with genus g. Then we can associate a signature (g | ni,n2,nr) ni, n 2 , n r to II where are the periods of II (see Definition 3.18) and the rij < oo for all j. Note: The geometry of II is spherical, Euclidean, or hyperbolic, according — > 1,= 1, or < 1. as Just as in [KS], we will write FL(g \ rii, n 2 , n r ) to denote a Fuchsian group of signature (g \ ni,n2, ...,nr). Note: If r — 0, then II is torsion free and we write II(g \ —). Fuchsian groups with r = 0 are the fundamental groups of Riemann surfaces. 5.2.2 Correspondence between Genus Zero Actions and Short Exact Squences of Groups Let G be a group acting on S. Then there is a correspondence between such actions (G x S -» S) and short exact sequences of groups i-+n->rAc7->i, where 1. T is a Fuchsian group with signature (h \ n\,n2, ...,nr) (h is the genus of U/T). 2. II = Ker 8 = { 7 e V \ 6(7) = 1} <T and II is a torsion free Fuchsian group with signature (g \ —) (II is the fundamental group of S). 3. S = U/U and the action of an element g G G on S is given by g[z] = [e(z)]r where brackets [ ] indicate the II equivalence class of points in U and e G V is any element such that 6(e) — g. 4. The orbit surface S/ G has genus h and is naturally isomorphic to U/T. See Sjerve and Yang [SY] for details of the above correspondence. The relationship between the genera g, h is given by the Riemann-Hurwitz formula: 2g-2 = \G\ 2h-2 + Y^ 1 The signature of F and the epimorphism 9 completely determines the action. Let us introduce the following definition. Chapter 5. Genus Zero Actions 59 Definition 5.5 The signature of the action ofGonS is defined to be the signature ofT. If G x S —> S is an action of genus zero then the signature of T will be (0 | n i , n ) and T = T(0 | n i , n ) has the presentation r r (X X ,x U 2 iX r n i x = X? = • • • = X? = XXX2 •••X = I). r See [TS1] or chapter 3 for details of the presentation. In this case, we let Tj = 6{Xj) and Gj = (Tj) < G, 1 < j < r. Then 1. T i , T , T 2 T generate G (since 9 is an epimorphism). 2. T J = T " = • • • = T?r = T T • • -T = 1 (since 6 must preserve the relations inT). 11 2 2 X 2 r 3. The order of Tj is rij, 1 < j <r (since the kernel of 6, Ker 6, is torsion free). Let us assume that the action G x S -> S is of genus zero. Let g € G such that |<?| = p where p is a prime. Let H = (g) = Z . Then f = e~ (H) is a 1 p Fuchsian group and there exists a short exact sequence i->n-»f Azp->i. The signature of f will be (0 | p ^ j j ) , t is the number of fixed points of g : t S —r S. The Riemann-Hurwitz formula ( A ) , stated above, gives 2g-2=p(-2 + t(l-^)=>g = ±(p-l)(t-2). Note: The key to studying actions G x S -> S satisfying S/G = F (C) is l the determination of the number of fixed points of the action. The RiemannHurwitz formula gives us the relationship between genus g of S and t, the number of fixed points. 5.3 Actions on Riemann Surfaces of Genus 0 and 1 In this section, we will discuss how to determine all genus zero actions on the Riemann sphere S (genus g = 0) and a torus T (genus g = 1). This is taken from [KS]. Chapter 5. Genus Zero Actions 60 Let G s be a finite group acting on S . It is a well known fact that the only finite groups G s acting on S are: 1. T(0 | n , n), the cyclic group of order n . 2. r ( 0 | 2,2, n), the dihedral group of order In. 3. T(0 | 2,3,3), the tetrahedral group of order 12. 4. r ( 0 | 2,3,4), the octahedral group of order 24. 5. T(0 | 2,3,5), the icosahedral group of order 60. See Farkas and Kra [FK92] for details. This gives us the possible signatures of our action G s x S - > E . Thus, the groups listed above admit a genus zero action on S . Let GT be a group. Assume that GT admits a genus zero action on a torus T . It is a well known fact that the group of automorphisms Aut(T) contains T as a normal subgroup. T acts on itself by translations. The quotient group is cyclic of order 2,4,or 6. Stated another way, there is a short exact sequence 1 -> T - » Aut{T) -> M -> 1, where M = Z , Z , or Z . The elements of the subgroup T act fixed point 2 4 6 freely and therefore a finite subgroup of T will not have genus zero. The quotient by any finite subgroup will again be a torus. So, GT is a subgroup of Z , 2 Z 4 or Z . Thus, G 6 T ^ Z , Z , Z or Z . 2 3 4 6 Now, we have the following theorem, which states the results above. Theorem 5.6 The groups G acting on a torus T such the action is of genus zero are: Z w/fJz signature (0 2,2,2,2), 2 Z3 ztfifTz signature (0 3.3.3) , Z wz'fTi signature (0 2.4.4) , 4 Z6 wz'riz signature (0 2,3,6). In all the cases above, the epimorphism 9 : V - » G is unique up to automorphisms of G . 5.4 The Zero Actions of ZM Groups G A{—1) P o Riemann Surface n Kallel and Sjerve [KS] describe the actions involved in Theorem 5.2. In this section, we will discuss the action of the Zassenhaus metacyclic groups (ab- Chapter 5. Genus Zero Actions 61 breviated Z M groups) G ^(—V), where p is an odd prime, on a Riemann surp S of genus g. Recall from Definition 5.4 that G ^(-1) is the metacyclic face p group presented by A p BAB' = A' 1 1 = 1, B = 1, 4 where p is an odd prime. The following theorem describes the action of G ,4(-1) on a Riemann surp face S. The theorem is due to Kallel and Sjerve [KS]. Theorem 5.7 All genus-zero actions of the ZM group G ( - 1 ) where p is an odd prime, have signature (0 | 4 , 4 , p ) . The corresponding genus g = p-l(gis the genus of Riemann surface S on which G is acting on). P ) 4 Let us state the following proposition taken from Farkas and Kra [FK92]. Let M be a compact Riemann surface of genus g. Then M is hyperelliptic 4=> there exists a conformal involution J (J £ AutM such that J = 1) on M that fixes 2g + 2 points. Proposition 5.8 2 In the proof of Theorem 5.7, Kallel and Sjerve [KS] proved that the involution B 2 fixes 2{p — 1) + 2 = 2p points. Thus, by Proposition 5.8, S is hyperel2p liptic. So, S is given by an equation of the form y = JJ(x — ej) where ej € C 2 and e / s are distinct. 5.5 Genus Zero Action of G P ) 4 (-1) and Dessin D'Enfants Recall that G (—1) (where p is an odd prime) is the Zassenhaus metacyclic P)4 group. Let S be some Riemann surface of genus g. Let us assume that the action G i(—1) Pt x S -> S has genus zero. In section 5.4, we stated that the Riemann surface S is given by an equation of the form: 2p y 2 = Y[( x - o)> e where ej £ C and e / s are distinct. For the remainder of Chapter 5, we will denote S to be the Riemann surface described above. In this section we will look at the Belyi functions 0 : S -> S and the dessins d'enfants associated to the Belyi pair (S, 0). Belyi's Theorem 4.3 states Chapter 5. 62 Genus Zero Actions that an algebraic curve X is defined over Q there exists a Belyi function / : X -> £ . Thus, there exists a Belyi function 3 on S if S is defined over Q . In our case, S is defined over Q ej G Q . (*) Thus, we have the existence of a Belyi function, /3 : S ->• £ if we pick the e / s such that ej G Q V j . Our Belyi function 3 is dependent on our choice of the e/s. Thus, for each particular Belyi function 3 : S -> £ that we find, we must specify our choice of Before we prove e/s corresponding it. We will prove (Jfr). (Jfr), we should mention that the Uniformisation theorem allows us to restate Belyi's Theorem 4.3 as follows (see Belyi [Bel79] for the proof). Let H be the upper half-plane. Theorem 5.9 If X is a compact Riemann surface then the following are equivalent: a) X is defined over Q ; b) X = M/Kfor some subgroup K of finite index in a triangle group A < PSL(2, R); c) X = M/Lfor some subgroup L of finite index in the modular group A; d) X = M./M for some subgroup M offiniteindex in the principal congruence subgroup A (2). The principal congruence subgroup A(2) = { T(z) = £ f J l ^ cz + d L : T G PSL(2,1) and a, d are odd, and b, c are even Thus, if the Riemann surface S is defined over Q , then the Belyi function 3 we obtain is of the form (in terms of triangle groups): 3 : S =• M/K -> H / r , where T is the triangle group with signature (0 | 4,4, p) (see Theorem 5.7) and the Belyi function is just the natural projections induced by the inclusion K < r . The degree of the Belyi function 3 is given by index [r : K}. First let us state some definitions and results needed to prove (•). An element a G C is called an algebraic integer and we write a G Z if it is a root of monk polynomial with coefficients from Z . Definition 5.10 Lemma 5.11 Given a G Q , there exists m G Z such that ma is an algebraic integer. Chapter 5. Genus Zero Actions 63 Proof. Let a £ Q then a is a root of some non-zero polynomial g(x) E Q[x], where deg(g(x)) = N. Without loss of generality, we can assume that g(x) is a monic polynomial. Then g(a) = a N + fc^-ic^ + • • • + ha + b = 0, -1 0 where bj E Q . Let m be the common denominator for all coefficients bj then m g(a) = (ma) + b ^m{ma) N N N + ••• + b - (ma) x N N l im +mb . N 0 Thus, ma is an algebraic integer. • Definition 5.12 M is a finitely generated Z-module if there exists afiniteset A = { a i , a , a } , where ai E M, such that any element of M can be expressed as a linear combination of the a/s with integral coefficients, that is 2 r r 7 = JEM i i> c a i=l for some CJ 6 Z , i = 1 , r and ai E A C M. Proposition 5.13 Let a EC. Then a is an algebraic integer if and only if all the non-negative powers of a are elements of M where M is afinitelygenerated Z-module. See Dummit and Foote [DF91] for proof of Proposition 5.13. Fact 5.13 Riemann surface S corresponding to an algebraic function of the form: 2p y = Y\( 2 x ~ j)> where ej E C for all j and e / s are distinct, is defined over Q e ej E Q , for all j E { 1 , 2 p ) . Proof. (<=) obvious n (=>) Let n = 2p and let f(x) = J\( ~ i ) X e A s s u m e f( ) x is defined over 3=1 Q . Then f(x) can be rewritten as f(x) = x n + a -ix ~ n n l H h a\x + a such 0 Chapter 5. Genus Zero Actions 64 that a G Q , for all k = 0 , 1 , n - 1. By Lemma 5.11, we can assume without loss of generality that f(x) + a _ix - k = x n n H 1 n + a x + a such that a x 0 (i.e. a is an algebraic integer), where ej, for j = 1 , 2 , n fe Thus, it suffices to show that ej for all j = 1 , 2 , n k are the roots 6 Z oif(x). are algebraic integers. Figure 5.1: Fix j e {1,2, ...n}, then ej is a root of /(a;) G Z[x]. Thus e^--(a _ie^ n 1 + --- + a ) 0 and so e" (and subsequently all higher powers of ej) can be expressed as Z[ao, a i , a _ i ] - l i n e a r combinations of e " n M = {bo + hej H h6«-ie" _ 1 _ 1 , e j , 1. Let | 6 ,-,6n-i e Z[a ,ai,-,a _i]}. 0 0 n Then all the non-negative powers of ej are in M . (Note: we say that M is finitely generated by the set { e ™ - 1 , e j , 1} over Z[ao, « i , Fix ^/ °fc is an algebraic integer, then by Proposition 5.13 a € M is a finitely generated k k Z-module. Thus, M is a finitely generated Z-module (see Figure 5.1) and all the powers of ej are in M. Therefore, by Proposition 5.13, ej is an algebraic integer. • Now, we look at Belyi functions 0 on the Riemann surface S. By Fact 5.13 and Belyi's Theorem 4.3, there exists a Belyi function B : S —> E ej € Q. For the remainder of chapter 5, we try to obtain Belyi functions on the Riemann surface S. Chapter 5. Genus Zero Actions 65 Let II be the projection map: TT: S ->£ (x,y)>-+ x. Let C n denote the set of critical values of II. II is not a Belyi function because C n = { e i , ...,e2 }. To obtain a Belyi function 0, we must compose IT with P functions P : £ ->• £ . We will now try to determine functions P : £ ->• £ that will send into {0,1}. Thus, to find P : E —• £ , we must specify our choice of the e / s are because P has to map the e / s into the set {0,1}. So, 0 = P o II will be our Belyi function. Theorem 4.18, tells us there is a correspondence between equivalence classes of Belyi pairs and abstract dessins. A s a result, in Section 4.4.1, we discussed ways of obtaining a dessin from a Belyi pair (X, 0), one of which was using bipartite maps. In Section 4.4.3, we stated that the dessin obtained via bipartite maps were plane trees if 0 was a Shabat polynomials (that is polynomials which has at most two critical values in C) and X — £ . In fact, Shabat and Zvonkin [SZ94] proved that there is a bijection between the set of bicoloured plane trees and the set of equivalence classes of Shabat polynomials (the coefficients of the polynomials are algebraic numbers). This is a special case of Grothendieck's correspondence (Theorem 4.18). In our case, to determine P : E —> S , we will use the approach given by Shabat and Zvonkin [SZ94]. We will use the notation given in section 4.4.3. Our approach is we start with a plane tree of type (7; 8) and try to find a Shabat polynomial R(x) associated to it. Here, 7 = ( 7 1 , 7 , . ) and 8 = (61,8 ) q are two partitions of 2p, and r, q are the number of black and white vertices, respectively, satisfying r + q = 2p +1. The black and white vertices denote the elements in sets i ? ( c o ) and i ? ( c i ) respectively. The valencies of our black _1 _ 1 and white vertices is specified by 7 and 8 respectively. See section 4.4.3 for further details. If the set of critical values of R, CR = {co, c\} is not a subset of {0,1} then as stated in Section 4.4.3, we can choose an affine transformation of C which sends Co »-)• 0 and c\ ^ 1. Then P = aR + b, for some a, b € C , is the Belyi polynomial that we are looking for (see Figure 5.2). Chapter 5. Genus Zero Actions 66 Figure 5.3: Plane tree of type {2p; 1 , 1 ) . Case A : Plane tree of type (2p; 1,..., 1). See Figure 5.3 for a picture of the plane tree. Let co = 0. Then the co-ordinates of the black vertices are the roots of R(x) and the sheets of the covering R come together in cycles of 2p over 0, so R(x) = X P . Let ci = 1 then 2 = 1 R(x) =x 2p Note: these are the co-ordinates of our white vertices. So, P(x) x 2p and our Belyi function isd = PoU: S-^-T, and our choice of e/s corresponding to this Belyi function is: ej = e (2 7_1 ) 7ri/ ' , for j = 2p 1,2p. Thus, our Riemann surface S is given by the equation: y = x ? - 1. 2 Case B: Plane tree of type 2 (p,p;2,1,1). 2p-2 = R(x) — Chapter 5. Genus Zero Actions 67 There is only one plane tree of this type. See Figure 5.4. Let CQ = 0 then the co-ordinates of the black vertices are the roots of R(x). Let the black vertex of valency p be at x = 0 and the other at x = 2. Then R(x) = x (x p - 2) . N o w to p determine the co-ordinates of the white vertices, we compute R\x) = px^ix R'(x) - 2) ~ {2x - 2), p l has one root at x = 1. Thus, x = 1 is the position of the white vertex of valency 2. To get the co-ordinates of the other white vertices, we compute the critical value c\: c\ = R(l) = 1 ( — l ) = —1 since p is an odd prime. p N o w to get the co-ordinates of the remaining white vertices, we have to solve the equation x (x-2) p = -1. p The solutions are x = 1± x/TTa" where a = j^+^lv j for j = 0 , 1 , ...,p - 1. Note: these are the co-ordinates of the white vertices with valency 1. It turns out that P(x) = -R(x) = -x {x p - 2) p and our Belyi function is 0 = P o II : S —> E . We choose our e / s to be the solutions of the equation x (x p — 2) p = —1. This is our choice of the e / s corresponding to the above Belyi function. Thus, Riemann surface S is given by the equation: P-I y = 2 H(x -2x- ). 2 aj Chapter 5. Genus Zero Actions 68 In conclusion, we see that Belyi functions on a Riemann surface S corresponding to an equation of the form 2p y = 2 J[( x - j)> e will have the following form: 0 = P o n, where II is the projection map from S to S , whose critical values are {e^} ^ 2 and P is a Belyi polynomial of degree 2p (see Definition 4.31). We had stated that P — aR + b, where a, b G C, and R is a Shabat polynomial (see Definition 4.32) with finite critical values {CQ,CI}. The Shabat polynomials are defined over Q. We computed only cases A and B because in these cases, the co-ordinates of all the black and white vertices can be determined explicitly. In cases A and B, we are able to describe explicitly our choice of the e / s corresponding to the particular Belyi function that we found (Note: Belyi functions 0 on S is dependent on the e / s because the Belyi polynomial P must send the e / s into the set {0,1}). Cases A and B are easy cases that demonstrates the nature of Belyi functions on Riemann surface S. In finding the specific Belyi functions in cases A and B, we were able to explicitly give examples of equations corresponding to Riemann surface S. Bibliography [Bea84] A l a n F. Beardon. A Primer on Riemann Surfaces. Number 78 in London Mathmematical Society Lecture Notes. Cambridge University Press, Cambridge, 1984. Cited on page(s) 20 [Bel79] G . V. Belyi. Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Sen Mat., 43(2):267-276, 479,1979. Cited on page(s) 31, 32, 62 [DF91] David S. Dummit and Richard M . Foote. Abstract Algebra. Prentice Hall, Inc, Englewood Cliffs, NJ, 1991. Cited on page(s) 63 [FK92] H . M . Farkas and I. Kra. Riemann surfaces. Springer-Verlag, N e w York, second edition, 1992. Cited on page(s) 60, 61 [Gro85] A . Grothendieck. l'Esquisse d'un Programme. Preprint, 1985. Cited on page(s) 1 Complex Functions: An algebraic and geometric viewpoint. Cambridge University Press, N e w York, 1987. Qon87] D. Jones, G . A . & Singerman. Cited on page(s) 4, 7, 8,10,11,12,13,16,17,19,20,22,24,26,28,40 [Jon97] Gareth A . Jones. Maps on surfaces and Galois groups. Math. Slovaca, 47(l):l-33,1997. Graph theory (Donovaly, 1994). Cited on page(s) 46,47 QS78] G . Jones and D. Singerman. Theory of maps on orientable surfaces. Proc. London Math. Soc. (3), 37(3):273-307,1978. Cited on page(s) 31, 36,47 [JS97] Gareth A . Jones and Manfred Streit. Galois groups, monodromy groups and cartographic groups. In Geometric Galois actions, 2, pages 25-65. Cambridge Univ. Press, Cambridge, 1997. Cited on page(s) 29,32,41,42,44,46,47, 51, 54 [KS] Sadok Kallel and Denis Sjerve. Genus zero actions on Riemann surfaces. University of British Columbia Math Dept. Cited on page(s) 2,55,56,57,58,59, 60,61 69 Index 70 [KSS] S. Kallel, D. Sjerve, and Y. Song. Equations for a class of Riemann surfaces. University of British Columbia Math Dept. Cited on page(s) 2 [Leh64] Joseph Lehner. Discontinuous groups and automorphic functions. American Mathematical Society, Providence, R.I., 1964. Mathematical Surveys, N o . VIII. Cited on page(s) 28 [Mas67] William S. Massey. Algebraic Topology: An Introduction. Springer Verlag, N e w York, 1967. Cited on page(s) 4,15,17,18, 39 [Sch94] Leila Schneps. Dessins d'enfants on the Riemann sphere. In The Grothendieck theory of dessins d'enfants (Luminy, 1993), pages 47-77. Cambridge Univ. Press, Cambridge, 1994. Cited on page(s) 29,31, 32,33,36,40,41,48 [Ser89] Jean-Pierre Serre. Lectures on the Mordell-Weil theorem. Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. Cited on page(s) 31,32 [SY] D. Sjerve and Q. J. Yang. The eichler trace of Z p actions on Riemann surfaces. University of British Columbia University. Cited on page(s) 58 [SZ94] George Shabat and Alexander Zvonkin. Plane trees and algebraic numbers. In Jerusalem combinatorics '93, pages 233-275. Amer. Math. S o c , Providence, RI, 1994. Cited on page(s) 65 [VM77] Christine Voisin and Jean Malgoire. Cartes cellulaires. U.E.R. de Mathematiques, Universite des Sciences et Techniques d u Languedoc, Montpellier, 1977. Cahiers Mathematiques Montpellier, No. 12. Cited on page(s) 31, 36,47 [Wey55] Hermann Weyl. Die Idee der Riemannscher Tlache. Teubner, Leibzig, fourth edition, 1955. Cited on page(s) 4 Index TJ-neighbourhood, 8 critical value, 11 G n(r) 57 cycles, 26 Belyi function, 30 dessin d'enfant, 29 Belyi pair, 31 dessins d'enfants, 45 Belyi polynomial, 49 discontinuously, 16 Euler characteristic, 13 elliptic cycle, 26 Fuchsian group, 21 elliptic vertices, 26 Grothendieck dessin, 33 equivalent, 42 Shabat polynomial, 49 finitely generated Z-module, 63 abstract dessins, 34 flag set, 34 algebraic function, 12 function element, 8 algebraic integer, 62 fundamental group TTI(S, a), 14 analytic, 6 fundamental region, 24 atlas, 4 generalized Chebyshev polynomial, mt r automorphism, 16, 20 49 block, 47 generalized quaternion group, 57 branch-point, 11 genus, 13 branched Riemann surface of m, 11 genus zero, 56 chart, 4 germ, 8 clean, 30 holomorphic, 7 clean dessins, 30 homomorphism, 15 compatible, 5 homotopy classes, 14 conformal equivalence, 19 isomorphic, 50 conformal homeomorphism, 19 isomorphism, 15 conformally equivalent, 19 local coordinate, 4 congruent, 25 marking, 34 congruent sides, 26 meromorphic, 7 coordinate transition function, 4 meromorphic continuation of [/] along 7,10 covering map, 10 covering transformations, 16 monodromy group, 39 critical points, 12 monodromy group of a dessin, 41 71 0 Index 72 open, 8 oriented flag set, 34 path, 9 periods, 26 pre-clean, 30, 34 pre-clean dessins, 30 primitive, 47 properly discontinuously, 22 ramification order, 11 ramified, 11 regular points, 12 sides, 26 signature, 28 signature of the action, 59 simply connected, 15 stellation, 45 surface, 4 triangle groups, 24 type, 50 unbranched Riemann surface, 10 unbranched Riemann surface of m , 9 vertex, 25
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Dessins d’enfants and genus zero actions Duong, Linh Ton 1999
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Title | Dessins d’enfants and genus zero actions |
Creator |
Duong, Linh Ton |
Date Issued | 1999 |
Description | A Dessin D'Enfant is a cellular map on a Riemann surface ramified over {0, 1, ∞}. We will describe Grothendieck's correspondence between the set of isomorphism classes of dessins and the set of ismorphism classes of algebraic curves defined over Q. We also describe the action of the Galois group of the algebraic numbers, Gal(Q/Q), on the dessins. Finally, we also talk about groups that admit genus zero actions on Riemann surfaces and investigate the nature of Belyi functions on these Riemann surfaces. |
Extent | 3337137 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080080 |
URI | http://hdl.handle.net/2429/9669 |
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Master of Science - MSc |
Program |
Mathematics |
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Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-11 |
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Scholarly Level | Graduate |
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