Arithmetic aspects of Triangle Groups Luiz Kazuo Takei Department of Mathematics and Statistics McGill University March 2014 A thesis submitted to McGill University in partial fulfillment of the requirements for a Ph.D. degree (cid:2)c Luiz Kazuo Takei 2014 Abstract We study some arithmetic properties of Triangle Groups, a family of Fuchsian groups that generalize the modular group. We first recall the ba- sic theory of Fuchsian groups and define precisely a Triangle Group. Sec- ondly, we define congruence subgroups and compute the genus of the curves they uniformize. Thirdly, we characterize the normalizers of certain Triangle Groups and corresponding congruence subgroups. Finally, we study a family of curves that is closely related to Triangle Groups. In particular, we study modular embeddings defined by those curves and their ordinary locus. Résumé Cette thèse est consacrée à l’étude de certains aspects arithmétique d’une famille de groupes fuchsiens engendrés par des reflections par rapport aux arrêtes d’un triangle hyperbolique. On commence par rappeler la théorie de groupes fuchsiens et la définition de cette famille de groupes. On définit ensuite les sous-groupes de congruence et on calcule le genre des courbes qu’ils définissent. Ensuite, on décrit les normalisateurs de ces groupes. Les dernières sections sont consacrées à l’étude d’une famille de courbes al- gébriques étroitement liée à ces groupes. On étudie notamment son image dans certains espaces de modules et son lieu non-ordinaire. Acknowledgements Iwouldliketothankmysupervisors, HenriDarmonandEyalGoren, fortheir support, patience and guidance throughout my studies at McGill University. They have also, together with the Department of Mathematics and Statistics at McGill, provided financial support during that time. I would also like to mention John Voight and Paulo Ribenboim: the former greatly helped me with his knowledge of triangle groups and related areas, while the latter was a source of kind support and encouragement, especially in difficult times. I would also like to thank my fellow students, including, but not limited to, Marc Masdeu, Mike Musty, Philip Rempel, Victoria de Quehen, Andrew Fiori, FrancescCastella, JuanIgnacioRestrepo, andBahareMirza, whohave all helped me at different times. Finally, I thank my parents, Mitsuca Miyashita and Suguio Takei, my wife, Phạm Nguyễn Hồng Phúc, my siblings, Andrea Mary Takei and Linus Jun Takei, and my in-laws, Phạm Hữu Cương, Nguyễn Thị Đao, Phạm Nguyễn Hữu Ân, Phạm Nguyễn Hồng Quang and Phạm Nguyễn Hữu Thuận: without their moral support I would not have been able to finish this thesis. 1 Contents Acknowledgements 1 Introduction 5 0 Fuchsian Groups and Triangle Groups 9 0.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . 9 0.2 Fuchsian Groups . . . . . . . . . . . . . . . . . . . . . . . . . 11 0.2.1 The topological space Γ\H∗ . . . . . . . . . . . . . . . 15 0.2.2 Γ\H∗ as a Riemann surface . . . . . . . . . . . . . . . 17 0.2.3 Signature of a Fuchsian Group . . . . . . . . . . . . . . 19 0.3 Triangle groups . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1 Congruence subgroups and modular curves 31 1.1 Basic definitions and notations . . . . . . . . . . . . . . . . . . 33 2 1.2 Computing the genus of X (p) . . . . . . . . . . . . . . . 36 q,∞,∞ 1.2.1 Special linear groups over finite fields . . . . . . . . . . 41 1.2.2 A bit of algebraic number theory . . . . . . . . . . . . 47 (cid:2) (cid:3) 1.2.3 Computing Γ : Γ (p) . . . . . . . . . . . . . 51 q,∞,∞ q,∞,∞ 1.3 Computing the genus of X(0) (p) . . . . . . . . . . . . . . . 55 q,∞,∞ 2 Action of PSL (F ) on mock modular curves 60 2 p 2.1 Character table of PSL (F ) for p ≡ 3 (mod 4) . . . . . . . 62 2 p2n+1 2.2 Statement of the main result . . . . . . . . . . . . . . . . . . . 66 2.3 A formula of Chevalley-Weil . . . . . . . . . . . . . . . . . . . 71 2.4 Proof of the main result . . . . . . . . . . . . . . . . . . . . . 73 3 Normalizers of triangle groups 82 3.1 Basic facts about normalizers of Fuchsian groups . . . . . . . . 84 3.2 (q,r,∞)-triangle groups . . . . . . . . . . . . . . . . . . . . . 87 3.3 (q,∞,∞)-triangle groups . . . . . . . . . . . . . . . . . . . . . 90 3.3.1 Congruence subgroups . . . . . . . . . . . . . . . . . . 98 3.3.2 Final remarks . . . . . . . . . . . . . . . . . . . . . . . 118 4 The TTV family of curves 121 4.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3 4.1.1 Igusa-Clebsch invariants . . . . . . . . . . . . . . . . . 123 4.1.2 Computing the Igusa-Clebsch invariants . . . . . . . . 124 4.2 Modular embedding . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2.1 Moduli space of Abelian surfaces with RM . . . . . . . 127 4.2.2 Jacobians of the TTV curves . . . . . . . . . . . . . . . 132 5 The ordinary locus of the TTV family of curves 145 5.1 The Hasse-Witt and Cartier-Manin matrices . . . . . . . . . . 146 5.1.1 Hasse-Witt matrix . . . . . . . . . . . . . . . . . . . . 146 5.1.2 Cartier-Manin Matrix . . . . . . . . . . . . . . . . . . 147 5.1.3 Jacobian of C . . . . . . . . . . . . . . . . . . . . . . . 149 5.1.4 Curves with real multiplication . . . . . . . . . . . . . 150 5.2 Studying C±(t) for q = 5 . . . . . . . . . . . . . . . . . . . . . 153 q 5.2.1 The Cartier-Manin matrix of the curve C− . . . . . . . 154 5.2.2 The Cartier-Manin matrix of the curve C+ . . . . . . . 156 5.2.3 A relation between X(0) (p) and the family C− . . . 159 5,∞,∞ 6 Future Directions 167 4 Introduction The theory of modular forms has been studied for more than one hundred years, dating back, at least, to Felix Klein and his contemporaries. In the previous century, modular forms for the classical modular group SL (Z) and 2 its congruence subgroups were extensively studied. It is now possible to say that we have a polished theory of modular forms. Good textbooks like [Miy06] and [DS05] are evidences of this fact. The recent interest in modular forms is related to its connection with numbertheoryquestions. Inparticular, motivatedbytheTaniyama-Shimura conjecture and Fermat’s Last Theorem, the second half of the twentieth cen- tury witnessed a gigantic effort to understand the relation between modular forms and elliptic curves, culminating in the celebrated proof of Fermat’s Last Theorem, more than 350 years after its first appearance in Pierre de Fermat’s notes. Since the modular forms of interest for this particular prob- 5 lem are those for SL (Z) and its congruence subgroups, much of the focus 2 was directed toward those groups. Although much less understood, non-congruence subgroups of SL (Z) 2 have also been studied in recent years. The pioneering article in this field is probably [ASD71], where Atkin and Swinnerton-Dyer proved some results and conjectured some other results about congruences involving the Fourier coefficients ofmodularforms fornon-congruence subgroups of SL (Z). Scholl 2 ([Sch85], [Sch86]) and Winnie-Li ([LLY05b]) have also contributed to this area. A slightly disappointing but (maybe exactly because of that) inter- esting fact proved by Serre-Thompson in [Tho89] and by Berger in [Ber94] states that Hecke operators, which play a prominent role in the theory of modular forms for congruence subgroups, yield no new information for non- congruence subgroups. A survey article about this area can be found in [LLY05a]. Even less understood than the non-congruence subgroups of SL (Z) are 2 the so called triangle groups. These form a special class of Fuchsian groups which includes SL (Z) as a particular example. From a number-theoretic 2 point of view, it is an interesting family of Fuchsian groups because, via Bely˘ı’s Theorem ([Bel79]), every algebraic curve defined over a number field 6 is uniformized, when viewed as a Riemann surface, by a triangle group. More recently, Darmon speculated in [Dar04] that triangle groups can be used to study the so called generalized Fermat’s equation. An example of this strat- egy can be seen in [DG95] and [Dar97]. Among others, Y. Yang [Yan] and Doran-Gannon-Movasati-Shokri [DGMM] have studied automorphic forms for triangle groups. A valuable resource to learn about triangle groups and facts of interest to number theory is [CV], by Clark and Voight. We present in the following chapters a study of triangle groups, the al- gebraic curves they uniformize and, in particular, their relations to number theory. Chapter 0 recalls the basic theory of Fuchsian groups that will be necessary for the later chapters and defines a triangle group. In Chapter 1, we define subgroups of triangle groups in analogy to the congruencesubgroupsofSL (Z). Thefirstnaturalquestionthenarises: what 2 are the genera of the curves uniformized by those subgroups? We answer this question as well as another interesting question, related to the nature of the quotient of a triangle group by one of these subgroups. In Chapter 2, we recall a relation found by Hecke in 1928 ([Hec28]) be- tween the class number of some quadratic number fields and the represen- tation of certain quotient groups on the space of holomorphic differentials 7