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Galois actions on Dessins D’Enfants Burggraf, David S. 2002-12-31

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GALOIS A C T I O N S O N DESSINS D ' E N F A N T S by DAVID S. BURGGRAF B.Sc. Carleton University, 1994 M.Sc. The University of British Columbia, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 2002 © David S. Burggraf, 2002  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 Date  MrVfcCri ~? /O-QOS  Abstract Grothendieck's theory of Dessins d'Enfants is an investigation of the many connections between permutations, maps on Riemann surfaces, algebraic curves and Galois groups. Some of these connections were studied long ago by Hamilton [Ham67] and Klein [Kle56], and some are quite new and continue to be analyzed today by researchers in various mathematical fields ranging from number theory to conformal field theory. One topic of great interest that emerges is the_ surprising fact, which was noticed by Grothendieck, that the absolute Galois group Gal(Q/Q) acts faithfully on certain combinatorial objects known as Dessins d'Enfants. The first conference on Grothendieck's theory of Dessins d'Enfants took place at Luminy in April 1993 where many of Grothendieck's ideas were explored and shared [Sch94a]. A paper in the conference proceedings [Sch94b], shows that Gal(Q/Q) acts faithfully on Dessins d'Enfants with genus 1 and later in the same paper, another proof shows that Gal(Q/Q) acts faithfully on plane trees in genus 0. Faithful actions of Gal(Q/Q) on such simple objects motivated a systematic study of the relationships between plane trees, polynomials and Galois groups, where computer algebra systems were often used for the calculations [Sha90], [And94], [Sch94b], [Cou97], [Zvo98], [MagOO]. Work in this direction, however, has not lead to many simply defined, explicit classes of Dessins, especially where more complicated Galois groups are acting. This lead to the concluding comment by Gareth Jones in [Jon97]: "It would be interesting to produce further classes of Dessins on which Gal(Q/Q) induces more complicated groups, such as nonsolvable groups or solvable groups with unbounded derived length". One of the main aims of this thesis is to address this comment. A method is provided herein that defines a faithful Galois action of the symmetric groups S , for infinitely primes p, on a special class T>(f3 $) of elliptic and hyperelliptic Dessins d'Enfants. Furthermore, the corresponding Galois orbits of S and its subgroups in V(^ ^) are made explicit. p  p  p  v  ii  Table of Contents Abstract  ii  Table of Contents  iii  Chapter 1. Introduction 1.1 Background and Overview 1.2 Notation and Terminology  1 1 4  Chapter 2. Belyi functions and permutations 2.1 Maps 2.2 Hypermaps 2.3 Belyi functions 2.4 Belyi functions on £ 2.5 Belyi's Theorem 2.6 Elliptic and hyperelliptic Belyi surfaces. 2.7 Monodromy groups 2.8 Calculating Monodromy groups of Belyi functions 2.9 Triangle Groups  :  5 5 8 10 10 12 15 21 24 26  Chapter 3. Galois Actions on Dessins d'Enfants 3.1 The Absolute Galois group 3.2 Galois actions on Elliptic and Hyperelliptic Dessins  31 31 32  Chapter 4. A special class of Dessins d'Enfants 4.1 The family of polynomials f 4.2 A class of Dessins of genus > 1  36 36 37  Chapter 5. The Arithmetic of f 5.1 The ring of integers OK 5.2 Decomposition and Inertia groups 5.3 Discriminants 5.4 Proof of Theorem 3 from Section 4.1  44 44 48 50 51  Chapter 6. Faithful Galois Actions of arbitrary finite groups 6.1 Faithful actions on higher genus Dessins  54 55.  Chapter 7. Galois Actions of PSL (p) 7.1 A Galois representation of PSL2 (5) 7.2 Galois representations of PSL2O?) 7.3 Examples: PSL (11) and PSL (13)  57 57 62 65'  Appendix A. Full-size PSL (p) orbits on n-tuples  71  p  p  2  2  2  2  iii  Appendix B. Mondodromy generators of a hyperelliptic Belyl function  74  Appendix C. Automating the Galois action of PSL2CP)  83  Appendix D. Displaying Galois orbits using JavaScript  89  Bibliography  90  iv  Chapter 1 Introduction  1.1  Background and Overview  The interplay between algebra and topology is a fascinating study that emerges in many branches of mathematics; one being the Grothendieck theory of Dessins d'Enfants. The algebraic object of study here is the absolute Galois group and its faithful action on certain topological objects known as Dessins d'Enfants. The absolute Galois group Gal(Q/Q) is the Galois group of thefieldof all algebraic numbers Q over thefieldof rational numbers Q and is one of the most complicated and mysterious groups studied in mathematics. Dessins d'Enfants (in English: children's drawings) are graphs embedded in Riemann surfaces and were named by Alexander Grothendieck for their similarity in appearance to a child's "stick-drawings". At the center of Grothendieck's theory is Belyi's theorem [Bel79], which states that a compact Riemann surface X is defined over Q if and only if there exists a function 3 : X —* E = CU{oo}, which is branched over at most three points. Inspired by Belyi's theorem, Grothendieck [Gro97] observed that the absolute Galois group Gal(Q/Q) acts naturally on these pairs (X,3), also known as Belyi pairs, and induces a faithful action on the corresponding Dessins. Indeed, in [Sch94b] the author proves that Gal(Q/Q) has a faithful action on Dessins D'Enfants of genus g = 1 and also on Dessins of genus g = 0. Geometrical insight into the intractable group Gal(Q/Q) can be gleaned from the action of Gal(Q/Q) on the Dessins and is the basis of an ambitious program started by Grothendieck aiming at a complete description of the absolute Galois group. To begin, a few basic definitions will be given. A Belyi function 3 denned on a compact Riemann 1  surface X is a meromorphic covering B : X -> £ of the Riemann sphere S = C U {00} with at most three branch points, which by composing with a suitable linear fractional transformation, can be taken to be a subset of {0,1,00}. Belyi functions B that satisfy the additional condition that all points in the preimage of 1 are ramified with ramification index equal to 2, are called clean  Belyi functions.  One can associate a graph Q{8) to the Belyi pair (X, B) by using 8 to lift the bipartite graph J consisting of the closed unit interval I = [0,1] C CU {00} with 0 labelled by a black vertex and  1 labelled by a white vertex. The points in Q(B)  = B~ {£) x  lying above 0 and  1, respectively, are  labelled by black and white vertices, respectively. Naturally, Q(B) has a bipartite structure since each connected component in the inverse image /?(0,1) (called an edge segment) joins two -1  vertices of opposite type. Several examples of bipartite graphs obtained this way are illustrated in Chapter 2. Each connected component of X \ B~ (T) is a simply connected region, which l  will be referred to hereafter as a face. A bipartite map M. on X consists of a bipartite graph Q lying on X together with the faces of X \ Q. Moreover, a bipartite map on X obtained by lifting I by a clean Belyi function 8 is denoted V(8), and belongs to an important class of combinatorial objects known as Dessins d'Enfants. In the survey papers ([Sch94b], [Jon97], [JS96]) examples of nontrivial Galois actions on classes of Dessins lying on: the Riemann sphere, elliptic curves, and hyperelliptic curves are given. In these examples, the Galois groups acting are usually solvable and in most cases abelian. In [Sch94b], the author describes the Galois action of A4 (and 54) on trees of genus 0, by first choosing (i.e. drawing) a representative Dessin T in the Galois orbit. The Belyi functions corresponding to the Dessins in the orbit of A4 were then computed using a method described in that paper. Although this method can be computationally difficult to use, requiring a computer algebra system to give an explicit description of the Belyi functions, it is quite effective for displaying Galois orbits if the Dessins in the Galois orbit are sufficiently small for the algorithm to work. In [Jon97], explicit Galois actions are given on classes of elliptic and hyperelliptic Dessins, byfirstdescribing the Belyi pairs (X,8), defined over somefieldextension K over Q, corresponding to the Dessins in the Galois orbits. The Galois action then corresponds to 2  that of Gal(K/Q). Several abelian examples and one non-abelian example, namely an action of AGLi (p) on a class of elliptic Dessins is given in [Jon97]. This leads to the natural generalization suggested by Gareth Jones in the last sentence of the paper [Jon97]: to produce examples of faithful Galois actions on explicitly defined classes of Dessins where more complicated Galois groups are acting, such as nonsolvable groups or solvable groups with unbounded derived length. This thesis addresses the more general case where the Galois action is of an arbitraryfinitegroup G and its action is on a special family ^(/^p.o) of Dessins d'Enfants, which include hyperelliptic curves of any genus. Chapter 2 gives a survey of some of the important connections between Maps, Belyi functions and permutations as it is related to Grothendieck's theory of Dessins d'Enfants. In that chapter some specific examples of Belyi pairs are given that are important in later chapters. The examples of Galois actions given in Chapter 3 lead to the construction of the class of Dessins d'Enfants V(^ ^) p  introduced in Chapter 4. In particular, the action of AGLi(p) on elliptic and hyperelliptic Dessins is revisited in Section 3.2 to motivate the more general approach. The main results of this thesis are contained in Chapters 4 through 7. In Chapter 4, the arithmetic properties of a family of polynomials f , which axe a component of the Belyi functions /3 $, are analyzed. p  P  Lemma 2 in Section 4.1 shows that f is irreducible for all primes p > 3. Theorem 3 in Section p  4.1 states that the Galois group Gal(/ (x)/Q) = S , the full symmetric group on p letters, for P  p  infinitely many primes p. The proof of Theorem 3 is given later in Section 5.4 after the relevant facts from algebraic number theory are developed in Sections 5.1-5.3. Chapter 6 describes a general method for producing faithful Galois actions on 2>(/3,a). Theorem 6 in Section 6.1 p  shows that anyfinitegroup G has a faithful Galois action on 2>(/? ,3). In Chapter 7 a more p  practical method for describing faithful Galois orbits is given once a group G is specified. In that chapter, the focus is on the group PSL2(p). Faithful Galois actions of PSL2(5), PSL2(11), and PSL2(13) are made explicit in sections 7.1 and 7.3'. The full Galois orbits of some of the examples given here, namely that of AGLi(ll), PSL2(11), PSL2(13) are displayed, using JavaScript and Maple, at www.math.ubc.ca/~burggraf/research.html. The explicit and simply defined nature of the Belyi pairs constructed here allows for the automatic generation of the Dessins (GIF  3  images) in the Galois orbits using Maple 7, and JavaScript enabled the interactive display of the Galois orbits. The Maple code for producing faithful orbits of PSL2(p) on unordered ntuples is given in Appendix A. The Maple code for determining the monodromy generators of a rational Belyi function defined on a hyperelliptic curve is given in Appendix B. The Maple code for automatically generating the images in the Galois orbits is given in Appendix C, and the JavaScript used for the display of the orbits is given in Appendix D.  1.2 Notation and Terminology Throughout this thesis X is a connected Riemann surface and S = C U {co} is the Riemann sphere. An n-sheeted branched covering / : X -* S has a branch point p E £ if f~ (p) contains l  fewer than n points. There is at least one ramified point x in the preimage / (p) of a branch -1  point p, where k > 2 sheets come together. The integer k in this case is called the ramification index of x  and k — 1 is its ramification order. The full set of branch points of / is denoted Br(/)  and ram(/) is the set of all ramified points. As mentioned in section 1.1,1 is a trivial bipartite graph embedded in S and Q(f) = f~ {T) denotes the bipartite graph embedded in X obtained l  by lifting T via / .  4  Chapter 2 Belyi functions and permutations  Some important connections between Maps, Hypermaps, Belyi functions and permutations axe given in the following sections of this chapter. At the center of these connections is Belyi's Theorem, which is stated as Theorem 1 in Section 2.5. The proof of Belyi's Theorem, or at least an outline of it, has appeared in various forms in: [Bel79], [Gro97], [Wol97], [Sch94b], [JS96], and [Jon97]. Another such outline is given here, in this case some of the details have been expanded upon as they are relevant to the construction the Belyi functions used later in Chapter 4. Some specific examples of Belyi pairs axe given in Sections 2.4 and 2.6 that axe used later in the Galois actions considered in Chapters 3 and 4.  2.1 Maps Definition 1 A map M is a connected graph Q (possibly with loops and multiple edges), which is embedded into a Riemann surface X such that: (i) the edges do not intersect; (ii) each connected component (called a face) in X \Q is simply connected.  A map At. can be described algebraically by two permutations mo, mi on its set A of directed edges or darts. Around each vertex v of M, the orientation of X determines a cyclic order on the darts pointing towards v, and mo is the permutation with these as its disjoint cycles.. The permutation m\ transposes pairs of darts that share a common edge and mi = (ra mi ) cycles - 1  0  each dart around one of its cobounding faces according to the orientation of X. Thus the cycles of mo, mi, and m-x correspond to the vertices, edges, and faces of M. Since Q is connected, the triple mo,mi,rri2 generate a transitive group G(mQ,m\,mi) < 5  SA  = Sym(A) called the  cartographic group of M, following the terminology of Alexander Zvonkin [Zvo98]. It is clear that G(mo, mi, m ) has the relations: ml — momim2 = 1 and is therefore an epimorphic image 2  of the universal cartographic group  Ct =  <Po,Pi,P2 | Pi = P0PXP2 = 1) •  Thus every map M determines a finite transitive permutation representation $ : C£ —* G(mo,mi,m2),  defined by # : pj i-> rrij, called an algebraic map. The cartographic group C?i(mo,mi,m2) of the map Mi in Figure 2.1 projected stereographically from the sphere to the plane is generated by the following permutations, which act on the right (and hence the product m ^ m Q is read 1  from left to right.) m = (l,2,3)(4,5,6)(7)(8) 0  mi = (l,4)(2,3)(5,7)(6,8) m = (momi)- = (1,6,8,5,7,4,3)(2). 1  2  Since the set of darts can be indexed in an arbitrary way, the map M. determines the carto-  Figure 2.1: The map Mi (orientation here is counter-clockwise) with cartographic group G i ( m , m , m ) = PSL (7). 0  1  2  2  graphic group only up to conjugation in 5a- It is not true that every algebraic map determines a map M because in general, an element #(pi) may fix a dart 6, and the edge underlying 5 6  will not correspond to a transposition of extended to that of a pre-clean  map,  However, the definition of a map can always be  which allows the graph Q to have free  edges consisting of  Then every algebraic map 1? determines a pre-clean map  a single dart, which isfixedby  M. Figure 2.2 shows a pre-clean map M i obtained from Mi by removing the two vertices associated to the darts 7 and 8. The cartographic group G(SQ, SI, s ) of M2 is now a permutation 2  group on only six darts: S o  = (l,2,3)(4,5,6)  si = (l,4)(2,3) 5 = (5 si)- = (l,6,5,4,3)(2). 1  2  0  As a third example, we start with an algebraic map # and construct the corresponding (pre-  Figure 2.2: A pre-clean map  Mi  with cartographic group Gi{s§,  s\, si) =  PSL (5). 2  clean) map. The map Mz corresponding to the permutations tf(po) = *o = (1,2,3,4,5,6,7)(8,10)(9,11) tf(p ) = ti = (l,8)(3,9)(5 10)(7,H) 1  >  0fo)= *2 = (*o*i) = (1,10,4,3,11,6,5,8 7,9,2). _1  V  has (counting the nontrivial cycles of U, i = 0,1,2) three vertices, four edges, and one face, which determines the Euler characteristic x = 3 — 4 + 1 = 0. Thus M% is the torus map shown in Figure 2.3.  7  19  11 Figure 2.3: The pre-clean torus map Mz-  2.2 Hypermaps A hypergraph is a coloured graph whose basic components are termed hypervertices, hyperedges, and hyperfaces (also referred to as j-faces, for j = 0,1,2), which are represented here by black, grey, and white polygons, respectively. The hypervertices are mutually disjoint, and so are the hyperedges and hyperfaces. A hypermap is an embedding of a hypergraph in a Riemann surface X. In the James representation [Jam88] of a hypermap, which is used here, each hypervertex and hyperedge intersect along a directed edge called a hyperdart, which terminates at a black vertex called a brin. The orientation of X induces cyclic orderings of the hyperdarts around each j-face, which correspond, respectively, to the cycles of the permutations: hj, j = 0,1,2. For example, the hypermap H stereographically projected from the sphere to the plane in Figure 2.4 determines the following permutations, ho, hi, hi on the set A of 11 hyperdarts satisfying the relation hohihi = 1: /i = (l,2,3,4,5,6)(7,8,9,10,ll) 0  hi = (1,7) h = (hohi)~ = (1,11,10,9,8,7,6,5,4,3,2). l  2  The permutation group G(ho,hi,hi) generated by the triple (ho,hi,hi) is called the hypercartographic group  of H and determines an algebraic hypermap or permutation representation 8  Figure 2.4: A hypermap H stereographically projected from the sphere to the plane. -& :  —* G(ho,hi,h2)  of the universal hypercartographic group (obtained by removing the  relation pi from C^) "ri-t ~ (P0,Pl,/>2 | A)PlP2 = 1>-  •  Conversely, every algebraic hypermap i? : po >-* ho, pi i—• hi, p2  l _ +  ^2, determines a hy-  permap H with hypercartographic group G(ho,hi,h2)- For example, the permutation group G(ho, hi, hi) with ho = hi = fi2 = (1,2,3) satisfies the relation /in/ii/12 = 1 and corresponds to the torus hypermap in Figure 2.5.  Figure-2.5: The torus hypermap corresponding to hypercartographic group G(HQ, h\, hi), where ho = hi = hi = (1,2,3). The torus is represented topologically by identifying opposite sides of the hexagon.  9  2.3  Belyi functions  A Belyi function 8 denned on a compact Riemann surface X is a branched covering 8 : X —* E of the Riemann sphere £ with at most three branch points. The automorphism group Aut(S) = PSL2(C) is triply transitive on £, meaning that any set of three distinct points in E can be mapped to any other set of three distinct points by some element of Aut(E). Thus, by composing 8 with an element of PSL2(C), the branch set can always be forced into a subset of {0, l,oo}. For example, if the distinct branch points of 8 are  {21,22,23}  C C, the element  4> € PSL2(C) represented by the linear fractional transformation: <p(z) =  22  —  23  22  —  21  2-21  2  , —  23  sends: 21 t-> 0, 22 •-»• 1, and 23 i-+ 00. Thus the composition J3 = </> o 8 is a new Belyi function whose branch points belong to the set {0,1, co}. Henceforth, it will be assumed that the branch points of all Belyi functions axe contained in {0,1,00}. If the Belyi function 8 is ramified above 1, with ramification index < 2 at each point in / 3 ( l ) , then 8 is called a pre-clean Belyi -1  function. In the case that the ramification index = 2 at each point in the preimage of 1, 8 is called a clean Belyi function. The following examples introduce some important Belyi functions E —• E, which are used repeatedly in later sections.  2.4  Belyi functions on £  Example 1. The polynomial P : E —>• E , denned by P (z) = z , has 0 as its only finite ramification n  n  n  point with ramification index n, and is a Belyi function with branch points {0,00}. The graph embedded in E consisting of the closed unit interval I = [0,1] with 0 labelled by a black vertex and 1 labelled by a white vertex is denoted by X. P lifts X to the graph n  Q(P ) = Pn (X) C E, which is an n-fold X-bouquet centered at 0, shown in figure 2.6 for l  n  the case n = 10. One can see that the graph Q(P ) has a bipartite structure, i.e. each n  open 1-cell of Q(P ) is bounded by 2 vertices of opposite color. Let 5 be the dart lying n  on.I that begins at 1 and ends at 0, then the darts of G{P ) are just the lifts of 5 by P . n  10  n  -Sa-  O 1  •  1  O  0  1  Figure 2.6: The graph G(P ) associated to the pre-clean Belyi function P\Q(Z) = z branched above {0, oo} c S. 10  N  Example 2.  The following rational maps 0 , : E -» £ form an important class of Belyi functions: m n  Pm,n^J -  mn  m  n  2  ^  Z  J  .  where m, n 6 Z — {0} are relatively prime and m + n^0. In the case that both m,n> 1, the ramification points of /?  m>n  are 2 = 0,1,  and oo of ramification indices m,n,2,  and m + n, respectively, lying over the branch points 0,0,1, oo € £. All of the Belyi functions /? , , for (m, n) ^ (1,1), are pre-clean. The graph G(0 ,n) — Pm}n(^) C E is m n  m  shown infigure2.7 for the case m = 6, n = 5. The corresponding hypermap H(0 , ) m n  associated to 0 , is obtained by "thickening" the black vertices and darts of G(8m,n) m n  and for the case m = 6, n = 5 it is the hypermap in Figure 2.4. A pre-clean Belyi function 3 can always be transformed into a clean one by composing' B with 8\,\ to get: 81,1 o 3 = 4/3(1 — 3). The resulting composition is a clean Belyi function because Bx i is ramified with ramification index 2 at 871(1) = \, which is not a branch point of 8, hence t  8i i o 8 has ramification index 2 at all points in the preimage of 1. For example, the pre-clean t  Belyi function 0e,5 is transformed into the clean one Bi^ o /3 as shown infigure2.8. 65  11  o  ft,!  1  Figure 2.7: The graph G(Pe^) associated to the pre-clean Belyi function /?6,s : above {0,1, co}.  2  —*  -0 1  -0-  l  2  Figure 2.8: The graph branched above {0,1, oo}.  2.5  £ branched  o /? ) associated to the clean Belyi function B\ \ o /3,5 : £ —> £ 6i5  t  6  Belyi's Theorem  Every nonsingular complex projective curve C defined by an irreducible homogeneous polynomial A(x, y, z) e C[x, y, z] denoted C = {[x : y : z] € P (C) | A(x,y,z) = o} 2  defines a compact Riemann surface. Conversely, Riemann proved that every compact Riemann surface is conformally equivalent to one obtained in this way and from now on, such Riemann surfaces and algebraic curves will be regarded as identical. The only algebraic curves considered in this thesis will be elliptic or hyperelliptic, each of which is birationally equivalent to a curve  12  of the form X = {[x : y : z) € P (C) | y V ~ 2  2  = B(x, z)} ,  where B(x, z) is a homogeneous polynomial of degree d > 3. It is customary, when defining a Belyi function on X, to "affinize" the curve by considering only the the part of X contained in the open neighbourhood of P (C) where 2 ^ 0 . Thus, only homogeneous coordinates of the 2  form (x, y, 1) in X are considered, and identifying these with the affine coordinates (x, y), yields the associated affine curve X ^. The projective curve is then easily recovered from the affine a  one by "homogenizing" its defining polynomial. Therefore, X will be identified with X«f = {(x,y)eC  2  y = /(*)}, 2  suitably completed at oo, where f(x) has distinct roots and degree d > 3. The Riemann surface X is said to be defined over a subfield K C C, if X is birationally equivalent to a curve denned by a polynomial equation A(x, y) = 0 with A(x, y) € K[x,y]. Furthermore, the smallest such field K is called the field of definition of X. The following theorem is due to Belyi. Theorem 1 [Bel79] A compact Riemann surface X is defined over the field Q if and only if there exists a Belyi function 8 : X —» S.  Proof For the direction (<=), Belyi quoted Weil's rigidity theorem [Wei56] and recently, J. Wolfart [Wol97] clarified Weil's proof. Belyi's proof in the direction (=*>) is relatively straightforward and is highly instructive for the construction of Belyi functions to come, so the relevant details will be included here. Thefirststep is to choose a suitable meromorphic function ir: X —>• S, then compose this with a sequence of polynomials S —• S which successively restricts the points of ramification from Q, to Q, andfinallyto{0,1, oo}. If X is defined over Q then a branched cover n : X —• S with branch points contained in P (Q) = Q u oo can always be found. For example, if X is defined by 1  A(x, y) = a (x) + a (x)y + ••• + On(x)y = 0, n  0  x  13  a (x) £ 0, n  for an irreducible polynomial A(x, y) projection ir: {x,  y) i-> x.  G  Q[x,y],  then  7r can  be chosen to be thefirstfactor  In this case, the branch points Br(7r) of 7T is the set of critical values  of A corresponding to the critical points satisfying one or more of the following conditions: (i) dn(x) = 0; (ii) A(x, y) has a repeated root y; (iii) x = oo. Being a set offinitesize, Br(7r)nQ has a minimal polynomial Hi G Q[x], i.e. a monic polynomial (not necessarily irreducible over Q in this case) of least degree which vanishes at each point in Br(7r).  for  i >  Likewise, a sequence of polynomials fj,2, M3> • • •> °f polynomials can be defined such that 2, Hi is the minimal polynomial of the set:  Br(/Zi_i)  n Q, and it can be shown that  the degrees of these polynomials in are strictly decreasing. To see this, decompose  into a  product of irreducibles i  where the degree of  pj  G  Q[x]  is dj with Y^j  dj — deg(/Xj)  1. The  —  dj  roots of each polynomial  Pj form a complete set of conjugate algebraic numbers, i.e. the Galois group Gal(pj/Q) of the splittingfieldof pj over Q permutes these roots transitively. Any root XQ of pj is mapped to a branch point jUi(xn) of m, which has (an irreducible) minimal polynomial automorphism cr G G&\(pj/Q) fixes the coefficients of m and qj{Hi(x )) = 0 Q  <r(qj(lM(x ))) = 0 & qj{a(ni(x )))  By the transitivity of Gal(pj/Q),  0  Q  each root of  pj  qj  qj  G  Q[x]. Each  and thus  = 0 <S> ^(/^(<7(x ))) = °-  C - ) 2  0  has the form <T(XO), for some  a  € Gal(pj/Q)  and by (2.1), each resulting branch point Hi(a(xo)) is a root of qj. Similarly, each root of has the form T(//J(XO)), for some r G  Gal(fy-/Q)  1  qj  and since r(//j(xo)) = IM(T(XQ)), each root of  qj is one of the branch points of m. Therefore, the degree of qj is at most dj. Now m i is the +  product, without repetitions, of the distinct polynomials qj, so deg(/ii+i) < J^rfj < deg0*i), j  as required. It follows that the sequence / i i , M2, • • • > terminates with a linear polynomial Hk G Q[x] and thus the composition H = Hk o • • • o H2 o / i ! : £ -»• £  14  hasfinitedegree with afinitenumber of branch points in Q. If | B r ( / z o 7 r ) | < 3, then composing fi o 7r with a suitably chosen linear fractional transformation <j> forces all the branch points into {0,1, oo} and < £ o u - o 7 r : X - + E i s a Belyi function. Otherwise, |Br(/i o 7r)| > 4, and composing with a suitably chosen linear fractional transformation <j> guarantees that the set \ 0,1, , oo \ C Br(0 o n o T T ) , ( m+n J where  is some rational number in lowest terms. Since B , : jo, 1, J ^ J , co j H-» {0,1, oo}, m n  composing <(>Qfj,oir with 8 , , reduces the size of the resulting branch set Br(/? ,n 0 ° M ° ) by 0  m n  7r  m  one fewer element than Br(</>ouo7r). Proceeding inductively, if necessary, by further composing /  with Belyi functions of the form /3 , reduces the size of the branch set at each stage by one. m>n  Eventually this leads to a branch set contained in {0,1, oo} and Belyi function 8 = bo<j>o/zo7r of finite degree, where 6 is a composition offinitelymany Belyi functions of the form 8 ,n- D m  In light of Belyi 's Theorem, D. Singerman [SinOl] gave the following definition of a Belyi surface.  Definition 2 A Belyi Surface is Riemann surface X defined overQ.  2.6 Elliptic and hyperelliptic Belyi surfaces. An elliptic curve X is an algebraic curve of genus 1. It is birationally equivalent to a curve in Weierstrass normal form X = {(x, y) e C | y = 4x - g x - <?}, 2  2  3  2  3  (2.2)  where the discriminant: g% — 21g\ is non-zero, which ensures that the cubic polynomial in x has distinct roots. Given any g , gz € C with g% - 27g% ^ 0, there is a lattice 2  A = {m + nr | m, n € Z}, r  Im(r) > 0,  in C for which the Weierstrass elliptic function. z  2  \(z — w)  2  15  w  2  satisfies the differential equation: (p') = ^p -gip-gz- Since p is doubly periodic, the mapping 2  z  from C -* X, defined by z i-» (p(z), p'{z)) induces an isomorphism C/A —• X. Two elliptic T  curves are isomorphic if and only if their corresponding lattices are similar, or equivalently if their moduli r are in the same orbit of the modular group PSL2(Z) = SL2(Z)/{±/}, where SL (Z) =  a b  a, b,c,dE Z, ad — be = 1  2  c d  As a discrete subgroup of PSLi2(R) = Aut(H), the automorphism group of the upper half plane H={r€C|lm(r)>0}, PSL2(Z) is a discrete subgroup of PSL2(M) and hence a Fuchsian group, which acts on H via  the linear fractional transformations ar + b cr + a  The elliptic modular function  '<'>-'<*>-IFW  (2  -  3)  is PSL2 (Z)-invariant and hence is an automorphic function (invariant under the action of a Fuchsian group) H —»C, which induces an isomorphism H/PSL2(Z) = C, so two elliptic curves axe isomorphic if and only if their j-invariants j(X) axe equal. Furthermore, an elliptic curve X is denned over a subfield if C C if and only if j(X) € K. To see this, note that if X is defined over K, then it is birationally equivalent to a curve in Weierstrass normal form (2.2) with <72)03 € K. Since j(X) is a rational function of #2, #3,  € Q(<72,53) C K.  Conversely,  if j(X) 6 K, then we canfindgi, g% 6 K satisfying (2.3) so X is isomorphic to a curve of the form (2.2), which is defined over K. By an aifine change of coordinates, X can be converted from Weierstrass form to Legendre form: X = {(x,y)eC  2  x  \y = x(x-l)(x-X)}, 2  where A satisfies 4(1-A + A ) ~ 27A (1-A) ' 2  J {  x )  2  16  2  3  A ^ 0,1  Example 3.  Let X\ be an elliptic curve in Legendre form X = {(x,y) € C \y 2  = x(x — l)(x  2  x  — A)},  where A ^ 0,1 is a zero of the polynomial f(x) = x — a, for some prime p, and a is a p  rational number which can be written as the following fraction in lowest common terms: a = ^^j- As in the proof of Belyi's Theorem, thefirstfactor projection IT : (x, y) H-+ X is a double covering of £ with branch points: 0,1, A, oo. Composing ir with P (x) = x pushes p  p  the branch points into Br(P o -K) = {0,1, ^^,oo}, which in turn get sent to {0, l,oo} p  by 8 ,n- Thus 8 ,n Pp 0  m  m  0  is a (preclean) Belyi function and 8\ — 8\ i o 8 , o Pp o 7r y  m n  is clean Belyi function. The graph Q(8\) is obtained by successively lifting the J by the components functions in the composition: B^i o 8 ,n o P oir. In the case p = 11, a = ^ m  p  the lift of the bipartite graph G(Pi,i o 83$) by P\\ : x ^ x  is shown in Figure 2.9. To  n  illustrate the graph Q(8\) (Figure 2.10), one must distinguish between the unique real root and the other complex roots of f(x) = x — a. p  Figure 2.9: The bipartite graph G(8^i o /3 o P ) is obtained by lifting the bipartite graph 3)8  G(p\,i Ps,a) by P :x^> 0  n  u  x  Example 4.  x(x - l)(x - X )(x kl  17  A  fe2  )},  Figure 2.10: The bipartite graph Q{8\) is obtained by using IT to lift the graph G(8i,i°0z,&-°-Pii) embedded in E. The case where A is.the unique real root of f(x) = x — a, is shown.on the left and the case where A is the complex root a / ^ "/ ! j shown on the right. The Riemann surface X\ is represented topologically by identifying the opposites sides of the rectangle. p  1  11  01  1  s  where A*^, Afc are distinct zeros of the polynomial f(x) = x —a, as in example 3. X\ \^ p  2  k  is a Belyi surface of genus 1, with clean Belyi function 8x ,x = B^i o 0 ^ kl  m  k2  n  In  OP O-K. P  this case, the graph G{8\ \ ), is represented more suitably on a hexagon with opposite kv  k2  sides identified as shown in Figure 2.11. Example 5.  Let Xi\ y be a class of hyperelliptic curves defined by k  0 < k < p — lare distinct  where N > 3 is a positive integer and Afc = a / e / , 1 p  2km p  zeros of the polynomial f(x) = x — a, as in the previous examples, with p > N. The p  corresponding class of clean Belyi functions  8s\ } k  :--^{A } fc  -*• £  c  a  n  be defined the same  way as in example 3: 8{X } = 01,1 Pm,n oPpOTT. 0  k  The Dessins d'Enfants V(8s\ \) are obtained by lifting the bipartite graph X by 0{\ y to k  k  the Belyi surface Xs\ y of genus g = [^f^J, which can be represented topologically by k  either a 4g-sided or a Ag + 2-sided regular polygon with opposite sides identified. If N is 18  Figure 2.11: The bipartite graph Q(3\) is obtained by using ir to lift the graph G(8i,i°&3,&°Pu) embedded in S. The case where one of A^, Afc is the unique real root (the choice Ao, A5 is used here) of f(x) = x — a, is shown on the left and the case where A^, Afc are not real: (A3, A7), is shown on the right. 2  p  2  odd, each of the 2g nonzero ramification points in  can be placed on one of the pairs of identified sides of the 4gr-sided polygon (see Figure 2.12). In the case that N is even, there are 2g + 1 nonzero ramification points contained in  and each of these can be conveniently placed on one of the 2g + 1 pairs of identified sides on the 4^ + 2-sided polygon. In the case p = 11, o = ^ the graphs £(/^A2,A5,A ,Aio) 8  3 X 6  shown on Belyi surfaces of genus 2 in Figure 2.12.  19  £(/3A ,A ,A8) 2  5  and  Figure 2.12: The hyperelliptic graphs G(3{\ y), where {A^} is a set of distinct roots of f(x) = x — a, are obtained by using w to lift the graph G{3i,i o 3$$ o P ) (see Figure 2.9) embedded in E. The graph G(3 ,\ \ ) is shown on the left and G(8 ,Xs,\ ,Xw) Sk  p  n  o  X2  s<  a  X2  20  8  n t h e  ri  nt  2.7 Monodromy groups In this section the monodromy group G(B) is defined and is shown to have an important connection to the graph Q(B) associated to B. Let / : X —• £ be a branched covering of the Riemann sphere, and suppose Br(/) = {bo,..., bk} C £ is its set of branch points. Fix a base point p E £ - Br(/), and let ft = f~ {p) denote thefiberabove p in C. The fundamental group l  fl = 7ri(E — Br(/),p) acts naturally on ft and has a presentation given by  where each generator a* is the homotopy class of a suitably chosen loop based at p, that winds once around 6j. For any element x Eft,each element a* E II lifts to a unique homotopy class r  of paths in X, such that any representative path in the homotopy class starts at x and ends r  at some common point x E ft. This defines a bijective mapping gi : ft —»ft,which gives a 3  permutation representation 9 : U —• Sym(ft) = 5n, defined by 6(ai) = g% (0 < i < k) has  The permutation  = \f~ (bi)\ disjoint cycles, each having cycle length k (1 < r < rii). l  r  Each disjoint cycle of gi corresponds to a ramification point of index k on C lying above r  where k sheets of C come together. The resulting permutation group G(f) = 0(11) is called the r  monodromy group of the branched covering / . In the case that / is replaced by a Belyi function B, we have Br(/3) C {0,1, oo}, II = H2 and G(B) can generated by the two permutations go and gi. Moreover, if 8 is a clean Belyi function, all ramification points above 1 have ramification order equal to 2, hence all disjoint cycles of g\ have length l\ = 2 (1 < r < ni), and the T  monodromy group G(8) has the following partial presentation G(j9) = (50,511 ^o = ^ = l , ETC),  where I is the least common multiple of {Zor}"=iSuppose the base point p E 2 — B is chosen to lie in the open interval (0,1). Then each edge segment e of Q(8) contains exactly one point p E ft and is incident to a unique black vertex r  r  at one end and a white vertex at the other. Each disjoint cycle of the monodromy generator go describes how the point p is cyclically permuted in ft around a ramification point v lying r  21  above 0, and thus describes how the edge segment e is cycled (according to the orientation of r  X) around its incident black vertex v. The valency of v is then given by the length lor of the disjoint cycle. This describes the connection between the black vertices of Q(8) and the disjoint cycles of go, and similarly the disjoint cycles of g\ can be associated with the white vertices. In the case that 3 is a clean Belyi function, the edge segments come in pairs with a white vertex of valency equal to 2 between them. The union of a white vertex together with its pair of incident edge segments is called an edge and each edge corresponds to a unique disjoint 2-cycle of g\. Thus, the black vertices and edges of Q(8) are determined by the disjoint cycles of go and gi, respectively, and vice versa. In the case that 8 : X —• S is a clean Belyi function, the graph Q(8) together with the faces in X \ Q(8) denoted V(8). is a special map called a Dessin d'Enfants and the monodromy group G(8) is just the cartographic group G(go,gi, (5o0i) ) of -1  V{8).  Figure 2.13 displays the graphs £0#i,i) and Q(ip), where ip =  °/3io,i> obtained by lifting the  bipartite graph J. The graphs £(/?i,i) and G(if>) are shown in relation to the lifts of rotations about 0 and 1 based at p= \, represented by the dashed and dotted loops, respectively. Using the labelling of the points in Cl = 8^\{j)) and Q, =  as shown in the figure, the monodromy  generators can be written explicitly: G(p\, ): x  <?o=(l)(2),  5i =(1,2). G(Ai &o,i) 0  :  50 =(1,2,3,4,5,6,7,8,9,10)(11,21), 5i =(1,11)(2,12X3,13)(4,14)(5,15)(6,16) (7,17)(8,18)(9,19)(10,20)(21,22).  The Belyi function 8\^\ is clean and has only one ramification point located at the white vertex w = \ E G{8\j), corresponding to the only disjoint 2-cycle of g\. The ramification order at w is the cycle length  = 2, which accounts for the two copies of 1 emanating from w.  The Belyi function ifr is also clean and G(ij)) has two copies of J emerging from every white vertex. In addition, i/> has two ramification points at the black vertices located at 0 and ^ in 22  Figure 2.13: The graphs  and G(ip) obtained by lifting the bipartite graph 1.  23  G(ip), corresponding to the two non-trivial cycles of go. The black vertex located at 0 in Q(ip) corresponds to the 10-cycle in go and has valency foi = 10. This accounts for the 10 copies of £(/?i,i) emanating from 0, forming afloralpattern in to as a 10-fold ^(/?ii)-bouquet. The black vertex at i  G(tjj) centered  at 0, which will be referred  € G(tp) corresponding to the 2-cycle of  go, has valency Z02 = 2 and is the center of a 2-fold <5(/?i4)-bouquet.  2.8  Calculating Monodromy groups of Belyi functions  Constructing a Dessin from a Belyi function Q :: X — > £ reduces to determining the pre-image which is equivalent (as illustrated in Figure 2.13 of Section 2.7) to determining the monodromy generators go, and g\ corresponding to path classes ao, and ct\ going once around 0 and 1, respectively. The case X = £ is considered here, where 8 : X —• £ is a rational function of degree d and a procedure is described for calculating G(0). In the more general case that X is an elliptic or hyperelliptic curve, a Belyi function @ = 8oir can be found by composing with the double covering TT : X — > £ denned by ir(x, y) — x. Computing the monodromy generators of ft involves lifting the path classes ao and ct\ first by 8 to £ and then by IT to X. The algorithm for the more general case is implemented by a sample Maple worksheet given in Appendix B. Choose V\ C £ to be a small open neighbourhood of A, where A € {0,1, oo}, such that ViDVj = •0,-for i,j e {0,1, oo}, i ^ j and let U\ = 0~ (V\) C X. Now define l  X~  = X  \ (U U Ui U t/oo) 0  and £~ = £ \  (VQ  U  VI  U  V^).  The following lemma (given by Leila Schneps in [Sch94b]) uses the fact that \8"(x)\ is bounded on £ and shows how to choose a neighborhood U of xo € X~ such that 8 maps U injectively -  onto V = /?([/) c £ ~ . Lemma 1 Suppose x e X~, po = 8(XQ) € £ ~ and \8"(x)\ < C. Choose open balls U = Q  B(x ,r ) 0  xo  C X~ and V = B(po,r ) C £ ~ , such that r po  <  XQ  and Tpg  2C  <  2  Define  where peV.  Then <j> (U) C U and for all xeU, p  24  \<j> (x) - <j> (x )\ < \\x - x | p  p  0  0  1  XCT  Proof  By the mean value theorem \<p (x) -<j)p(xo)\ <sup|<£p(a:)||z-a;o|, u p  where x € U and  Once again, by the mean value theorem \8'{x) - 8'(x )\ < sup\8"(x)\\x - sol < C\x - x \. u 0  0  Thus  sup|^(x)|<% =^i<-g\. 7  So CT  ,x  \<pp(x) - (j) {x )\ < ip,'^\ XQ  p  0  , , 1,  ~xo\<^\x-  XQ\.  Next, to prove <f> (U) C U it is sufficient to show \<j> (x) — XQ\ < r . p  p  Xo  \<f> (x)-xo\ < \<t>p{x) -#p(xo)| + \<t> ( o) -xo\ x  p  P  <L _ 2'  K  x  X o  | b-^o)l +  \3'(x )\  Ul  Q  2 \8>(x )\ +  0  by definition of r  po  ~ ' rx0  •.  The function 8 is injective on U since U C X , which does not contain any ramification points. -  It follows from Lemma 1 that 8 maps U onto V because 4> is a contraction mapping and for v  any x € U the sequence of points {^(x)} converges to afixedpoint of <f> in U, namely the p  unique point x € U such that 3(x) — p. Now choose a base point po in S~ and let XQ, ..., x<i-i be the points in thefiberfi= /3 (po)-1  Suppose 70 is representative loop in the path class OJO going clockwise around 0, that lies entirely in S~. Afinitenumber of distinct points: po,pi, • • • ,Pk on 70 can be chosen which partition 70 25  into curve segments of length less than r . Next consider the lift 70 of the loop 70 by 3 starting Po  at xo- As 70 is traversed counterclockwise from po to p\, 70 goes from XQ to some point p~\ in the fiber 3~ (p{) above p\. Iterating the function 0 starting at XQ, yields the sequence {^(x)} l  Pl  which converges to pi. Now repeated application of this procedure around 70 gives the first cycle of the permutation go and similarly all of the cycles of 30 and g\ can be determined.  2.9 Triangle Groups 7T TC  7T  Let r, s,t be positive integers and let T(r,s,t) be a triangle with angles —,—,— lying in the r s t  simply connected domain U, where U=<  if F + 7 + T > 1 C, ifI + I + I = l [ EI- iff + 7 + T < l 2TT 2TT  2ir  The triangle group T(r, 3, t) is generated by the three rotations A, B, C by angles —, —, —, r  s  t  respectively around the vertices a, b, c of T. T(r, s, t) has the following group presentation T(r, s, t) = (A, B, C  IA  r  = B = C* = ABC = 1) 3  and is a subgroup of index 2 in the group T*(r,s,t) generated by the reflections i?i,i?2,-R3 in the sides of T. The images under the action of T(r, s, i) on the fundamental region shown in Figure 2.14 give a tessellation of U in the case that U = EL T(r,s,t) is a cocompact group, meaning that if T is a subgroup offiniteindex in T(r,s,t), the quotient space U/T is a compact Riemann surface. The quotient U/T(r,s,t) is isomorphic to 2 and the natural projection 7r : U/T -* U/T(r,s,t) = S is branched over at most 3 points corresponding to projections of thefixedpoints a, b, c, of the generators of T(r, s, t). Applying a linear fractional transformation <f>, which maps the branch points into {0,1,00} yields a Belyi function 3 = <£OTT  :U/T  ^U/T(r,s,t)^H.  The periods r, s, t of a triangle group are not restricted to justfinitevalues. If T is a hyperbolic triangle with a vertex on R U { 0 0 } , then the angle is zero at that vertex, which corresponds 26  Figure 2.14: T U Ri(T) forms a fundamental region for T(r, s, t) in H, where ± + ± + i < 1. a parabolic element with infinite period. For example, the modular group PSL (Z) can be 2  generated by the elements Si  0 -1  = .  1  ,  0 1  Sp — .  0  _  1  ,  Soo —  1  1 1 _ 0 1  and can be given the presentation PSL (Z) = (Si, S , Soo\Sf = S = SiSpSoc = 1) 3  2  p  p  =•  T(2,3, oo).  Note that Si, S , and Soo act on H via the Hnear fraction transformations: Si(z) = —-, p  S (z) = jrj, Soo(z) = z +1, whichfixthe points: i, p = e ™/ , and oo, respectively. Thus Si, S 1  3  p  p  are elliptic elements and Soo is a parabolic element. The fundamental region for PSL (Z) is 2  shown in Figure 2.15 Any subgroup H of PSL (Z) acts properly discontinuously on U, so U/H is a Riemann surface. 2  If H has finite index in PSL (Z), then U/H has finite hyperbolic area and can be extended to 2  a compact Riemann surface U/H by adding finitely many ff-orbits in the extended rationals p!(Q) = Qu {oo}, i.e. U/H is just the quotient space (U U ¥ ( Q ) ) / H . 1  The Riemann surface  W/PSL (Z) can be visualized topologically by identifying pairs of sides of the fundamental 2  region (Figure 2.15) in the same PSL (Z) orbit. Thus the pairs of sides in the Si-orbit and the 2  Soo-orbit are identified and W/PSL (Z) is homeomorphic to a sphere with a single puncture. 2  27  Figure 2.15: The diagrams show from left to right, the fundamental regions in H for PSL2(Z), r(2), and TQ(2), respectively. Each fundamental region is the union of the white and shaded region shown. PSL (Z) is transitive on P^Q) and so W/PSL (Z) = S is a one point compactification of 2  2  the punctured sphere W/PSL (Z), the single point corresponding to the PSL (Z)-orbit P (Q). X  2  2  Furthermore, by defining j(t) = oo on this single orbit, the modular function j can be extended to an isomorphism j : W/PSL (Z) —* S. 2  The principal congruence subgroup T(2) of level 2 which is a normal subgroup of index 6 in PSL (Z) can also be regarded as a triangle group with infinite periods. T(2) is defined by 2  r(2) = 1<  ±  a b c d  € PSL (Z) 2  a b  1 0  c d  0 1  mod (2)  and can be generated by the following representative elements in SL (Z): 2  T = 0  -1  0  -1 2  1 2  2  -1  -2 3  0 1  where To,Ti,Too are parabolic elements withfixedpoints 0,1, oo, respectively. Identifying the sides of the fundamental region of T(2) (see Figure 2.15) paired by To and U/T(2)  shows that  is a sphere punctured by three distinct points, which correspond to the T(2)-orbits [0],  [1], and [oo], respectively, in P (Q). Since the generators all have infinite order, T(2) has the 1  28  following group presentation T(2) = (T ,Ti, 0  | ToTiToo = 1)  3  T(oo, oo, oo).  1?(2) is a free group of rank 2 and has the same group presentation as the universal hypercartographic group H2 (and the fundamental group IIo of Eo = £ \ {0, l,oo}). Thus T(2) is naturally isomorphic to TC^ and these will be treated as identical hereafter. Now each algebraic hypermap # : Ti^ —* G determines a conjugacy class of point stabilizers H < TC^ and the hypercartographic group G determines a hypermap H with underlying Riemann surface U/H. Another triangle group having infinite periods is the congruence subgroup To (2) of index 3 in PSL (Z) defined by 2  a b  To(2)  €PSL (Z) c==0mod(2) 2  c d  and can be generated by the following representative elements in SL (Z): 2  -1 0 2  -1 1  -1  -2  ,  U  00  1  =  11 0 1  where UQ, U , and Uoafixthe vertices {0,w = ^(1 + i),oo}, respectively, of the (non-shaded) u  triangle in the fundamental region of To (see Figure 2.15). To(2) has the following group presentation T (2) = 0  (U  0T  U, U  Uoc\U  2  =  UQUUUOC  = 1)  =  T(oo, 2,00).  Thus To (2) is isomorphic to the universal cartographic group Clf. Each algebraic map 1? : —» G determines a conjugacy class of point stabilizers M < Clf and the cartographic group G determines a map M with underlying Riemann surface U/M. The following theorem is a generalization of Belyi's Theorem, the proof of which can be found in [SinOl]. Theorem 2 The following statements are equivalent: (i) A compact Riemann surface X is defined overQ, (ii) There exists a Belyi function 3 : X -> £, (iii) X = U/H for some subgroup  29  H of finite index in PSI (Z), (iv) X = M/H for some subgroup H of finite index in T(2), (v) 2  X = M/H for some subgroup H of finite index in To (2), (vi) X =U/L for some subgroup L of finite index in T(r, s, t), for some integers r, s, t.  30  Chapter 3 Galois Actions on Dessins d'Enfants  A field K is Galois extension of Q if it is afiniteextension of < Q > such that the Galois group GK  —  Gal(K/Q)  has order equal to the degree [K : Q]. In Section 3.1 the absolute Galois Group  Gal(Q/Q) is described as a projective limit of thefiniteGalois groups GK- The examples of Galois actions given in Section 3.2 of this Chapter motivate the construction of the class V(P £$) V  of Dessins d'Enfants introduced in Chapter 4. Later in Chapter 6, the action of any GK (and its subgroups) are shown to have a faithful action on V(8 ^). V  3.1 The Absolute Galois group Every algebraic number is contained in the splittingfieldof its minimal polynomial, which is a Galois extension of Q so we have Q= U > K  KeK.  where K, is the set of all Galois extensions of Q in C. If K and F are Galois extensions with K > F, then every automorphism of GKfixesF set-wise, so restricting the automorphisms of GK to F defines an epimorphism PK,F '• GK —* GF, since every automorphism of F extends to K. The set of Galois groups {GK\K e K} together with the epimorphisms PK,F form an inverse system in the sense that PK,L ° PL,F — PK,F whenever K > L > F. Since Q is the union of all Galois extensions of Q, an element a e Gal(Q/Q) is a choice of automorphisms <JK € GK, one for each K 6 JC, such that PK,F sends OK to op whenever K > F. In this way, Gal(Q/Q) is the subgroup of the direct product YIKZK.  defined by the restriction epimorphisms, so  31  Gal(Q/<Qj) is the projective limit Gal(Q/Q) = lim GK of the Galois groups GK- This shows that Gal(Q/Q) is a profinite group, that is, a projective limit of finite groups. The subfields of Q correspond to the closed subgroups of Gal(Q/Q) in the Krull topology. By the Galois connection between subfields and subgroups, the algebraic numberfieldscorrespond to the subgroups offiniteindex in Gal(Q/Q) and the Galois extensions K € K correspond to the normal subgroups of finite index with quotient group isomorphic to GK- In this sense, the whole of classical Galois theory is contained in the single group Gal(Q/Q) and this, together with Belyi's theorem, leads to the surprising conclusion that one can do Galois theory by drawing pictures.  3.2  Galois actions on Elliptic and Hyperelliptic Dessins  Suppose X is a Belyi surface defined over K  C Q  with defining polynomial A(x,y)  K[x,y]  G  = 0}.  X = {(x,y)eC \A(x,y) 2  An automorphism a of the Galois group Gal(-ftyQ) acts in a natural way on X by acting on the coefficients of A(x, a :X -  y) = ao(x) + a\(x)y  X" = {(ar, y) € C  2  H  (-  | A°{x, y) = a(a (x)) 0  On(x)y : n  + a(a (x))y l  + ••• + a{a {x))y  n  n  =  0}.  Since X is a Belyi surface, there exists a clean Belyi function 8 : X — > S with corresponding Dessin T>(0). As 8 is also defined over K, the action of a on X induces an action T>(8)  —*• V(8 ) a  on the corresponding Dessins d'Enfants. It is not difficult to show that Gal(Q/Q) acts faithfully on elliptic curves. The isomorphism classes of elliptic curves are invariant under the elliptic modular function  as described in  section 2.6, where an elliptic curve X is shown to be defined over Q if and only if j(X) is. For every a 6 Gal(Q/Q) there exists a € Q such that cr does not act trivially on a. Now let X be the elliptic curve with j(X) = a, then there exists a Belyi function 8 : X —• S and both X and 8 are defined over afieldcontaining Q(a). Thus a acts nontrivially on both X and the 32  corresponding Dessin V{3). For an explicit example of a Galois action on the elliptic Dessins V(8x),fixA € <Q and let K/Q be the splittingfieldof the minimal polynomial of A. Gal(Q/Q) then induces an action of the quotient Gal(K/Q) and the orbit is contained in a subclass of T>(3\) defined over K. In example 3 of section 2.6, the elliptic curve X\ , where A = a ^ V 5  5  splittingfieldK = Qia / , 1 11  zeros Afe = /Pe / , l  2kvi p  a  2 e  "/ ) of x u  11  0  ™/  1 1  is defined over the  ,  - a. The Galois group G = Gal(K/Q) permutes the  0<fc<p-lof f(x) by inducing the affine transformations k  ak + b  on the index k, where a 6 Z * and b € Z . These transformations form the 1-dimensional p  affine group AGI_a(p) over Z , a split extension of a cyclic normal subgroup of order p (the p  transformations k (-»• k + b) by a cyclic group of order p — 1 (the transformations k i-> afc). The action of G is transitive (in fact doubly transitive) on the zeros Afc so the Dessins V(3\ ) are k  all conjugate under G. The size of the largest G-orbit in V(8\ ) is p, the number of distinct k  Dessins in this class, one corresponding to each zero of f(x). Figure 3.1 shows the action of the element cr: k t-» k + 1 € AGLi(p) on V(3\ ). 5  In example 4 of section 2.6 the elliptic curve X\ x kit  is also defined over K — <Q(a / , e ™/ ). 1 11  k2  27  11  The Galois group G = AGLi(p) is doubly transitive on the zeros A^ so the Dessins T)(8\ ^x ), k  with Afcj 7^ Afc are all conjugate under G. The size of the largest G-orbit in V(3\ 2  k  k2  is  \p{p — 1), the number of distinct Dessins in this class. The Galois group G also acts on the family of hyperelliptic curves Xtxky in example 5 of section 2.6, by acting on the indexed set {Afc}. For example, cr : {A2, A5, As} i-> {A3, A6, A9} and sends the corresponding Dessin £>(/3A ,A ,A )  H->  Dessins in the class ^(/?{A  where the unordered triples {A^, Afc, Afc} are distinct  2  FCL)  5  8  A ,A })) FC2  FC3  T)(3X ,X ,^9) 3  6  3 3  shown in Figure 3.2. The hyperelliptic 2  3  are clearly non-isomorphic, but certainly not all conjugate under G, because G is not triply transitive on the zeros of f(x). The action of AGLi(ll) is faithful on the orbit containing T>(P{X2,x M}) 5  s i n c e t n e s i z e  °f  t n e  orbit is |AGLi(ll)| = 110, as any computer algebra system  such as Maple can verify. The full orbit of AGLi(ll) on 2?CS{A  FCL  ,A ,A }) FC2  JavaScript and Maple, at www.math.ubc.ca/~burggraf/reseaxch.html.  33  FC3  i s  displayed, using  Examples of Galois actions on explicit and simply denned elliptic and hyperelliptic Dessins such as these have been considered in [Jon97], although the Galois groups acting are usually solvable and in most cases abelian. This leads to the natural generalization suggested by the author in the last sentence of the paper [Jon97]: to produce examples of faithful Galois actions on explicitly defined classes of Dessins where more complicated Galois groups are acting, such as nonsolvable groups or solvable groups with unbounded derived length. In chapter 4 classes of elliptic and hyperelliptic Dessins are defined over the splittingfieldK(f ) of a polynomial p  f (x), where p is an odd prime. The criteria for choosing the polynomials f are that they p  p  admit "maximal" Galois groups and "minimal" monodromy groups. The polynomial f (x), p  having only two critical values, is a Shabat polynomial and is chosen so that the corresponding Belyi function of the form j3 = b o p o f , as constructed in the proof of Belyi's Theorem, p  p  has "minimal" degree. The polynomial f also has Gal(/ /Q) = Sym(p), when p — 2 is not a p  P  perfect square, which yields faithful Galois actions by arbitraryfinitegroups and the explicit definition of the Belyi pairs (X, B ) on which they act allows the faithfulness of the action to p  be seen upon visual inspection as in the examples of this section.  Figure 3.1: The action oia :k^k + le AGLi(p) on the elliptic Dessin V(3\) from example 3 (section 2.6), where k = 5 and A = A =. o / ^ **/ . 1  1  5  34  10  11  Figure 3.2: the action of the element a : k '•-»• k + 1 6 AGLi(p) on the hyperelliptic Dessin  35  Chapter 4 A special class of Dessins d'Enfants  In this chapter certain Belyi pairs (X ^, 3 ^) are described, each defined over the splitting p  P  field K of an irreducible polynomial f p  pi  of prime degree p. The Galois action of  Ga\(K /Q), P  which is isomorphic to Sym(p) for infinitely many primes by Theorem 3 in Section 4.1, is later shown to be faithful in Chapter 6 on the corresponding Dessins  V(B Q). PI  4.1 The family of polynomials f  p  Lemma 2 For every odd prime p, the polynomial f (x) = x  p  p  —  ^ryX ~ p  l  +  1 is irreducible over  the rational numbers Q.  Proof The polynomial g (x) = (p - l)(x - p - l) f p  p  p  (jZJZ^)  = (P -  " P " 1) P  " P ~ 1) + P "  1  is.irreducible by Eisenstein's irreducibility criterion applied to the prime p and f (x) is irrep  ducible if and only if g (x) is. p  Theorem 3 Let hp be the monic polynomial defined by h (x) = (p - If p  ) fp  =x -p{j>-  P  p  lf~ x + {p- If. 2  For all primes p such that p — 2 is not a perfect square, the Galois group Gal(f (x)/Q) p  Gal(h (x)/Q) = Sp, the full symmetric group onp letters. p  36  =  Proof see Section 5.4 It should be noted that Galois groups of trinomials of the form x + ax + b, with certain n  conditions on n,s,a,b  3  have been considered in [Osa87] and [Coh99], however the trinomials  hp do not satisfy any of these specified conditions on n, s, a, b. One can verify using computer algorithms that f = x - ^x ~ n  n  n  l  + 1 is irreducible for many values of n > 3. In fact, this  has been done by the author using Maple 7 for the values: 3 < n < 500. The software package, KANT, can verify that Gal(/ /Q) = S , for 3 < n < 15. Furthermore, by reducing /„ mod p, n  n  for various primes p, one can extend these results a little further, although the calculations are somewhat tedious.  Conjecture 1 full symmetric  For every integer n  > 3,  the Galois group Gal(f (x),Q)  is isomorphic  n  to the  group Sn-  The result of Theorem 3 allows for the construction of a special class V of Dessins d'Enfants on which an arbitraryfinitegroup G can be shown to have a faithful Galois action. The remainder of this chapter will be devoted to the properties of this class V, which contains the Galois orbits of G, for all G. The idea is to define a suitable class of Belyi surfaces, on which Belyi functions can be chosen quite naturally; then the corresponding Dessins are induced by lifting the bipartite graph J. The Belyi surfaces underlying the Dessins can be defined using the zeros (j of the polynomial f , for some suitable prime p such that p — 2 is not a perfect square. p  4.2  A class of Dessins of genus > 1  Suppose Xp# is a compact Riemann surface of genus g > 0, defined by  where S = {Cfc, • • •, 0fc -i} is a subset of 2g — 1 distinct zeros of f , and 2g — 1 < p ^ n + 2, t  2g  v  for any n.  37  2  Proposition 1 A clean Belyi function d # : X $ p  p  £ of degree 4(p -p) can be defined by 2  4(p - I) ?" 2  - 9^-2  2  Proof The function 8p$(x, y) can expressed as the composition d^i o /3 _2,i o / o p  TT  :  X c j —* P i  p  where  IT,  £ is the first factor projection: tr(x, y) = x. The branch points of 8 ^ result from p  all branching and ramification of the composite functions: TT, f , 6 - ,i, p  p  2  and to show that  /3 $ is a Belyi function, it is sufficient to show that these branch points are contained in the P)  set {0,1, oo}. The degree 2 projection TT is branched over 0,1, oo, and the zeros Oc € X; these in turn, are sent to 1, j£f, oo, and 0 by the polynomial f . The ramification points of f are: p  p  oo, and those points that satisfy f'p(x) = 0, namely x — 0 and x = 1, all of which are already branch points of r. The Belyi function 0 -2,i sends 1, p^EfjOo, and 0 to the points 0,1, oo, p  and 0, respectively, and is ramified at 0, |pf, and oo, which again, are all branch points of the preceding function f o n. At this stage 0 -2,i fp ° TT is a preclean Belyi function, and further p  composition with  p  0  ensures that all ramification points over the point 1 have ramification  order equal to 2, hence 8 ^ is a clean Belyi function. • P  Thefinaloutcome is a clean Belyi function /? s, which is naturally defined on X $ in the P)  p  sense that all ramification points of each composite function of @ $ are contained in the set of p  branch points of the previous one. In this fashion, the desired Belyi function 0 $ is defined in P  a relatively uncomplicated way having minimal ramification and degree. The Dessin T>(3 ^) is p  now obtained as a lift of the bipartite graph 1. Figures 4.2 and 4.3 illustrate the succession of lifts by the composite functions required to obtain the Dessin T>(0 ^) in the case that p = 7, p  genus 3=1, and Q = {Ci}, where £i is thefirstzero of ft in the labelled set of zeros shown in Figure 4.1'.. The ramification point of fp at v = 1 lies above the black vertex / (l) = jj^f € G(Pi,i °0 -2,i), P  p  which joins a (p - 2)-fold ^(/?ii)-bouquet to the left and a 1-fold ^(/3ii)-bouquet to the right. )  i  Since v has ramification order equal to 2, a lift of a neighbourhood of  € G(Pi,i°dp-2,i) by f  p  to a neighborhood of v yields two (p-2)-fold £(/3i,i)-bouquets and two 1-fold ^(5 i)-bouquets /  38  1)  Im(z) Cl  C2  ,O  O  C7  Ca  , •  , R*M  -1  1  o  Ce o  ...  C4  "  Cs  Figure 4.1: The labelling of the zeros of the polynomial fa emanating from v. This accounts for two copies of G(Pi,i 3p-2,i) ha G(3i,i Pp-2,i ° fp), which 0  0  intersect at v; one starts at 0, which passes through v and proceeds downward, and the other which passes through v and proceeds upward. The ramification point of f at  starts at  p  0.6 G(/3i,x 3p-2,i ° fp) has ramification index equal to p — 1 and lies above / (0) = 1, giving 0  p  rise to the (p — l)-fold G(3\ \ o /? _ i)-bouquet centered at 0. The central black vertices of the t  p  2i  p [p — 2)-fold ^(/3ii)-bouquets in G{3\,\ Pp-2,1 fp) are the zeros {Ci, • • •, Cr} 0  i  0  Figure 4.1. The projection ir is ramified over only three points of  0I  *  fp shown in  °/5p-2,i fp)' 0, 1, and 0  Ci, each having ramification order equal to 2. Hence, it lifts the (p — 2)-fold 0(/?ii)-bouquets i  centered at £i to a 2p - 4-fold 5(^ii)-bouquets in G(P{Ci})- Since ir is a degree 2 covering (  )  and is unbranched away from {0,1, Ci}, the remaining p — 1 (p — 2)-fold ^(/3ii)-bouquets in i  ^(^1,1 /3p-2,i /p) are each lifted to two distinct 0  0  (p - 2)-fold ^(/3i i)-bouquets giving a total >  of 2p - 2 [p - 2)-fold ^(/3i,i)-bouquets in £(/%})• The Dessin V(3^ \) consists of the graph Cl  Gifirci}) and a single face lying on the torus obtained by identifying the opposite sides of a square, as shown at the top of Figure 4.3. It will be convenient and sometimes necessary to represent V(8 $) more abstractly using suitV  able diagrammatic notation, especially when the value of the prime p is large. To this end, graphical index notation is introduced in Figures 4.4 and 4.5, and is used to illustrate Galois 39  actions on V(3 Q). In Figure 4.4 a dashed circular arc indexed by i represents i copies of G(Q\,i) Pj  emanating from the center of the arc. In Figure 4.5 a double-dashed circular arc indexed by j represents j copies of G{p\,\ ° p\-2,i) emanatingfromthe center of the arc. This notation is used later in Figures 6.1 and 6.2 to illustrate the Galois action of an arbitrary finite group on  40  41  42  Figure 4.4: The (p — 2)-fold ^(/?ii)-bouquet on the left has the equivalent graphical index notation to the right. Each dashed arc together with the index i = 4 represents 4 copies of Q emanating from the black vertex !  Figure 4.5: A 5-fold Q o /^ij-bouquet is represented by a double dashed arc indexed by i = 5 and two 4-fold Q (^ii)-bouquets are represented by single dashed arcs, each indexed by )  i = i.  43  Chapter 5 The Arithmetic of f  p  The proof of Theorem 3 requires some detailed analysis of the family of polynomials f and p  a closely related family of monic polynomials h . In section 5.1, the relevant definitions and v  propositions are introduced and developed in the language of algebraic number theory; the main focus being: ramified primes in the ring of integers OK of afieldK. Section 5.3 is devoted to calculating the polynomial discriminant disc(h ), which contains important information about p  the ramified primes in OK, where K is the splittingfieldof the polynomial hp.  5.1 The ring of integers OK Let if be afinitefieldextension of Q and let thefinitefieldZ/pZ be denoted by Z . p  Definition 3 An element a € K is integral if there exists a monic polynomial p(x) G Z[x] such that p(a) = 0.  Proposition 2 The following statements are equivalent: (i) x is an integral element of K (ii) Z[x] is a finitely generated Z-module  (iii) A finitely generated Z-module M C K exists such that xM C M  Proof (i)  (ii) 44  Suppose x + a _ix m  m  m_1  + • • • + a x + a = 0. Then we have Z[x] = Z + Zx + • • • + Zx ~ m  x  l  0  since x e span{l, x,..., x } for k > m. k  m_1  (ii) => (iii)  Let M = Z + Zx H  h Zx  m_1  , then xM C M by the previous argument.  (iii) => (i)  By hypothesis, we know M = Zv\ + • • • + Zv with xM c M. Thus each element of n  {xv\,... ,xv } can be decomposed as aZ-linear combination of the elements of {v\,... ,v }. n  n  This means there is a matrix A with integer coefficients such that (A — xl)v = 0, where v — [v\,..., v ] and I is the n x n identity matrix. Finally, since v ^ 0, x is a root of T  n  p(x) = det(.4 - x7) € Z[x]. •  Lemma 3 The set OK of all integral elements in K forms a ring (called the ring of integers).  Proof Let x, y G OK, then by condition (ii) of the proposition above there exists finitely generated Z-modules M and iV (hence MN is afinitelygenerated), such that xM c M and yN C N. It follows that (x + y)MN C MN and (xy)MN C MN, therefore x + y and xy are elements of OK- • Properties of OK (i) Every ideal 7 C OK is a free Z-module, i.e. there exist elements yi,...,y El n  I = Zy + ...+Zy . 1  n  (ii) Every prime ideal P C OK is maximal, hence OK/P is a field. (iii) Every ideal / C OK can be factored uniquely into a product of prime ideals: I =  where P\,...,P  R  P^---P£ , R  are distinct prime ideals and n\,..., n are positive integers. r  45  such that  Definition 4 Consider a prime ideal p Z c Z c OK- A prime ideal P C OK lies above p if P fl Z = pZ. fn £/iis case, we also say that P divides p and use the notation P\p.  By property (iii) above, we can factor the ideal POK uniquely in the form  Proposition 3 The only prime ideals in OK lying above pZ are the primes Pi that occur in the unique factorization: POK = P\ • • • P? • r  1  Proof Suppose that Q C OK lies above pZ. Then QflZ = pZ  =•  peQ POK  =•  c QO = Q K  Pf ---PfcQ  =*.  1  Pi C Q, for some « It follows that P{ = Q for some i, since both Pj and Q are maximal ideals by property (ii) of Proposition 5.1. •  Definition 5 If P divides p, the ramification index e = e(P\p) is the largest exponent of P that divides pO , i.e. P \pO e  K  but P  e+1  K  \pO K  A prime p is called inert if the prime ideal (p) remains prime in OK- If the ideal POK factorizes into exactly n = [K : Q] primes: POK =  ITj=i Pji where each ramification index ej =  1,  then  p is said to split completely. On the other hand, a prime p is ramified if there is a prime ideal P C O K  dividing p such that e(P|p) >  1.  Definition 6 If P divides p, the residue degree f = f[P\p) is the dimension of the Z vector p  space O /P, i.e. f = [0 /P • Zp]. K  K  46  Proposition 4 Let pbe a prime in Z and suppose pOx factorizes as pOx = Pf • • • Pf • Then 1  r  r  where fx,...,f  r  are the residue degrees.  Proof By the Chinese Remainder Theorem r  = O /P?  OK/POK  where OK/POK  and  = n Ojr/if,  K  are Z -vector spaces. Thus  OK/P*  1  p  r  dim (0*/pOtf) = £ dim ( 0 * / P f ) . Zp  Zp  Now it is sufficient to show: (i) dim  Zp  (0*:/p0A:) = [K  : Q], and (ii) dim (C^/P^) = e^/,-. Zp  For the proof of (i) consider the map Zxi © * • • © Zx^ —• Zp © • • • © Z ,  c  denned by m\xi H  p  1- mnX >-> (mi mod p,...,  rrin mod p), where n = [K : Q]. The kernel of  n  this map is just pOjc and therefore OK/POK  — Z © • • • © Z , which has dimension n. p  p  For the proof of (ii), first consider the map $ : OK —* P where P|p and a € P  i _ 1  i - 1  I  4 1  1  I  i _ 1  K  /P ' , I  K  1 - 1  The latter implies P  \ P \ so the former is true and O /P = P  for each i > 1, define a map Otf/P* -> 0 P - /!*,  / P , defined by x  \ P . $ is surjective since a 0 K + P = P  maximality of P, ker($) = P or ker(<3?) = a €P  l _ 1  1  and P C ker($). By the  1 - 1  = P , which contradicts 1  / P as Z -vector spaces, for i > 1. Now i  p  where x + P*  i-»  x + P* . The kernel is just _1  so {OK/P^HP^/P*)  .=  OK/P ' 1 1  Taking dimensions of both sides gives dim (C?^/P ) - dim (P - /P ) = dim (0^/P - ). i  Zp  i  1  Zp  i  i  Zp  47  ax + P*,  1  Now substituting in the values i — 2,..., e and eliminating terms appropriately using OK/P — pi-l/pi  yie^g  dim (0 /P )  =  e  Zv  K  edim (0 /P). Zp  By definition / = dim (0jr/P), therefore dimz (0 /P )  = ef. •  e  Zp  5.2  p  K  K  Decomposition and Inertia groups  Now suppose K / Q is afiniteGalois extension and G = Gel(K/Q).  G acts on a prime ideal  P C OK over p by P° = (P) = {cr(x) : x € P}. CT  Note that P is also a prime ideal lying above p and furthermore, the following equalities of a  ramification indices and residue degrees hold e{P\p) = e{P°\p),f{P\p)  = f{P°\p)  Proposition 5 G acts transitively on the set of the primes above p.  Proof Suppose Pi and Pi are two primes in OK lying above p and Pf # PJ, for all a, r e G. Now consider the system of congruences: x = lmodP?,  for all cr 6 G  (5.1)  x = 0 mod P , for all r € G  (5.2)  T  2  Since Pf ^ PJ,forall cr, r € G, the Chinese Remainder Theorem implies that there exists an element y € OK satisfying the above congruences. Next, consider the norm N (y) K  =n  <y)  e  °K  creG  Since thefixedfieldof G is Q and NK{V) isfixedby G, it must be a rational. Furthermore NK{V) € OK =*•  € Z. Thefirstset of congruences (5.1) implies NK(y) = 1 mod Pi,  and the second set of congruences (5.2) gives Nx/k(y) = 0 mod P . Since Pi DZ = P flZ = pZ, 2  2  both NK/k{y) and N /k{y) - 1 are elements of pZ, which gives the contradiction: 1 e pZ. • K  48  The transitivity of G and the equality of ramification indices implies that all of the ramification indices tj are the same and likewise the residue degrees fj are all the same. Thus r  [tf:Q] = 2>/i  [K:Q\=ref,  =•  3=1  where r is just the number of distinct primes above p. Definition 7 For any P\p the decomposition group is D = {aeG\aP P  = P}  Proposition 6 |Dp| | = ef p  Proof G acts transitively on the set {Pi,..., P } of distinct primes lying above p and Dp is the r  stabilizer of the action. Thus \G/Dp\ = |orbit(P)| = r, the number of distinct primes above p. Therefore \D \ = \G\/r = ref/r = ef. • P  acts naturally on  D\  P p  OK/P,  since D \ fixesP, and trivially on P p  OK/POK-  SO reduction  mod  P induces a homomorphism D  P]p  - Gal(K(P)/Z ), p  where K(P) denotes OK/P- The kernel of this homomorphism is called the inertia group. Definition 8 For any P\p the inertia group is I = {cr G Dp | (TX = x mod P, for all x € OK} P  It turns out that the homomorphism D \  P p  Gal(F(P)/Z ) is surjective [Lan94] so there is a p  short exact sequence 1- I  Plp  - D  Plp  -> Gal(]?(P)/Zp) - 1.  It follows that \Dp\ /Ip\ \ = |Gal(]?(P)/Zp)| = / , which combined with |£> | | = ef gives p  p  P p  |i>| | = e. This means that p is a ramified prime if and only if I \ is non trivial for some P\p. p  P p  49  The following two theorems, stated without proof (see [FT91]), are well known and lead to a complete description of: (i) the primes p that ramify in OK; and (ii) the factorization in OK of ramified primes.  Theorem 4 The prime p ramifies in OK if and only ifp divides thefielddiscriminant  D(K/Q).  Theorem 5 (Rummer's Criterion) Let K be a numberfieldwith ring of integers OK and let 9 € OK be a primitive element. Suppose f(x) is the minimal polynomial of 9 then the reduction f of f mod p has the following factorization:  7(xT = p(xT---p7(xT, ei  r  1  where the exponents ej, j = 1,..., r are the ramification indices and fj = deg(Pj(x)) are the residue degrees corresponding to the prime factorization ofpOK C OK-  5.3 Discriminants The polynomial discriminant disc(h ) of the polynomial hp v  = n?=i (  ~ i)> defined in Theorem  x  a  3, Section 4.1 can be calculated using the resultant formula (-l)* T R{hp,ti ),  disc(hp) =  11  iE  p  were R(hp, h' ) is the resultant of hp and h' . The resultant R(fi, fi) of any two polynomials of p  p  degrees n and m, respectively: fx(x) = a x + an-ix " H n  71  1  n  and h{x) = b x + 6m-ix  m_1  m  m  50  1- a\x + ao,  + • • • + bx +b x  0  is the determinant of the (n + m) x (n + m) matrix: a  a _i n  n  a  =  b  n  ai  oo  a _i • • •  ai  n  n  RUuh)  • ••  a  n  a -i  •••  bi  bo  b  b -i  •••  b\  n  n  •••  n  b -i n  ao  > m a i ao  bo > n  K  b -i  bi bo  n  n  m  Thus R(h , h ) is the following determinant p  p  1 0 ••• 1  0  0  0  1  0  0  0 •••  p  (p-l)  2  •••  p 0 ••• p  -p(p-l)P-  p  -p(p-l)P~  (p-l)  2  p  o  -pip - i)p~  -p{p-i) -  2  o-  ly  -p(p-l)P~  2  0  -p(p-l)P~  o  ••••  o  P  o  •••  2  p  o  2  -p{p-iy-  2  which can be calculated directly, but the result for trinomials such as hp is well known (see [Swa62]) and is used here: disc(h ) = (-l)^r^pP(p p  - l)P-P- {p L  - 2).  5.4 Proof of Theorem 3 from Section 4.1 We will assume in what follows that p > 3. The zeros of the polynomial M*)  = (P - i )  p  (^T)  ?  fp (  £  r ) i  51  = ^  P  - P ( P - i ) " ^ + (P - ! ) p  2  P  axe rational multiples of the zeros of f (x) and hence Gal(hp(x), Q) = Gal(/ (x), Q). Therefore, p  P  it is sufficient to prove that Gal(/ip(x), Q) = S . Gal(hp(x), Q) acts on the set of p distinct zeros p  of h (x) by permutations and is thus isomorphic to a permutation subgroup G < S . To showp  p  that G = S it is sufficient to show that G contains both a p-cycle and a 2-cycle. Since h is p  p  irreducible (if and only if f is), Gal(h /Q) is transitive on the set of zeros of hp, and since p is p  p  prime, G must contain a p-cycle. It remains to show that G contains a 2-cycle. First we will attempt tofinda prime q that divides thefielddiscriminant disc(K/Q) of the splittingfieldK of hp. Then q will be a ramified prime when viewed as an element of the ring of integers OK of K. Thefielddiscriminant disc(K/Q) is related to the polynomial discriminant disc(h ) as follows: p  disc(K/Q) = —disc(h ), p  Th  where m and n axe relatively prime integers. Thus, the field discriminant is ,2  disc(K/Q) = ^  R  [(-lJ^'rVfr - rf-v~ (p p(p-i)  - 2)  l  The Prime Number Theorem implies that there axe infinitely many choices of primes p such that p — 2 ^ n , thus a ramified prime can be found by taking the prime q to be a prime factor 2  of p — 2 which is not a prime factor of n so that q\dK2  Next, observe that GCD(/ip,/i ) divides xh'(x) -php(x) =p(p-l) p  p_1  (x-p-l), and therefore  d = GCD(/ip, h'p) must divide p(jp — l) (a: - p — 1) mod q, where the notation d, hp, and h' p-1  p  represents the reduction modulo q of the polynomials d, hp, and h' , respectively. Note that p  p(p—l) (x— p— 1) mod q is not identically zero because q is relatively prime to both p and p—1 p-1  (q is relatively prime to p — 1 because q is a prime divisor of p — 2, which is relatively prime to p — 1). It follows that either d = 1 mod q, in which case h is separable, ord = x— p — 1 mod p  Thefirstpossibility can be ruled out because hp separable 23 [FT91]), which is a contradiction. Thus h = a<P € ¥ [x] p  q  52  qis not ramified in K (Theorem  where a 6 Fjx] is some polynomial and ad is separable in Fjx]. Now let a\,..., a denote the p  distinct zeros of h in K and suppose that P is a prime ideal of OK above q. Then p  v  hp = J J (x — ai) = acP mod P. i=l  Reordering the zeros if necessary, we can write d = (x — ai)(x — a ) = (x — ai) mod P 2  2  2  p a = J J (x ai) mod P. —  The existence of the ramified prime q ensures that there is a non-trivial element a in the inertia group Ip C Gal(/i ,Q) and under the action of <r, the zeros p  0:2,...,a , p  are pairwise  non-congruent mod P, since ad is separable. Thus we must have a(ot\) = ai, cr(a>i) = a\ and cr(ai) = a^, for i > 2. Hence as a permutation, the element a = (1,2), and therefore Gal(/ip(x), Q) contains a 2-cycle. •  53  Chapter 6 Faithful Galois Actions of arbitrary finite groups  Any element a of an arbitrary finite group G has a natural left action on the elements of G, yielding a permutation representation r : G —• S\G\, namely, the regular representation of G. S\Q\, in turn, can be injected into the Galois group Gal(/ /Q), for some prime p > \G\, such P  that p — 2 is not a perfect square, thanks to Theorem 3. This yields a Galois representation p : G —• Gal(/ /Q) given by the composition p = c6 o i o r, where 4> is the isomorphism of P  Theorem 3 and i is inclusion of permutation groups. G  S \ iG  S -t Gal(/ Q) p  PI  The Galois group p{G) acts on the Belyi surface X ^ , by acting on the coefficients of the P  defining polynomial of X $. P  More explicitly, an automorphism a €  p(G)  takes X Q PI  to  X  P  ^  defined by  r y = x(x-l)n(*-^(0) a  y = x(x-l) fj ( ^ - 0 2  C€<r(3)  So the element a € p(G), regarded as a permutation of the zeros of f , acts on the family X ,Q, p  by acting on the index set  P  consequently, a induces an analogous action on the corresponding 54  family of Dessin d'Enfants denned by cr (£>(/? )) = ©(/SL^cj)). Thus, a necessary condition pQ  for p(G) to act faithfully on the class of Dessins £>(/3 $>) with genus g > 1 is that p(G) act P)  faithfully on the subsets S of 2g - 1 zeros of f . It will also be shown that this is a sufficient p  condition.  6.1 Faithful actions on higher genus Dessins Theorem 6 p(G) acts faithfully on the class of Dessins d'Enfants P(/? CJ) with genus g ranging PJ  over all integers > 1.  Proof For any nontrivial element a € p(G) we will show that cr(9) ^ Of for some set of 2g — 1 :  zeros S of / . Choose a set 3? such that one and only one element £fc satisfies k <\G\ and for p  each zero Q of the remaining 2g — 2 elements in Of — {Ofc}> I > \G\. Notice that this choice of 9 requires that the prime p > \G\ + 2g + 2, which can easily be arranged since p can be taken to be arbitrarily large. Evidently cr((k) i fa because otherwise p 1  -1  (er) G  G would have a fixed  point x € G => P~ {&) = 1, which contradicts the fact that a is nontrivial. Thus, cr(fa = fa, l  where k' ^ k and k' < \G\, which implies fa £ that the Dessins V(B^ y) and  and therefore cr(9f) ^ 9?. Finally, it is clear  are non-isomorphic (see Figure 6.1) if k' =fi k. •  k  The action of an automorphism a € p{G) on a Dessin V(8p^) can be described once the genus g is chosen. Figure 6.1 illustrates the action of a on the Dessin V(3p$) of genus equal to 1, where Of = {£fc}, and Figure 6.2 displays the action of cr on a Dessin V(d ^) of genus equal to v  g, where J = {fa fa,..., Q _ }. 2g  2  55  \  \  -p-2  :.-,,.p-k-2  ^..p-<^;)-2 P^  p-a{k)-i%  P-2  p-k-fcP-2...  P-2  2  -.P-2  -,P-2  Figure 6.1: The action of an automorphism <7 € p(G) on a Dessin V(8 $) of genus 5 = 1, represented by a square with opposite sides identified. The dotted lines in this Figure and the next are as described in Figures 4.4 and 4.5. P  Figure 6.2: The action of an automorphism a 6 p(G) on a Dessin V(8 $) of genus g > 1, represented by a 4g-gon with opposite sides identified. P  56  Chapter 7 Galois Actions of P S L ( p ) 2  Every element, of afinitegroup G has a natural (left) action on the left cosets of any subgroup H < G, yielding an injective permutation representation r : G —• S[G-.H\, provided that core(iJ) = 1. By identifying each coset with a zero of f , where p is a prime such that p  p > [G : H], G can be identified with a subgroup of Gal(/ /Q) and r becomes a Galois repP  resentation. In the case that H = {1}, the Galois representation is the regular representation on G, which was used in chapter 6 to determine a faithful Galois action of an arbitrary finite group G on V. If G contains a proper subgroup H ^ {1}, the degree of the representation r is lowered by a factor of \H\, leading to Belyi functions Q and Dessins V(0 ) of lower degree; v  P  clearly, a desirable situation for the purposes of computing faithful Galois actions. Furthermore, a subgroup H < G of minimal index in G corresponds to actions of G on Dessins of minimal degree.  7.1 A Galois representation of PSL (5) 2  The alternating group A§ = PSL2(5) is the orientation preserving symmetry group of the regular dodecahedron and its action on the faces yields a permutation representation A$ —• S12. With the faces labelled as in Figure 7.1, A5 can be given the following presentation: A5 = (a, 61 a = 6 = (at-) = 1), 5  3  57  2  where a =(2,3,4,5,6X7,8,9,10,11), 6=(1,2,3)(4,6,7)(5,11,8)(9,10,12), ab = (1,2)(3,6)(4,11)(5, 7)(8,10)(9,12). It can be verified using a computer algebra program such as [GAP] or KANT that the separable  Figure 7.1: Two views of the dodecahedron are shown; the view on the right is a rotation of the view on the left by TT radians about the vertical axis. In both views, 12 is assigned to the hidden face at the bottom. polynomial fn(x) = x — -jf z +1 is irreducible over Q, and the Galois group G = Gal(/i2/Q) 12  11  of its splittingfieldover Q is the full symmetric group Su- The zeros of fu(x) are displayed in Figure 7.2 and labelled for later reference. C4 o  0.8-  C5 o  ° C2  0.60.40.2-  0  -0.5  1  0.5  -0.2-0.4C8 o.  -0.6-  0  Cu  -0.8o-Cio  C9o  Figure 7.2: The indexed zeros of'the polynomial /i (x) 2  58  Consider the elliptic curve Xi,j,k  where C»>0>  = {(x, y)eC \y 2  2  = (x- Q)(x -  -<*)}.  CA; are distinct zeros of the polynomial fu(x). The projection TT : Xij k —• S t  defined by ir(x, y) = x is branched over  Ot, and oo, which are in turn sent to 0,0,0, and oo,  respectively, by the polynomial f\2(x). In addition, fi2(x) has two ramification points 0 and 1 lying above the branch points 1 and  hence foir has a total of four branch points {0, j j , 1, oo}.  The Belyi function /?io,i sends {0, ^ , 1, co} into {0,1, oo}. Therefore, the composition dij^ — A , i ° /3io,i o / o 7T is a clean Belyi function. The lift of the unit interval X c S by Q%j,k gives rise to an elliptic Dessin P(A,j,ifc)> which is shown as thefirstDessin in various orbits at the top left corner of Figures 7.3, 7.4, and 7.5. By taking the ordering of the roots {Ci}l£i of f(x) in Figure 7.2, and identifying the i root th  with the i face of the dodecahedron, we get a Galois representation As —• G. th  The Galois group G = Gal(//Q) acts on X by permuting the triple of zeros in the set {Q, Ck} By identifying the i  th  root Ci with its index i, we have an induced action of G on the triple  {i,j, k}. For example, a : X2j,n ^ -^3,8,7 = -^3,7,8 which induces the action a : D(&2j,ii) ^ (  D(03,7,$)-  Moreover, the orbits containing the triple {2,7,11} under the action of the subgroups  <a>, <6>, and <a6>C As are as follows: <a> : {2,7,11} i-+ {3,7,8} i-» {4,8,9} ^ {5,9,10} ^ {6,10,11}, <b> : {2,7,11}  {3,4,8} ^ {1,5,6},  <ab> : {2,7,11} >-• {1,4,5}. The orbits of the corresponding Dessins are illustrated in Figures 7.3, 7.4, and 7.5.  59  Figure 7.3: The orbit of the Dessin V(d2j,u) under the action of <a>C G, shown from left to right as successive powers of a act on X>(/32,7,ii)-  60  Figure 7.4: The orbit of the Dessin V(p\,r,\\) under the action of <6>C G, shown from left to right as successive powers of b act on V(J32,7,u)-  7.2 Galois representations of PSL (p) 2  The special linear group SL2O?) (p > 2 here) consists of all 2 x 2 matrices with elements in F  p  having determinant = 1. The projective special linear group PSL (p) is the quotient group of 2  SL (p) by its center Z(SL2{p)), where the center is a subgroup consisting of elements of SL (p) 2  2  which commute with every other element of SL (p). In this case 2  <  2(SL (p)) = {<  PSL2O?) has order l ^ l = ^ ~^ SL  (p)  p  1  0  0  1  ±  2  and has maximal subgroups of order ^f -. PSLiip) can  2  2  be generated by as few as three elements (01,02,03), where one choice can be represented by the following three matrices in SL.2(p): 1  Ol =  0  0  , 1  1  ,  0-2 =  1  -1  and a =  x  0  0  x-  3  0  1  where x is a primitive element (mod p). The orders of the generators are: p— 1  ord(a2) = 2,  ord(ai) = p,  and ord(a$) = —-—.  Moreover, PSL2(p) can be given the following group presentation [CM65] PSL (p) = ^ai, a , a 2  2  1 = a = a , = (aia ) = (a a ) = 2  3  3  2  2  x  2  3  (afa a3) = aj^axazaT^ • 3  2  PSL2(p) has a natural (transitive) action on P (F ), where the elements of P (F ) consist of 1  1  P  P  p + 1 homogeneous coordinates of the form: [a : b] E F x F , which will be represented (using p  p  the notation =) here by column vectors: [0:6]  =  a b  Recall that homogeneous coordinates are invariant under scalar multiplication by F , that is, p  [a : b] = [aa, ab] for all a € F . Thus, each element of P (F ) can be represented by a 1  p  P  homogeneous coordinate of the form: [0 : 1] if a = 0(modp), or [1 : 6/0], otherwise. The  62  generator ai of PSL (p) acts on these homogeneous coordinates as follows: 2  1  0  0  0  _ 1  1  1  1  1  0  1  1  , and  1  1  , for j = 0 , . . . , p - l .  . 3 .J . +1  From this we see that the onlyfixedpoint of a\ is [0 : 1] and a\ cycles through the remaining p elements of the form [1 : j], j = 0 , . . . ,p - 1. The element a acts on P^Fp) as follows: 2  0  1  -1  0  0  1  =  1  0  , and  0  .  _  1  1  3  1  .3 .  0  1  , j = 1, . . . , p - l .  —  . -r .  -1  1  Thefixedpoints of a are the elements [1 : j], for which j = —j (mod p) or j = —l(mod p), _1  2  2  which, can only happen if p = l(mod 4), by an elementary result of number theory. The element 03 acts on P (F ) as: :  P  x  0  0  x-  1  0  0  1  1  , and  x  0  0  x~  x  1  1  . 3 .. ~ 3 . . ~ i  l  x  l  x  .  2  ,3=  1,  Thefixedpoints of 03 are [0 : 1] and the elements: [1 : j] for which j = x j(mod p) which _2  implies j = 0(mod p) or x~ = l(mod p). Notice that the primitive element x has order p — 1 2  and thus ord(x~ ) = 2  p-1  Hence, a has fourfixedpoints:  x = l(modp), 2  onlyifp = 3.  [ 0 : 1 ] , [1 : 0], [1 : 1], [1 : 2],  3  if p =  3  and twofixedpoints:  [0 : 1], [1 : 0], if p 7^ 3. The p + 1 elements of P (F ) can be identified with the elements of the 1  P  set  { 0 , 1 , 2 , . . . ,p}  as follows: , for j = 0 , . . . , p — 1,  and p  0 1  Enumerating the elements of P (F ) in this way, gives a permutation representation r of 1  p  PSL (p) in the symmetric group S i = 2  p+  p  ,p}. It is clear that PSL (p) acts  Sym{0,1,2,...  2  transitively on P (F ) and the stabilizer Stab[ ^ in PSL (p) of any homogeneous coordinate 1  P  a:  2  [a : b] has index p + 1. Thus r becomes a Galois representation of PSL (p) of degree p + 1 p  2  (which is minimal if p > 11) by identifying the left cosets of Stab^ 63  in PSL (p) with a set, 2  denoted by £(p+i, ) of p + 1 distinct zeros of f , where q is a prime such that q > p + 1 and q  q  3 - 2 is not a perfect square. This in turn induces an action of PSLiip) on the family of curves: = { (x,y) eC \y 2  = x(x - 1) J J (s - Q  2  where S is a subset of any n zeros of / . n  g  The generators: a\, ai, 03 of PSL2(p) can now be interpreted as permutations on the indices of the set  S(p  + l g  ) = {Co, Ci> C2, • • •, Cp}  of zeros of f , and have the following cycle decompositions: q  ai = (0,1,...,P-1) 2  = (0,p)Uj (j,-j mod p)  a  = (x fc, x fc,..., x fc) (xk, x k, x f c , x ~ k ) , k €F2  a  -1  2  3  4  p-1  3  5  p  2  2=1 2  2^1 2  Notice that in the case p = l(mod 4), one of the 2-cycles of 02 collapse to 1-cycles, which correspond to the twofixedpoints of ai. Thus, ai is a product of ^ 2-cycles if p = l(mod 4), and a product of ^ 2-cycles if p = 3 (mod 4). The element 03 is a product of two cycles of length  for any primitive element x G F*.  The Galois action of each element of PSLi(p) on the curve X ,3 is reduced to a permutation g  n  on the indices of the unordered set S„ = {Q ..., Q } of n zeros of f , where the genus of the i;  curve is g =  L^y^J-  n  q  So a necessary condition for a faithful Galois action of PSLi2(p) is that it  act faithfully on the unordered n-tuples of zeros in ~(p+i, )- It is sufficient to produce an orbit q  containing an n-tuple of size |PSL2(p)| in which case there must be at least |PSL.2(p)| unordered n-tuples, i.e. _ (p + l)l ^p(g -l) ' - nl(p-n + l)! " ~ T — 2  C p + l  n  ( 7  , . ' 1 }  Since Cp+i = P^" ) < |PSL<2(p)|, n cannot be smaller than 4. The three smallest primes 1  i3  satisfying inequality (7.1) are: p — 11, p = 13 and p = 17, where n may take values in: {5,6,7}, {5,6,7,8,9} and {4,5,6,..., 14}, respectively.  64  7.3 Examples: PSL (11) and PSL (13) 2  2  The two examples we consider are: p = 11 (= 3 mod 4) and p = 13 (= 1 mod 4), where in both cases x = 2 is the chosen primitive element. The generators of PSL (11) and PSL (13) 2  2  are written explicitly in cycle notation: PSL (11) : 2  = (1,2,3,4,5,6,7,8,9,10,11)  fll  a  2  = (1,12)(2,11)(3,6)(4,8)(5,9)(7,10)  a  3  = (2,4,10,6,5)(3,7,8,11,9)  PSL (13) : a  x  = (1,2,3,4,5,6,7,8,9,10,11,12,13)  a  2  = (1,14)(2,13)(3,7)(4,5)(8,12)(10,11)  a  3  = (2,11,10,13,4,5)(3,8,6,12,7,9)  2  A search algorithm, which computes and counts the size of orbits on all unordered n-tuples can be used to determine the existence of a full sized orbit of PSL (p), i.e. an orbit of length 2  |PSL (p)|. One such algorithm carried out by Maple 7, is used here to determine various pairs 2  (p, n) satisfying condition (7.1) that admit full sized orbits of PSL (p) on the class V(3 $) of 2  P  Dessins consisting of curves C ^ . A sample Maple worksheet containing the search algorithm q  n  for any pair (p, n) is given in Appendix A. Full sized orbits of PSL (11) on all unordered n-tuples are summarized in the following table. 2  PSL (11) (order = 660) 2  n  Maximum orbit size Representative n-tuple contained in orbit of size 660  5,7  660  {0,1,2,3,5}, {0,1,2,3,4,6,10}  6  330  none found  In the case p = 13, the condition (7.1) allows for five possibilities for n, only two of which admit  65  full sized orbits of PSL (13): 2  PSL (13) (order = 1092) 2  n  Maximum orbit size Representative n-tuple contained in orbit of size 1092  5,9  546  none found  6,8  1092  {0,2,5,9,12,13}, {0,1,2,3,4,5,6,8}  7  546  none found  Similarly, full sized orbits of PSL (17) and PSL (19) on all unordered n-tuples are summarized 2  2  in the following tables. PSL (17) (order = 2448) 2  n  Maximum orbit size Representative n-tuple contained in orbit of size 2448  4,14  1224  none found  5,13  1224  none found  6,12  2448  {0,1,2,3,4,6}, {0,1,2,3,4,5,6,7,8,9,10,12}  7,11  2448  {0,1,2,3,4, 5,7}, {0,1,2,3,4,5,6,7,8,9,11}  8,10  2448  {0,1,2,3,4, 5,6,8}, {0,1,2,3,4,5,6,7,8,11}  9  2448  {0,1,2,3,4,5,6,7,9} PSL (19) (order = 3420) 2  n  Maximum orbit size Representative n-tuple contained in orbit of size 3420  4,16  1710  none found  5,15  3420  {0,1,2,3,4}, {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}  6,14  3420  {0,1,2,3,4,7}, {0,1,2,3,4,5,6,7,8,9,10,11,12,14}  7,13  3420  {0,1,2,3,4,5,6}, {0,1,2,3,4,5,6,7,8,9,10,11,12}  8,12  3420  {0,1,2,3,4,5,6,8}, {0,1,2,3,4,5,6,7,8,9,10,11}  9,11  3420  {0,1,2,3,4,5,6,7,8}, {0,1,2,3,4,5,6,7,8,9,10}  10  3420  {0,1,2,3,4,5,6,7,8,9}  In. the case of PSL (11), the 5-tuple: {0,1,2,3,5} corresponds to the subset 2  2*5 = {Co, Ci) C2, C3, Cs} G S5 C H( 66  12)13  )  of indexed zeros of /13, which is in a full-size orbit of PSL2(11). The corresponding Dessin is obtained by lifting the graph G(0i,i o /3n,i o / ) in Figure 7.7 by 7^,1,2,3,5 to the genus 3 curve 13  -^i3.{Co,Ci.C2.C3,C5}  e  -^13,^5  i Figure 7.8. The class of curves on which PSL (11) acts, belong to n  2  the family: X\z^ and the Galois action of the element a in PSL2(11) sends -Xi3,{c ,Ci,C2,C3.C5} 2  s  to ^i3,{< ,C5.C7,Cio>Cii}' h i w  c n  0  is illustrated in Figure 7.9. The full orbit of PSL2(11) on 2>(/3i3,9 ) 5  2  is displayed, using JavaScript and Maple, at www.math.ubc.ca/~burggraf/research.html. Likewise, in the case of PSL2(13), the 6-tuple: {0,2,5,9,12,13} corresponds to the set ^6 = {Co, C2, C5, C9> Cl2, Cl3} € 3?6 C 5(14,17)  of six zeros of /17. The corresponding Dessin is obtained by lifting the graph G(3i,i °/3i5,i  0/17)  in Figure 7.10 by ^0,2,5,9,12,13 to the genus 3 curve *i7,{C6,Ca,0j,C»,Cia,Cis} ir,&» ™ Figure 7.11. 6  The Galois action of a in PSL (13) sends -^i7,{Co,C2,Cs,C9,Ci ,Ci } 2  x  2  t  3  o  x  • i7,{<b,Ci,C3,C(i,Cio,Cia}> y  w  h  i  c  h  is shown infigure7.12. The full orbit of PSL2(13) on T>(8n^) is displayed, using JavaScript and Maple, at www.math.ubc.ca/~burggraf/research.html.  1  C2, C3  Im(z)  I  C4  Ci  Im(z) C2  Cs  Co  C4  C3  Co  Ce . C12  Cs  C7  Re(z)  JBsiz)  "C11  °Cl5  Cs'  Ce  C10  Cs-i  J  °C  Co*  C9  °Cl3 C10  Cri  fJ  °Cl2  Figure 7.6: The indexed set of zeros of /13 on the left and of /17 on the right.  67  14  Figure 7.9: The Galois action of a € PSL (11) on G(Pi,i o Bu,i ° hz 2  2  ° ^0,1,2,3,5)-  Figure 7.11: The graph G(Pi,i ° /?i5,i ° fi7 ° ^0,2,5,9,12,13) lies on a curve of genus 3 represented topologically by identifying opposite sides of the 12-gon.  f o r j l from 1 to numcombs do i f num >= numperms then print("Exhausted a l l permutations of n-tuples",num); p r i n t ( " F a i t h f u l a c t i o n s correspond to o r b i t s of s i z e =",N);, break; fi; orbit:=array(1..N); C:=nextstruct(allcombs): P:=convert(C,list); flag:=0; for j2 i n orblist do i f member(P,orbp[j2]) then flag:=l; p r i n t ( j l . C , " d o n e i n ",j2,"skipped"); break; od; i f flag=l then next; fi; f o r j from 1 t o N do yCj]:=list(l..n): # y[j][k] i s an integer modulo p+1 r e p r e s e n t i n g the # homogeneous coordinate [ l : y [ j ] [ k ] ] r e s u l t i n g from the # a c t i o n of PSL2p[j] on x[k] . # I f z i s an integer mod p+1, i t represents the # f o l l o w i n g homogeneous coordinate: # z <—> [ l : z ] , i f i<p # z <—> od: f o r j from 1 t o N do  [0:1] , i f i=p  f o r k from 1 to n do A:=PSL2p[j] [1] [ i ] * x [ C [ k ] ] [1] [l]+PSL2p[j] [1] [2] *x[C[k]] [2] [1] ; B:=PSL2p[j] [2] [1] *x[C[k]] [1] [1]+PSL2p [j] [2] [2] *x[C[k]] [2] [1] ; i f (A mod p<>0) then y [ j ] [k] :=B/A mod p; e l s e y [ j ] [k] :=p;  # . B/A <—> #  fi; od:  72  p <—>  [1:A/B] = [A:B] [0:B] = [0:1]  orbitc[j]:=convert(y[j],set); orbitp[j]:=convert(y[j],list); od: orbc:=convert(orbitc,set); orbp[j1]:=convert(orbitp,set); num:=num+nops(orbc); orblist:=[op(orblist),jl]: i f nops(orbc)=N then f a i t h l i s t : = [ o p ( f a i t l i l i s t ) ,C] : print(jl.C.nops(orbc),"faithful!!!",num/numperms*100.0) else print(j1,C,nops(orbc),num/numperms*100.0) fi: od: > i f nops(faithlist)>0 then p r i n t ( " F a i t h f u l o r b i t s contain the f o l l o w i n g n - t u p l e s : " ) ; f o r j from 1 t o n o p s ( f a i t h l i s t ) do print(j,faithlist[j]); od; else printC'No f u l l - s i z e o r b i t s found.")  73  Appendix B Mondodromy generators of a hyperelliptic Belyi function  The elliptic curve *2,7,n =  {(x,y) eC \y 2  2  = (x- Q)(x - Cj)(x  -fa},  where Ci,Cj,Cfc are distinct roots of fn(x) = x - 12/llx 4- 1, admits a Belyi function 0 = 0\o,i fu ° TT, where ir(x,y) = x and 0\Q,I(X) = ll /10 x (l - x). This worksheet computes the monodromy generator go of 0. The monodromy generator g\ of 0 is easily determined from the monodromy generator hi about 1 of the Belyi function /3io,i(/i2)- This is illustrated in the worksheet bl32MonCycles.mws. The monodromy generator g\ is represented by the following permutation on 264 letters. 12  11  n  0  10  10  g =[l, 12] [2,22] [3,32] [4,42] [5,52] [6,62] [7,72] [8,82] [9,92] [10,102] [11,112,132,122] [133,144] [134,154] [135,164] [136,174] [137,184] [138,194] [139,204] [140,214] [141,224] [142,234] [143,244,264,254]. x  Note that 51 has two -^cycles and twenty 2-cycles. The following diagram displays the inverse image in the y-plane of two circles Co and C\ centered about 0 and 1, of radius 1/2, and based at 1/2, under the Belyi function 0. The blue dots are in the inverse image of Co and the red dots are in the inverse image of C\. The the points in inverse image of 1/2 are marked by black squares. Any path on the circle Co (or C\) travelling counterclockwise beginning and ending at 1/2 lifts to a path in the y-plane beginning at some point p\ in the inverse image of 1/2 and ending at another point p in the inverse image of 1/2. This defines the monodromy generator about 0 (or 1) in the y-plane. The monodromy in the y-plane will yield the monodromy on the elliptic curve via the correspondence y <-> (x, y). The cycles of the monodromy generators go and g\ correspond to the blue and red paths, respectively, in Figure B.l. 2  74  Figure B.l: The inverse lift of the loops CQ and Ci by 82,7,11 to the y-projection of  Xij^i-  In the following program code of this worksheet, the 264 points in the inverse image of 1/2 axe, numbered and the blue and red paths are followed between these points to determine the cycles of the monodromy generators go and g'i.  > _MaxSols:=132;  # By d e f a u l t the solve f u n c t i o n w i l l find' only 100 # s o l u t i o n s but 132 w i l l be needed below.  >#Find and d i s p l a y roots of polynomial >  the i r r e d u c i b l e  f l 2 ( x ) = x~12-12/ll*x-11+1.0  r:=solve(x~12-12/ll*x~ll+1.0,x):  f o r i from 1 t o 12 do p r i n t ( i , r [ i ] ) od; >#The s o l u t i o n s t o the equations below give the inverse image of p o i n t s on the two  c i r c l e s CO and C l , under the B e l y i f u n c t i o n b(10,1)(f12(x)). The p o i n t s  i n the inverse image of CO are stored i n the array trO and the points i n the inverse image of C l are s t o r e d i n the array t r l .  Note that a l l the p o i n t s i n  trO and t r l l i e i n the x-plane. > eqnO := l l - l l / 1 0 - 1 0 * ( x " 1 2 - 1 2 / l l * x " l l + l ) - 1 0 * ( l - ( x ~ 1 2 - 1 2 / l l * x - l l + l ) )  75  = 0.5*exp(2.0*Pi*k*I/n); eqnl : < r i l / 1 0 ~ 1 0 * ( x ~ 1 2 - 1 2 / l l * x ~ l l + l ) ~ 1 0 * ( l - ( x ~ 1 2 - 1 2 / l l * x ~ l l + l ) ) = l-0.5*exp(2.0*Pi*k*I/n); > f o r j from 1 t o 5 do tr0[j]:=solve(eval(eqn0,{k=j-l,n=5})); . i f j=l then t r l [1] :=trO[l] ; e l s e t r l [j] :=solve(eval(eqnl,-Ck=j-l,n=5})); fi; p r i n t ( j , " out of 5") od: >#The f o l l o w i n g loops solve f o r y i n the expression y " 2 = ( x - r [ 2 ] ) * ( x - r [ 7 ] ) * ( x - r [ l l ] ) , where the x-values are the p o i n t s i n the inverse image of b ( 1 0 , l ) ( f 1 2 ( x ) ) found above. T h i s l i f t s the p o i n t s on the c i r c l e s CO and C l t o the y-plane, which are s t o r e d i n the arrays yO and y l . > f o r i from 1 t o 5 do f o r j from 1 t o 132 do y O [ i ] [ j ] := ^ s q r t ( ( t r O [ i ] [ j ] - r [ 2 ] ) * ( t r 0 [ i ] [ j ] - r [ 7 ] ) ' * ( t r 0 [ i ] [ j ] - r [11])); if i=l then y l [ i ] :=yO[i] ; e l s e y l [ i ] [ j ] := s q r t ( ( t r l [ i ] [ j ] - r [ 2 ] ) * ( t r l [ i ] [j] - r [7] ) * ( t r l [i] [ j ] - r [11])); fi; od: f o r j from 133 t o 264 do yO[i][j]  := - y 0 [ i ] [ j - 1 3 2 ] :  yl[i][j]  := - y l [ i ] [ j - 1 3 2 ] :  od: p r i n t ( i , " out of 5"): od: >#In the loop below, a s t a r t i n g point p i = 0.5897322923+0.4356541991*1 i n the array t r 0 [ l ] i s s e l e c t e d f o r the monodromy generator  about 0  i n t h e x-plane and i s denoted by t r 0 [ l ] [ o r d 0 [ l ] ] . > f o r j from 1 t o 132 do d0[j] :=abs(Re(trO[l] [ j ] ) - 0 . 6 ) : . i f d0[j]<0.1 and I m ( t r 0 [ l ] [ j ] ) > 0 then o r d 0 [ l ] :=j : o r d l [ l ] : = j : p r i n t (j , t r 0 [ l ] [ j ] ) ; fi; od:. >#The f o l l o w i n g code determines the c y c l e g o [ l ] i n gO s t a r t i n g and f i n i s h i n g at p i = 0.5897322923+0.4356541991*1. The symmetry y ->• -y i n the y-plane y i e l d s a second c y c l e g0[2] beginning and ending a t - p i . > f o r j from 1 t o 26 do g 0 [ j ] : = a r r a y ( l . . 2 0 ) : od:  76  gO[l] [1]:=1:  # Let the c y c l e gO[l] begin with 1  g0[2][1]:=1+132:  # Let the c y c l e gO[2] begin w i t h 133  ordO[1+132]:=ord0[l]+132: # ordO[i] s t o r e s the index of t r O [ j ] that # corresponds t o the p o i n t on the path # # # #  i n the x-plane l a b e l l e d by i i n the c y c l e of the monodromy generator hO about 0. For example, t r 0 [ l ] [ord0[6]] i s the p o i n t on the path corresponding t o the 6 i n the  # c y c l e [1,2,3,4,5,6,7,8,9,10,11] of h O [ l ] . p r i n t (gO [1] [1] ,yO[l] [ordO[l]] ,g0[2] [1] ,y0[l] [ordO [1+132] ] ) : yflag:=ord0[1]: # y f l a g i s the index out of the 264 # i n d i c e s of y 0 [ j ] , j = l t o 5, that corresponds to the # j - t h p o i n t on the path between two b l a c k squares, # where j = l corresponds t o a b l a c k square, f o r m from 1 to 22 do i f y f l a g >132 then flag:=yflag-132: # f l a g i s the index out of the 132 i n d i c e s of else flag:=yflag: fi:  # t r 0 [ j ] , j = l to 5, that corresponds t o the j - t h # p o i n t on a path above CO i n the x-plane, where # j=lcorresponds to a p o i n t i n the inverse # image of 1/2 under b ( p ) .  f o r i from 1 t o 5 do. fix:=flag: f o r j from 1 t o 132 do # The f o r loop c a l c u l a t e s d i s t a n c e s d [ j ] : = s q r t ( ( R e ( t r 0 [ i ] [ f i x ] - t r 0 [ ( ( i ) m o d 5 ) + l ] [ j ] ) ) ~ 2 # between + (Im(trO[i] [ f i x ] - t r 0 [ ( ( i ) m o d 5)+l] [ j ] ) ) * 2 ) ; # t r 0 [ i ] [ f i x ] and od:  # t r 0 [ ( ( i ) m o d 5 ) + l ] [ j ] , where ((i)mod 5)+l # runs through the sequence 2,3,4,5,1 as # i runs from 1 t o 5.  dmin:=d[l]: f o r j from 1 t o 132 do  # F i n d the minimum d i s t a n c e of the # 132 d i s t a n c e s found above and set  i f d[j]<=dmin then dmin:=d[j]; od:  flag:=j; f i ;  # f l a g to t h i s  # minimum d i s t a n c e . T h i s determines the # next point t o proceed t o i n a path above # CO i n the x-plane.  dstl:=sqrt((Re(yO[i][yflag]-yO[((i)mod 5)+l][flag]))~2 + (Im(yO[i] [yflag]-y0[((i)mod 5)+l] [ f l a g ] ) ) ~ 2 ) ; dst2:=sqrt((Re(yO[i][yflag]+y0[((i)mod 5 ) + l ] [ f l a g ] ) ) - 2 +(Im(yO[i][yflag]+y0[((i)mod 5 ) + l ] [ f l a g ] ) ) ~ 2 ) ; i f dstl<dst2 # To proceed from one p o i n t on a blue path to then y f l a g : = f l a g : # the next i n the y-plane, the c l o s e s t of the e l s e yflag:=flag+132: # two p o i n t s t o y 0 [ i ] [ y f l a g ] i n the i n v e r s e 77  fi:  # image of the "next"point found above i s chosen.  if dstl=dst2 then printO'Error:  cannot proceed to next y-value"):  fi: od:  i f yflagOordOCl] then  # y f l a g <> ordO[l] means the c y c l e w i l l continue # and y f l a g = ordO[l] i n d i c a t e s t h a t the c y c l e ends.  ordOCm+1]:=flag: ordO[m+1+132]:=flag+132: gO[l][m+1]:=m+l:  # Label next entry of the c y c l e gO[l]  g0[2][m+1]:=m+l+132:  # Label next entry of the c y c l e g0[2]  p r i n t (gO [1] [m+1] ,y0[l] [yflag] ,g0[2] [m+1] ,-yO[l] [ y f l a g ] ) : else f o r i from 1 t o 2 do  # The f o r loops format the p r i n t output  f o r j from 0 t o m+1 do i f j=0 then p r i n t f ("\ngO['/.d] = [",!): else i f j<m+l then i f j<m then p r i n t f 07.A," ,gO[i] [ j ] ) : e l s e p r i n t f ("7.A" ,g0 [i] [ j ] ) : fi: else p r i n t f ( " ] " ) : fi: fi: od: od: printf("/n"): m:=22:  # End f o r m from 1 t o 22 loop since c y c l e has ended,  fi: od: >#The f o l l o w i n g code determines the remaining c y c l e s of gO. > f o r s from 1 t o 12 do if  s=12  then f l a g : = o r d O [ l l ] : e l s e flag:=ordO[s] : fi: f o r i from 1 t o 5 do  # Follow the r e d path above C l i n the y-plane  fix:=flag:  # s t a r t i n g at yO[l] [ o r d O [ j ] ] , j = 1 t o 11.  f o r j from 1 t o 132 do  # The endpoint of t h i s path determines the  i f s=12 then  # s t a r t i n g point of a c y c l e i n gO. 78  dCj]:=sqrt((Re(trl[((6-i)mod 5 ) + l ] [ f i x ] - t r l [ 6 - i ] [ j ] ) ) ~ 2 +(Im(trl[((6-i)mod 5 ) + l ] [ f i x ] - t r l [ 6 - i ] [ j ] ) ) ~ 2 ) ; else d[j] : = s q r t ( ( R e ( t r l [ i ] [ f i x ] - t r l [ ( ( i ) m o d 5 ) + l ] [ j ] ) ) ~ 2 + ( I m ( t r l [ i ] [ f i x ] - t r l [ ( ( i ) m o d 5)+l] [ j ] ) ) ~ 2 ) ; fi; od: dmin:=d[l]: f o r j from 1 t o 132 do i f d[j]<=dmin then dmin:=d[j]; od: od:  flag:=j; f i ;  ordO[12+10*(s-1)]:=flag; ordO[12+10*(s-1)+132]:=flag+132; yflag:=ordO[12+10*(s-1)]: g0[2*s+l] [1]:=12+10*(s-l): # Label f i r s t entry of c y c l e g0[2*s+l] g0[2*s+2] [1]:=12+10*(s-1)+132: # Label f i r s t entry of c y c l e g0[2*s+l] print(g0[2*s+l][1],y0[l][yflag] , g0[2*s+2][1],y0[l][yflag+132]): f o r m from 1 t o 10 do i f ordO[12+10*(s-1)+m-l] >132 then flag:=ord0[12+10*(s-l)+m-l]-132: e l s e flag:=ord0[12+10*(s-l)+m-l]: fi: f o r i from 1 t o 5 do fix:=flag:~ f o r j from 1 t o 132 do d[j] :=sqrt((Re(tr0[i] [ f i x ] - t r 0 [ ( ( i ) m o d 5 ) + l ] [ j ] ) ) * 2 + (Im(tr0[i] [ f i x ] - t r 0 [ ( ( i ) m o d 5)+l] [j] ) ) ~ 2 ) ; od: dmin:=d[l]: f o r j from 1 t o 132 do i f d[j]<=dmin then dmin:=d[j]; od:  flag:=j; f i ;  dstl:=sqrt((Re(y0[i][yflag]-y0[((i)mod 5)+l][flag]))~2 + (Im(y0[i] [yflag]-y0[((i)mod 5)+l] [ f l a g ] ) ) ~ 2 ) ; dst2:=sqrt((Re(yO[i][yflag]+y0[((i)mod 5 ) + l ] [ f l a g ] ) ) ~ 2 . +(Im(y0[i] [yflag]+y0[((i)mod 5 ) + l ] [ f l a g ] ) ) ~ 2 ) ; i f dstl<dst2  79  then  yflag:=flag:  else yflag:=flag+132: fi: i f dstl=dst2 then p r i n t ( " E r r o r : cannot proceed to next y - v a l u e " ) : fi: od: i f yflagOordO[12+10*(s-1)] then  # True i n d i c a t e s c y c l e i s not  complete  ordO [12+10*(s-1)+m]:=yflag: if  yflag>132  then ordO[12+10*(s-l)+m+132]:=yflag-132: e l s e ordO[12+10*(s-l)+m+132]:=yflag+132: fi: g0[2*s+l][m+1]:=12+10*(s-l)+m: g0[2*s+2][m+1]:=12+10*(s-1)+m+132: i f m=10  # m=10  then  and c y c l e not complete means the c y c l e w i l l be  # of length 20. In f a c t the c y c l e i s the concatenation  f o r j from 1 t o 10 do  # of two symmetric  g0[2*s+l][10+j]:=g0[2*s+l][j]+132:  10-cycles of hO under  # the symmetry y ->  -y.  od: g0[2*s+2]:=g0[2*s+l] : f o r j from 0 t o 2*m+l do if  # The f o r loop formats the p r i n t output  j-0  then p r i n t f ("\ng0[%d] = [",2*8+1): else , i f j<2*m+l then i f j<2*m then  printf("%A,",gO[2*s+l][j]):  e l s e p r i n t f 07,A",gO[2*s+l] [ j ] ) : fi: else p r i n t f C ' ] " ) : fi: fi: od: printf("\ng0[%d] =  g0[°/,d]\n",2*s+2,2*s+l):  else print(gO[2*s+l][m+1],y0[1][yflag], g0[2*s+2][m+1],y0[l][ordO[12+10*(s-1)+m+132]]); fi: else f o r i from 1 to 2 do  # The f o r loops format the p r i n t output  80  f o r j from 0 t o m+1 do i f j-0  .  then p r i n t f (" \ngO [7.d] = [", 2* s+i): else i f j<m+l then i f j<m then else fi:  printf("%A,",g0[2*s+i][j]): printf("%A\gO[2*s+i][j]):  else p r i n t f ( " ] " ) : fi: fi: od: od: printf("\n"): od: od: The following Maple output gives the cycles of go'gO[l]  = [1,2,3,4,5,6,7,8,9,10,11]  gO[2] = [133,134,135,136,137,138,139,140,141,142,143] gO[3] = [12,13,14,15,16,17,18,19,20,21] g0[4] = [144,145,146,147,148,149,150,151,152,153] g0[5] = [22,23,24,25,26,27,28,29,30,31] g0[6] = [154,155,156,157,158,159,160,161,162,163] g0[7] = [32,33,34,35,36,37,38,39,40,41] g0[8] = [164,165,166,167,168,169,170,171,172,173] g0[9] = [42,43,44,45,46,47,48,49,50,51] g0[10] = [174,175,176,177,178,179,180,181,182,183] g 0 [ l l ] = [52,53,54,55,56,57,58,59,60,61,184,185,186,187,188,189, 190,191,192,193] g0[12] = g 0 [ l l ] g0[13] = [62,63,64,65,66,67,68,69,70,71] g0[14] = [194,195,196,197,198,199,200,201,202,203] g0[15] = [72,73,74,75,76,77,78,79,80,81] g0[16] = [204,205,206,207,208,209,210,211,212,213] g0[17] = [82,83,84,85,86,87,88,89,90,91] g0[18] = [214,215,216,217,218,219,220,221,222,223] g0[19] = [92,93,94,95,96,97,98,99,100,101,224,225,226,227,228, 229,230,231,232,233] g0[20] = gO[19] g0[21] = [102,103,104,105,106,107,108,109,110,111] g0[22] = [234,235,236,237,238,239,240,241,242,243] g0[23] = [112,113,114,115,116,117,118,119,120,121,244,245,246, 247,248,249,250,251,252,253]  81  gO [24] = g0[23] gO [25] = [122,123,124,125,126,127,128,129,130,131] g0[26] = [254,255,256,257,258,259,260,261,262,263]  82  Appendix C Automating the Galois action of PSL (p) 2  > restart: > with(plottools) : > p:=ll:  ####  > N:=p*(p~2-l)/2; ####  The o r b i t of PSL(2,p) on c l a s s of Dessins i s computed i n t h i s work sheet. N i s the order of PSL(2,p)  > q_:=13: #### q_ i s a prime > p+1 > 11:=2; c l : = [ l l / 2 , 0 ] ; c2:=[ll,0]; r1:=Pi/(q_-2)*sqrt(q_)/IO; r2:=2*rl; r3:=4*rl; thick:=1; thick2:=l;. Ll:=2*q_/3; Cl:=[Ll/2,0]; short:=ll/3+2*Ll/3-ll/6; > h a l f c e n t r e : = P L O T ( e l l i p t i c A r c ( [ 0 , 0 ] , r 2 , r 2 , O..Pi, f i l l e d = t r u e , > centre:=PL0T(disk([0,0] , r l , f i l l e d = t m e , c o l o r = b l a c k ) ) : > centre2:=PL0T(disk([0,0], > centre3:=PL0T(disk([0,0],  color=black))  r2, filled=true,color=black)): r3, filled=true,color=black)):  > petal.short:=PL0T(line([0,0], [ l l / 5 - r l , 0 ] , color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( [ 1 1 / 5 , 0 ] , r l , color=black, t h i c k n e s s = t h i c k ) , line([11/5+r1,0], [ l l / 3 - r l , 0 ] , color=black, thickness=thick, disk([11/3,0], r l , c o l o r = b l a c k ) ) :  linestyle=l),  > petal_long:=PL0T(line([0,0], [ l l / 2 - r l , . 0 ] , color=black,. thickness=thick, l i n e s t y l e = l ) , c i r c l e ( [ 1 1 / 2 , 0 ] , r l , color=black, t h i c k n e s s = t h i c k ) , line([11/2+r1,0], [ l l - r l , 0 ] , color=black, thickness=thick,. l i n e s t y l e = l ) , d i s k ( [ l l , 0 ] , r l , color=black)): > l i n e l : = l i n e ( [ 0 , 0 ] , [ L l / 2 - r l , 0 ] , color=black, t h i c k n e s s = t h i c k ) : > c i r c : = c i r c l e ( [ L l / 2 , 0 ] , r l , color=black, thickness=thick):. > f o r j from 1 t o (q_-2) do  83  petal[j]:=rotate(petal_short,2*Pi*j/(q_-2),[0,0], scaling=constrained); od: > bouquet:=plots[display](seq(petal[i],i=l..q_-2),centre): > f o r j from 3 t o q_-3 do long:=short+2*4/(j+1)*ll/3+2*r1: bouquet_scaled[j-2]:=scale(bouquet,4/(j+l),4/(j+l), [0,0]): bouquet.close[j-2]:=translate(rotate(bouquet.scaled[j-2], Pi,[0,0]).short,0): bouquet_far[j-2]:=translate(rotate(bouquet_scaled[j-2], Pi,[0,0]),long,0): stem.short [j-2] : = l i n e ( C l + [ r l , 0 ] , [ s h o r t - 4 / ( j + l ) * l l / 3 , 0 ] , color=black, thickness=thick): stem_long[j-2]:=line(Cl+[rl,0],[long-4/(j+l)*ll/3,0], color=black,thickness=thick): flower[-(j-2)] :=plots[display](line1,circ,stem_short[j-2], bouquet_close[j-2]): flower [ j - 2 ] : = p l o t s [ d i s p l a y ] ( l i n e l , c i r c , s t e m _ l o n g [ j - 2 ] , bouquet.f a r [ j - 2 ] ) : od: > PSL2p: = [] : f o r a from 1 to ( p - l ) / 2 #### generate elements with a>0 do f o r b from 0 t o p-1 do f o r c from 0 t o p-1 do PSL2p:=[op(PSL2p),[[a,b],[c,(l+b*c)/a mod p ] ] ] ; od; od; od;. f o r b from 1 t o ( p - l ) / 2 #### generate elements w i t h a=0 do ' . . f o r d from 0 t o p-1 do PSL2p:=[op(PSL2p),[[0,b],[-l/b mod p , d ] ] ] ; od; od;, >• C:={0, 1, 2, 3, 5}: ####  C i s a r e p r e s e n t a t i v e unordered n-tuple i n the o r b i t of PSL(2,p)  > n:=nops(C):  n i s the number of elements i n C  ####  > f o r i from 0 t o p-1 do x[i]: = [[l],[i]]; od: x[p]: = [ [ 0 ] , [ l ] ] :  84  > f o r j from 1 t o N do y[j]:-list(i..n): # # #  y C j ] [k] i s an i n t e g e r modulo p+1 r e p r e s e n t i n g the homogeneous coordinate [ l : y [ j ] [ k ] ] r e s u l t i n g from the a c t i o n of PSL2p[j] on x[k] .  # # # #  I f z i s an i n t e g e r mod p+1, i t represents the f o l l o w i n g homogeneous coordinate: z <—> [ l : z ] , i f i<p z <—> [0:1] , i f i=p  od: f o r j from 1 t o N do f o r k from 1 t o n do A:=PSL2p[j] [1] [ l ] * x [ C [ k ] ] [1] [l]+PSL2p[j] [1] [2]*x[C[k]] [2] [1] ; B:=PSL2p[j] [2] [ l ] * x [ C [ k ] ] [1] [l]+PSL2p[j] [2] [2]*x[C[k]] [2] [1]; i f (A mod p<>0) then y [ j ] [k] :=B/A mod p; # B/A <—> [1:A/B] = [A:B] e l s e y [ j ] [k] :=p; # p <—> [0:B] = [0:1] fi; od: orbit[j]:=convert(y[j],set); od: > orbsize:=nops(convert(orbit,set)); > f o r k from 1 t o o r b s i z e do tuplel:=orbit[k]; num_sides:=nops(tuplel)+l:  # h a l f the number of s i d e s of the polygon  special_case:=false: i f member(p,tuple1) then special_case:=true fi: tuple:={op(tuplel),-1,q_-2}; ### The f o l l o w i n g commands p a r t i t i o n the flowers with non-ramified bouquets count:=0: f o r j l from 1 t o nops(tuple)-1 do num:=tuple[j1+1]-tuple[j1]-1: i f num<3 then increment:=Pi/2/num_sides: e l s e increment :=Pi/num_sides/(l*+num): fi:  85  i f num=2 then  offset:=Pi/4/num_sides:  e l s e offset:=Pi/num_sides/(1+num): fi: f o r j 2 from 1 t o num do count:=count+l; , angle:=increment*(j2-l)+offset+Pi/num_sides*(jl-l); i f num<4 then amplitude:=1: e l s e amplitude:=num-2: fi: indx:=amplitude*(-l)"trunc(Heaviside(num-2.9)*(j2-D); vase[count] :=rotate(flower[indx].angle,[0,0]): od: od: half_flowers:=plots[display](seq(vase[i],i=l;.count)): ql:=PL0T(line([0,0], [ l l / 4 - r l , 0 ] , color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( [ 1 1 / 4 , 0 ] , r l , color=black, thickness=thick), l i n e ( [ 1 1 / 4 + r l , 0 ] , [11/2,0], color=black, thickness=thick, l i n e s t y l e = l ) , disk([11/2,0], r l , color=black)): petal_extended:=PLOT(line([0,0] , [11/5-rl,0], color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( [ 1 1 / 5 , 0 ] , r l , color=black, t h i c k n e s s = t h i c k ) , l i n e ( [ l l / 5 + r l , 0 ] , [3*11/4,0], color=black, thickness=thick, l i n e s t y l e = D ) : f o r j from 0 t o (q_-2) do long_petal[j]:=rotate(petal_long,Pi*j/(q_-2)-Pi/(2*q_-4), [0,0]); od: bouquet_special:=plots[display](petal_extended,seq(petal[i], i = l . . q _ - 3 ) , centre,axes=none): flower.special:=translate(bouquet_special,-3*11/4,0): i f special_case then ramified_flower_clip:=plots[display](rotate(flower_special, - P i / 5 , [0,0]), e l l i p t i c A r c ( [ 0 , 0 ] , r 2 , r 2 , Pi/2...3*Pi/2, filled=true,color=black),rotate(petal_long,9*Pi/16, [0,0])): else ramified_flower.clip:=plots[display](rotate(flower_special, Pi/5, [0,0]), r o t a t e ( f l o w e r _ s p e c i a l , - P i / 5 , [ 0 , 0 ] ) , e l l i p t i c A r c ( [ 0 , 0 ] , r 2 , r 2 , Pi/2..3*Pi/2, f i l l e d = t r u e , color=black), rotate(petal_long,9*Pi/16, [0,0])) :• fi:  86  raMfied_flower_stem:=PLOT ( l i n e ([0,0], Cl-[rl,0], color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( C l , r l , color=black, t h i c k n e s s = t h i c k ) , line(Cl+[rl,0], [Ll+11,0], color=black, thickness=thick, l i n e s t y l e = D ) : i f special_case then petal_special:=PLOT(line([0,0], cl-[rl,0], color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( c l , r l , color=black, t h i c k n e s s = t h i c k ) , l i n e ( c l + [ r l , 0 ] , c2, color=black, thickness=thick, l i n e s t y l e = D ) : fi: half_bouquet:=plots[display](seq(long_petal[i],i=l..q_-2), halfcentre): i f special_case then long.petal[ceil((q_-2)/2)]:=rotate(petal.special,Pi/2,[0,0]) fi: i f special.case then half_bouquet_special:=plots[display] ( s e q ( l o n g _ p e t a l [ i ] , i=l..q_-2), h a l f c e n t r e ) : f i :  long_stem:=PL0T(line([0,0], Cl-[rl,0],  color=black, thickness=thick, l i n e s t y l e = l ) , c i r c l e ( C l , r l , color=black, t h i c k n e s s = t h i c k ) , line(Cl+[rl,0], [Ll-rl,0], color=black, thickness=thick, l i n e s t y l e = D ) : stem_special:=plots[display](curve([[-(Ll+11),0], [(LI-(num_sides-l)*rl)*cos(Pi*(num_sides-l)/num.sides), (Ll-5*rl)*sin(Pi*(num_sides-l)/num_sides)], [(LI)*cos(Pi*(num_sides-l)/num_sides), (LI)*sin(Pi*(num_sides-l)/num_sides)]])): ramified_flower_head:=plots[display] ( t r a n s l a t e ( r o t a t e ( half_bouquet, Pi/2,[0,0]), 11+L1.0),long_stem): i f special_case then ramified_bouquet_special:=plots[display](stem_special, r o t a t e ( t r a n s l a t e ( r o t a t e ( h a l f _bouquet_special, Pi/2,. [0,0]), 11+L1.0), Pi*(num_sides-l)/num_sides,[0,0])): fi: f o r j from 1 t o num_sides-l do ramified_bouquet[j]:=rotate(ramified_flower_head, Pi*j/num_sides,[0,0]): od: i f special_case then ramified_bouquet[5]:=ramified_bouquet_special fi: ramified_flower:=plots[display](translate( r a m i f i e d . f l o w e r _ c l i p , Ll+11,0),ramified_flower.stem): 87  p o l y l : t r a n s l a t e (PLOT ( l i n e ([0, - (Ll+11) * tan(Pi/2/num_sides)3,[0,(Ll+ll)*tan(Pi/2/num_sides)], color=black, thickness=thick2, linestyle=D),Ll+11,0): f o r j from 1 t o num.sides do poly[j]:=rotate(polyl,Pi*(j-l)/num_sides): od: half_dessinl:=plots[display](seq(ramified_bouquet[i], i=l..num_sides-l), seq(poly[i],i=l..num_sides), ramified_flower, h a l f _ f l o w e r s ) : h a l f _ d e s s i n 2 : = r o t a t e ( h a l f _ d e s s i n l , P i , [0,0]): str:="": f o r j from 1 t o 4-length(convert(k,string)) by 1 do s t r : = c a t ( s t r , " 0 " ) : od: str:=cat(str,convert(k,string),"-"): f o r j from 1 to nops(tuplel) do s t r : = c a t ( s t r , c o n v e r t ( t u p l e l [ j ] . s t r i n g ) ) : od: file_name:=cat("c:\\docume~l\\dsbl\\mydocu~l\\thesis\\ PSL2-ll-orbit-gifs\\ ",str,".gif"): strl:=cat("PSL2p[",convert(k,string),"] = ",. convert(PSL2p[k], s t r i n g ) ) : str2:=cat(convert(C,string)," -> " , c o n v e r t ( t u p l e l , s t r i n g ) ) plotsetup(gif,plotoutput=file_name, plotoptions='portrait, noborder'); t e x t . :=PL0T(TEXT([0,Ll+7*ll/5], strl,C0L0R(RGB, 0,0,0), FONT(TIMES,ROMAN,22)), T E X T ( [ 0 , - L l - 8 * l l / 5 ] , Str2, COLOR(RGB, 0,0,0), FONT(TIMES,ROMAN,22))): print(plots[display](text_,half_dessinl,half_dessin2, centre3, font=[TIMES,ROMAN,20],axesfont=[TIMES,ROMAN,16], l a b e l s = [ " " , " " ] , axes=none, tickmarks=[0,0], scaling=constrained));  88  Appendix D Displaying Galois orbits using JavaScript  <html> <body> <form> <center> <script type=text/javascript> var current = 0 ; var g i f L i s t = new A r r a y ( ) ; f u n c t i o n addPic(_p) { gifList[gifList.length?gifList.length:0] gifList[gifList.length-1].src=_p;  = new Image();  >  f o r ( i = l ; i<661 ; i++) { addPic( orb"+i+" . g i f " ) ; ,,  } function checklt(val) { current = Math.abs((current+parseInt(val))7.gifList.length); document.myimg.src = g i f L i s t [ c u r r e n t ] . s r c ; >  '  '  '  </script> <script type=text/javascript> document.write("<img name=myimg a l t = ' ' src='"+gifList[current].src+"'>"); </script> <br><input  type=button value="Back" o n c l i c k = " c h e c k I t ( - l ) " >  <input type=button value="Forward" onclick="checkIt(l)"> </center> </form> </body> </html>  89  Bibliography [And94] Shabat, George and Nikolai Andrianov. 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