Algebraic and Geometric Properties of the Boundary of Orthogonal Shimura Varieties Dylan Attwell-Duval Doctor of Philosophy Department of Mathematics and Statistics McGill University Montr´eal, Qu´ebec February, 2015 A thesis submitted to McGill University in partial fulfillment of the requirements for a Ph.D. degree Copyright c Dylan Attwell-Duval, 2015 Acknowledgements First and foremost I would like to thank my supervisor, Eyal Goren. His advice and tutelage have made me into the mathematician I am today. With- out his guidance, helpful suggestions on reference material, as well as the numerous hours put into editing and proof checking, the completion of this project would not have been possible. I would also like to thank the numerous professors that have taught courses I have attended during my time at McGill and the University of Toronto. I would particularly like to thank Prof. Adrian Iovita for going above and be- yond on numerous occasions to deliver excellent courses on algebraic geometry, despite administrative impediments. I would also like to thank Prof. Donald James for providing me with a portion of his former student’s thesis. Another individual who is due recognition is the anonymous referee of my first submitted paper. My submission was initially returned with a number of insightful suggestions and comments, indicative of the amount of effort this person but forth in proofreading. I would also like to thank the Mathematics and Statistics Department at McGill, both for providing financial support as well as an excellent environ- ment for developing as a mathematician. Other sources of financial support during my time at McGill include the National Science and Engineering Re- search Council of Canada as well as the generous philanthropists who donate to McGill, in particular Lorne Trottier and the estate of Max E. Binz. Finally, I would again like to thank all four of my parents for their dedi- cation and support during my entire life. iii Abstract In this thesis we study the boundary of orthogonal Shimura varieties with a focus on those arising from maximal lattices lying inside a rational vector space. The goal is to provide generalizations of boundary structure results that are known in lower dimensional or special cases, for example Hilbert modular surfaces, Siegel threefolds, and the unimodular cases. We begin by recalling the general theories of lattices and Shimura vari- eties, emphasizing the Baily-Borel boundary structure of these spaces. Upon recalling the theory of PEL-type Shimura varieties, we take a quick detour to discuss the structure of the PEL-Shimura varieties that are associated to orthogonal Shimura varieties. We return to our study of the boundary by cal- culating explicit formulas for the number of 0 and 1-dimensional cusps in the Baily-Borel compactification of orthogonal Shimura varieties. We also provide algebraic and geometric results regarding the structure and configuration of each component within the boundary. Finally, we study the closure of special divisors in the Baily-Borel compactification of these spaces, providing results on the intersection of these divisors with the various boundary components. iv Abr´eg´e Cette th`ese est d´edi´ee `al’´etude de la fronti`ere des vari´et´es de Shimura or- togonales avec une emphase particuli`ere sur les vari´et´es attach´ees `a des r´eseaux maximaux dans des espaces vectoriels rationnelle. L’objectif est de g´en´eraliser certains r´esultats sur les fronti`eres de vari´et´es de Shimura ´etablis pour des dimensions inf´erieures ou des cas particuliers comme par exemple les surfaces modulaires de Hilbert, les vari´et´es de Siegel de dimensi´on 3 et les cas unimod- ulaires. Nous ´evoquons d’abord la th´eorie g´en´erale des r´eseaux et des vari´et´es de Shimura en nous attardant sur leurs fronti`eres de type Baily-Borel. Nous abor- dons ensuite la th´eorie des vari´et´es de Shimura de type PEL et d´ecrivons la structure de telles variet´es associ´ees ´a des variet´es de Shimura orthogonales. Nous poursuivons notre ´etude de la fronti`ere en fournissant des formules ex- plicites pour le nombre de points de rebroussement de dimension 0 et 1 dans la compactification de Baily-Borel des vari´et´es de Shimura orthogonales. Nous obtenons des r´esultats alg´ebriques et g´eom´etriques sur la structure et configu- ration de chacune des composantes de la fronti`ere. Enfin, nous ´etudions la fer- meture de certains diviseurs sp´eciaux dans la compactification de Baily-Borel de ces espaces obtenant ainsi des r´esultats sur l’intersection de ces diviseurs avec les composantes de la fronti`ere. v TABLE OF CONTENTS Acknowledgements .............................. iii Abstract .................................... iv Abr´eg´e..................................... v List of Tables .................................viii 1 Introduction ............................... 1 2 Quadratic Forms, Lattices and Spin Groups ............. 6 2.1 Lattices and associated structures ............... 6 2.2 Clifford algebras and spin groups ............... 15 2.3 Signatures and connectivity .................. 24 2.4 Unimodular lattices ....................... 27 2.5 The image of GSpin(L)..................... 31 3 General Theory of Shimura Varieties ................. 34 3.1 Reductive and semisimple algebraic groups over perfect fields 34 3.2 Connected Shimura varieties .................. 40 3.3 The construction and structure of Baily-Borel compactifica- tions . ............................. 52 3.4 General Shimura varieties ................... 67 3.5 An embedding of orthogonal Shimura varieties ........ 78 4 The Hermitian Symmetric Spaces of Even Clifford Algebra PEL- type Shimura Varieties ........................ 82 4.1 Case 1: G2,n,n≡ 1, 3 mod8 ................. 84 4.2 Case 2: G2,n,n≡ 0, 4 mod8 ................. 88 4.3 Case 3: G2,n,n≡ 5, 7 mod8 ................. 91 4.4 Case 4: G2,n,n≡ 2 mod8 .................. 92 4.5 Case 5: G2,n,n≡ 6 mod8 .................. 96 5 Counting Cusps for Maximal Lattices .................101 5.1 Counting the orbits of isotropic lines .............101 5.2 Counting the orbits of isotropic planes . .........105 5.3 Examples ............................114 5.3.1 Unimodular .......................114 vi 5.3.2 Cyclic ...........................116 5.3.3 Lattices from CM-fields .................117 5.4 Scaled lattices and level structure ...............119 6 Algebraic and Topological Properties of the Boundary .......133 6.1 Connected components and their field of definition ......133 6.2 Boundary components and their field of definition ......137 6.3 The geometry of the boundary .................142 7 Special Divisors and their Compactifications .............147 7.1 Special divisors at the boundary ................147 7.1.1 0-dimensional cusps ...................153 7.1.2 1-dimensional cusps ...................155 7.2 Intersections with the cusps ..................159 7.3 Discussion and examples . ...................166 8 Conclusion ................................172 Index of Notation ...............................176 References ...................................177 vii List of Tables Table page 4–1 Classification of G2,n over R ..................... 99 5–1 Cusps for Unimodular Lattices ...................115 viii CHAPTER 1 Introduction The main motivating factor in this thesis is understanding various aspects of a general orthogonal Shimura variety that does not depend on any specific Q- form of SO(2,n). The theory of Shimura varieties of any type has numerous applications throughout number theory, notably abelian varieties, automorphic forms, representations of algebraic groups, and moduli problems, just to name a few. The Langlands program is an example of an area of considerable research interest where Shimura varieties have successfully aided in our understanding. The study of orthogonal Shimura varieties is relatively new to number theory research but has sparked a lot of recent interest, due in large part to the Fields medal winning work of Borcherds [8]. In his seminal works, Borcherds fixes a lattice L of signature (2,n) and defines a lift of certain vector- valued modular forms on H to an orthogonal Shimura variety associated to the lattice L. These so-called Borcherds lifts are modular forms with an infinite product expansion and they contain a plethora of arithmetic and geometric information. One aspect of note is that the zeroes and poles of these modular forms are prescribed linear combinations of special divisors that arise naturally from the inclusion of quadratic subspaces of signature (2,n− 1) inside L ⊗ Q. Furthermore, these divisors are equipped with a Green’s function, Φβ,m, in the sense of Arakelov geometry ([9], pg. 124), and their study has aided in the development of the Kudla Program [32], comparing the arithmetic intersection numbers of these divisors to coefficients of modular forms. 1 We now discuss the scope of this project in terms of the layout of each chapter. For the reader’s convenience, we have used Chapters 2 and 3 to re- call some general theory that will be used in the later chapters. Chapter 2 focuses on the development of quadratic forms from the perspective of Z and Zp-lattices. Here we introduce important notation and definitions associated to a lattice L, including the discriminant group Δ(L)=L∨/L and the discrim- + inant kernel ΓL, which is the subgroup of SO (L) that acts trivially on Δ(L). The group ΓL is the arithmetic subgroup that gives rise to Shimura varieties where Borcherds lifts are possible. We also recall the Clifford algebras