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Galois actions on Dessins D’Enfants 2002
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Title | Galois actions on Dessins D’Enfants |
Creator |
Burggraf, David S. |
Date Created | 2009-11-07 |
Date Issued | 2009-11-07 |
Date | 2002 |
Description | 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. |
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Thesis/Dissertation |
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File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2009-11-07 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080063 |
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Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2003-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/14690 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0080063/source |
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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 refer- ence 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 Sp, for infinitely primes p, on a special class T>(f3p$) of elliptic and hyperelliptic Dessins d'Enfants. Furthermore, the corresponding Galois orbits of Sp and its subgroups in V(^v^) are made explicit. ii Table of Contents Abstract ii Table of Contents iii Chapter 1. Introduction 1 1.1 Background and Overview 1 1.2 Notation and Terminology 4 Chapter 2. Belyi functions and permutations 5 2.1 Maps : 5 2.2 Hypermaps 8 2.3 Belyi functions 10 2.4 Belyi functions on £ 10 2.5 Belyi's Theorem 12 2.6 Elliptic and hyperelliptic Belyi surfaces. 15 2.7 Monodromy groups 21 2.8 Calculating Monodromy groups of Belyi functions 24 2.9 Triangle Groups 26 Chapter 3. Galois Actions on Dessins d'Enfants 31 3.1 The Absolute Galois group 31 3.2 Galois actions on Elliptic and Hyperelliptic Dessins 32 Chapter 4. A special class of Dessins d'Enfants 36 4.1 The family of polynomials fp 36 4.2 A class of Dessins of genus > 1 37 Chapter 5. The Arithmetic of fp 44 5.1 The ring of integers OK 44 5.2 Decomposition and Inertia groups 48 5.3 Discriminants 50 5.4 Proof of Theorem 3 from Section 4.1 51 Chapter 6. Faithful Galois Actions of arbitrary finite groups 54 6.1 Faithful actions on higher genus Dessins 55. Chapter 7. Galois Actions of PSL2(p) 57 7.1 A Galois representation of PSL2 (5) 57 7.2 Galois representations of PSL2O?) 62 7.3 Examples: PSL2(11) and PSL2(13) 65' Appendix A. Full-size PSL2(p) orbits on n-tuples 71 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 al- gebraic 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 the field of all algebraic numbers Q over the field of 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 /?-1(0,1) (called an edge segment) joins two vertices of opposite type. Several examples of bipartite graphs obtained this way are illustrated in Chapter 2. Each connected component of X \ B~l(T) is a simply connected region, which 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, by first describing the Belyi pairs (X,8), defined over some field extension 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 arbitrary finite group 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 fp, which axe a component of the Belyi functions /3P$, are analyzed. Lemma 2 in Section 4.1 shows that fp is irreducible for all primes p > 3. Theorem 3 in Section 4.1 states that the Galois group Gal(/P(x)/Q) = Sp, the full symmetric group on p letters, for 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>(/3p,a). Theorem 6 in Section 6.1 shows that any finite group G has a faithful Galois action on 2>(/?p,3). In Chapter 7 a more 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 n- tuples 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~l(p) contains fewer than n points. There is at least one ramified point x in the preimage / -1(p) of a branch 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~l{T) denotes the bipartite graph embedded in X obtained 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 = (ra0mi ) - 1 cycles 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) < S A = Sym(A) called the 5 cartographic group of M, following the terminology of Alexander Zvonkin [Zvo98]. It is clear that G(mo, mi , m 2) has the relations: ml — momim2 = 1 and is therefore an epimorphic image 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 1 is read from left to right.) m 0 = (l,2,3)(4,5,6)(7)(8) mi = (l,4)(2,3)(5,7)(6,8) m 2 = (momi)- 1 = (1,6,8,5,7,4,3)(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 Gi (m 0 ,m 1 ,m 2 ) = PSL 2(7). 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 However, the definition of a map can always be extended to that of a pre-clean map, which allows the graph Q to have free edges consisting of a single dart, which is fixed by Then every algebraic map 1? determines a pre-clean map M. Figure 2.2 shows a pre-clean map M i obtained from Mi by removing the two vertices as- sociated to the darts 7 and 8. The cartographic group G(SQ, SI, s2) of M2 is now a permutation group on only six darts: S o = (l,2,3)(4,5,6) si = (l,4)(2,3) 5 2 = (5 0si)- 1 = (l,6,5,4,3)(2). 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) = PSL2(5). clean) map. The map Mz corresponding to the permutations tf(po) = *o = (1,2,3,4,5,6,7)(8,10)(9,11) tf(p1) = ti = (l,8)(3,9)(5>10)(7,H) 0 fo) = *2 = (*o*i)_1 = (1,10,4,3,11,6,5,8V7,9,2). 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 0 = (l,2,3,4,5,6)(7,8,9,10,ll) hi = (1,7) h2 = (hohi)~l = (1,11,10,9,8,7,6,5,4,3,2). The permutation group G(ho,hi,hi) generated by the triple (ho,hi,hi) is called the hyper- cartographic 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 _ + 2̂, 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: 22 — 23 2 - 2 1 <p(z) = , 22 — 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 - 1 ( l ) , then 8 is called a pre-clean Belyi 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 Pn : E —>• E, denned by Pn(z) = zn, has 0 as its only finite ramification 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. Pn lifts X to the graph Q(Pn) = Pnl(X) C E, which is an n-fold X-bouquet centered at 0, shown in figure 2.6 for the case n = 10. One can see that the graph Q(Pn) has a bipartite structure, i.e. each open 1-cell of Q(Pn) is bounded by 2 vertices of opposite color. Let 5 be the dart lying on.I that begins at 1 and ends at 0, then the darts of G{Pn) are just the lifts of 5 by Pn. 10 O -Sa- • 1 O 1 0 1 Figure 2.6: The graph G(PN) associated to the pre-clean Belyi function P\Q(Z) = z10 branched above {0, oo} c S. Example 2. The following rational maps 0m,n : E -» £ form an important class of Belyi functions: P m , n ^ J - mmnn 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 /?m,n, for (m, n) ^ (1,1), are pre-clean. The graph G(0m,n) — Pm}n(^) C E is shown in figure 2.7 for the case m = 6, n = 5. The corresponding hypermap H(0m,n) associated to 0m,n is obtained by "thickening" the black vertices and darts of G(8m,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 Bxti is ramified with ramification index 2 at 871(1) = \, which is not a branch point of 8, hence 8iti o 8 has ramification index 2 at all points in the preimage of 1. For example, the pre-clean Belyi function 0e,5 is transformed into the clean one Bi^ o /365 as shown in figure 2.8. 11 ft,! o 1 Figure 2.7: The graph G(Pe^) associated to the pre-clean Belyi function /?6,s : 2 —* £ branched above {0 ,1 , co}. - 0 - l 2 - 0 1 Figure 2.8: The graph o /?6i5) associated to the clean Belyi function B\t\ o /36,5 : £ —> £ branched above { 0 ,1 , oo}. 2.5 Belyi's Theorem Every nonsingular complex projective curve C defined by an irreducible homogeneous polyno- mial A(x, y, z) e C[x, y, z] denoted C = {[x : y : z] € P2(C) | A(x,y,z) = o} 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) € P2(C) | y V ~ 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 P2(C) where 2 ^ 0 . Thus, only homogeneous coordinates of the form (x, y, 1) in X are considered, and identifying these with the affine coordinates (x, y), yields the associated affine curve Xa^. The projective curve is then easily recovered from the affine one by "homogenizing" its defining polynomial. Therefore, X will be identified with X«f = {(x,y)eC2 y2 = /(*)}, 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. The first step 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, and finally to{0,1, oo}. If X is defined over Q then a branched cover n : X —• S with branch points contained in P1(Q) = Q u oo can always be found. For example, if X is defined by A(x, y) = a0(x) + ax(x)y + ••• + On(x)yn = 0, an(x) £ 0, 13 for an irreducible polynomial A(x, y) G Q[x,y], then 7r can be chosen to be the first factor projection ir: {x, y) i-> x. 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 of finite size, 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). Likewise, a sequence of polynomials fj,2, M3> • • •> °f polynomials can be defined such that for i > 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 splitting field of 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 qj G Q[x]. Each automorphism cr G G&\(pj/Q) fixes the coefficients of m and qj and thus qj{Hi(xQ)) = 0 <r(qj(lM(xQ))) = 0 & qj{a(ni(x0))) = 0 <S> ^(/^(<7(x0))) = °- C 2 - 1 ) By the transitivity of Gal(pj/Q), each root of pj 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 qj has the form T(//J(XO)), for some r G Gal(fy-/Q) 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 , M 2 , • • • > terminates with a linear polynomial Hk G Q[x] and thus the composition H = Hk o • • • o H2 o / i ! : £ -»• £ 14 has finite degree with a finite number of branch points in Q. If |Br ( / zo7r ) | < 3, then composing fi o 7r with a suitably chosen linear fractional transformation <j> forces all the branch points into {0,1, oo} and <£ou -o7r :X-+Eisa 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 Bm,n : jo, 1, J ^ J , co j H-» {0,1, oo}, composing <(>Qfj,oir with 8m,n, reduces the size of the resulting branch set Br(/?m,n 0 0 ° M ° 7 r ) by one fewer element than Br(</>o/uo7r). Proceeding inductively, if necessary, by further composing with Belyi functions of the form /3m>n, reduces the size of the branch set at each stage by one. 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 of finitely many Belyi functions of the form 8m,n- D 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 2 | y2 = 4x3 - g2x - <?3}, (2.2) where the discriminant: g% — 21g\ is non-zero, which ensures that the cubic polynomial in x has distinct roots. Given any g2, gz € C with g% - 27g% ^ 0, there is a lattice A r = {m + nr | m, n € Z}, Im(r) > 0, in C for which the Weierstrass elliptic function. z2 \(z — w)2 w2 15 satisfies the differential equation: (p')2 = ^pz-gip-gz- Since p is doubly periodic, the mapping from C -* X, defined by z i-» (p(z), p'{z)) induces an isomorphism C/AT —• X. Two elliptic 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 SL2(Z) = a b c d a, b,c,dE Z, ad — be = 1 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 can find gi, 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: Xx = {(x,y)eC2 \y2 = x(x-l)(x-X)}, A ̂ 0,1 where A satisfies 4(1-A + A2)3 J { x ) ~ 27A2(1-A)2 ' 16 Example 3. Let X\ be an elliptic curve in Legendre form Xx = {(x,y) € C2\y2 = x(x — l)(x — A)}, where A ̂ 0,1 is a zero of the polynomial f(x) = xp — a, for some prime p, and a is a rational number which can be written as the following fraction in lowest common terms: a = ^^j- As in the proof of Belyi's Theorem, the first factor projection IT : (x, y) H-+ X is a double covering of £ with branch points: 0,1, A, oo. Composing ir with Pp(x) = xp pushes the branch points into Br(Pp o -K) = {0,1, ^^,oo}, which in turn get sent to {0, l,oo} by 8m,n- Thus 8m ,n 0 Pp 0 is a (preclean) Belyi function and 8\ — 8\yi o 8m,n o Pp o 7r 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 8m,n o Ppoir. In the case p = 11, a = ^ the lift of the bipartite graph G(Pi,i o 83$) by P\\ : x ^ xn is shown in Figure 2.9. To illustrate the graph Q(8\) (Figure 2.10), one must distinguish between the unique real root and the other complex roots of f(x) = xp — a. Figure 2.9: The bipartite graph G(8^i o /33)8 o Pu) is obtained by lifting the bipartite graph G(p\,i 0 Ps,a) by Pn:x^> x Example 4. x(x - l)(x - Xkl)(x - A f e 2 )} , 17 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) = xp — a, is shown.on the left and the case where A is the complex root a1/1 1^0 1"/1! j s shown on the right. The Riemann surface X\ is represented topologically by identifying the opposites sides of the rectangle. where A*̂ , Afc2 are distinct zeros of the polynomial f(x) = xp—a, as in example 3. X\k \^ is a Belyi surface of genus 1, with clean Belyi function 8xkl,xk2 = B^i o 0m^n OPPO-K. In this case, the graph G{8\kv\k2), is represented more suitably on a hexagon with opposite sides identified as shown in Figure 2.11. Example 5. Let Xi\ky be a class of hyperelliptic curves defined by where N > 3 is a positive integer and Afc = a1/pe2km/p, 0 < k < p — lare distinct zeros of the polynomial f(x) = xp — a, as in the previous examples, with p > N. The either a 4g-sided or a Ag + 2-sided regular polygon with opposite sides identified. If N is corresponding class of clean Belyi functions 8s\k} :--^{Afc} -*• £ c a n be defined the same way as in example 3: 8{Xk} = 01,1 0 Pm,n oPpOTT. The Dessins d'Enfants V(8s\k\) are obtained by lifting the bipartite graph X by 0{\ky to the Belyi surface Xs\ky of genus g = [^f^J, which can be represented topologically by 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^, Afc2 is the unique real root (the choice Ao, A5 is used here) of f(x) = xp — a, is shown on the left and the case where A^, Afc2 are not real: (A3, A7), is shown on the right. 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 4̂ + 2-sided polygon. In the case p = 11, o = ^ the graphs £(/3A 2,A 5,A8) and £ ( / ^ A 2 , A 5 , A 8 , A i o ) 3 X 6 shown on Belyi surfaces of genus 2 in Figure 2.12. 19 Figure 2.12: The hyperelliptic graphs G(3{\ky), where {Â } is a set of distinct roots of f(x) = xp — a, are obtained by using w to lift the graph G{3i,i o 3$$ o Pn) (see Figure 2.9) embedded in E. The graph G(3X2,\s<\a) is shown on the left and G(8X2,Xs,\8,Xw) o n t h e riSnt- 20 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~l{p) denote the fiber above p in C. The fundamental group 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 xr E ft, each element a* E II lifts to a unique homotopy class of paths in X, such that any representative path in the homotopy class starts at xr and ends at some common point x3 E ft. This defines a bijective mapping gi : ft —» ft, which gives a permutation representation 9 : U —• Sym(ft) = 5n, defined by 6(ai) = The permutation g% (0 < i < k) has = \f~l(bi)\ disjoint cycles, each having cycle length kr (1 < r < rii). Each disjoint cycle of gi corresponds to a ramification point of index kr on C lying above where kr sheets of C come together. The resulting permutation group G(f) = 0(11) is called the 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\T = 2 (1 < r < ni), and the monodromy group G(8) has the following partial presentation where I is the least common multiple of {Zor}"=i- Suppose the base point p E 2 — B is chosen to lie in the open interval (0,1). Then each edge segment er of Q(8) contains exactly one point pr E ft and is incident to a unique black vertex at one end and a white vertex at the other. Each disjoint cycle of the monodromy generator go describes how the point pr is cyclically permuted in ft around a ramification point v lying G(j9) = (50,511 ô = ^ = l , ETC), 21 above 0, and thus describes how the edge segment er is cycled (according to the orientation of 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)-1) of 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 0 &o,i) : 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 a floral pattern in G(tjj) centered at 0, which will be referred to as a 10-fold (̂/?iii)-bouquet. The black vertex at € 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~l(V\) C X. Now define X~ = X \ (U0 U Ui U t/oo) 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 £ ~ . 2C and Tpg < 2 1 XCT Lemma 1 Suppose xQ e X~, po = 8(XQ) € £~ and \8"(x)\ < C. Choose open balls U = B(x0,rxo) C X~ and V = B(po,rpo) C £ ~ , such that rXQ < Define where peV. Then <j>p(U) C U and for all xeU, \<j>p(x) - <j>p(x0)\ < \\x - x 0 | 24 Proof By the mean value theorem where x € U and \<pp(x) -<j)p(xo)\ <sup|<£p(a:)||z-a;o|, u Once again, by the mean value theorem Thus \8'{x) - 8'(x0)\ < sup\8"(x)\\x - sol < C\x - x0\. u sup|^(x)|<%7=^i<-g\. CTXQ , , , 1, So \<pp(x) - (j)p{x0)\ < ip,'^\x ~xo\<^\x- XQ\. Next, to prove <f>p(U) C U it is sufficient to show \<j>p(x) — XQ\ < rXo. \<f>p(x)-xo\ < \<t>p{x) -#p(xo)| + \<t>P(xo) -xo\ < L x _ X o | + b - ^ o ) l 2' U l \3'(xQ)\ K 2 +\8>(x0)\ ~rx0' by definition of rpo •. 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>v is a contraction mapping and for any x € U the sequence of points {̂ (x)} converges to a fixed point of <f>p in U, namely the 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 the fiber fi = /3-1(po)- Suppose 70 is representative loop in the path class OJO going clockwise around 0, that lies entirely in S~. A finite number of distinct points: po,pi, • • • ,Pk on 70 can be chosen which partition 70 25 into curve segments of length less than rPo. Next consider the lift 70 of the loop 70 by 3 starting at xo- As 70 is traversed counterclockwise from po to p\, 70 goes from XQ to some point p~\ in the fiber 3~l(p{) above p\. Iterating the function 0Pl starting at XQ, yields the sequence {^(x)} 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 if F + 7 + T > 1 U=< 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 I Ar = B3 = C* = ABC = 1) 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 of finite index 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 the fixed points 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 = < £ O T T :U/T ^U/T(r,s,t)^H. The periods r, s, t of a triangle group are not restricted to just finite values. 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 PSL2(Z) can be generated by the elements Si 0 -1 0 1 1 1 = , Sp — , Soo — . 1 0 . _ 1 1 _ 0 1 and can be given the presentation PSL2(Z) = (Si, Sp, Soo\Sf = S3p = SiSpSoc = 1) =• T(2,3, oo). Note that Si, Sp, and Soo act on H via the Hnear fraction transformations: Si(z) = —-, Sp(z) = jrj, Soo(z) = z +1, which fix the points: i, p = e1™/3, and oo, respectively. Thus Si, Sp are elliptic elements and Soo is a parabolic element. The fundamental region for PSL2(Z) is shown in Figure 2.15 Any subgroup H of PSL2(Z) acts properly discontinuously on U, so U/H is a Riemann surface. If H has finite index in PSL2(Z), then U/H has finite hyperbolic area and can be extended to 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¥ 1 (Q)) /H. The Riemann surface W/PSL2(Z) can be visualized topologically by identifying pairs of sides of the fundamental region (Figure 2.15) in the same PSL2(Z) orbit. Thus the pairs of sides in the Si-orbit and the Soo-orbit are identified and W/PSL2(Z) is homeomorphic to a sphere with a single puncture. 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. PSL2(Z) is transitive on P^Q) and so W/PSL2(Z) = S is a one point compactification of the punctured sphere W/PSL2(Z), the single point corresponding to the PSL2(Z)-orbit PX(Q). Furthermore, by defining j(t) = oo on this single orbit, the modular function j can be extended to an isomorphism j : W/PSL2(Z) —* S. The principal congruence subgroup T(2) of level 2 which is a normal subgroup of index 6 in PSL2(Z) can also be regarded as a triangle group with infinite periods. T(2) is defined by r(2) = 1 a b a b 1 0 < ± € PSL2(Z) c d c d 0 1 mod (2) and can be generated by the following representative elements in SL2(Z): -1 0 -1 2 1 2 T0 = 2 -1 -2 3 0 1 where To,Ti,Too are parabolic elements with fixed points 0,1, oo, respectively. Identifying the sides of the fundamental region of T(2) (see Figure 2.15) paired by To and shows that U/T(2) is a sphere punctured by three distinct points, which correspond to the T(2)-orbits [0], [1], and [oo], respectively, in P1(Q). Since the generators all have infinite order, T(2) has the 28 following group presentation T(2) = (T0,Ti, | 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 hyper- cartographic 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 PSL2(Z) defined by To(2) a b c d €PSL2(Z) c==0mod(2) and can be generated by the following representative elements in SL2(Z): -1 0 -1 1 1 1 , U00 = 2 -1 -2 1 0 1 where UQ, Uu, and Uoa fix the vertices {0,w = (̂1 + i),oo}, respectively, of the (non-shaded) triangle in the fundamental region of To (see Figure 2.15). To(2) has the following group presentation T0(2) = (U0T UU, Uoc\U2 = 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 PSI2(Z), (iv) X = M/H for some subgroup H of finite index in T(2), (v) 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 a finite extension 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 the finite Galois groups GK- The examples of Galois actions given in Section 3.2 of this Chapter motivate the construction of the class V(PV£$) 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(8V^). 3.1 The Absolute Galois group Every algebraic number is contained in the splitting field of 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 GK fixes F 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 number fields correspond to the subgroups of finite index 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) G K[x,y] X = {(x,y)eC2\A(x,y) = 0}. An automorphism a of the Galois group Gal(-ftyQ) acts in a natural way on X by acting on the coefficients of A(x, y) = ao(x) + a\(x)y H (- On(x)yn: a : X - X" = {(ar, y) € C 2 | A°{x, y) = a(a0(x)) + a(al(x))y + ••• + a{an{x))yn = 0}. Since X is a Belyi surface, there exists a clean Belyi function 8 : X —> S with corresponding Dessin T>(0). As 8 is also defined over K, the action of a on X induces an action T>(8) —*• V(8a) on the corresponding Dessins d'Enfants. It is not difficult to show that Gal(Q/Q) acts faithfully on elliptic curves. The isomorphism classes of elliptic curves are invariant under the elliptic modular function as described in section 2.6, where an elliptic curve X is shown to be defined over Q if and only if j(X) is. For every a 6 Gal(Q/Q) there exists a € Q such that cr does not act trivially on a. Now let X be the elliptic curve with j(X) = a, then there exists a Belyi function 8 : X —• S and both X and 8 are defined over a field containing Q(a). Thus a acts nontrivially on both X and the 32 corresponding Dessin V{3). For an explicit example of a Galois action on the elliptic Dessins V(8x), fix A € <Q and let K/Q be the splitting field of the minimal polynomial of A. Gal(Q/Q) then induces an action of the quotient Gal(K/Q) and the orbit is contained in a subclass of T>(3\) defined over K. In example 3 of section 2.6, the elliptic curve X\5, where A5 = a ^ V 0 ™ / 1 1 , is defined over the splitting field K = Qia1/11, e 2"/ u) of x11 - a. The Galois group G = Gal(K/Q) permutes the zeros Afe = al/Pe2kvi/p, 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 p . These transformations form the 1-dimensional affine group AGI_a(p) over Z p , a split extension of a cyclic normal subgroup of order p (the 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\k) are all conjugate under G. The size of the largest G-orbit in V(8\k) is p, the number of distinct 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\kitxk2 is also defined over K — <Q(a1/11, e27™/11). The Galois group G = AGLi(p) is doubly transitive on the zeros Â so the Dessins T)(8\k^xk2), with Afcj 7^ Afc2 are all conjugate under G. The size of the largest G-orbit in V(3\k 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 £>(/3A2,A5,A8) H-> T)(3X3,X6,^9) 3 3 shown in Figure 3.2. The hyperelliptic Dessins in the class ^(/?{A F C L ) A F C 2 ,A F C 3 })) where the unordered triples {A ,̂ Afc2, Afc3} are distinct 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,x5M}) 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{AFCL,AFC2,AFC3}) i s displayed, using JavaScript and Maple, at www.math.ubc.ca/~burggraf/reseaxch.html. 33 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 splitting field K(fp) of a polynomial fp(x), where p is an odd prime. The criteria for choosing the polynomials fp are that they admit "maximal" Galois groups and "minimal" monodromy groups. The polynomial fp(x), having only two critical values, is a Shabat polynomial and is chosen so that the corresponding Belyi function of the form j3p = b o p o fp, as constructed in the proof of Belyi's Theorem, has "minimal" degree. The polynomial fp also has Gal(/P/Q) = Sym(p), when p — 2 is not a perfect square, which yields faithful Galois actions by arbitrary finite groups and the explicit definition of the Belyi pairs (X, Bp) on which they act allows the faithfulness of the action to 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 = A5 =. o 1/ 1^ 1 0**/ 1 1. 34 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 (Xp^, 3P^) are described, each defined over the splitting field Kp of an irreducible polynomial fpi of prime degree p. The Galois action of Ga\(KP/Q), 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(BPIQ). 4.1 The family of polynomials fp Lemma 2 For every odd prime p, the polynomial fp(x) = xp — ^ryXp~l + 1 is irreducible over the rational numbers Q. Proof The polynomial gp(x) = (p - l)(x - p - l)pfp (jZJZ^) = (P - " P " 1)P - " P ~ 1) + P " 1 is.irreducible by Eisenstein's irreducibility criterion applied to the prime p and fp(x) is irre- ducible if and only if gp(x) is. Theorem 3 Let hp be the monic polynomial defined by hp(x) = (p - If )P fp =xp-p{j>- lf~2x + {p- If. For all primes p such that p — 2 is not a perfect square, the Galois group Gal(fp(x)/Q) = Gal(hp(x)/Q) = Sp, the full symmetric group onp letters. 36 Proof see Section 5.4 It should be noted that Galois groups of trinomials of the form xn + ax3 + b, with certain conditions on n,s,a,b 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 fn = xn- ^xn~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(/n/Q) = Sn, for 3 < n < 15. Furthermore, by reducing /„ mod p, for various primes p, one can extend these results a little further, although the calculations are somewhat tedious. Conjecture 1 For every integer n > 3, the Galois group Gal(fn(x),Q) is isomorphic to the full symmetric group Sn- The result of Theorem 3 allows for the construction of a special class V of Dessins d'Enfants on which an arbitrary finite group 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 fp, for some suitable prime p such that p — 2 is not a perfect square. 4.2 A class of Dessins of genus > 1 Suppose Xp# is a compact Riemann surface of genus g > 0, defined by where S = {Cfct, • • •, 0fc2g-i} is a subset of 2g — 1 distinct zeros of fv, and 2g — 1 < p ^ n2 + 2, for any n. 37 Proposition 1 A clean Belyi function dp# : Xp$ £ of degree 4(p2 -p) can be defined by 4(p - I)2?"2 - 9^-2 Proof The function 8p$(x, y) can expressed as the composition d^i o /3p_2,i o / p o IT, where TT : X P i cj —* £ is the first factor projection: tr(x, y) = x. The branch points of 8p^ result from all branching and ramification of the composite functions: TT, fp, 6p-2,i, and to show that /3P)$ is a Belyi function, it is sufficient to show that these branch points are contained in the 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 fp. The ramification points of fp are: 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 0p-2,i sends 1, p̂ EfjOo, and 0 to the points 0,1, oo, and 0, respectively, and is ramified at 0, |pf, and oo, which again, are all branch points of the preceding function fp o n. At this stage 0p-2,i 0 fp ° TT is a preclean Belyi function, and further composition with ensures that all ramification points over the point 1 have ramification order equal to 2, hence 8P^ is a clean Belyi function. • The final outcome is a clean Belyi function /?P)s, which is naturally defined on Xp$ in the sense that all ramification points of each composite function of @p$ are contained in the set of branch points of the previous one. In this fashion, the desired Belyi function 0P$ is defined in a relatively uncomplicated way having minimal ramification and degree. The Dessin T>(3p^) is 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>(0p^) in the case that p = 7, genus 3=1, and Q = {Ci}, where £i is the first zero 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 /p(l) = jj^f € G(Pi,i °0P-2,i), which joins a (p - 2)-fold (̂/?i)i)-bouquet to the left and a 1-fold (̂/3iii)-bouquet to the right. Since v has ramification order equal to 2, a lift of a neighbourhood of € G(Pi,i°dp-2,i) by fp to a neighborhood of v yields two (p-2)-fold £(/3i,i)-bouquets and two 1-fold (̂/51)i)-bouquets 38 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,i03p-2,i) ha G(3i,i0Pp-2,i ° fp), which intersect at v; one starts at 0, which passes through v and proceeds downward, and the other starts at which passes through v and proceeds upward. The ramification point of fp at 0.6 G(/3i,x 0 3p-2,i ° fp) has ramification index equal to p — 1 and lies above /p(0) = 1, giving rise to the (p — l)-fold G(3\t\ o /?p_2i )-bouquet centered at 0. The central black vertices of the p [p — 2)-fold (̂/3ii )-bouquets in G{3\,\ 0 Pp-2,1 0 fp) are the zeros {Ci, • • •, Cr} 0 I* fp shown in Figure 4.1. The projection ir is ramified over only three points of °/5p-2,i 0 fp)' 0, 1, and Ci, each having ramification order equal to 2. Hence, it lifts the (p — 2)-fold 0(/?ii )-bouquets centered at £i ( to a 2p - 4-fold 5(̂ i)i)-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 (̂̂ 1,1 0 /3p-2,i 0 /p) are each lifted to two distinct (p - 2)-fold (̂/3i>i)-bouquets giving a total of 2p - 2 [p - 2)-fold (̂/3i,i)-bouquets in £(/%})• The Dessin V(3^Cl\) consists of the graph 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(8V$) more abstractly using suit- 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(3PjQ). In Figure 4.4 a dashed circular arc indexed by i represents i copies of G(Q\,i) 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) emanating from the 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 (̂/?i!i)-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 (̂ i)i)-bouquets are represented by single dashed arcs, each indexed by i = i. 43 Chapter 5 The Arithmetic of fp The proof of Theorem 3 requires some detailed analysis of the family of polynomials fp and a closely related family of monic polynomials hv. In section 5.1, the relevant definitions and 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 a field K. Section 5.3 is devoted to calculating the polynomial discriminant disc(hp), which contains important information about the ramified primes in OK, where K is the splitting field of the polynomial hp. 5.1 The ring of integers OK Let if be a finite field extension of Q and let the finite field Z/pZ be denoted by Zp. 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 xm + am_ixm _ 1 + • • • + axx + a0 = 0. Then we have Z[x] = Z + Zx + • • • + Zxm~l since xk e span{l, x,..., xm_1} for k > m. (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\ + • • • + Zvn with xM c M. Thus each element of {xv\,... ,xvn} can be decomposed as aZ-linear combination of the elements of {v\,... ,vn}. This means there is a matrix A with integer coefficients such that (A — xl)v = 0, where v — [v\,..., vn]T and I is the n x n identity matrix. Finally, since v ^ 0, x is a root of 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 a finitely generated), 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,...,ynEl such that I = Zy1 + ...+Zyn. (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 = P^---P£R, where P\,...,PR are distinct prime ideals and n\,..., nr are positive integers. 45 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\1 • • • P?r • Proof Suppose that Q C OK lies above pZ. Then QflZ = pZ =• peQ =• POK c QOK = Q =*. Pf1---PfcQ 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 pOK, i.e. Pe\pOK but Pe+1 \pOK- 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 Zp vector space OK/P, i.e. f = [0K/P • Zp]. 46 Proposition 4 Let pbe a prime in Z and suppose pOx factorizes as pOx = Pf1 • • • Pfr • Then r where fx,...,fr are the residue degrees. Proof By the Chinese Remainder Theorem r OK/POK = OK/P? = n Ojr/if, where OK/POK and OK/P*1 are Zp-vector spaces. Thus r dimZp (0*/pOtf) = £ dimZp (0*/Pf) . Now it is sufficient to show: (i) dimZ p(0*:/p0A:) = [K : Q], and (ii) dimZp (C^/P^) = ê /,-. For the proof of (i) consider the map c Zxi © * • • © Zx̂ —• Zp © • • • © Z p, denned by m\xi H 1- mnXn >-> (mi mod p,..., rrin mod p), where n = [K : Q]. The kernel of this map is just pOjc and therefore OK/POK — Z p © • • • © Z p, which has dimension n. For the proof of (ii), first consider the map $ : OK —* P l _ 1 / P 1 , defined by x ax + P*, where P|p and a € P i _ 1 \ P I . $ is surjective since a 0 K + PI = P 1 - 1 and P C ker($). By the maximality of P, ker($) = P or ker(<3?) = The latter implies P 1 - 1 = P1, which contradicts a € P i - 1 \ P\ so the former is true and OK/P = P i _ 1 / P i as Zp-vector spaces, for i > 1. Now for each i > 1, define a map Otf/P* -> 0 K / P I ' 1 , where x + P* i - » x + P*_1. The kernel is just P4-1/!*, so {OK/P^HP^/P*) .= OK/P1'1- Taking dimensions of both sides gives dimZp(C?^/Pi) - dimZp(P i-1/P i) = dimZ p(0^/P i- 1). 47 Now substituting in the values i — 2,..., e and eliminating terms appropriately using OK/P — pi-l/pi yie^g dimZv(0K/Pe) = edimZp(0K/P). By definition / = dimZp (0jr/P), therefore dimzp(0K/Pe) = ef. • 5.2 Decomposition and Inertia groups Now suppose K / Q is a finite Galois extension and G = Gel(K/Q). G acts on a prime ideal P C OK over p by P° = CT(P) = {cr(x) : x € P}. Note that P a is also a prime ideal lying above p and furthermore, the following equalities of 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 P2T, for all r € G (5.2) Since Pf ^ PJ, for all cr, r € G, the Chinese Remainder Theorem implies that there exists an element y € OK satisfying the above congruences. Next, consider the norm NK(y) = n <y)e °K- creG Since the fixed field of G is Q and NK{V) is fixed by G, it must be a rational. Furthermore NK{V) € OK =*• € Z. The first set of congruences (5.1) implies NK(y) = 1 mod Pi, and the second set of congruences (5.2) gives Nx/k(y) = 0 mod P2. Since Pi DZ = P2 flZ = pZ, both NK/k{y) and NK/k{y) - 1 are elements of pZ, which gives the contradiction: 1 e pZ. • 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 DP = {aeG\aP = P} Proposition 6 |Dp|p| = ef Proof G acts transitively on the set {Pi,..., Pr} of distinct primes lying above p and Dp is the stabilizer of the action. Thus \G/Dp\ = |orbit(P)| = r, the number of distinct primes above p. Therefore \DP\ = \G\/r = ref/r = ef. • DP\p acts naturally on OK/P, since DP\p fixes P, and trivially on OK/POK- SO reduction mod P induces a homomorphism DP]p - Gal(K(P)/Zp), 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 IP = {cr G Dp | (TX = x mod P, for all x € OK} It turns out that the homomorphism DP\p Gal(F(P)/Zp) is surjective [Lan94] so there is a short exact sequence 1 - IPlp - DPlp -> Gal(]?(P)/Zp) - 1. It follows that \Dp\p/Ip\p\ = |Gal(]?(P)/Zp)| = / , which combined with |£>P|p| = ef gives |i>|p| = e. This means that p is a ramified prime if and only if IP\p is non trivial for some 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 the field discriminant D(K/Q). Theorem 5 (Rummer's Criterion) Let K be a number field with 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 = p1(xTei---p7(xTr, 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(hv) of the polynomial hp = n?=i (x ~ ai)> defined in Theorem 3, Section 4.1 can be calculated using the resultant formula disc(hp) = (-l)*iET11R{hp,tip), were R(hp, h'p) is the resultant of hp and h'p. The resultant R(fi, fi) of any two polynomials of degrees n and m, respectively: fx(x) = anxn + an-ix71"1 H 1- a\x + ao, and h{x) = bmxm + 6m-ixm_1 + • • • + bxx + b0 50 is the determinant of the (n + m) x (n + m) matrix: RUuh) = an a n _ i • •• a i oo an a n _ i • • • a i ao an an-i • • • a i ao bn bn-i ••• bi bo bn bn-i ••• b\ bo K bn-i bi bo > m > n n m Thus R(hp, hp) is the following determinant 1 0 ••• 0 -p(p-l)P-2 (p-l)p 1 0 ••• 0 -p(p-l)P~2 (p-l)p 1 0 p 0 ••• 0 -p(p-l)P~2 p 0 ••• 0 -p(p-l)P~2 o -pip - i)p~2 o - ly p o •••• o -p{p-i)p-2 P o ••• o -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(hp) = (-l)^r^pP(p - l)P-P-L{p - 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 ) = ^ P - P (P - i ) p " 2 ^ + (P - ! ) P 51 axe rational multiples of the zeros of fp(x) and hence Gal(hp(x), Q) = Gal(/P(x), Q). Therefore, it is sufficient to prove that Gal(/ip(x), Q) = Sp. Gal(hp(x), Q) acts on the set of p distinct zeros of hp(x) by permutations and is thus isomorphic to a permutation subgroup G < Sp. To show- that G = Sp it is sufficient to show that G contains both a p-cycle and a 2-cycle. Since hp is irreducible (if and only if fp is), Gal(hp/Q) is transitive on the set of zeros of hp, and since p is prime, G must contain a p-cycle. It remains to show that G contains a 2-cycle. First we will attempt to find a prime q that divides the field discriminant disc(K/Q) of the splitting field K of hp. Then q will be a ramified prime when viewed as an element of the ring of integers OK of K. The field discriminant disc(K/Q) is related to the polynomial discriminant disc(hp) as follows: disc(K/Q) = —disc(hp), Th where m and n axe relatively prime integers. Thus, the field discriminant is ,2 R disc(K/Q) = ̂ [(-lJ^'rVfr - rf-v~l(p - 2) p ( p - i ) The Prime Number Theorem implies that there axe infinitely many choices of primes p such that p — 2 ^ n2, thus a ramified prime can be found by taking the prime q to be a prime factor of p — 2 which is not a prime factor of n2 so that q\dK- Next, observe that GCD(/ip,/ip) divides xh'(x) -php(x) =p(p-l) p _ 1(x-p-l), and therefore d = GCD(/ip, h'p) must divide p(jp — l)p-1(a: - p — 1) mod q, where the notation d, hp, and h'p represents the reduction modulo q of the polynomials d, hp, and h'p, respectively. Note that p(p—l)p-1(x— p— 1) mod q is not identically zero because q is relatively prime to both p and 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 hp is separable, ord = x— p — 1 mod The first possibility can be ruled out because hp separable qis not ramified in K (Theorem 23 [FT91]), which is a contradiction. Thus hp = a<P € ¥q[x] 52 where a 6 Fjx] is some polynomial and ad is separable in Fjx]. Now let a\,..., ap denote the distinct zeros of hp in K and suppose that P is a prime ideal of OK above q. Then v hp = J J (x — ai) = acP mod P. i=l Reordering the zeros if necessary, we can write d2 = (x — ai)(x — a2) = (x — ai)2 mod P 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(/ip,Q) and under the action of <r, the zeros 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) = â , 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(/P/Q), for some prime p > \G\, such that p — 2 is not a perfect square, thanks to Theorem 3. This yields a Galois representation p : G —• Gal(/P/Q) given by the composition p = c6 o i o r, where 4> is the isomorphism of Theorem 3 and i is inclusion of permutation groups. G SiG\ Sp -t Gal(/PIQ) The Galois group p{G) acts on the Belyi surface X P ^ , by acting on the coefficients of the defining polynomial of XP$. More explicitly, an automorphism a € p(G) takes XPIQ to X P ^ defined by r ya = x (x - l )n (* -^ (0 ) y2 = x(x-l) fj ( ^ - 0 C€<r(3) So the element a € p(G), regarded as a permutation of the zeros of fp, acts on the family XP,Q, by acting on the index set consequently, a induces an analogous action on the corresponding 54 family of Dessin d'Enfants denned by cr (£>(/?pQ)) = ©(/SL^cj)). Thus, a necessary condition for p(G) to act faithfully on the class of Dessins £>(/3P)$>) with genus g > 1 is that p(G) act faithfully on the subsets S of 2g - 1 zeros of fp. It will also be shown that this is a sufficient condition. 6.1 Faithful actions on higher genus Dessins Theorem 6 p(G) acts faithfully on the class of Dessins d'Enfants P(/?PJCJ) with genus g ranging 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 / p . Choose a set 3? such that one and only one element £fc satisfies k <\G\ and for 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) i1 fa because otherwise p - 1(er) G G would have a fixed point x € G => P~l{&) = 1, which contradicts the fact that a is nontrivial. Thus, cr(fa = fa, where k' ^ k and k' < \G\, which implies fa £ and therefore cr(9f) ^ 9?. Finally, it is clear that the Dessins V(B^ky) and are non-isomorphic (see Figure 6.1) if k' =fi 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$) of genus equal to 1, where Of = {£fc}, and Figure 6.2 displays the action of cr on a Dessin V(dv^) of genus equal to g, where J = {fa fa,..., Q2g_2}. 55 \ -p-2 :.-,,.p-k-2 p-k-fc- P-2... P-2 -.P-2 P-2 \ 2 .̂.p-<̂ ;)-2 p-a{k)-i% P ^ -,P-2 Figure 6.1: The action of an automorphism <7 € p(G) on a Dessin V(8P$) 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. Figure 6.2: The action of an automorphism a 6 p(G) on a Dessin V(8P$) of genus g > 1, represented by a 4g-gon with opposite sides identified. 56 Chapter 7 Galois Actions of PSL 2 (p) Every element, of a finite group G has a natural (left) action on the left cosets of any sub- group H < G, yielding an injective permutation representation r : G —• S[G-.H\, provided that core(iJ) = 1. By identifying each coset with a zero of fp, where p is a prime such that p > [G : H], G can be identified with a subgroup of Gal(/P/Q) and r becomes a Galois rep- resentation. In the case that H = {1}, the Galois representation is the regular representation on G, which was used in chapter 6 to determine a faithful Galois action of an arbitrary finite group G on V. If G contains a proper subgroup H ^ {1}, the degree of the representation r is lowered by a factor of \H\, leading to Belyi functions Qv and Dessins V(0P) of lower degree; clearly, a desirable situation for the purposes of computing faithful Galois actions. Furthermore, a subgroup H < G of minimal index in G corresponds to actions of G on Dessins of minimal degree. 7.1 A Galois representation of PSL2(5) The alternating group A§ = PSL2(5) is the orientation preserving symmetry group of the regular dodecahedron and its action on the faces yields a permutation representation A$ —• S12. With the faces labelled as in Figure 7.1, A5 can be given the following presentation: A5 = (a, 61 a5 = 63 = (at-)2 = 1), 57 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) = x12 — -jf z 1 1 +1 is irreducible over Q, and the Galois group G = Gal(/i2/Q) of its splitting field over Q is the full symmetric group Su- The zeros of fu(x) are displayed in Figure 7.2 and labelled for later reference. C5 o C4 o 0.8- 0.6- ° C2 0.4- 0.2- -0.5 0 -0 .2 - -0 .4 - 0.5 1 C8 o. -0.6- -0.8- C9o o-Cio 0 C u Figure 7.2: The indexed zeros of'the polynomial /i2(x) 58 Consider the elliptic curve Xi,j,k = {(x, y)eC2\y2 = (x- Q)(x - -<*)}. where C»>0> CA; are distinct zeros of the polynomial fu(x). The projection TT : Xijtk —• S 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 the first Dessin 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 ith root with the ith face of the dodecahedron, we get a Galois representation As —• G. The Galois group G = Gal(//Q) acts on X by permuting the triple of zeros in the set {Q, Ck} By identifying the ith 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,$)- Moreover, the orbits containing the triple {2,7,11} under the action of the subgroups <a>, <6>, and <a6>C As are as follows: <a> : {2,7,11} i-+ {3,7,8} i -» {4,8,9} ^ {5,9,10} ^ {6,10,11}, <b> : {2,7,11} {3,4,8} ^ {1,5,6}, <ab> : {2,7,11} >-• {1,4,5}. The orbits of the corresponding Dessins are illustrated in Figures 7.3, 7.4, and 7.5. 59 Figure 7.3: The orbit of the Dessin V(d2j,u) under the action of <a>C G, shown from left to right as successive powers of a act on X>(/32,7,ii)- 60 Figure 7.4: The orbit of the Dessin V(p\,r,\\) under the action of <6>C G, shown from left to right as successive powers of b act on V(J32,7,u)- 7.2 Galois representations of PSL2(p) The special linear group SL2O?) (p > 2 here) consists of all 2 x 2 matrices with elements in F p having determinant = 1. The projective special linear group PSL2(p) is the quotient group of SL2(p) by its center Z(SL2{p)), where the center is a subgroup consisting of elements of SL2(p) which commute with every other element of SL2(p). In this case 2(SL2(p)) = { < 1 0 < ± 0 1 PSL2O?) has order l S L^ ( p )l = p^2~^ and has maximal subgroups of order ^f2-. PSLiip) can be generated by as few as three elements (01,02,03), where one choice can be represented by the following three matrices in SL.2(p): 1 0 0 1 x 0 Ol = , 0-2 = , and a 3 = 1 1 - 1 0 0 x-1 where x is a primitive element (mod p). The orders of the generators are: p — 1 ord(ai) = p, ord(a2) = 2, and ord(a$) = —-—. Moreover, PSL2(p) can be given the following group presentation [CM65] PSL2(p) = ^ai, a2, a 3 1 = ax = a2, = (aia2)3 = (a2a3)2 = (afa 2a3) 3 = aj^axazaT^ • PSL2(p) has a natural (transitive) action on P1(FP), where the elements of P1(FP) consist of p + 1 homogeneous coordinates of the form: [a : b] E F p x F p, which will be represented (using the notation =) here by column vectors: [ 0 : 6 ] = a b Recall that homogeneous coordinates are invariant under scalar multiplication by F p, that is, [a : b] = [aa, ab] for all a € F p . Thus, each element of P1(FP) can be represented by a homogeneous coordinate of the form: [0 : 1] if a = 0(modp), or [1 : 6/0], otherwise. The 62 generator ai of PSL2(p) acts on these homogeneous coordinates as follows: , for j = 0 , . . . , p - l . 1 0 0 0 , and 1 0 1 1 _ 1 1 1 1 1 1 . 3 . J + 1 . 0 1 0 1 0 1 1 3 1 = , and — - 1 0 1 0 . _ 1 0 . 3 . - 1 . - r 1 . From this we see that the only fixed point of a\ is [0 : 1] and a\ cycles through the remaining p elements of the form [1 : j], j = 0 , . . . ,p - 1. The element a2 acts on P^Fp) as follows: , j = 1, . . . , p - l . The fixed points of a2 are the elements [1 : j], for which j = —j_1(mod p) or j2 = —l(mod p), which, can only happen if p = l(mod 4), by an elementary result of number theory. The element 03 acts on P:(FP) as: , 3 = 1, The fixed points of 03 are [0 : 1] and the elements: [1 : j] for which j = x_2j(mod p) which implies j = 0(mod p) or x~2 = l(mod p). Notice that the primitive element x has order p — 1 and thus x 0 0 0 , and x 0 1 x 1 0 x-1 1 1 0 x~l . 3 . . x~l3 . . x~2i . ord(x~2) = p - 1 x 2 = l(modp), onlyifp = 3. Hence, a3 has four fixed points: [0 :1] , [1 : 0], [1 : 1], [1 : 2], if p = 3 and two fixed points: [0 : 1], [1 : 0], if p 7̂ 3. The p + 1 elements of P1(FP) can be identified with the elements of the set {0 ,1 ,2 , . . . ,p} as follows: , for j = 0 , . . . , p — 1, and p 0 1 Enumerating the elements of P1(Fp) in this way, gives a permutation representation rp of PSL2(p) in the symmetric group Sp+i = Sym{0,1,2,... ,p}. It is clear that PSL2(p) acts transitively on P1(FP) and the stabilizer Stab[a:^ in PSL2(p) of any homogeneous coordinate [a : b] has index p + 1. Thus rp becomes a Galois representation of PSL2(p) of degree p + 1 (which is minimal if p > 11) by identifying the left cosets of Stab^ in PSL2(p) with a set, 63 denoted by £(p+i,q) of p + 1 distinct zeros of fq, where q is a prime such that q > p + 1 and 3 - 2 is not a perfect square. This in turn induces an action of PSLiip) on the family of curves: = { (x,y) eC2\y2 = x(x - 1) J J (s - Q where S n is a subset of any n zeros of / 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 fq, and have the following cycle decompositions: ai = (0,1,...,P-1) a 2 = (0,p)Uj (j,-j -1mod p) a3 = (x 2fc, x4fc,..., xp-1fc) (xk, x3k, x 5 f c ,x p ~ 2 k ) , k €F2 2^1 2=1 2 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 two fixed points 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 0 3 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 Xg,3n is reduced to a permutation on the indices of the unordered set S„ = {Q i ;..., Qn} of n zeros of fq, where the genus of the curve is g = L̂ ŷ J- 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,q)- It is sufficient to produce an orbit 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 2-l) , . C p + l ' n - nl(p-n + l)! " ~ T — ( 7 ' 1 } Since Cp+ii3 = P^" 1) < |PSL<2(p)|, n cannot be smaller than 4. The three smallest primes 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: PSL2(11) and PSL2(13) 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 PSL2(11) and PSL2(13) are written explicitly in cycle notation: PSL2(11) : fll = (1,2,3,4,5,6,7,8,9,10,11) a2 = (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) a3 = (2,4,10,6,5)(3,7,8,11,9) PSL2(13) : ax = (1,2,3,4,5,6,7,8,9,10,11,12,13) a2 = (1,14)(2,13)(3,7)(4,5)(8,12)(10,11) a3 = (2,11,10,13,4,5)(3,8,6,12,7,9) 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 PSL2(p), i.e. an orbit of length |PSL2(p)|. One such algorithm carried out by Maple 7, is used here to determine various pairs (p, n) satisfying condition (7.1) that admit full sized orbits of PSL2(p) on the class V(3P$) of Dessins consisting of curves Cq^n. A sample Maple worksheet containing the search algorithm for any pair (p, n) is given in Appendix A. Full sized orbits of PSL2(11) on all unordered n-tuples are summarized in the following table. PSL2(11) (order = 660) n Maximum orbit size Representative n-tuple contained in orbit of size 660 5,7 6 660 330 {0,1,2,3,5}, {0,1,2,3,4,6,10} 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 PSL2(13): PSL2(13) (order = 1092) 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 PSL2(17) and PSL2(19) on all unordered n-tuples are summarized in the following tables. PSL2(17) (order = 2448) 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} PSL2(19) (order = 3420) 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 PSL2(11), the 5-tuple: {0,1,2,3,5} corresponds to the subset 2*5 = {Co, Ci) C2, C3, Cs} G S5 C H(12)13) 66 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 /13) in Figure 7.7 by 7 ,̂1,2,3,5 to the genus 3 curve -^i3.{Co,Ci.C2.C3,C5} e -^13,^5 i n Figure 7.8. The class of curves on which PSL2(11) acts, belong to the family: X\z^s and the Galois action of the element a2 in PSL2(11) sends -Xi3,{c0,Ci,C2,C3.C5} to ^i3,{<2,C5.C7,Cio>Cii}' w h i c n is illustrated in Figure 7.9. The full orbit of PSL2(11) on 2>(/3i3,95) 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 0̂,2,5,9,12,13 to the genus 3 curve *i7,{C6,Ca,0j,C»,Cia,Cis} 6 xir,&» ™ Figure 7.11. The Galois action of ax in PSL2(13) sends -^i7,{Co,C2,Cs,C9,Ci2,Ci3} t o •yi7,{<b,Ci,C3,C(i,Cio,Cia}> w h i c h is shown in figure 7.12. The full orbit of PSL2(13) on T>(8n^) is displayed, using JavaScript and Maple, at www.math.ubc.ca/~burggraf/research.html. C3 C4 Cs Ce C 2 , 1 Im(z) Cs-i J C i C9 Co . C12 Re(z) "C11 C10 C4 Cs Ce C7 Cs' Co* C3 I Im(z) C10 C r i f J C2 Co ° C JBsiz) °Cl5 14 °Cl2 °Cl3 Figure 7.6: The indexed set of zeros of /13 on the left and of /17 on the right. 67 Figure 7.9: The Galois action of a2 € PSL2(11) on G(Pi,i o Bu,i ° hz ° ^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. for 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 actions correspond to orbits of size =",N);, break; f i ; orbit:=array(1..N); C:=nextstruct(allcombs): P:=convert(C,list); flag:=0; for j2 in o r b l i s t do i f member(P,orbp[j2]) flag:=l; print(jl.C,"done i n ",j2,"skipped"); break; od; i f flag=l then next; f i ; for j from 1 to N yCj]:=list(l..n): # y[j][k] i s an integer modulo p+1 representing the # homogeneous coordinate [l :y [ j][k]] resul t i n g from the # action of PSL2p[j] on x[k] . # If z i s an integer mod p+1, i t represents the # following homogeneous coordinate: # z <—> [ l : z ] , i f i<p # z <—> [0:1] , i f i=p od: for j from 1 to N do for k from 1 to n 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 do do then y[j] [k] :=B/A mod p; else y[j] [k] :=p; f i ; # . B/A <—> [1:A/B] = [A:B] # p <—> [0:B] = [0:1] od: 72 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); or b l i s t : = [ o p ( o r b l i s t ) , j l ] : i f nops(orbc)=N then f a i t h l i s t : = [op ( 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) f i : od: > i f nops(faithlist)>0 then p r i n t ( " F a i t h f u l orbits contain the following n-tuples:"); for j from 1 to nops(faithlist) do p r i n t ( j , f a i t h l i s t [ j ] ) ; od; else printC'No f u l l - s i z e orbits found.") 73 Appendix B Mondodromy generators of a hyperelliptic Belyi function The elliptic curve *2,7,n = {(x,y) eC2\y2 = (x- Q)(x - Cj)(x -fa}, where Ci,Cj,Cfc are distinct roots of fn(x) = x12 - 12/llx11 4- 1, admits a Belyi function 0 = 0\o,i 0 fu ° TT, where ir(x,y) = x and 0\Q,I(X) = ll n/10 1 0x 1 0(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. gx=[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]. 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 p2 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. 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 default the solve function w i l l find' only 100 # solutions but 132 w i l l be needed below. >#Find and display roots of the irreducible polynomial fl2(x) = x~12-12/ll*x-11+1.0 > r:=solve(x~12-12/ll*x~ll+1.0,x): for i from 1 to 12 do p r i n t ( i , r [ i ] ) od; >#The solutions to the equations below give the inverse image of points on the two c i r c l e s CO and C l , under the Belyi function b(10,1)(f12(x)). The points i n the inverse image of CO are stored i n the array trO and the points i n the inverse image of Cl are stored i n the array t r l . Note that a l l the points i n trO and t r l l i e i n the x-plane. > eqnO := ll-ll/10-10*(x"12-12/ll*x"ll+l)-10*(l-(x~12-12/ll*x-ll+l)) 75 = 0.5*exp(2.0*Pi*k*I/n); eqnl : < r i l/10~10*(x~12-12/ll*x~ll+l)~10*(l-(x~12-12/ll*x~ll+l)) = l-0.5*exp(2.0*Pi*k*I/n); > for j from 1 to 5 do tr0[j]:=solve(eval(eqn0 , {k=j-l,n=5})); . i f j=l then t r l [1] :=trO[l] ; else t r l [j] :=solve(eval(eqnl , -Ck=j-l,n=5})); f i ; p r i n t ( j , " out of 5") od: >#The following loops solve f o r y i n the expression y"2=(x-r[2])*(x-r[7])*(x-r[ll]), where the x-values are the points i n the inverse image of b(10,l)(f12(x)) found above. This l i f t s the points on the c i r c l e s CO and Cl to the y-plane, which are stored i n the arrays yO and y l . > for i from 1 to 5 do for j from 1 to 132 do yO[i][j] := ^ sqrt((trO[i] [ j ] - r [ 2 ] ) * ( t r 0 [ i ] [j]-r[7])'*(tr0[i] [ j ] - r [11])); i f i = l then y l [ i ] :=yO[i] ; else 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])); f i ; od: for j from 133 to 264 do yO[i][j] := -y0[i][j-132]: y l [ 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 starting point p i = 0.5897322923+0.4356541991*1 i n the array t r 0 [ l ] i s selected for the monodromy generator about 0 i n the x-plane and i s denoted by t r 0 [ l ] [ o r d 0 [ l ] ] . > for j from 1 to 132 do d0[j] :=abs(Re(trO[l] [j])-0.6):. i f d0[j]<0.1 and Im(tr0[l][j])>0 then ord0[l] :=j : ordl[l]:=j: print (j ,tr0[l] [ j ] ) ; f i ; od:. >#The following code determines the cycle go[l] i n gO starting 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 yields a second cycle g0[2] beginning and ending at - p i . > f or j from 1 to 26 do g0[j]:=array(l..20): od: 76 gO[l] [1]:=1: # Let the cycle gO[l] begin with 1 g0[2][1]:=1+132: # Let the cycle gO[2] begin with 133 ordO[1+132]:=ord0[l]+132: # ordO[i] stores the index of trO[j] that # corresponds to the point on the path # i n the x-plane la b e l l e d by i i n the cycle of # the monodromy generator hO about 0. For # example, t r 0 [ l ] [ord0[6]] i s the point on # the path corresponding to the 6 i n the # cycle [1,2,3,4,5,6,7,8,9,10,11] of hO[l]. pri n t (gO [1] [1] ,yO[l] [ordO[l]] ,g0[2] [1] ,y0[l] [ordO [1+132] ] ) : yflag:=ord0[1]: # yflag i s the index out of the 264 # indices of y0[j], j=l to 5, that corresponds to the # j - t h point on the path between two black squares, # where j=l corresponds to a black square, for 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 indices of else flag:=yflag: # t r 0 [ j ] , j = l to 5, that corresponds to the j - t h f i : # point on a path above CO i n the x-plane, where # j=lcorresponds to a point i n the inverse # image of 1/2 under b(p). for i from 1 to 5 do. fix:=flag: f o r j from 1 to 132 do # The for loop calculates distances d[j]:=sqrt((Re(tr0[i][fix]-tr0[((i)mod 5)+l][j]))~2 # between + (Im(trO[i] [fix]-tr0[((i)mod 5)+l] [j]))*2); # t r 0 [ i ] [fix] and od: # tr0[((i)mod 5) + l ] [ j ] , where ((i)mod 5)+l # runs through the sequence 2,3,4,5,1 as # i runs from 1 to 5. dmin:=d[l]: # Find the minimum distance of the for j from 1 to 132 do # 132 distances found above and set i f d[j]<=dmin then dmin:=d[j]; flag:=j; f i ; # f l a g to t h i s od: # minimum distance. This determines the # next point to proceed to 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] [flag]))~2); dst2:=sqrt((Re(yO[i][yflag]+y0[((i)mod 5)+l][flag]))-2 +(Im(yO[i][yflag]+y0[((i)mod 5)+l][flag]))~2); i f dstl<dst2 # To proceed from one point on a blue path to then yflag:=flag: # the next i n the y-plane, the closest of the else yflag:=flag+132: # two points to y0[i][yflag] i n the inverse 77 f i : # image of the "next"point found above i s chosen. if dstl=dst2 then printO'Error: cannot proceed to next y-value"): f i : od: i f yflagOordOCl] # yflag <> ordO[l] means the cycle w i l l continue then # and yflag = ordO[l] indicates that the cycle ends. ordOCm+1]:=flag: ordO[m+1+132]:=flag+132: gO[l][m+1]:=m+l: # Label next entry of the cycle gO[l] g0[2][m+1]:=m+l+132: # Label next entry of the cycle g0[2] print (gO [1] [m+1] ,y0[l] [yflag] ,g0[2] [m+1] ,-yO[l] [yflag]) : else for i from 1 to 2 do # The for loops format the print output for j from 0 to m+1 do i f j=0 then printf ("\ngO['/.d] = [",!): else i f j<m+l then i f j<m then pr i n t f 07.A," ,gO[i] [ j ] ) : else p r i n t f ("7.A" ,g0 [i] [j]) : f i : else p r i n t f ( " ] " ) : f i : f i : od: od: p r i n t f ( " / n " ) : m:=22: # End for m from 1 to 22 loop since cycle has ended, f i : od: >#The following code determines the remaining cycles of gO. > for s from 1 to 12 do i f s=12 then flag:=ordO[ll] : else flag:=ordO[s] : f i : for i from 1 to 5 do fix:=flag: for j from 1 to 132 do i f s=12 then # Follow the red path above Cl i n the y-plane # starting at yO[l] [ordO[j]], j = 1 to 11. # The endpoint of t h i s path determines the # starting point of a cycle 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] :=sqrt((Re(trl[i] [f i x ] - t r l [ ( ( i ) m o d 5)+l ] [ j]))~2 + (Im(trl[i] [ f i x ] - t r l [ ( ( i ) m o d 5)+l] [ j]))~2); f i ; od: dmin:=d[l]: for j from 1 to 132 do i f d[j]<=dmin then dmin:=d[j]; flag:=j; f i ; od: od: 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 cycle g0[2*s+l] g0[2*s+2] [1]:=12+10*(s-1)+132: # Label f i r s t entry of cycle g0[2*s+l] print(g0[2*s+l][1],y0[l][yflag] , g0[2*s+2][1],y0[l][yflag+132]): for m from 1 to 10 do i f ordO[12+10*(s-1)+m-l] >132 then flag:=ord0[12+10*(s-l)+m-l]-132: else flag:=ord0[12+10*(s-l)+m-l]: f i : for i from 1 to 5 do fix:=flag:~ for j from 1 to 132 do d[j] :=sqrt((Re(tr0[i] [fix]-tr0[((i)mod 5)+l][j]))*2 + (Im(tr0[i] [fix]-tr0[((i)mod 5)+l] [j] ))~2); od: dmin:=d[l]: for j from 1 to 132 do i f d[j]<=dmin then dmin:=d[j]; flag:=j; f i ; od: dstl:=sqrt((Re(y0[i][yflag]-y0[((i)mod 5)+l][flag]))~ 2 + (Im(y0[i] [yflag]-y0[((i)mod 5)+l] [flag]))~2); dst2:=sqrt((Re(yO[i][yflag]+y0[((i)mod 5)+l][flag]))~2 . +(Im(y0[i] [yflag]+y0[((i)mod 5)+l][flag]))~2); i f dstl<dst2 79 then yflag:=flag: else yflag:=flag+132: f i : i f dstl=dst2 then print("Error: cannot proceed to next y-value"): f i : od: i f yflagOordO[12+10*(s-1)] # True indicates cycle i s not complete then ordO [12+10*(s-1)+m]:=yflag: i f yflag>132 then ordO[12+10*(s-l)+m+132]:=yflag-132: else ordO[12+10*(s-l)+m+132]:=yflag+132: f i : 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 and cycle not complete means the cycle w i l l be then # of length 20. In fact the cycle i s the concatenation for j from 1 to 10 do # of two symmetric 10-cycles of hO under g0[2*s+l][10+j]:=g0[2*s+l][j]+132: # the symmetry y -> -y. od: g0[2*s+2]:=g0[2*s+l] : for j from 0 to 2*m+l do # The for loop formats the print output i f j-0 then pr 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]): else p r i n t f 07,A",gO[2*s+l] [ j ] ) : f i : else p r i n t f C ' ] " ) : f i : f i : 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]]); f i : else for i from 1 to 2 do # The for loops format the print output 80 for j from 0 to m+1 do i f j-0 . then printf (" \ngO [7.d] = [", 2* s+i): else i f j<m+l then i f j<m then printf("%A,",g0[2*s+i][j]): else printf("%A\gO[2*s+i][j]): f i : else p r i n t f ( " ] " ) : f i : f i : od: od: p r i n t f ( " \ 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] g0[ll] = [52,53,54,55,56,57,58,59,60,61,184,185,186,187,188,189, 190,191,192,193] g0[12] = g0[ll] 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 PSL2(p) > restart: > with(plottools) : > p:=ll: #### The orbit of PSL(2,p) on class of Dessins i s computed i n t h i s work sheet. > N:=p*(p~2-l)/2; #### N i s the order of PSL(2,p) > q_:=13: #### q_ i s a prime > p+1 > 11:=2; cl:=[ll/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; > halfcentre:=PLOT(ellipticArc([0,0], r2, r2, O..Pi, filled=true, color=black)) > centre:=PL0T(disk([0,0] , r l , filled=tme,color=black)): > centre2:=PL0T(disk([0,0], r2, filled=true,color=black)): > centre3:=PL0T(disk([0,0], 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 ) , circle([11/5,0], r l , color=black, thickness=thick), line([11/5+r1,0], [ l l / 3 - r l , 0 ] , color=black, thickness=thick, linestyle=l), disk([11/3,0], r l , color=black)): > petal_long:=PL0T(line([0,0], [ll/2-rl,.0], color=black,. thickness=thick, li n e s t y l e = l ) , circle([11/2,0], r l , color=black, thickness=thick), 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)): > linel:=line([0,0], [ L l / 2 - r l , 0 ] , color=black, thickness=thick): > circ:=circle([Ll/2,0] , r l , color=black, thickness=thick):. > for j from 1 to (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): > for j from 3 to 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] :=line(Cl+[rl,0], [short-4/(j+l)*ll/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]:=plots[display](linel,circ,stem_long[j-2], bouquet.f a r [ j - 2 ] ) : od: > PSL2p: = [] : for a from 1 to (p-l)/2 #### generate elements with a>0 do for b from 0 to p-1 do for c from 0 to p-1 do PSL2p:=[op(PSL2p),[[a,b],[c,(l+b*c)/a mod p]]]; od; od; od;. for b from 1 to (p-l)/2 #### generate elements with a=0 do ' . . for d from 0 to 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 representative unordered n-tuple i n the orbit of PSL(2,p) > n:=nops(C): #### n i s the number of elements i n C > for i from 0 to p-1 do x [ i ] : = [ [ l ] , [ i ] ] ; od: x[p]: = [ [ 0 ] , [ l ] ] : 84 > for j from 1 to N do y [ j ] : - l i s t ( i . . n ) : # y C j ] [k] i s an integer modulo p+1 representing the # homogeneous coordinate [ l : y [ j ] [ k ] ] resulting from the # action of PSL2p[j] on x[k] . # If z i s an integer mod p+1, i t represents the # following homogeneous coordinate: # z <—> [ l : z ] , i f i<p # z <—> [0:1] , i f i=p od: for j from 1 to N do for k from 1 to 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] else y[j] [k] :=p; # p <—> [0:B] = [0:1] f i ; od: orbit[j]:=convert(y[j],set); od: > orbsize:=nops(convert(orbit,set)); > for k from 1 to orbsize do tuplel:=orbit[k]; num_sides:=nops(tuplel)+l: # half the number of sides of the polygon special_case:=false: i f member(p,tuple1) then special_case:=true f i : tuple:={op(tuplel),-1,q_-2}; ### The following commands pa r t i t i o n the flowers with non-ramified bouquets count:=0: for j l from 1 to nops(tuple)-1 do num:=tuple[j1+1]-tuple[j1]-1: i f num<3 then increment:=Pi/2/num_sides: else increment :=Pi/num_sides/(l*+num): f i : 85 i f num=2 then offset:=Pi/4/num_sides: else offset:=Pi/num_sides/(1+num): f i : for j2 from 1 to num do count:=count+l; , angle:=increment*(j2-l)+offset+Pi/num_sides*(jl-l); i f num<4 then amplitude:=1: else amplitude:=num-2: f i : 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, line s t y l e = l ) , circle([11/4,0], r l , color=black, thickness=thick), line([11/4+rl,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, lin e s t y l e = l ) , circle([11/5,0], r l , color=black, thickness=thick), l i n e ( [ l l / 5 + r l , 0 ] , [3*11/4,0], color=black, thickness=thick, linestyle=D): for j from 0 to (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, -Pi/5, [0,0]), ellipticArc([0,0] , r2, r2, 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]), rotate(flower_special,-Pi/5,[0,0]), e l l i p t i c A r c ( [ 0 , 0 ] , r2, r2, Pi/2..3*Pi/2, filled=true, color=black), rotate(petal_long,9*Pi/16, [0,0])) :• f i : 86 raMfied_flower_stem:=PLOT ( l ine ([0,0], Cl-[rl,0], color=black, thickness=thick, l ines ty le= l ) , c i r c l e ( C l , r l , color=black, thickness=thick), line(Cl+[rl,0], [Ll+11,0], color=black, thickness=thick, linestyle=D): i f special_case then petal_special:=PLOT(line([0,0], cl-[rl,0], color=black, thickness=thick, l ines ty le= l ) , c i r c l e ( c l , r l , color=black, thickness=thick), line(cl+[rl,0], c2, color=black, thickness=thick, linestyle=D): f i : half_bouquet:=plots[display](seq(long_petal[i],i=l..q_-2), ha l fcentre) : i f special_case then long.petal[ceil((q_-2)/2)]:=rotate(petal.special,Pi/2,[0,0]) f i : i f special.case then half_bouquet_special:=plots[display] (seq(long_petal[ i], i=l..q_-2), ha l fcentre) : f i : long_stem:=PL0T(line([0,0], Cl-[rl,0], color=black, thickness=thick, l ines ty le= l ) , c i r c l e ( C l , r l , color=black, thickness=thick), line(Cl+[rl,0], [Ll-rl,0], color=black, thickness=thick, linestyle=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] (translate(rotate( half_bouquet, Pi/2,[0,0]), 11+L1.0),long_stem): i f special_case then ramified_bouquet_special:=plots[display](stem_special, rotate (translate (rotate(half _bouquet_special, Pi/2,. [0,0]), 11+L1.0), Pi*(num_sides-l)/num_sides,[0,0])): f i : for j from 1 to 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 f i : ramified_flower:=plots[display](translate( ramif ied.f lower_cl ip, Ll+11,0),ramified_flower.stem): 87 p o l y l : t r a n s l a t e (PLOT (line ([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): for j from 1 to 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, half_flowers): half_dessin2:=rotate(half_dessinl,Pi, [0,0]): str:="": for j from 1 to 4-length(convert(k,string)) by 1 do str:=cat(str,"0"): od: str:=cat(str,convert(k,string),"-"): for j from 1 to nops(tuplel) do str:=cat(str,convert(tuplel[j].string)): od: file_name:=cat("c:\\docume~l\\dsbl\\mydocu~l\\thesis\\ PSL2-ll-orbit-gifs\\ " , s t r , " . g i f " ) : strl:=cat("PSL2p[",convert(k,string),"] = ",. convert(PSL2p[k], st r i n g ) ) : str2:=cat(convert(C,string)," -> ",convert(tuplel,string)) plotsetup(gif,plotoutput=file_name, plotoptions='portrait, noborder'); te x t . :=PL0T(TEXT([0,Ll+7*ll/5], strl,C0L0R(RGB, 0,0,0), FONT(TIMES,ROMAN,22)), TEXT([0,-Ll-8*ll/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], labels=["",""], 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 Array(); function addPic(_p) { gifList[gifList.length?gifList.length:0] = new Image(); gifList[gifList.length-1].src=_p; > f o r ( i = l ; i<661 ; i++) { addPic( , ,orb"+i+" .gif " ) ; } function checklt(val) { current = Math.abs((current+parseInt(val ) )7.gifList.length); document.myimg.src = gifList[current].src; > ' ' ' </script> <script type=text/javascript> document.write("<img name=myimg alt='' src='"+gifList[current].src+"'>"); </script> <br><input type=button value="Back" onclick="checkIt(-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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