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 varieties, 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 calculating 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 thèse est dédiée àl’étude de la frontière des variétés de Shimura or- togonales avec une emphase particulière sur les variétés attachées à des réseaux maximaux dans des espaces vectoriels rationnelle. L’objectif est de généraliser certains résultats sur les frontières de variétés de Shimura établis pour des dimensions inférieures ou des cas particuliers comme par exemple les surfaces modulaires de Hilbert, les variétés de Siegel de dimensión 3 et les cas unimod- ulaires. Nous évoquons d’abord la théorie générale des réseaux et des variétés de Shimura en nous attardant sur leurs frontières de type Baily-Borel. Nous abor- dons ensuite la théorie des variétés de Shimura de type PEL et décrivons la structure de telles varietés associées á des varietés de Shimura orthogonales. Nous poursuivons notre étude de la frontière 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 variétés de Shimura orthogonales. Nous obtenons des résultats algébriques et géométriques sur la structure et configuration de chacune des composantes de la frontière. Enfin, nous étudions la fer- meture de certains diviseurs spéciaux dans la compactification de Baily-Borel de ces espaces obtenant ainsi des résultats sur l’intersection de ces diviseurs avec les composantes de la frontière.

v TABLE OF CONTENTS

Acknowledgements ...... iii Abstract ...... iv Abrégé...... 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 compactifications ...... 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 recall 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 and Spin groups, including constructing an explicit representation for the Cliﬀord algebra associated to L0 := H ⊥ H (Theorem 2.2.9). Finally, we reference a number of key results that will be used in the sequel, including providing an

+ adapted proof from [34] that relates ΓL to the image of Spin(L)inSO (L). In Chapter 3, we recall the theory of Shimura varieties, with a particular eye towards the orthogonal case. We discuss reductive Q-groups, Hermitian symmetric domains, and both connected and general Shimura datum. We explicitly construct the Baily-Borel compactiﬁcation of a general orthogonal Shimura variety and discuss the representation of the associated Hermitian symmetric domain as both a real Grassmannian and as an open subset of a complex projective variety. We use the latter interpretation to realize the symmetric space as a Siegel domain of the third kind via the theory of [1]. We conclude the chapter by realizing every orthogonal Shimura variety as a cycle inside a canonical PEL-Shimura variety. Chapter 4 is somewhat self-contained compared to the rest of the thesis. The purpose is to provide a representation of the R-forms of the algebraic groups associated to the PEL-Shimura varieties mentioned at the end of the previous chapter. By understanding these groups, we are able to deduce

2 the dimensions of the associated moduli spaces. In the future, these moduli spaces could provide interesting applications to the generalization of known results pertaining to low dimensional orthogonal Shimura varieties (for example, the aforementioned Kudla program) because the presence of a PEL-structure (which does not hold for general orthogonal Shimura varieties) enables one to more easily construct integral canonical models and therefore discuss arithmetic geometry. The bulk of the original results of this thesis are contained in Chapters 5 and 6, and focus on better understanding the boundary structure in the Baily- Borel compactiﬁcations of orthogonal Shimura varieties. Much of Chapter 5 was originally published in [2] and a number of the results pertaining to scaled lattices are to appear in [3] which is currently in preparation. The main purpose of Chapter 5 is determining explicit formulas for the number of 0 and 1-dimensional cusps in an orthogonal Shimura variety associated to a lattice L that splits two hyperbolic planes. In the case of 0- dimensional cusps, we compare the orbits of 0-dimensional isotropic subspaces of L with such cusps in order to derive the following result (Corollary 5.1.2).

Theorem 1.0.1. Let (L, q) be a lattice of signature (2,n) that splits two hyperbolic planes. Then the number of 0-dimensional cusps in the associated

ΓL-Shimura variety is equal to the cardinality of the set

{x ∈ Δ(L): q(x)=0}/{x ∼−x}.

Here q denotes the quadratic form q :Δ(L) → Q/Z induced from q.An immediate corollary to the above result is that if L is a maximal lattice, the associated Shimura variety contains only a single 0-dimensional cusp.

3 Including the assumption that L is maximal, we go on to prove a counting result for the number of 1-dimensional cusps in the associated ΓL-Shimura variety. This is done by comparing the number of such cusps to the ΓL-orbits of isotropic planes, and then comparing this to elements of the genus class of a maximal negative deﬁnite sublattice of L. The result is as follows (Theorem 5.2.9).

Theorem 1.0.2. Let (L, q) be a maximal lattice of signature (2,n) that splits two hyperbolic planes and choose a decomposition

L = H ⊥ H ⊥ A.

Let Y denote the connected Hermitian space associated to SO+(L) and let

+ ΓL be the discriminant kernel of SO (L). Then the number of 1-dimensional cusps in the Baily-Borel compactiﬁcation of SO+(L)\Y is equal to

2 . [O(A):SO(A)] A∈gen(A)

In the Baily-Borel compactiﬁcation of ΓL\Y , the number of 1-dimensional cusps is equal to

2 |O(Δ(A))| · , [O(A):SO(A)] |ρ(SO(A))| A∈gen(A) where ρ : O(A) → O(Δ(A)) is the projection map.

Combining these two results leads to the description of the boundary as a bouquet of projective lines, all intersecting at a common point. We complete the chapter by providing some explicit examples, and then deriving formulas analogous to those above for the case when L is replaced by a scaled lattice

4 N · L. We show that this can also be thought of as reflecting a certain level structure on our original space. The purpose of Chapter 6 is to apply some of the theory recalled in Chap- ters 2 and 3 to the results of the previous chapter. In particular, we study the fields of definition of the various connected components and boundary components of the orthogonal Shimura varieties associated to the scaled lattice N ·L.

This is mainly done through the embedding L0 → L and the representation of the Clifford algebra of L0 previously mentioned. We also discuss the graphical configuration of the boundary components for a scaled lattice, analogous to the aforementioned bouquet in the case when N =1. Chapter 7 serves to discuss the relationship between the special divisors discussed earlier and the boundary components of our space. In particular, we discuss which boundary components lie in the compactification of a given special divisor. Furthermore, we consider the divisor of a Borcherds lift ϕ restricted to this boundary. In the case when the boundary is not completely contained in the support of Div(ϕ), we give a formula for recovering the degree of ϕ by calculating a certain intersection pairing. We use Chapter 8 to discuss the potential to further develop the theory contained within this thesis, as well as possible future research problems to pursue.

5 CHAPTER 2 Quadratic Forms, Lattices and Spin Groups

This chapter of the thesis serves to familiarize the reader with the theory of quadratic forms on lattices (familiarity with quadratic forms over fields is assumed). The main purpose is to fix definitions and notation that will be used throughout the remainder of the paper. We will also recall the Clifford algebra and Spin group associated to a quadratic form, and finally we shall prove some results on lattices that will be used later in the thesis. Good references for the material within this chapter include Lam [33] and O’Meara [41].

2.1 Lattices and associated structures

Recall that a ﬁnite dimensional vector space V over a ﬁeld F is a quadratic space if it is equipped with a map

q : V → F

that is represented by a homogeneous polynomial of degree 2 in F [x1,...,xn] for some choice of basis V ∼= F n (different bases may give different polynomi- als). Such a map is called a quadratic form. Suppose we replace the field F in the definition above with a ring R, where R is either Z or Zp, and V with a free R-module L of finite rank. Then we say that L is an R-lattice if it is equipped with a quadratic form q as above, where now q is represented by a homogenous polynomial of degree 2 over R. The lattice (L, q) can be thought of as sitting inside the quadratic space (L ⊗ F, q), where F is the field of fractions of R.

6 Remark 2.1.1. Throughout this thesis, R will always denote Z or Zp as above. When discussing R-lattices or related structures, F will denote the ﬁeld of fractions of R. Otherwise F simply denotes a generic ﬁeld. Remarks will be made to avoid any potential confusion.

We begin by listing a number of constructions that apply both to R-lattices and quadratic spaces. The deﬁnitions will be given with respect to lattices, but the same notation and terminology will be used when they apply to quadratic spaces as well.

Deﬁnition 2.1.2. The bilinear form B associated to q is the map

B(x, y):L × L → R, B(x, y)=q(x + y) − q(x) − q(y).

Remark 2.1.3. Since q(λ.x)=λ2q(x), it follows easily from the definition that B(x, x)=2q(x). In many references, a lattice or a quadratic space is defined by specifying a bilinear form B, and then a quadratic form Q is constructed through B by the equation Q(x)=B(x, x). Thus there is often a factor of 2 that differs between various texts (as Q(x)=2q(x) in our notation). This is especially relevant for R-lattices where 2 ∈/ R× or quadratic spaces over fields of characteristic 2. When dealing with such ambiguities, one says that L is even if B(x, x) ∈ 2R for all x ∈ L. Such a bilinear form necessarily comes from a quadratic form q as defined above. Thus all lattices considered in this text are even.

Deﬁnition 2.1.4. A sublattice of a given R-lattice (L, q)isanR-lattice (L,q) such that L is a submodule of L and q restricted to L is equal to q.A

7 sublattice L ⊂ L splits L if there is another sublattice (L,q) ⊂ L such that L = L ⊕ L and q = q ⊕ q. We write this as L = L ⊥ L.

Deﬁnition 2.1.5. An R-lattice is called non-degenerate or regular if the Gram matrix of B has non-zero determinant (this is the discriminant, det(L), of q or L, and is unique up to an element of (R×)2). Given a non-degenerate lattice L, the associated dual lattice L∨ is the R-module

L∨ := {x ∈ L ⊗ F : B(x, y) ∈ R, ∀y ∈ L}. R

The ﬁnite group Δ(L)=L∨/L is called the discriminant group of L. Since L∨ is a subset of L ⊗ F, the quadratic form q extends to it naturally, although q(L∨) need not be contained in R. Nevertheless, the extended q leads to the quotient map on Δ(L) called the discriminant form:

q :Δ(L) → F/R, q(x):=q(x)modR.

Remark 2.1.6.

∨ 1. The dual lattice L is naturally isomorphic to HomR(L, R) via the map v → B(v, ·). Therefore it makes sense to refer to the dual lattice as the dual group of L. 2. If R = Z, then |Δ(L)| is equal to the absolute value of the discriminant of L. Indeed, given any basis β of L, a basis for L∨ can be found by applying B−1 to β, where B is the matrix representation of the bilinear

form with respect to β. Similarly, when R = Zp, the size of the dis-

1 criminant group is equal to N(det(B)) , where N denotes the p-adic norm a −a sending u · p ∈ Zp to p (here u is a unit in Zp).

8 Deﬁnition 2.1.7. A map between R-lattices ϕ :(L, q) → (M,q)isanR-linear map that respects the quadratic forms, ie.

q(ϕ(x)) = q(x).

Two R-lattices are called isomorphic if there is an invertible map between them. The group of automorphisms of (L, q) is denoted O((L, q)), or simply O(L) when no confusion will arise. The subgroup of automorphisms of determinant 1 is likewise denoted SO(L).

Remark 2.1.8. The group O(L) is a subgroup of the group of automorphisms of O(L ⊗ F ) where F is the ﬁeld of fractions of R. More generally, if R → K is any map of rings, then there is an induced map O(L) → O(L ⊗ K). In particular, if L is a Z-lattice then O(L) embeds uniquely into O(L ⊗ Zp)and O(L ⊗ R).

Remark 2.1.9. Any element α ∈ O(L) extends uniquely to an F -linear automorphism of L ⊗ F. This extension preserves the dual group L∨ because it preserves the bilinear form associated to L. In particular we get an induced map

α :Δ(L) → Δ(L) that preserves the discriminant form q. If K ⊂ O(L) is any subgroup of the automorphism group of L, we say that the discriminant kernel of K (with respect to L) is the subgroup of all α ∈ K such that α acts trivially on Δ(L). Since Δ(L) is ﬁnite, the discriminant kernel will be of ﬁnite index in K. We will

+ use ΓL to denote the discriminant kernel of the group SO (L) (See Corollary 2.5.3).

9 Deﬁnition 2.1.10. An R-lattice L is unimodular if its discriminant is a unit in R. More generally, L is called maximal if it is regular and

L∨ ∩{x ∈ L ⊗ F : q(x) ∈ R} = L.

Equivalently, L is maximal if it is regular and it is not properly contained in any lattice L ⊂ L ⊗ F.

Remark 2.1.11. A lattice is unimodular precisely when L = L∨ and hence all unimodular lattices are maximal.

The most important example of a unimodular lattice is the rank 2 lattice

01 whose bilinear form B has a matrix representation ( 10) for some basis. Such a lattice is called a hyperbolic plane and denoted H. This notation and terminology can also apply to a quadratic space with the same conditions on B.

Deﬁnition 2.1.12.

1. A non-zero vector v ∈ L is called isotropic if q(v)=0. A lattice is called isotropic if it contains isotropic vectors and it is called anisotropic otherwise.

2. A vector v ∈ L is called primitive if n.v ∈ L for some n ∈ F implies n ∈ R.

Remark 2.1.13. A lattice L is isotropic iﬀ its associated quadratic space L ⊗ F is. H is the prototypical example of a lattice (or quadratic space) that contains isotropic vectors. By a well-known theorem of Gauss, a vector v ∈ L can be

10 extended to a basis of L iﬀ v is primitive. Every 1-dimensional lattice is generated by a primitive vector that is unique up to multiplication by R×.

Lemma 2.1.14. Let L be a maximal R-lattice and let V = L ⊗ F be the encompassing quadratic space, where F is the ﬁeld of fractions of R. Suppose there exists a hyperbolic plane H ⊂ V that splits V . Then there exists a sublattice H ⊂ L that splits L.

Proof. Assume H splits V . Then there exists an isotropic vector u ∈ V and some F -multiple of u must be a primitive element of L. We may therefore assume that u itself is primitive. Consider the ideal

(a):={B(u, x): x ∈ L} R.

u The R-module L + a contains L and q still takes R-values on it, so by maximality of L, a is a unit. Therefore there exists v ∈ L such that B(u, v)=1. Replace v with v − q(v)u,soq(v) = 0. But now ∀ w ∈ L,

w =(B(w, u)v + B(w, v)u)+(w − B(w, u)v − B(w, v)u).

This splits L into an orthogonal sum u, v⊥ u, v⊥ and gives the result.

Finding sublattices that split a given Z-lattice is relatively diﬃcult compared to the equivalent problem over Q. Over a ﬁeld of characteristic not equal to 2, any non-degenerate subspace V ⊂ V splits V . In particular, any quadratic space has a diagonal basis. This is not true over Z. For example,

01 Z H the matrix ( 10) is not diagonalizable over and hence the integral lattice does not split into two 1-dimensional sublattices. In order to ﬁnd sublattices that split a given lattice, we need to investigate the idea of modularity which is deﬁned below.

11 Deﬁnition 2.1.15. Let L be an arbitrary lattice. The scale of L is deﬁned as the ideal sL := { B(x, y): x, y ∈ L},

while the norm, nL,ofL is the ideal generated by the norms q(x), ∀x ∈ L.

Remark 2.1.16.

1. Given the equation B(x, y)=q(x+y)−q(x)−q(y), it follows easily that

nL ⊃ sL ⊃ 2nL. In particular, the scale and norm of Zp-lattices are the same, for p =2. 2. The discriminant of a rank m lattice, det(L), is contained in the ideal (sL)m, as can be seen by writing out the formula for the determinant of the matrix B(x, y). If det(L) generates (sL)m, then L is called sL- modular or simply modular. Note that this does not depend on choice

of det(L)mod(R×)2. 3. A unimodular lattice is precisely as (2) implies: it is a lattice of unit discriminant, hence also of scale 1.

Theorem 2.1.17. If J is an sL-modular sublattice of L (meaning it is modular with sJ = sL), then J splits L.

Proof. [41] 82:15.

Modularity is very important over local rings as it leads to the so-called

Jordan decomposition of any non-degenerate L. For such an L, one of the following must occur due to the non-Archimedean metric on Zp (See §91 of [41] for details):

12 (a) There exists x ∈ L with (2q(x)) = sL so that J = R · x is an sL-modular sublattice of L. (b) There exists a binary lattice J ⊂ L that is sL-modular. In either case we see that we are guaranteed a 1 or 2-dimensional sublattice that splits our given lattice L. By applying this method inductively, we get a Jordan splitting. In the case when p = 2, only case (a) occurs, and the Jordan Decomposition Theorem can be stated as follows:

Theorem 2.1.18. Let p =2 be a prime. Suppose L is a non-degenerate

Zp-lattice. Then L has a decomposition into an orthogonal sum

n L = Li i=0 where each Li is either {0} or has a basis such that the restriction of B to Li is represented by the matrix ⎛ ⎞ pi ⎜ . ⎟ ⎜ .. ⎟ B(x, y)i = ⎜ ⎟ ⎝ pi ⎠ i i.p

∈ Z× where i p . If m L = Li i=0 is another such decomposition, then m = n, rk(Li) = rk(Li), and i = i Z× 2 mod ( p ) .

Proof. [41] 91:9 and 92.

Theorem 2.1.18 tells us that when p is odd, every p-adic lattice has an orthogonal basis. The uniqueness properties of the Jordan decomposition in this case leads to a Witt-type Theorem (Compare with Witt’s Theorem for quadratic spaces).

13 Theorem 2.1.19. Let p =2 be a prime. Let L, M and N be non-degenerate p-adic lattices and suppose L ⊥ M ∼= L ⊥ N. Then M ∼= N.

Proof. Any Jordan decomposition of the lattice L ⊥ M must have the property ⊥ that rk(L M)i = rk(Li) + rk(Mi) and (L⊥M)i must be the product of the corresponding i’s for L and M. Since the same holds for L ⊥ N, it follows that the Jordan decomposition M and N are the same. Therefore the two lattices are isomorphic.

When p =2, Theorem 2.1.19 does not hold in general. This is mainly a result of the non-uniqueness of a Jordan splitting when p =2, arising from the fact that a Z2-lattice need not have an orthogonal basis. For example, any unimodular Z2-lattice cannot be diagonalizable because the entries on the diagonal must be even, and 2 is not a unit in Z2 (note that this same remark holds for Z). As a counter example to Theorem 2.1.19 when p = 2 (and also for Z-lattices), consider the two lattices L1 and L2 represented by the bilinear forms

02 22 B1(x, y)=(20) and B2(x, y)=(20) .

It is easy to see that L1 does not contain a vector of norm 1, while L2 does. Hence the two lattices are non-isomorphic. On the other hand, the change of basis formula

10 1 020 10 0 220 01 0 · 200 · 01 1 = 200 01−1 002 10−1 002 ∼ shows that L1 ⊥ 2 = L2 ⊥ 2.

Remark 2.1.20. As in the case of quadratic forms over ﬁelds, the notation

2a1,...,2an is used to represent a rank n lattice (L, q) with a given basis

14 such that q takes the form

n 2 · q(x1,...,xn)= xi ai. i=1 Despite the lack of a general Witt-type theorem in the 2-adic case, we still have the following cancelation law that will be useful to us in the sequel.

Theorem 2.1.21. Let L be a non-degenerate Z2-lattice such that L has the splittings L = H ⊥ J and L = H ⊥ K. Then J ∼= K.

Proof. [41] 93:14.

2.2 Cliﬀord algebras and spin groups

In this section we recall some of the theory of Cliﬀord algebras associated to a quadratic space or lattice. The main result of this section is Theorem 2.2.9, which gives a useful representation of the Cliﬀord algebra associated to H ⊥ H. The representation of this algebra and its associated Spin group will be useful later in the text. A good reference for this section is Chapter 5 of [33].

Definition 2.2.1. Let (V,q) be a finite dimensional quadratic space over a field F . A pair (A, i) consisting of an F -algebra A and an inclusion of vector spaces i : V→ A is said to be q-compatible if it satisfies

i(x)2 = q(x) · 1, ∀x ∈ V.

The Cliﬀord algebra (C(V ),i)of(V,q) is the unique (up to canonical isomorphism) q-compatible pair with the following universal property: if (A, j) is another q-compatible pair, then there exists a unique F -algebra map ϕ : C(V ) → A such that ϕ(i(x)) = j(x) for all x ∈ V .

15 Remark 2.2.2. In general we will suppress the notation for the inclusion map in the pair (A, i) and simply view V as a subspace of the vector space A. Of course it requires proof to show that Cliﬀord algebras exist in general.

This is well-known however: let T (V ) be the tensor algebra of V , and let C(V ) denote the quotient algebra T (V )/I(q), where I(q) is the 2-sided ideal generated by elements of the form

x ⊗ x − q(x) · 1,x∈ V.

Proposition 2.2.3. C(V ) is the Cliﬀord algebra of (V,q). If {x1,...,xn} is a

{ e1 · · en ∈{ }} basis for V , then the image of the elements x1 ... xn : ei 0, 1 serve as a basis for C(V ).

Proof. [33] Theorem 1.8.

From the relations deﬁning the ideal I(q), one gets the important identity

x.y = B(x, y) − y.x, for all x, y ∈ V ⊂ C(V ).

In particular, if x ⊥ y in V, then x.y = −y.x in C(V ). It follows that if q ≡ 0 then C(V ) is the exterior tensor algebra V .

Since T (V ) is a graded algebra and I(q) is generated by elements that lie in the even degree components, it follows that C(V )isaZ/2Z-graded algebra. The even Cliﬀord algebra C(V )+ is the subalgebra of C(V ) consisting of the “even” components only.

Deﬁnition 2.2.4. Let (V,q) be a quadratic space with Cliﬀord algebra C(V ). The canonical anti-involution of C(V ) is the map ι : C(V ) → C(V )opp constructed from the universal property of C(V ) applied to the inclusion map V→ C(V )opp.

16 We will forget that ι is a map to the opposite algebra of C(V ) and instead consider it as an anti-homomorphism of C(V ). If {e1,...,en} is an orthogonal basis for V , then ι is deﬁned on the corresponding basis for C(V )by

s(s−1) f1 f1 f1 · · fn fn · · − 2 · · fn ι(e1 ... en )=en ... e1 =( 1) e1 ... en where s = fi and the fi ∈{0, 1}. Suppose now that v ∈ V is an anisotropic vector, so that v ∈ C(V )×.

Conjugation by v induces an automorphism of C(V ), say ψv. Consider the restriction of ψv to the subspace V ⊂ C(V ):

−1 ψv(w)=v · w · v v = v · w · q(v) B(w, v) − v · w = v · ( ) q(v) B(w, v) = −w + v. q(v)

We see from the resulting formula that not only does conjugation by v preserve V, but it actually deﬁnes a well-known map: ψv restricted to V is the composition of reﬂecting in v⊥ and multiplication by −1.

Theorem 2.2.5 (Cartan-Dieudonn´e). Every element of O(V ) can be written as the composition of reﬂections, ie. maps of the form −ψv as above.

Proof. [49] Theorem 3.5.

Theorem 2.2.5 tells us that the Cliﬀord Group

G(V ):={x ∈ C(V )× : x.V.x−1 = V } surjects onto the full automorphism group O(V ) when n is even. To determine the image when n is odd, observe that the center of C(V ) (whose units are the

17 kernel of the map from G(V )toO(V )) is either F or F ⊕ F · e1 ...en (where the ei are an orthogonal basis of V as above) depending on whether n is even or odd respectively ([33] Theorem 2.4). If n is odd, then for any element t of

−1 G(V ), t.e1 ...en.t = e1 ...en. On the other hand, if conjugation by t induced

−1 n the map −1 ∈ O(V ), then t.e1 ...en.t =(−1) ·e1 ...en = −e1 ...en. Clearly this is a contradiction, so −1 does not lie in the image of G(V ). Hence G(V ) is mapped surjectively into SO(V ) when V is of odd dimension.

Definition 2.2.6. The General Spin Group, GSpin(V ), of (V,q) is the intersection of the Clifford Group G(V ) and the even Clifford algebra C(V )+ :

GSpin = {x ∈ C(V )+ : x.V.x−1 = V }.

By the discussion preceding Theorem 2.2.5 we see that GSpin(V ) contains the set of elements of C(V ) that can be expressed as the product of an even number of non-isotropic elements of V . By [49] Theorem 3.7 (i), these two sets are in fact equal. Observe that the inverse of such elements can be expressed in terms of the canonical anti-involution:

−1 −1 ι(v) v =(v1 · ...· vn) = . i q(vi)

It follows that the group GSpin can be expressed as the set

GSpin(V )={x ∈ C(V )+ : x · ι(x) ∈ F ×, x.V.ι(x)=V }.

Deﬁnition 2.2.7. The Spin Group, Spin(V ), of (V,q) is the group

Spin(V ):={x ∈ C(V )+ : x · ι(x)=1, x.V.ι(x)=V }.

18 We see from the remarks above that Spin(V ) ⊂ GSpin(V ) and the quotient is equal to the image of GSpin in F × under the map

v(x)=x · ι(x).

In fact Spin is the derived group of GSpin (See Deﬁnition 3.1.3). As GSpin surjects onto SO(V ) with kernel F × via the conjugation map (denoted ϕ), there is an induced map

τ : SO(V ) → F ×/(F ×)2 deﬁned by sending σ to v(x), where σ is equal to conjugation by x. The map τ is called the spinor norm and elements of spinor norm 1 are exactly those σ such that x can be chosen to lie in Spin. In particular, if F is algebraically closed then Spin(V )isa2to1covering of SO(V ). In any case, we have the exact sequences

ϕ 1 →{±1}→ Spin(V ) → SO(V ) → F ×/(F ×)2 (2.1) ϕ 1 → F × → GSpin(V ) → SO(V ) → 1.

In the sequel, we will be interested in considering Spin(V ) as an algebraic group, ie. an F -scheme with a group structure. For any F -algebra K, we shall denote the K-rational points of Spin(V ) by Spin(V )(K) or simply Spin(K) if V is understood. However, to simplify notation we will use Spin(V ) for Spin(V )(F ) when no confusion will arise. The same remark holds for other F -group schemes (eg. SO(V ), etc.). As algebraic groups, it is often preferable to work with Spin(V ) over O(V ) due to the following theorem.

19 Theorem 2.2.8. Let (V,q) be a quadratic space over a perfect ﬁeld F of dimension n ≥ 3. Then the algebraic group Spin(V ) is simply connected as an F -group.

Proof. It is well-known that the group Spin(Wn) is a simply connected F - group, where (Wn,qn) is the quadratic space

n−1 n 2 2 Wn := H ⊥ 2 or H .

For example, such a group is given by the Chevalley construction ([52]) applied to the root system B n−1 or D n (depending on whether n isoddoreven 2 2 respectively) and is therefore a simply connected, semisimple, F -split group (see Chapter 3 for details on these properties). On the other hand, (V,q)is a Galois twist of (W, qn) over some ﬁnite extension of F and so Spin(V )isa twist of Spin(Wn) over the same ﬁeld. Since Galois twists of simply connected groups are simply connected (a central isogeny over F is trivial if and only if it is trivial after base extension), it follows that Spin(V ) is simply connected.

So far in this section we have been discussing Cliﬀord algebras from the point of view of quadratic spaces over a ﬁeld F . Suppose now that our quadratic space (V,q) comes from an R-lattice L. Then L induces an R-order on C(V ): It is the R-algebra generated by R ⊂ F and L ⊂ V inside C(V ).

Denote this order C(L). If {e1,...,en} is a basis for L, then the set

{ f1 · · fn ∈{ }} e1 ... en : fi 0, 1 is a basis for C(L)asanR-module. Observe that C(L) is closed under ι and the Z/2Z-grading of C(V ) de- scends to one on C(L). Therefore we can deﬁne Spin and GSpin groups on

20 C(L):

GSpin(L):={x ∈ C(L)+ : x · ι(x) ∈ R×, x.L.ι(x)=L},

Spin(L):={x ∈ C(L)+ : x · ι(x)=1, x.L.ι(x)=L}.

Note that since C(L) is an order, Spin(L) = Spin(V ) ∩ C(L). Furthermore, we have the exact sequences:

ϕ 1 →{±1}→ Spin(L) → SO(L), ϕ 1 → R× → GSpin(L) → SO(L).

We shall see later (Theorem 2.4.6) that in the case of unimodular lattices, the exact sequences can be extended to the right as in the case over F .

We conclude this section by producing explicit models for the Cliﬀord algebra and Spin group associated to the R-lattice L0 := H ⊥ H. These structures will be useful to us in Chapter 6.

Theorem 2.2.9. Let L0 := H ⊥ H. There is an isomorphism of algebras ψ

+ between C(L0 ⊗ F ) and M2(F ) × M2(F ) that identiﬁes:

adj adj ι((A1,A2)) = (A1 ,A2 ),

+ 2 C(L0) = M2(R) ,

2 GSpin(L0 ⊗ F )={(A1,A2) ∈ M2(F ) :det(A1) = det(A2) =0 },

2 Spin(L0 ⊗ F )={(A1,A2) ∈ M2(F ) :det(A1) = det(A2)=1}, where Aadj denotes the classical adjoint matrix of A (ie. the transpose of the cofactor matrix).

21 Proof. It suﬃces to prove the theorem for R = Z and F = Q, as the associated structures for Zp and Qp are obtained through base change and hence the same formulas hold. Thus we assume L0 is a Z-lattice throughout the proof. { } Choose a basis e1,e2,e3,e4 of L0 such that the bilinear form B on L0 1 1 is represented by the matrix 1 . We also deﬁne a diagonal basis for 1 L0 ⊗ Q as follows:

E1 = e1 + e4,E2 = e2 + e3,

F1 = e1 − e4,F2 = e2 − e3. There is a well-known isomorphism (see for example [33] Chapter 5, Corollary

2.10) of Q-algebras between the full Cliﬀord algebra C(Q · E1,E2,F1) and

+ the even Cliﬀord algebra C(L0 ⊗ Q) deﬁned by

E1 → E1.F2,

E2 → E2.F2,

F1 → F1.F2.

Therefore we will work with the full Cliﬀord algebra C(Q · E1,E2,F1) and

+ then use the isomorphism above to deduce the desired map on C(L0 ⊗ Q) .

2 The following gives an embedding of Q · E1,E2,F1 into M2(Q) that

2 satisﬁes the condition A = q(A).(I2 × I2):

→ 01 × 0 −1 E1 ( 10) ( −10)=:A1,

→ 10× −10 E2 ( 0 −1 ) ( 01)=:A2,

→ 01 × 0 −1 F1 ( −10) ( 10)=:A3.

By the universal property of Cliﬀord algebras, this map extends uniquely to a

2 map from C(Q · E1,E2,F1)intoM2(Q) . Since the image of the three gen-

2 erators of the Cliﬀord algebra generate M2(Q) , this map is an isomorphism

22 + 2 by dimension considerations. Let ψ : C(L0 ⊗ Q) → M2(Q) be the composition of this map with the earlier isomorphism between C(Q · E1,E2,F1) and

+ C(L0 ⊗ Q) . It remains to show the ψ makes the identiﬁcations described in the theorem.

First observe that ι on the generating set {E1.F2,E2.F2,F1.F2} acts as multiplication by −1 and this property along with the fact that ι is an anti- adj − homomorphism uniquely determines it. On the other hand, Ai = Ai and since sending matrices to their adjoint is also an antihomomorphism, it follows

adj adj that ψ(ι(x)) = ψ(x) . Since A.A = det(A) · I2, the representation of Spin and GSpin induced by ψ is also as claimed. It only remains to calculate the

+ + image of C(L0) under ψ. Observe that C(L0) is spanned by the basis

{1,e1e2,e1e3,e1e4,e2e3,e2e4,e3e4,e1e2e3e4} and therefore it suﬃces to calculate the image of these elements under ψ.For example,

1 ψ(e e )=ψ(− (E + F )F · F (E + F )) 1 2 4 1 1 2 2 2 2 1 = ψ((E1 + F1)F2) · (ψ(E2F2)+ψ(1)) 4 00 0 −1 = × , 00 00 and similar calculations yield the following for the other basis vectors:

0 −1 × 00 10 × −10 ψ(e1e3)=(00) ( 00) ,ψ(e1e4)=(00) ( 00) , 00 × −10 00 × 00 ψ(e2e3)=(01) ( 00) ,ψ(e2e4)=(−10) ( 00) , 00 × 00 00 × 10 ψ(e3e4)=(00) ( −10) ,ψ(e1e2e3e4)= (00) ( 00) ,

10 × 10 ψ(1) = ( 01) ( 01) .

23 2 To see that these vectors span M2(Z) , it suﬃces to show that the inverse map Z ab × a b is deﬁned over . But this map is given by sending ( cd) c d to

−(b e1e2 + be1e3 + c e3e4 + ce2e4)+d · 1+(a − d )e1e4

+(d − d )e2e3 +(a − 3d + a + d)e1e2e3e4.

Theorem 2.2.9 gives us a very clear picture of the structures associated to

L0 that have been discussed in this section. Using the inverse map constructed at the end of the proof, we can consider the conjugation map

ϕ : Spin(L0) → SO(L0)

2 as a map from SL2(R) to SO(L0) that can be extended to the slightly larger group

GSpin(L0)={(A1,A2) ∈ GL2(R) : det(A1) = det(A2)}.

Calculating the map explicitly gives the following formula:

→ ϕ : GSpin(L0) SO(L0) aa −ba −ab −bb ab a b ϕ 1 −ca da cb db ( ) , −→ . (2.2) cd c d − −ac bc ad bd (ad bc) −cc dc cd dd

2.3 Signatures and connectivity

In this section we brieﬂy discuss some of the geometrical considerations that arise from structures associated to quadratic forms. In particular, we consider the real Lie groups Spin(V ) and SO(V ), where V is a quadratic space over R. We also deﬁne the genus of a lattice. References for this section include Nikulin [40] and Serre [47].

24 Over R, every non-degenerate quadratic form is equivalent to exactly one of the following:

r s 2 − 2 qr,s := xk xi+k. (Sylvester’s Law of Inertia) k=1 k=1 If (V,q) is a quadratic space over R we say the signature of q is (r, s)ifq is equivalent to qi,j. The same terminology is used for a Z-lattice or a Q- quadratic space by considering their completions to R. A quadratic form is called positive (resp. negative) definite if its signature is (r, 0) (resp. (0,s)). When an R-quadratic space (V,q) is definite, the Lie group SO(V ) is compact and connected in the real topology, assuming dim(V ) ≥ 2. Compactness is clear by analyzing the restrictions on the matrix coefficients induced by the

T equation M .M = In. Connectedness can be seen in a number of ways, for example since the group is compact, every element is semisimple and hence belongs to a maximal R-torus ([51] Corollary 13.3.8). Such tori are isomorphic to direct products of the circle group and hence are connected.

If q is not deﬁnite, then the picture changes. Suppose q ≡ qr,s with r, s =0.

Let {e1 ...,er} be a set of linearly independent vectors that span a positive deﬁnite subspace W of V . For any σ ∈ SO(V ), the map W → W deﬁned by

πW ◦σ|W is invertible, where πW is the orthogonal projection map from V to W . This induces a continuous map from SO(V )to{±1} which can be shown to be surjective. Thus SO(V ) contains at least 2 connected components in this case. On the other hand, the following lemma shows that the Spin group of an isotropic quadratic form is connected in the real topology. By the exact sequences (2.1), this shows that SO(V ) has exactly 2 connected components. We denote the connected component containing the identity by SO+(V ).

25 Lemma 2.3.1 (Cartan). Let G be a simply connected algebraic group over R that contains a copy of Gm. Then G(R) is connected in the real topology.

Proof. [44] Proposition 7.6.

When we consider the local ﬁeld Qp, we are in a similar situation. Let (V,q) be a non-degenerate Qp quadratic space of dimension at least 3. As in the Case of R, if q is anisotropic then SO(V ) is compact in the analytic topology ([44]

Theorem 3.1). If q is isotropic, then the Qp points of Spin(V ) are connected in the Qp topology (this is an example of the Kneser-Tits conjecture holding, Q× Q× 2 see [44] pg. 409). If p =2, then p /( p ) has 4 elements, and so the image of

Spin(V )inSO(V ) for a Qp-quadratic space is the connected component and has index 4 in the group. If p =2, the index is 8. As in the real case, we let SO+(V ) denote the connected component of SO(V ) in the analytic topology.

Before moving to the next section, we would like to use the newly introduced signature to deﬁne an equivalence relation on Z-lattices. Observe that if L is a Z-lattice, then for every prime p, L induces a Zp-lattice via the tensor product L ⊗ Zp and the unique extension of q. In fact L satisﬁes the equality L = (L ⊗ Q ∩ L ⊗ Zp). p

∼ If L and L share the same dimension and discriminant, then L ⊗ Zp = L ⊗ Zp for almost all p by the Jordan Decomposition Theorem (2.1.18). Indeed, for odd primes not dividing the common discriminant d of the two lattices, by the

theorem we have both L ⊗ Zp and L ⊗ Zp represented by the bilinear form 1 . B = .. . 1 d

26 Deﬁnition 2.3.2. Let L, L be Z-lattices. We say that L lies in the genus

∼ class of L, denoted gen(L), if L ⊗ Zp = L ⊗ Zp for all primes p and L has the same signature as L.

Remark 2.3.3. Observe that isomorphic lattices share the same genus class and so genus class equivalence is a weaker relation than that of isomorphism. In fact genus class is a strictly weaker relation than isomorphism on the set of lattices of any ﬁxed rank greater than 1. We will see an example of this in the next chapter when we show that unimodular lattices of the same signature share the same genus class.

Remark 2.3.4. A Z-lattice L is maximal if and only if every lattice in the same genus class as L is also maximal. Indeed, this follows immediately from the fact that a Z-lattice is maximal if and only if its completion to Zp is maximal for every prime p.

2.4 Unimodular lattices

In this section we flesh out some of the theory we will need regarding unimodular lattices. The theory of such lattices is well understood in comparison to more general lattices. We will prove an embedding theorem that allows us to view an arbitrary lattice as a saturated sublattice of a unimodular one. In this section, we assume that all lattices and sublattices are non-degenerate and defined over R with field of fractions F . References for this section include [40] and [41].

27 Deﬁnition 2.4.1. A sublattice M ⊂ L is called saturated if the quotient group L/M is torsion free. Equivalently, M is saturated if any basis of M can be extended to a basis of L.

Remark 2.4.2. Clearly any direct summand of a lattice L is saturated, but the converse doesn’t hold in general. For example, the line e1 is a saturated

23 sublattice of L =(34) but it is not a direct summand as the determinant of L is −1.

Given any saturated sublattice M ⊂ L, one can consider the orthogonal complement N := M ⊥ ⊂ L. Note that N is also saturated, and N ⊥ = M (recall that both L and M are non-degenerate). The sublattice M ⊥ N ⊂ L is necessarily of maximal rank and so we get the chain of groups

M ∨ ⊥ N ∨ ∼= (M ⊥ N)∨ ⊃ L∨ ⊃ L ⊃ M ⊥ N.

The following result is Proposition 1.6.1 of [40] and is a special case of a more general result on embeddings of non-degenerate saturated sublattices inside arbitrary lattices, see §1.5 of loc. cit.. We provide a modiﬁed proof for this special case as some of the constructions given will be useful in the future.

Proposition 2.4.3. A saturated sublattice (M,qM ) with orthogonal complement (N,qN ) inside a unimodular lattice (L, q) determines an isomorphism → ◦ − of discriminant groups γ :Δ(M) Δ(N) such that qN γ = qM . Con- versely, any such triple (M,N,γ) determines an embedding of (M,qM ) into a unimodular lattice (L, q) with orthogonal complement N.

28 ∨ ∨ ∨ Proof. We have an inclusion of dual groups L = L → M ⊥ N , so let πM

∨ and πN denote the projection of L onto each component. We claim that the map

γ :Δ(M) → Δ(N)

◦ −1 γ(v)=πN πM (v)

is a well defined isomorphism of groups and satisfies the equality qN ◦γ = −qM . ∈ −1 Indeed, for any v M we have πN (πM (v)) = πN (v + N)=N because M and N are the kernels of the maps πM and πN by saturation and the fact that M and N are both non-degenerate. Thus γ is at least well defined on the discriminant groups. On the other hand, it is clearly invertible with inverse ◦ −1 πM πN , which is also well defined by saturation. Finally, since M is saturated ∨ and L is unimodular, Remark 2.1.6 shows that the map πM : L → M is

∨ ∨ surjective. Therefore, if we write u = πM (u)+πN (u) (viewing L ⊂ M ⊥ N ) and observe that

qM (πM (u)) + qN (πN (u)) = q(πM (u)+πN (u)) = q(u) ∈ R,

∨ it follows that the equality qN ◦ γ = −qM ∈ F/R holds on M /M as claimed. This completes one direction of the proof.

Conversely, given such lattices M, N and isomorphism γ, consider the lattice

∨ ∨ L := {(x, y) ∈ M × N : y = γ(x)}⊂(M ⊥ N,qM ⊥ qN ) ⊗ F.

Observe that q = qM ⊥ qN is indeed integral on L because of the condition on γ ◦ − that qN γ = qM . Note that M is saturated in L because it can be extended { }n { }n to a basis by adding in the elements (xi,yi) i=1, where yi i=1 runs through a

∨ ∨ −1 basis of N and xi is any lift to N of γ (yi). To show that L is unimodular,

The following theorem on unimodular lattices leads to an example of non- isomorphic Z-lattices that lie in the same genus class.

Theorem 2.4.4. Let L be an indeﬁnite unimodular Z-lattice. Then L is uniquely determined by its signature (i, j).

Proof. [47] §V, Theorem 5.

Corollary 2.4.5. All positive (or negative) deﬁnite unimodular Z-lattices of the same rank share the same genus class.

Proof. Suppose L and L are both positive (or negative) definite Z-lattices of the same rank. Clearly by definiteness, they are isomorphic over R, hence it suffices to consider the extension of each lattice to a finite prime Zp.By Theorem 2.4.4, H ⊥ L and H ⊥ L are isomorphic. The result now follows from Theorem 2.1.19 and 2.1.21.

The ﬁnal theorem of this section will be used in proving Theorem 2.5.1.

Theorem 2.4.6. Let L be a unimodular R-lattice. Then there exists an exact sequence

1 → ±1→Spin(L) → SO(L) → R×/(R×)2 where the last arrow sends an automorphism of L to its spinor norm in F ×/(F ×)2.

30 Proof. [30] Chapter III, exact sequence (3.2.5) and Chapter IV, Theorem 6.2.6.

Remark 2.4.7. If L is a unimodular lattice that splits a hyperbolic plane, it follows from the previous theorem that GSpin(L) surjects onto SO(L) (consider the spinor norm of the element 1 + λe1.e2, where e1,e2 is the standard basis of isotropic vectors of the hyperbolic summand).

2.5 The image of GSpin(L)

The purpose of this section is to prove Theorem 2.5.1 below. This theorem was ﬁrst brought to the author’s attention in [34]. We use the results of the previous sections to provide a complete proof of this result as much of the notation and language in loc. cit. is in reference to group schemes, as opposed to the traditional group structures we have been dealing with thus far.

Theorem 2.5.1. Let (L, q) be a non-degenerate R-lattice. Then the image of GSpin(L) in SO(L) under the natural map ϕ is the discriminant kernel of SO(L) (see Remark 2.1.9).

Proof. Observe that for a given L, it is easy to construct a lattice K and an isomorphism of discriminant groups γ :Δ(L) → Δ(K) such that the triple (L, K, γ) satisﬁes the conditions in Proposition 2.4.3 (for example, take K to equal L but replace q with −q). Therefore we can always embed L into a unimodular U as in Proposition 2.4.3 so that L is saturated, K = L⊥ ⊂ U and γ relates the two discriminant groups. Without loss of generality on L, we can assume U splits 2 hyperbolic planes. Note that since L is saturated, we have GSpin(L) → GSpin(U) by simply extending a basis of L to one of U.

31 If ϕU : GSpin(U) → SO(U), then we note that for any v ∈ GSpin(L), ϕU (v) acts trivially on K and its restriction to L is equal to ϕ(v).

Claim 2.5.2. The discriminant kernel of SO(L) has a unique extension to SO(U) that acts trivially on K. Conversely any element of SO(U) that acts trivially on K arises in this way.

Proof. Recall from the proof of Proposition 2.4.3 that we have the isomorphism

U ∼= {(x, y) ∈ L∨ × K∨ : y = γ(x)}.

Making this identiﬁcation, given α in the discriminant kernel of SO(L), we know that α also preserves L∨. Thus we can extend α to L∨ × K∨ by deﬁning

∨ it as the identity on K . Denoting this new map by α0, we claim that it preserves U. Indeed if u =(x, y) ∈ U, then

α0(u)=(α(x),y)

≡ (x, y)modL × K

and therefore γ(α(x)) = γ(x)=y.Thusα0 is an automorphism of U as required.

Conversely, if β ∈ SO(U) and acts trivially on K, then for all v ∈ L, β(v) ∈ L as L is the orthogonal complement of K in U.Thusβ restricts to an automorphism of L and if v ∈ L∨, then for some lift w ∈ K∨ of γ(v)we have v ≡ γ(w) ≡ β(v) because β ﬁxes w.

Thus by the claim, ϕ(GSpin(L)) is a subset of the discriminant kernel of SO(L). It remains to show that the map is surjective. From Theorem 2.4.6,

32 it follows immediately that we have the short exact sequence

1 → R× → GSpin(U) → SO(U) → 1 because the composition map GSpin(H ⊥ H) → R×/(R×)2 is surjective (by Theorem 2.2.9). Therefore, all of SO(U) lies in the image of GSpin(U). Hence viewing the discriminant kernel of SO(L) as a subgroup of SO(U) through the extension deﬁned in Claim 2.5.2, we can consider its preimage in Spin(U). We claim that the preimage lies in GSpin(L). This can be seen by passing to the ﬁeld F where clearly any element of GSpin(U ⊗ F ) whose image under

ϕU acts trivially on K ⊗ F must arise from an element of GSpin(L ⊗ F ),

× because over F , both ϕ and ϕU are surjective with kernel F (2.1). But GSpin(U)∩GSpin(L⊗F ) = GSpin(L). Hence the discriminant kernel of SO(L) lies in the image of GSpin(L) and this completes the proof.

Corollary 2.5.3. Let (L, q) be a non-degenerate R-lattice. Then the image of

Spin(L) in SO(L) under the natural map ϕ is ΓL, the discriminant kernel of SO+(L), where SO+(L) denotes the intersection of SO(L) with SO+(L ⊗ R)

+ if R = Z or SO (L ⊗ Qp) if R = Zp.

Proof. From the discussion in Section 2.3, we know that any element of SO+(L) that lies in the image of GSpin(L) must lie in the image of Spin(L). Therefore the result follows immediately from Theorem 2.5.1 above.

33 CHAPTER 3 General Theory of Shimura Varieties

This chapter will serve as an introduction to Shimura varieties, with a particular eye on those arising from orthogonal groups of signature (2,n). We begin with constructing connected Shimura varieties and their compactiﬁca- tions from the ground up by considering Hermitian symmetric domains and their automorphism groups. We will then move towards the ad`elic perspective and general Shimura varieties. Good references for this material include Chapter III of AMRT [1] and Milne [35]. The reader will be referred to these texts for the proofs of most theorems as well as other omitted details. In lieu of detailed proofs, this chapter will provide in-depth examples, particularly those that will be relevant in future chapters. General knowledge of Hodge structures is assumed in sections 5 and 6.

3.1 Reductive and semisimple algebraic groups over perfect ﬁelds

Basic familiarity with algebraic groups is assumed throughout the text. For a general reference see Borel [7] or Springer [51], particularly the second half of [51] for the theory over linear algebraic groups over non-algebraically closed ﬁelds which we will use extensively. This section can be considered a simple review or summary of facts we will need about linear algebraic Q-groups. As mentioned above, the focus will be on understanding the material in the case of orthogonal groups.

We always assume F is a perfect ﬁeld in what follows.

34 Deﬁnition 3.1.1. An algebraic group G deﬁned over F (henceforth an F - group) is connected if it is connected as a scheme in the Zariski topology. The largest connected component of G containing the identity is denoted G0.

It is an easy proof to show that G0 is an open (and closed!) subgroup of G with finite index. The important thing to note is that G0 is always defined over F if G is ([51] Proposition 12.1.1), hence the notion of connectedness is independent of base change. Compare this with an arbitrary scheme which can become disconnected after base change (for example Spec(Q[x]/(x2+1))). Also note that although G0 is defined over F , the remaining components of G ⊗ F need not be. For example, the group of pth roots of unity (Spec(Q[x]/(1−xp))) contains only 2 components over Q when p is a prime, but is isomorphic to the constant group scheme Z/pZ over Q[ζp].

Deﬁnition 3.1.2. Given a connected linear algebraic F -group G we call the maximal closed, connected, normal solvable F -subgroup its radical (R(G)) and the maximal closed, connected, unipotent F -subgroup its unipotent radical

(Ru(G)). G is called semisimple if R(G)={e} and reductive if Ru(G)={e}. If G contains no proper nontrivial normal connected closed subgroups then it is simple.

Note that this deﬁnition makes sense since being closed, connected, normal, solvable or unipotent are all properties of subgroups such that if A, B ≤ G both have them then so does their product AB.

Since Ru(G)=R(G)u, we have the chain of implications: simple ⇒ semisimple ⇒ reductive.

35 Unlike the property of connectedness, being semisimple or reductive need not be preserved under base change in general. However if F is a perfect field then in fact G is reductive if and only if it is over F ([51] Proposition 14.4.5). In general though, the more useful observation is that if G is reductive or semisimple after base change, then it was already semisimple or reductive as an F -group. This is convenient because over an algebraically closed field, semisimple groups are determined up to isogeny by a reduced root system of its Lie algebra and such systems are completely classified. In particular the classical groups like SLn,SOn,Spn, etc. are all semisimple (in fact simple!) and so any F -form of a direct sum of these groups is semisimple as an F -group.

Likewise the groups GLn,GOn,GSpn,...are all reductive. Since these classical linear groups will be of interest to us in future examples, this is something to keep in mind.

Recall that a sequence of linear F -groups 1 → A → B → C → 1is called exact if A → B is injective and B → C is faithfully ﬂat (henceforth: surjective or a quotient map) with kernel A.IfB → C is a surjective map of linear F -groups with kernel A, then every homomorphism from B to C whose kernel contains A factors through C uniquely. Whenever we have an injection of linear F -groups A→ B such that A is normal in B, there is a universal quotient group C making a short exact sequence as above ([36] Chapter 7, §7-8). We denote C by B/A.

Deﬁnition 3.1.3. For any reductive F -group G deﬁne the derived group GDer, of G to be the intersection of all closed normal subgroups N of G such that G/N is commutative.

36 It is easy to see that G/GDer is commutative because GDer must be defined by the intersection of a finite number of closed normal subgroups by the de- scending chain condition on closed subsets of G. Hence GDer is the smallest normal subgroup with this property. GDer respects base change and assuming G is smooth, then for a separably closed field k ⊃ F , GDer(k) is the usual derived group of G(k), ie. the subgroup generated by commutators, and this property characterize GDer. For details see [36] Chapter 16, §3. The key point for us is that the derived group of a reductive group G is semisimple with quotient isomorphic to the torus R(G) ([37]). Relevant examples of such exact sequences include

1 → SLn → GLn → Gm → 1

1 → SOn → GOn → Gm → 1 (3.1)

1 → Spin → GSpin → Gm → 1

Since F is perfect, the center of any F -group is deﬁned over F ([51] 12.1.7). Thus the following makes sense.

Deﬁnition 3.1.4. For F -group G, deﬁne the adjoint group Gad to be the quotient G/Z, where Z is the center of G.

The prototypical adjoint group is PGLn = GLn/Gm. Observe that PGLn

× is isomorphic to PSLn over F despite the fact that SLn(F )/{±1} → GLn(F )/F need not be surjective if F is not algebraically closed.

Deﬁnition 3.1.5. Suppose G is a connected, reductive F -group. If F is any ﬁeld containing F , we say that the F -rank of G is the dimension of any maximal, F -split torus in G(F ).

37 By Theorem 15.2.6 of [51], any two such tori in G(F ) are conjugate by an element of G(F ) and hence this definition makes sense. If the F -rank of G is 0, we say G is anisotropic. There is a theory of root data and F -parabolic subgroups of a connected, reductive F -group G that mimics the more well-known case when F is algebraically closed. Briefly, one fixes a maximal F -split torus T ⊂ G and decomposes the F -rational Lie algebra g by the adjoint action of T : g = g0 + gα α∈Φ(G,T ) where Φ(G, T ) consists of the non-trivial characters of T and gα is the subspace of g where s ∈ T acts by α(s). As in the algebraically closed case, the set Φ(G, T ) is a root system with Weyl group isomorphic to N(T )/Z(T ). Fixing a set of positive roots and then a set of simple roots corresponds to a minimal F - parabolic subgroup of G. Any subset K of simple roots corresponds to an F - parabolic subgroup PK of G containing the minimal parabolic and every such subgroup arises this way. Every F -parabolic subgroup is conjugate to a PK .

Example 3.1.6 (Special orthogonal groups over F ). Let V be an F -rational vector space of dimension m ≥ 3, equipped with a non-degenerate bilinear form q. As shown in [6], the F -rank of SO(V,q) is equal to the dimension of a maximal totally-isotropic subspace of V . Explicitly, suppose W ⊂ V is such a subspace, with dim(W )=n. Since q is non-degenerate, there exists a basis

{ei} for V such that the ﬁrst n basis vectors span W and the bilinear form B associated to q satisﬁes ⎧ ⎪ ⎨ 1ifi + j = 2n+1 B(ei,ej)=⎪ ⎩ 0ifi ≤ 2n, j =2 n +1− i.

38 A maximal F -split torus is then spanned by matrices of the form

−1 −1 diag(t1,...,tn,tn ,...,t1 , 1,...,1).

We remark that when F is a subﬁeld of R,n≤ min{i, j}, where (i, j) is the signature of q.

By analyzing the action of this torus on the Lie algebra g, one gets the eigenspace decomposition g = g0 + gα α∈Φ(G,T ) where the non-trivial characters appearing are either of the form

±1 ±1 α(±i,±j)(t1,...,tn)=ti tj ,i= j or

±1 α(±i)(t1,...,tn)=ti .

The dimension of an eigenspace corresponding to the ﬁrst type of character is always 1, while the dimension of an eigenspace corresponding to the second type of character is equal to m − 2n (and hence does not appear if V ∼=

n H ). The relative Weyl group W is generated by maps that transpose ti

−1 2 with tj or tj . The standard Euclidean metric xi with basis vectors αi therefore deﬁnes a W -invariant form on the root system, and one can easily show that a set of simple roots contains exactly n elements (for example,

∼ n either {αi} when V = H , or {α(1,j),α(1,−2)}). It follows from counting that the parabolic subgroups PK correspond to the stabilizers of elements inside various ﬂag varieties of totally-isotropic F -subspaces of V . Since every F - parabolic subgroup is conjugate to a PK , the same remark holds for all F - parabolic subgroups of SO(V,q). In particular, we see that the maximal proper

39 parabolic subgroups correspond to the stabilizers of isotropic F -subspaces of varying dimensions from 1 to n.

3.2 Connected Shimura varieties

Connected Shimura datum

Deﬁnition 3.2.1. The Deligne torus S is the real algebraic group ResC/R(Gm). A representation for S is the group

x −y 2 2 ( yx): x + y =0 , ∈ C× ∼ S R a −b and we identify a + ib = ( ) with the real matrix ba . The Deligne torus ﬁts into the short exact sequence of linear R-groups

ψ ϕ 1 → Gm −→ S −→ U1 → 1,

where U1 is the 1-dimensional unitary group, represented by matrices

x −y 2 2 U1 := {( yx): x + y =1}.

Here the algebraic maps ψ and ϕ are deﬁned by x2−y2 −2xy z 0 x −y x2+y2 ψ(z)=(0 z ) and ϕ(( yx)) = x2−y2 . 2xy x2+y2

The eﬀect of ϕ on S(R) ∼= C× is to send z = a + ib to z/z, where we identify

U1(R) with the complex points of norm 1.

The following well-known lemma will be useful in deﬁning maps from S and U1 to other algebraic groups over R.

40 Lemma 3.2.2. Every representation of U1(R) as a real Lie group arises from an algebraic representation. If the representation is real, then it arises as an algebraic representation of R-groups

Proof. Since U1 is abelian, every complex representation of it decomposes into 1-dimensional representations. Thus we can reduce to complex representations

× into GL1(C)=C . Since U1(R) is compact and connected, its image must be a compact and connected subgroup of C×. There are only 2 such subgroups, namely the trivial group and U1 itself. Obviously any map to the trivial group is algebraic, and any homomorphism of the circle group sends z to zn for some n ∈ N (this can be seen by considering the map on the universal covering space R which must also be a group homomorphism and send Z to Z). Such a map is clearly algebraic. If the representation is defined over R, then it is easy to show that for every 1-dimensional eigenspace spanned by v in the complexified vector space with eigenvalue zn, v spans an eigenspace for zn. The span of is then defined over R and there is a basis for the underlying real vector space for

abn which z = a + bi acts by ( −ba) .

Deﬁnition 3.2.3. A connected Shimura datum is a pair (G, X+), where G is a semisimple (hence connected) algebraic group over Q and X+ is a Gad(R)+-

ad conjugacy class of homomorphisms h : S → GR satisfying the following three conditions:

1. For h ∈ X+, only the characters 1,z/z, and z/z occur in the adjoint

ad representation of h(S) on Lie(G )C. Equivalently, h = u ◦ ϕ factors

through U1 and only the characters 1, z and z occur in the representation

ad of u on Lie(G )C.

ad 2. Conjugation by h(i)onGR is a Cartan involution (see below).

41 3. Gad has no Q-factor on which the projection of h is trivial.

Remark 3.2.4. Whenever discussing a real or complex linear group or Lie group, the notation G+ will denote the connected component of G as a real or complex manifold as the case may be. When discussing the Zariski topology, we will use G0 to denote the connected component of G.

Remark 3.2.5. A Cartan involution is an involution ϕ on a real Lie group G such that the set

{g ∈ G(C): ϕ(g)=g} is compact. A theorem of Satake states that a Cartan involution of G exists if and only if G is reductive, and any two such are conjugate by an element of G(R) ([46] Chapter I, 4.3). The prototypical example is the involution

t −1 g → g of GLn(C) whose ﬁxed set is the group of unitary matrices Un(C).

Remark 3.2.6. A semisimple Q-group decomposes into an almost direct product

G = G1 ...Gn of its minimal connected normal subgroups, meaning there is a surjective map from the direct product of the Gi to G with a ﬁnite kernel ([25] pg. 167). Each component is simple and they share a ﬁnite intersection

(hence trivial intersection in the adjoint quotient). If h satisﬁes (1) and (2)

ad ad R + above, then the projection of h onto Gi is trivial iﬀ Gi ( ) is a compact group ([35] Lemma 4.7). If none of the projections are trivial then none of the

Gi(R) are compact in which case we say G is of noncompact type.

Before getting into the signiﬁcance of the deﬁnition of connected Shimura varieties, it will be useful to provide some examples of pairs satisfying the

42 required axioms. By the ﬁnal remark above, it is clear that a pair (G, X+)is completely determined by the induced pair on each of its simple factors.

Example 3.2.7. Our ﬁrst example is perhaps the most well-known one. Let Q × G =Spn be the -group of 2n 2n matrices preserving the anti-symmetric

0 −In + form Jn := ( In 0 ). Since X is a conjugacy class, it suffices to define one map h : S → Gad and check conditions (1) and (2) just for h. To do this, define

ad the map from U1 to GR by a · In −b · In u : z −→ =: Mz b · In a · In

2 where (a + bi) = z ∈ U1. Note that this is a well deﬁned map between real

ad + Lie groups U1(R) and Spn(R)/{±1} = G (R) (because square roots are well deﬁned modulo ±1) and by Lemma 3.2.2, this is an algebraic representation of U1. The complex Lie algebra of Spn consists of 2n-square matrices of the form AB M = ,A= −DT ,B = BT ,C = CT . CD with Mz acting by conjugation. In particular it suﬃces to show that only the characters 1,z and z appear in the representation of U1 on M2n arising from conjugation by Mz. The following calculations are an easy, albeit tedious, check: A 0 −1 A 0 − T Mz. .Mz = (A = A ), 0 A 0 A 0 B −1 0 B T Mz. .Mz = (B = B ), − B 0 − B 0 A i.A −1 A i.A T Mz. .Mz = z. (A = A ), i.A − A i.A − A − − A i.A −1 A i.A T Mz. .Mz = z. (A = A ). −i.A − A −i.A − A

43 So clearly condition (1) holds for our map h = u ◦ ϕ. To check condition (2), note that h(i)=Jn and the set of complex matrices M satisfying both J.M.J −1 = M and M T .J.M = M is the group AB∈ C T T T T T T GL2n( ): A .A + B .B =1,B .A = A .B , −B A

2 which is clearly compact, because it is closed and bounded in C4n . Since condition (3) is satisfied trivially as G is simple, it follows that defining X+ as the Gad(R)+-conjugacy classes of h defines a connected Shimura datum.

Example 3.2.8. In our second example, we will consider the orthogonal group G = SO(V,Q) for some rational vector space (V,q) of signature (2,n). Note that this example also encompasses the work for Spin(V,Q) because the Shimura datum only depends on Gad and Spin is a 2 to 1 cover of SO with central isogeny, hence they share the same adjoint group.

Any orthogonal space (V,q) of signature (2,n) has a real basis where the bilinear form of q has matrix representation B := diag(1, 1, −1, −1,...,−1).

Again we deﬁne our representation of S through U1 by mapping

∈ → a −b ⊥ ∈ ad 1 u : z = a + bi U1 ( ba) In =: Nz GR .

We next look at the Lie algebra of GC as a subspace of M2+n(C). Recall A| B − T the matrix ( C| D ) lies in Lie(GC) only if we have the relations A = A ,

T T D = −D , and B = C (with A ∈ M2(C), D ∈ Mn(C),B∈ M2×n(C)). All

1 Here and in the rest of the thesis, given matrices M and N, the notation M ⊥ N M denotes the matrix ( N )

44 such matrices are included in the span of the eigenspaces below: ⎛ ⎞ ⎛ ⎞ 01 01 0 − 0 ⎝ − ⎠ 1 ⎝ − ⎠ Nz. 10 .Nz = 10 , 0 0 0 0

⎛ ⎞ ⎛ ⎞ 00 00 0 0 ⎝ ⎠ −1 ⎝ ⎠ Nz. 00 .Nz = 00 , 0 D 0 D

−1 Nz.M1.Nz = z.M1,

−1 Nz.M−1Nz = z.M−1, where for ∈{±1},M is a matrix of the form ⎛ ⎞ b11 ... b1n 0 ⎜ · i · b11 ... · i · b1n ⎟ ⎜ ⎟ b11 · i · b11 . ⎝ . . ⎠ . . 0

b1n · i · b1n

Therefore condition (1) holds once again. We check condition (2) for our matrix h(i)=−B.Wehave

{ A| B ∈ C − − } ( C| D ) SO(V )( ): A = A, B = B,C = C,D = D which is bounded because to lie in SO(V ), the submatrices must satisfy

T T T T A .A − C .C = I2 and B .B − D .D = −In.

Again condition (3) is satisﬁed trivially, so we conclude that the Gad(R)+- conjugacy class of h deﬁnes a Shimura datum for either SO(V ) or Spin(V ).

Hermitian Symmetric Domains

A Hermitian symmetric domain D is a connected complex manifold with a Hermitian metric g such that the automorphism group of holomorphic isometries Aut(D, g) satisﬁes the following 2 conditions:

45 (i) The group Aut(D, g) acts transitively on D (ie. D is homogenous).

(ii) At each point p ∈ D there exists an involution sp ∈ Aut(D, g) such that p is an isolated ﬁxed point.

The connection between Hermitian symmetric domains and connected Shimura varieties is described as follows.

Proposition 3.2.9. To give a connected Shimura datum is the same as to give

(i) a semisimple algebraic group G over Q of noncompact type, (ii) a Hermitian symmetric domain (D, g), and

(iii) an action of G(R)+ on D deﬁned by a surjective homomorphism G(R)+ → Aut(D, g)+ with compact kernel.

Proof. [35] Proposition 4.8.

Without going into complete details of the proof (see loc. cit. for that), we will describe the equivalence. The algebraic group in condition (i) is the same G in the Shimura datum (G, X+) (see Remark 3.2.6). Fixing a point h ∈ X+, let K ⊂ Gad(R)+ be the stabilizer of h. Then K is compact since conjugation by h(i) is Cartan and the set X+ obtains the structure of a real manifold through the bijective map

Gad(R)+/K → X+ defined by identifying g with the map g.h.g−1. The set X+ further inherits a complex structure with conjugation by u(i) acting as multiplication by i on the tangent space at 1=h (recall h = u ◦ ϕ in Definition 3.2.3). The compactness of K guarantees a positive-definite, K-invariant form on the tangent space at h which is therefore Hermitian. This can then be propagated to the entire space through conjugation.

46 Remark 3.2.10. A Hermitian symmetric domain is called irreducible if it is not isomorphic to a product of two Hermitian symmetric domains of lower dimension. This is equivalent to saying that its group of automorphisms Gad(R)+ is simple.

Let us construct the Hermitian symmetric domains associated to Examples 3.2.7 and 3.2.8 above.

Example 3.2.11. For G =Spn, we consider the Siegel upper half space of genus n, Hn :

t Hn = {Z = X + iY ∈ Mn(C): Z = Z ,Y is positive deﬁnite}.

The group G(R) acts on Hn via the map

AB −1 ( CD).Z =(A.Z + B)(C.Z + D) .

This action is well-deﬁned and transitive (for details see pg. 2 of [38]). The stabilizer of iIn ∈ Hn is the compact subgroup

{ A −B t t t t } ∼ K = ( BA): A .B = B .A, A .A + B .B = In = U(n).

Thus we get an isomorphism as real manifolds

ad + ∼ G(R)/K = G (R) /K = Hn.

We note from Example 3.2.7 that the Lie algebra of K, contained in Lie(G), is precisely the subspace on which the adjoint representation of u acts trivially; therefore, u induces a complex structure on Gad(R)+/K. Moreover, the tangent

47 space at 1ofGad(R)+/K is spanned by the matrices A 0 T 0 B t : A = A ∈ GLn(R) ∪ : B = B ∈ GLn(R) . 0 −A B 0

The map into Hn induces a map into the tangent space at i.In so that A 0 0 B → 2i.A, → 2.B. 0 −A B 0

Since 0 B B 0 u(i)−1. .u(i)= B 0 0 −B and A 0 0 −A u(i)−1. .u(i)= 0 −A −A 0 it follows that the complex structure induced on the tangent space of Gad(R)+/K at 1byu is equal to that induced by the identiﬁcation with Hn. Since the

ad + action of G (R) on Hn is holomorphic, the complex structures agree at every point and the two spaces can be identiﬁed as complex manifolds.

Example 3.2.12. As in Example 3.2.8, suppose G = SO(V ) is the group of proper orthogonal transformations of some rational vector space (V,q)of signature (2,n), and let us further assume that n ≥ 2. Let B denote the bilinear form arising from q. The Hermitian symmetric domain D will be an open subset of the projective variety

D∨ := {z ∈ P(V ⊗ C): q(z)=0}.

More precisely, ﬁx a real basis {x1,...,x4,y1,...,yn−2} of VR such that B is represented by the matrix 1 1 ⊥− B = 1 In−2. 1

48 Then the subset

D := {z ∈ P(V ⊗ C): q(z)=0,B(z,z) > 0}⊂D∨ can be identiﬁed as an open subset inside of Cn via the map 2 t yi t 1:x : x : −x x + : y : ...: y − → (x ,x ,y ,...,y − ) . 2 3 2 3 2 1 n 2 2 3 1 n 2

Remark 3.2.13. For a projective vector z = a + ib ∈ D,q(z) = 0 gives the pair of equalities

q(a)=q(b) and B(a, b)=0, while the inequality B(z,z) > 0 is equivalent to saying

q(a)+q(b) > 0.

Supposing that the coordinate x1 = a1 + ib1 =0, then replacing z with an appropriate scalar multiple, we can assume that x2 is totally real, ie. b2 =0.

2 − (yi) But this implies that q(a)=q(b)= 2 which is a clear contradiction of our inequality. Therefore x1 = 0 making the above map well-deﬁned.

The image of D in Cn contains two isomorphic connected components distin- guished by the sign of (z2). Choosing one (say (z2) > 0), we identify D with this set:

n 2 ∨ D = {(x2,x3,y) ∈ C :2.(x2).(x3) > (yi) , (x2) > 0}⊂D .

The group Gad(R) clearly acts on D, and the subgroup Gad(R)+ is precisely the subgroup that ﬁxes D. As a real manifold with G(R)+-action, D is isomorphic to the Grassmannian of positive deﬁnite planes in VR (denoted Gr(V ⊗ R)). In fact the map from D to Gr(V ⊗ R) sending projective vector

49 z = u + iv to the plane u, v is a 2 to 1 map that identiﬁes each connected component of D with Gr(V ⊗ R) and commutes with the G-action deﬁned on both spaces. In particular this shows the action of Gad(R)+ on D is transitive because it is transitive on the Grassmannian by Witt’s theorem.

2 ∈ − yi For an arbitrary vector x =(x2,x3,y) D, let λx = x2x3 + 2 . Then + the action of matrix M =(mij) ∈ G on x is given by:

⎛ ⎞ m21+x2m22+x3m23+λxm24+ yim 2(i+4) m11+x2m12+x3m13+λxm14+ yim1(i+4) ⎜ ⎟ m31+x2m32+x3m33+λxm34+ yim ⎜ 3(i+4) ⎟ ⎜ m11+x2m12+x3m13+λxm14+ yim1(i+4) ⎟ ⎜ ⎟ m51+x2m52+x3m53+λxm54+ yim M.x = ⎜ 5(i+4) ⎟ . ⎜ m11+x2m12+x3m13+λxm14+ yim1(i+4) ⎟ ⎜ ⎟ . ⎝ . ⎠ m +x2m +x3m +λxm + yim (n+2)1 (n+2)2 (n+2)3 (n +2)4 (n+2)(i+4) m11+x2m12+x3m13+λxm14+ yim1(i+4)

The vector w0 =(i, i, 0,...,0) lies in D and corresponds to the positive deﬁnite

ad + plane x1 + x4,x2 + x3. The stabilizer K of w0 in G (R) is precisely the stabilizer of the plane x1 + x4,x2 + x3 and hence is isomorphic to the group SO(2) × SO(n) (modulo {±1} if n is even). If we assume that the vectors x1 + x4,x2 + x3 correspond to the orthogonal basis vectors of positive norm given in the deﬁnition of u in Example 3.2.8, then u(z) ﬁxes w0 and has matrix representation 1 a−1 bba+1 · −ba−1 a+1 −b ⊥− u(z)= −ba+1 a−1 −b In−2 (z = a + ib). 2 a+1 bba−1

Differentiating the formula for M.w0 above, one gets that the Jacobian of u(i) at w0 is i.In and hence the complex structure of D on the tangent space at w0 identifies with the complex structure on the tangent space at 1ofGad(R)+/K. Since the G+-action on D is holomorphic, it follows that these two spaces are identified as complex manifolds.

Connected Shimura Varieties

50 A connected Shimura variety is constructed as a quotient space of the Hermitian symmetric domain associated to a Shimura datum. Recall that given a faithful representation ρ of a reductive Q-group G into GLV for a finite dimensional Q-vector space V and a maximal rank Z-module L ⊂ V ,an arithmetic subgroup ΓofG is any subgroup that is commensurable with the stabilizer of L in G(Q). This definition does not depend on ρ or L. Fixing a representation and an L, suppose Γ is the arithmetic subgroup equal to the stabilizer of L, and let Γ(N) denote the kernel of the map Γ → Aut(L/N · L). Then Γ(N) is called the principal congruence subgroup of level N and any subgroup containing Γ(N) for some N is called a congruence subgroup. All congruence subgroups arise as the intersection of a compact open K ⊂ G(Af ) with G(Q) ([35] Proposition 4.1). Hence the definition of congruence subgroup is independent of ρ and L even though principal congruence subgroups are not.

Deﬁnition 3.2.14. Let (G, X+) be a connected Shimura datum, and let D be the associated Hermitian symmetric domain (Proposition 3.2.9). The connected Shimura variety associated to an arithmetic group Γ is the quotient

D(Γ) := Γ\D, where Γ is the image of Γ in Aut(D)+.

It turns out that the use of the term “variety” in Shimura varieties is appropriate. Recall that Chow’s Theorem states that every projective complex manifold has a unique structure as a nonsingular projective variety and every holomorphic map between such manifolds is regular as a map of varieties ([48]

§VIII). If Γ is a torsion free arithmetic subgroup of G, then D(Γ) will be a complex manifold. In general, D(Γ) will be a manifold with singularities and will not necessarily be compact, let alone projective. The theory of Baily and Borel ([4]) remedies this situation by constructing a canonical compactiﬁcation

51 D(Γ)∗ of D(Γ) and endows it with an algebraic structure (even when Γ is not torsion free). As a prequel for the next section, we cite the following theorem.

Theorem 3.2.15 (Baily-Borel, [4] Theorem 10.4). The compactiﬁcation D(Γ)∗ carries an analytic structure with which it becomes a normal analytic space.

The normal analytic space D(Γ)∗ can be embedded as a projective normal algebraic variety by means of a set of automorphic forms for Γ, of some suitably high weight.

3.3 The construction and structure of Baily-Borel compactiﬁcations

We will spend this section discussing the aforementioned canonical compacti-

ﬁcation D(Γ)∗ of a Shimura variety D(Γ). As D(Γ) is a quotient space of the Hermitian symmetric domain D, it is perhaps to be expected that D(Γ)∗ is obtained by enlarging the space D to a new space D on which Γ acts and then taking the quotient. Deﬁning the boundary ∂D as the set D \ D, we will be interested in answering the following questions:

1. What is the structure of ∂D and what is the Γ action on it? 2. What is the topology of D(Γ)∗ in a neighbourhood of a point of ∂D?

We first discuss how to find the larger space D. Recall that a Hermitian symmetric domain D can be identified as the quotient of real Lie groups Gad(R)+/K with a certain complex structure. As it turns out, the com- plexification of the maximal compact group K is contained in a complex

ad parabolic subgroup P˜ ⊂ GC and the so-called Harish-Chandra embedding

ad + ad G (R) /K → GC /P˜ is an open holomorphic map. See [21] Chapter 8, §7 for details on this embedding in the general case, however the case of interest

52 where G = SO(2,n) is discussed below in Example 3.3.4. We refer to the

ad ∨ projective complex manifold GC /P˜ as D , the compact dual of D.

Note that D∨ obviously has a transitive G(C)-action on it, which extends the action of Gad(R)+ on D. Since Gad(R)+ preserves D ⊂ D∨, the action must also preserve the topological closure Dc. It turns out that Dc \ D is a disjoint union of Hermitian symmetric domains of strictly smaller dimension than D. Each such domain is referred to as a boundary component and is equal to an equivalence class under a certain relation ∼ on the points of Dc. The equivalence relation ∼ can be roughly2 described as identifying pairs of points p ∼ q if p and q can be connected by a ﬁnite number of holomorphic curves in Dc. The domain D itself is equal to an equivalence class under this deﬁnition, and hence D is often referred to as a boundary component, while the equivalence classes contained in Dc \ D are referred to as proper boundary components when potential confusion may arise.

Given a boundary component F, let P (F ) denote the stabilizer of this equivalence class in Gad(R)+. If F is proper and G is Q-simple, then P (F ) is the intersection of a proper maximal parabolic subgroup of Gad(R) with Gad(R)+ ([1] Proposition 3.6). Conversely, the restriction of every proper maximal parabolic subgroup to Gad(R)+ stabilizes a unique equivalence class ([1] Proposition 3.9). If G is not Q-simple, then the spaces D and Dc decompose into a product corresponding to each Q-simple factor of G, each proper boundary component decomposes into a product of (not necessarily proper)

2 More precisely, p ∼ q if there exists holomorphic maps σi,i∈{1,...,m}, mapping balls c B = {z ∈ C : |z| < 1} into D , such that σ1(0) = p, σm(0) = q and σi(B) ∩ σi+1(B) = φ.

53 boundary components for each factor, and each stabilizer decomposes into the corresponding product of parabolic subgroups.

We say that a boundary component F is rational if there is some rational parabolic subgroup P ⊂ Gad such that P (F )=P (R)∩Gad(R)+. It is important to note that any arithmetic subgroup Γ of G preserves the set of rational boundary components because P (γ · F )=γ.P(F ).γ−1 for any γ ∈ Gad(R)+. This union of all rational boundary components (including the improper boundary component D) is what we shall denote by D and the quotient of this set by Γ is denoted D(Γ)∗. We summarize this discussion with the following diagram:

∨ ∼ c D = GC/P˜ ⊃ D = D Fα ⊃ D = D Fα ⊃ D proper rtnl

π π D(Γ)∗ ⊃ D(Γ).

The association of a rational boundary component F with its stabilizing subgroup P (F ) is convenient as it allows us to understand the structure of the component as well as the structure of D near F. This is done through the so-called 5-factor decomposition of P (F ) which we will now explain. Let F be a proper rational boundary component and let W (F ) denote the unipotent radical of P (F )+. Then we can ﬁrst decompose P (F )+ into the semi-direct product

P (F )+ = P (F )+ W (F )

+ where P (F ) centralizes a certain cocharacter of GR ([1] Theorem 3.10) and is referred to as a Levi factor. The two subgroups split further:

+ W (F )=V (F ) × U(F ),P(F ) = Gh(F ) · Gl(F ) · M(F ).

54 Here U(F ) is the center of W (F ) and V (F ) ∼= W (F )/U(F ) is a vector group. P (F )+ is the direct product modulo ﬁnite intersections of its three factors, the product (Gl(F ) · M(F )) W (F ) is the centralizer of F (ie. acts trivially),

Gh is semi-simple, Gl(F ) is reductive with no compact factors and M(F )is compact ([1] pg. 144).

The signiﬁcance of the two groups Gh(F ) and Gl(F ) is directly linked to questions (1) and (2) above. Gh(F ) modulo its ﬁnite center is isomorphic to Aut(F )+ and it is the only part of P (F )+ that acts non-trivially on F . F being a Hermitian symmetric domain itself implies that F is isomorphic to Gh(F ) modulo some maximal compact subgroup. Furthermore, for any arithmetic

∗ subgroup Γ of G, the image of F in D(Γ) is isomorphic to F/(Γ ∩ Gh(F )).

The group Gl(F ) is also the automorphism group of a structure related to F . Using the fact that Gl(F ) normalizes the unipotent radical W (F ), one constructs an open, self-adjoint homogeneous cone C(F ) inside the vector

∼ + space U(F ) ([1] III.4.2) so that Gl(F ) = Aut (C(F )). The cone C(F )is important in understanding the topology of D(Γ)∗ around F . To explain this, we need the concept of Siegel domains of the third kind.

Deﬁnition 3.3.1.

(i) A Siegel domain of the ﬁrst kind is an open subset D(C)ofCm of the form

{z ∈ Cm : (z) ∈ C}

where C is an open convex cone of Rm (identiﬁed as the imaginary part of Cm).

55 (ii) A Siegel domain of the second kind is an open subset D(C, H)ofCm ×Cn of the form

{(z,w) ∈ Cm × Cn : (z) − H(w, w) ∈ C}

where C is as above and H is an anisotropic Cm-valued Hermitian form such that H(w, w) ∈ C for all w. (iii) A Siegel domain of the third kind is an open subset D(C, H, F, ϕ)of Cm × Cn × Ck of the form

{(z,w,t) ∈ Cm × Cn × Ck : t ∈ F, (z) −(H(w, (I + ϕ(t))−1w)) ∈ C}

where C and H are as above, F is a bounded domain of Ck and ϕ is a holomorphic map from F into a certain subset of the vector space of all complex anti-linear mappings from Cn to Cn (see [27] pg. 238 for the

precise deﬁnition of ϕ).

Remark 3.3.2. As any Hermitian symmetric domain is biholomorphic with a bounded domain ([1] Chapter III, Theorem 2.1). We will use such spaces to construct Siegel domains of the third kind.

Deﬁnition 3.3.3. Let D = D(C, H, F, ϕ) be as in (iii) above. Let O be an open subset of F .Acylindrical set with base O is

−1 Dr(O):={(z,w,t) ∈D: t ∈O, (z) −(H(w, (I + ϕ(t)) w)) − r ∈ C} for some r ∈ C.

For a given rational boundary component F , the group P (F )+ acts transitively on D, and the decomposition of P (F )+ into its ﬁve factors leads to a

56 decomposition of D as a real manifold ([1] III.4.3):

D ∼= C(F ) × W (F ) × F, which is simply the direct product of an open cone, some copies of the real line, and our lower dimensional Hermitian domain F . Further analysis leads to the holomorphic inclusion map

t ∼ ∨ D→ U(F )C × C × F = U(F )C · D ⊂ D

(here t is half the real dimension of V (F ) and in fact there is a natural action of V (F ) on the complex bundle Ct × F realizing each ﬁber as a V (F ) torsor). The image of D under this map is a Siegel domain of the third kind:

∼ t D = {(x, y, z) ∈ U(F )C × C × F : (x) −(Hz(y, y)) ∈ C(F )}.

Here Hz is a family of maps as in the deﬁnition above. This description is used in explaining the topology of D(Γ)∗ around the image of the rational boundary component F . Let E be the set of all proper rational boundary components that contain F in their closure (as a subset of D∨). Modulo the action of Γ, this set is ﬁnite ([1] Chapter III, Theorem

6.2), so let F, F1,...,Fs be distinct representatives. As F lies in the boundary of each Fi, each Fi has a realization as a Siegel domain with respect to F .

Let C(F )i denote the cone associated to F for each Fi. Let x be any point in F . Then as O ranges over a neighborhood basis for x ∈ F , r ∈ C(F ) and

57 s ∈ 3 (ri)i=1 C(F )i, the sets O O F1 O Fs O U = π Dr( ) Dr1 ( ) ... Drs ( ) form a neighbourhood basis around π(x) ∈ D(Γ)∗ ([27] pg. 240). This is known as the Piatetski-Shapiro topology on D(Γ)∗ and is equivalent to the quotient of the Satake topology originally used by Baily and Borel. In particular it is independent of any choice of basis.

Example 3.3.4. Recall from Example 3.2.8 the decomposition of the Lie algebra gC for G = SO(V,q) in terms of eigenspaces for conjugation by u(z):

gC = kC ⊕ p+ ⊕ p−,

where Lie(K)=k, and p+, p− are the eigenspaces where u(z) acts by z and z respectively. The Harish-Chandra embedding theorem ([1] Chapter III, Theo- rem 2.1) states that the map

G/K → GC/(KC · exp(p−)) is a holomorphic open immersion of the associated Hermitian symmetric do-

∨ main D into its compact dual D . Calculating exp(p−) explicitly, one checks that KC · (exp p−) is the stabilizer of the element [1 : i :0:... : 0] inside of {z ∈ P(V ⊗ C): q(z)=0} (using the basis given in Example 3.2.8). This justiﬁes the notation of D∨ used to denote this set in Example 3.2.12 as it is indeed the dual symmetric space. For simplicity, let us assume that n ≥ 2, so

3 DFi The notation ri simply denotes that we are looking at a cylindrical set in the Hermi- tian symmetric domain Fi, which makes sense since F is a boundary component of each of the Fi.

58 that we can change the basis so our notation coincides with that of Example

3.2.12, where the bilinear form B has matrix representation 1 1 ⊥− B = 1 In−2 1 and we write

n 2 ∨ D = {(x2,x3,y) ∈ C :2.(x2).(x3) > (yi) , (x2) > 0}⊂D .

Proposition 3.3.5. The proper boundary components of D consist of two types: closed points and upper half planes H.

Proof. We know from Example 3.1.6 that up to conjugation, Gad(R)+ has precisely two proper maximal parabolics, corresponding to either the stabilizers of real isotropic lines (denoted P1) or real totally isotropic planes (denoted P2) of V . It follows that the boundary of D in D∨ is composed of two distinct types of components, which we refer to as type F1 and F2 respectively. Since G(R)+ acts transitively on boundary components of a given type, all boundary components of a given type are isomorphic.

Looking at the description of D above, it is easy to see that the point (0,...,0) ∈ Cn lies in the topological closure of D. This point corresponds to

∨ the projective vector [1 : 0 : ... :0]∈ D ⊂ P(VC) and so is clearly stabilized by a parabolic of type P1. Therefore it is a boundary component of type F1, and all such boundary components must be points in the set

{z ∈ P(V ⊗ R): q(z)=0}⊂D∨.

59 The F2-boundary components are somewhat less obvious. We ﬁrst deﬁne the set D+ as

D+ := {z =[x + iy] ∈ P(V ⊗ C): q(z)=0, rank(x, y)=2}.

It is easy to check that D+ is a well-deﬁned open subset of D∨. It is composed of 2 connected components, determined by the orientation of the ordered pair

x, y, and so D is contained in one of these components. Next, recall from Example 3.1.6 that D is isomorphic (as a real manifold with G(R)+-action) to the Grassmannian Gr(V ⊗ R) of positive deﬁnite planes in V ⊗ R, by sending the point [a + ib] (written in projective coordinates) to the plane spanned by the real vectors a and b. Let ψ denote this map. Finally, we note that Gr(V ⊗ R) is an open subset of the Grassmannian of all planes in V ⊗ R (say Gr(˜ V ⊗ R)) and that ψ can be extended to a G(R)+-equivariant continuous map between the larger spaces D+ and Gr(˜ V ⊗ R):

D ⊂ D+

ψ ψ Gr(V ⊗ R) ⊂ Gr(˜ V ⊗ R).

where as before, ψ ([x+iy]) = x, y. Since the P2-parabolics are the stabilizers of totally isotropic planes, we are interested in the preimage of such a plane under ψ and how this intersects with the closure of D.

Claim 3.3.6. ψ−1( x, y) for isotropic plane x, y is isomorphic to HH and exactly one copy of the upper half plane lies in Dc ⊂ D∨.

Proof. Since G(R)+ acts transitively on isotropic planes, it is enough to assume

−1 that x, y = e1,e2 in our given basis. One checks that ψ ()is

60 equal to the set

{e1 +(a + ib)e2 ∈ P(V ): b =0 }.

Clearly this set is isomorphic to 2 disjoint copies of the complex upper half plane H. From our aﬃne description of D given above, it is clear that vectors e1 +(a + ib)e2 lie in the closure of D iﬀ b>0.

From the claim, we see that F2-boundary components are copies of the upper half plane, which is a Hermitian symmetric domain corresponding to the group SL2. Explicitly, if v, w is an oriented isotropic plane of V ⊗ R in the

+ G(R) -orbit of the oriented plane e1,e2, then the corresponding boundary component in Dc is the set

{[v + λw] ∈ P(V ): λ ∈ H}⊂D∨.

From the description of the boundary components given in the proof of the proposition, we are also able to explicitly determine which boundary components are rational:

Corollary 3.3.7. A boundary component of type F1 is rational iﬀ it lies in the set

{z ∈ P(V ): q(z)=0}⊂D∨.

A boundary component of type F2 is rational iﬀ the corresponding plane x, y has a basis over Q.

Proof. Immediate from the fact that the rational parabolic subgroups stabilize rational isotropic lines and planes.

61 Next we turn to the structure of the parabolics themselves. Since these groups are all conjugate under G(R), it suﬃces to look at one subgroup of each type. We begin by looking at a representative of P1. Suppose we choose { }{ } 01 ⊥ a basis e1,e2 f1,...,fn so that B is of the form ( 10) B and assume

4 that the norm of f1 is positive . Then a representative of P1 is P e1 , the stabilizer of e1. It is immediately clear that P e1 consists of matrices of the form ⎛ ⎞ ∗∗∗ ... ∗ ⎜ ⎟ ⎜ 0 ∗ ∗ ... ∗ ⎟ ⎜ ⎟ ⎜ ∗ ⎟ M = ⎜ 0 ⎟ ⎝ . . ⎠ . . M 0 ∗ since P e1 stabilizes the vector space spanned by e1. Making use of the equation M T .B.M = B, one deduces that M must be of the form ⎛ ⎞ λm1,2 m1,3 ... m1,n+2 ⎜ ⎟ ⎜ 0 λ−1 0 ... 0 ⎟ ⎜ ⎟ ⎜ 3,2 ⎟ M = ⎜ 0 m ⎟ . . ⎝ . . M ⎠

0 m(n+2),2 for λ ∈ R×,M ∈ SO(B). The unipotent radical of this group is ⎧⎛ ⎞⎫ ⎪ 1 m1,2 m1,3 ... m1,n+2 ⎪ ⎪⎜ ⎟⎪ ⎨⎪⎜ 01 0 ... 0 ⎟⎬⎪ ⎜ ⎟ W (F e1 ):= ⎜ 0 x1 ⎟ ⎪⎜ . . ⎟⎪ ⎪⎝ . . In ⎠⎪ ⎩⎪ ⎭⎪ 0 xn

4 We could once again simply assume that B = 1⊥−In−1, however in the sequel we will be interested in the Q-structure of the rational parabolics, so in order to refer to notation established here, we do not make this assumption.

62 with the xi arbitrary in R and the m1j satisfying

− m1,2 = q( j xjfj), (3.2) − m1,i+2 = B( j xjfj,fi).

This group is commutative and hence equal to its center, while the Levi subgroup ⎧⎛ ⎞⎫ ⎪ λ 0 0 ... 0 ⎪ ⎪⎜ ⎟⎪ ⎨⎪⎜ 0 λ−1 0 ... 0 ⎟⎬⎪ ⎜ ⎟ ⎜ ⎟ ⎪⎜ 00 ⎟⎪ ⎪ . . ⎪ ⎪⎝ . . M ⎠⎪ ⎩⎪ ⎭⎪ 00 has connected component isomorphic to GO(B, R)+. The unipotent radical can be identiﬁed with the vector space spanned by the fi in the obvious manner, by sending the vector x to xi.fi. The orbit of f1 under the action of R + GO(B , ) on W (F e1 ) via conjugation gives rise to the cone

{ ∈ Rn } C(F e1 )= x : B (x, x) > 0,x1 > 0 which is isomorphic under R to the light cone5 , ∈ Rn 2 2 (s1,...,sn) : s1 > s2 + ...+ sn .

We conclude that the 5-factor decomposition for P e1 is of the form

+ · · × P e1 =(Gh(F e1 ) Gl(F e1 ) M(F e1 ) (V (F e1 ) U(F e1 )) =( e·GO(B, R)+ · e) (0 × Rn).

5 See Footnote 4

63 Since F e1 is just a point, and V (F e1 ) is trivial, this leads to a realization of D as a Siegel domain of the ﬁrst kind:

∼ ∈ Cn ∈ D = x : (x) C(F e1 )

= {[−B (x, x):1:x1 : ...: xn] ∈ P(VC):

B ((x), (x)) > 0, (x1) > 0} and the cylindrical sets with respect to [1:0:...: 0] are the domains

D v = {[−B ((x), (x)):1:x1 : ...: xn] ∈ P(VC):

B ((x) − v, (x) − v) > 0, (x1) >v1}

∈ for v C(F e1 ).

It remains to consider the structure of a parabolic subgroup representing

P2. For ease of notation, we will again change our basis for V ⊗ R, written

{e1,...,e4}{f1,...,fn−2}, and assume B has matrix representation 1 1 ⊥ 6 B = 1 B . 1

Then the parabolic subgroup P e1,e2 is a representative of P2. As above, we ⊗ R see that P e1,e2 consists of matrices in SO(V ) of the form ⎛ ⎞ ∗∗∗∗∗ ... ∗ ⎜ ∗∗∗∗∗ ... ∗ ⎟ ⎜ ⎟ ⎜ 00∗∗∗ ... ∗ ⎟ ⎜ ⎟ M = ⎜ 00∗∗∗ ... ∗ ⎟ ⎜ 0 0 ∗ ∗ ⎟ ⎝ . . . . ⎠ . . . . M 0 0 ∗ ∗

6 As in Footnote 4, instead of assuming B = −In−2, we keep our notation general so that it applies in the sequel when we are studying rational parabolics.

64 and using the deﬁning equation M T .B.M = B, this simpliﬁes to ⎛ ⎞ a −bm1,3 m1,4 m1,5 ... m1,n+2

⎜ −cd m2,3 m2,4 m ... m ⎟ ⎜ 2,5 2,n+2 ⎟ ⎜ 00 ab 0 ... 0 ⎟ ⎜ ⎟ M = ⎜ 00 cd 0 ... 0 ⎟ (3.3) ⎜ 0 0 m5,3 m5,4 ⎟ ⎝ . . . . ⎠ . . . . M

0 0 mn+2,3 mn+2,4

ab ∈ R ∈ for ( cd) GL2( ) and M SO(B ). The unipotent radical of this group is ⎧⎛ ⎞⎫ ⎪ 10 m1,3 m1,4 m1,5 ... m1,n+2 ⎪ ⎪ ⎪ ⎪⎜ 01 m z m ... m ⎟⎪ ⎪⎜ 2,3 2,5 2,n+2 ⎟⎪ ⎨⎜ 00 1 0 0 ... 0 ⎟⎬ ⎜ ⎟ W (F )= 00 0 1 0 ... 0 , e1,e2 ⎪⎜ ⎟⎪ ⎪⎜ 00 x1 y1 ⎟⎪ ⎪ . . . . ⎪ ⎪⎝ . . . . ⎠⎪ ⎩⎪ . . . . I ⎭⎪ 00xn−2 yn−2 where the xi,yi and z are arbitrary and the remaining matrix coeﬃcients satisfy

− m1,j+4 = B( i yifi,fj),

− m2,j+4 = B( i xifi,fj),

− (3.4) m2,3 = q( i xifi)

− m1,4 = q( i yifi)

− m1,3 = (z + B( i xifi, j yjfj)).

The center of this group is a 1-dimensional vector space and consists of matrices of the form ⎧⎛ ⎞⎫ ⎪ 10−z 0 0 ... 0 ⎪ ⎪ ⎪ ⎪⎜ 01 0 z 0 ... 0 ⎟⎪ ⎪⎜ ⎟⎪ ⎨⎜ 00 1 00 ... 0 ⎟⎬ ⎜ 00 0 10 ... 0 ⎟ ⎪⎜ ⎟⎪ , ⎪⎜ 00 0 0 ⎟⎪ ⎪ . . . . ⎪ ⎪⎝ . . . . ⎠⎪ ⎩⎪ . . . . I ⎭⎪ 00 0 0 while the quotient of the unipotent radical by its center is isomorphic to R2n−4 and the cosets are determined by the xi and yi values of their representatives.

The Levi subgroup of P e1,e2 has connected component isomorphic to R · R×,+ · 01 ⊥ 01 ⊥ SL2( ) SO(B ) (observe that the matrix ( 10) ( 10) I lies in the Levi subgroup but not in the connected component). The orbit of any

65 non-trivial vector in the center of the unipotent radical under the action of the connected component of the Levi subgroup is isomorphic to the cone

{ ∈ R } C(F e1,e2 )= x : x>0 which clearly has automorphism group isomorphic to R×, + as expected. Thus we conclude that the 5-factor decomposition for P e1,e2 is of the form

+ · · × P e1,e2 =(Gh(F e1,e2 ) Gl(F e1,e2 ) M(F e1,e2 ) (V (F e1,e2 ) U(F e1,e2 ))

×,+ 2n−4 =(SL2(R) · R · SO(B )) (R × R).

Following the methods outlined in [1] III.4, this 5-factor decomposition leads to the representation of D as the Siegel domain of the third kind: ∼ n−2 B (w, w) D (z,w,t) ∈ C × C × H : (z)+ ∈ C(F ) = e1,e2 2 (t) B(w, w) = [−z.t − : z : t :1:w1 : ...: wn−2] ∈ P(VC): 2 B((w), (w)) (t) > 0, (z)(t)+ > 0 2 which is similar to the representation we found back in Example 3.2.12. Now however, we know that this description leads to the cylindrical sets for neigh- borhoods of the boundary component corresponding to P e1,e2 . In the given coordinates, this boundary component can be written as

{ − ∈ P } ∼ F e1,e2 := [ t :1:0:...:0] (VC) = H.

Taking some open set O⊂H and r ∈ R+, we then have B(w, w) Dr(O)= [−z.t − : z : t :1:w1 : ...: wn−2] ∈ P(VC): 2 B((w), (w)) t ∈O, (z)(t)+ >r . 2

66 3.4 General Shimura varieties

In this section we will recall the construction of Shimura varieties arising from a reductive Q-group. Although our main interest in this thesis is in connected Shimura varieties, it is important to understand the more general case because this will allow us to construct morphisms between connected Shimura varieties as well as construct models over subﬁelds of C.

Deﬁnition 3.4.1. A Shimura datum is a pair (G, X) , where G is a reductive algebraic group over Q and X is a G(R)-conjugacy class of homomorphisms h : S → GR satisfying the following three conditions: 1. For h ∈ X, only the characters 1,z/z, and z/z occur in the adjoint

representation of h(S) on Lie(G)C.

ad 2. Conjugation by h(i) induces a Cartan involution on GR . 3. Gad has no Q-factor on which the projection of h is trivial.

The similarities of Deﬁnition 3.2.3 and 3.4.1 are apparent, the main diﬀer- ences being that we are interested in reductive groups instead of semisimple ones and we take the entire G(R)-conjugacy class instead of only a connected component. The following result explains the relationship at a topological level.

Proposition 3.4.2. Let (G, X) be a Shimura datum, and let X+ beacon- nected component of X (topology induced from G(R)). Then X+ can be identified with a Gad(R)+-conjugacy class of maps S → Gad(R) and this identification gives (Gder,X+) the structure of a connected Shimura datum. In particular, X is a finite disjoint union of Hermitian symmetric domains.

Proof. See [35] Proposition 5.7 and Corollary 5.8.

67 Recall that we deﬁned a connected Shimura variety by taking the quotient of the associated Hermitian symmetric domain by an arithmetic subgroup of the associated semisimple group G. We also mentioned that every arithmetic subgroup can be viewed as arising from certain compact open subgroups of

G(AQ). For a general Shimura datum (G, X), it is this ad`elic perspective that we wish to start with, and similar to Proposition 3.4.2 above, what we end up with is a disjoint union of connected Shimura varieties. More precisely, let

K ⊂ G(Af ) be a compact open subgroup. Let G(R)+ denote the subgroup of

ad G(R) that projects to the connected component of G (R) and let G(Q)+ =

G(Q) ∩ G(R)+. Then the group G(Q)+\G(Af )/K is ﬁnite (Lemma 5.12 of [35]). Let C denote a set of representatives. Consider the double coset space

G(Q)\X × G(A)f /K, with action deﬁned by g · (x, t) · k =(gx, gtk). Then the map −1 + (g.K.g ∩ G(Q)+)\X → G(Q)\X × G(Af )/K (3.5) g∈C x → (x, g) is a homeomorphism of topological spaces. Such a double quotient space is called a Shimura variety and denoted ShK (G, X).

Remark 3.4.3. If K ⊃ K are aﬃne opens subgroups of G(Af ) then the inclusion map K → K induces a projection map on the double coset spaces

ShK (G, X) → ShK (G, X) and therefore we get a projective system of complex, quasi-projective varieties {ShK (G, X)}. Furthermore, there is a G(Af )- action on this system where g ∈ G(Af ) sends ShK (G, X)toShg−1.K.g(G, X) via (x, h) → (x, hg).

68 The Baily-Borel compactiﬁcation of ShK (G, X) is just the compactiﬁcation of the individual connected components in (3.5). The projection maps for

ShK (G, X) → ShK (G, X) extend to the associated projective varieties, as does the G(Af )-action of the resulting inverse system ([43] Chapter 6).

Example 3.4.4. Much like when we deﬁned connected Shimura varieties, we will now explain two well-known examples, both pertinent to the remainder of this thesis. We begin with a rational vector space V of dimension 2n, equipped with an alternating form ψ. All alternating forms on V are equivalent up to

0 −In conjugation, so we can assume ψ is represented by the matrix Jn := ( In 0 )as Q in Example 3.2.7. We saw in that example that the -group Spn(ψ) gives rise to a connected Shimura datum. Now we shall construct the general Shimura datum that gives rise to a Shimura variety whose connected components are associated to the aforementioned example. Let G = GSpn(ψ) be the group of symplectic similitudes of ψ. Using the aforementioned basis, we consider the map

h : S → G

a + ib → a · I + b · Jn.

ad Since the composition of this map with the quotient map GR → GR is the same map given in Example 3.2.7, it follows that the G(R)-conjugacy class X of maps deﬁned by h gives rise to a Shimura datum (note that the kernel of the quotient map is the center Z(G)ofG, and therefore h acts trivially on its Lie algebra). The resulting Hermitian symmetric space is Hn Hn. In particular, when n = 1, the space is the complex plane with the real line removed.

69 The importance of symplectic Shimura varieties can be largely ascribed to their interpretation as moduli spaces for complex abelian varieties. This can be brieﬂy described as follows (see §4 of [11] for complete details). By Riemann’s Theorem, there is an equivalence of categories between complex abelian varieties (up to isogeny) and polarizable rational Hodge structures of type {(−1, 0), (0, −1)} that sends an abelian variety A to its ﬁrst coho-

∼ n mology group H1(A, Z) ⊗ Q. Recall that A = C /Λ for maximal rank Z- module Λ, and so H1(A, Z) ∼= Λ, with Hodge structure arising from the fact that Λ ⊗ R = Cn (a Hodge structure of type {(−1, 0), (0, −1)} is equivalent to a complex structure on the associated real vector space). On the other hand, it is easy to check that the space X parameterizes rational Hodge structures of type {(−1, 0), (0, −1)} on VR such that ψ or −ψ is a polarization. By taking a compact subgroup K of G(Af ), one shows that the resulting space ShK (G, X) parameterizes abelian varieties with polarization information induced from ψ, along with certain integrality conditions imposed by K, up to isogeny. In particular, the set G(Q)\X × G(Af )/G(Z) is a coarse moduli space for principally polarized abelian varieties of dimension n such that the induced polarized Hodge structure on the ﬁrst homology group is isomorphic to (V,±ψ).

Example 3.4.5. Recall that earlier in Example 3.2.8 we deﬁned the connected

Shimura datum associated to a rational quadratic space (V,q) of signature

(2,n). Fix some real basis {ei} of V that diagonalizes q and such that q(e1)=

ad + q(e2)=1. In this setting we studied the SO (R) -conjugacy class of the map

h : S → SOad(R) a2−b2 a2+b2 2ab h(a + ib)= 2 2 ⊥ In. − a −b 2ab a2+b2

70 Observe, however, that the map h factors through the projection of SO to SOad (in fact these groups are equal if n is odd). Therefore we can simply view h as a map into SO(R) instead, and clearly it satisfies the three necessary conditions for the conjugacy class SO(R) · h to define a Shimura datum, since the corresponding map into SOad satisfied the conditions for a connected Shimura datum. The stabilizer of h lies in SO+(R) and the group SO(R) has two connected components, projecting disjointly to the two connected components of SOad(R). Therefore the Hermitian symmetric space X can be viewed as 2 copies of the connected Shimura variety X+. Following the discussion of X+ and the Hermitian domain D in examples 3.2.12 and 3.3.4, we see that X can be represented by the space

D = {z ∈ P(V ⊗ C):q(z)=0,B(z,z) > 0}.

A related construction is the Shimura datum associated to the Q-group

GSpin(V ). Here we deﬁne the map

h : S → GSpin(R)

h (a + ib)=a + be1 · e2

for the basis {ei} chosen above. Using the fact that the projection of h to SO is equal to h and that the kernel of the projection is the center of GSpin, we conclude that the GSpin-conjugacy class of h also deﬁnes a Shimura datum. The corresponding Hermitian symmetric space is the same as that of SO because the center of GSpin(R) lies in its connected component.

Example 3.4.6. Suppose (G, X) is a Shimura datum and G is commutative. Since G is reductive it follows that G is a torus and X = {h} is a single point. Furthermore, the three conditions on a map h : S → GR are vacuously

71 true (note that Gad = {e} has no Q-factors at all). Therefore any group homomorphism from S to GR deﬁnes a 0-dimensional Shimura datum. If K is a compact open subset of G(Af ),

ShK (G, X)=G(Q)\G(Af )/K is a finite set. One can generalize this by replacing X = {h} with an arbitrary finite set Y equipped with a transitive G(R)/G(R)+-action. Then we define

ShK (G, Y )=G(Q)\Y × G(Af )/K = (G(Q) ∩ K)\Y

where the union is taken over the representatives of G(Q)\G(Af )/K. Such a set is also referred to as a 0-dimensional Shimura variety.

Deﬁnition 3.4.7. Let (G, X) and (G,X) be Shimura data. A morphism of Shimura varieties is a morphism between the corresponding inverse systems of algebraic varieties that respects the G(Af )-action.

Remark 3.4.8. A Q-map of algebraic groups ϕ : G → G induces a morphism

of Shimura varieties whenever ϕ maps X into X . In this case, if K ⊂ G(Af ),

K ⊂ G (Af ) are compact open subgroups such that ϕ(K) ⊂ K then we get

an obvious induced map from ShK (G, X)toShK (G, X ). It is an easy check that the collection of these maps respects the G(Af )-action and commutes with projections.

Finally, we remark that the induced map from ShK (G, X)toShK (G, X ) extends uniquely to a regular map between the corresponding projective varieties arising from the Baily-Borel compactiﬁcation (see for example [45]).

72 Example 3.4.9. From Example 3.4.5 and the remark above, the projection map π from GSpin(V )toSO(V ) induces a map of Shimura varieties. If ⊂ K GSpin(VAf ) is a compact and open subgroup, then π(K) is also a compact open subgroup of SO(VAf ). Since GSpin(V )toSO(V ) is surjective over

ﬁelds and with respect to unimodular lattices over Zp that split a hyperbolic plane (Remark 2.4.7), it follows that π(GSpin(VAf )) = SO(VAf ). Further- more, the kernel of π acts trivially on the GSpin(VR)-conjugacy class of maps deﬁning the Shimura datum (say X). We conclude that the induced map

ShK (GSpin(V ),X )toShπ(K)(SO(V ),X) is an isomorphism of complex varieties. We will come back to this map in the future as the fact that the algebraic group Spin(V ) is simply connected makes certain calculations in GSpin(V ) easier than in SO(V ).

Recall that the group S is a two-dimensional torus that decomposes over C and therefore splits into the product C× × C×. The identiﬁcation z1+z2 z1−z2 → 2 2i ∈ S (z1,z2) z2−z1 z1+z2 C 2i 2 realizes S(R) ⊂ C× × C× as the set of points {(z,z)}, ie. complex conjugation acts by

(z1,z2)=(z2, z1).

Suppose now that we are given a Shimura datum (G, X). Let hx ∈ X be an arbitrary map. Then hx induces a cocharacter of GC, denoted μx,

μx(z):=hxC(z,1)

and diﬀerent hx induce conjugate maps. It turns out that this conjugacy class of cocharacters always contains a map deﬁned over Q, and the G(Q)-conjugacy

73 class (denoted c(X)) of any such map is independent of choice ([35] Lemma 12.1). The reflex field E(G, X) is the fixed field of the subgroup of Gal(Q/Q) that fixes c(X).

Remark 3.4.10. Let (G, X) and (G,X) be Shimura data. If ϕ : G → G is a Q- map as in Remark 3.4.8, then E(G, X) ⊃ E(G,X). Indeed, ϕ(X) ⊂ X and the Galois action on any appropriate cocharacter in c(X) commutes with ϕ.

Example 3.4.11. We return to the Shimura datum (GSpin(V ),X) given in Example 3.4.5. Recall that even over Q, there still exists a rational basis

β = {ei} of V such that q is diagonalizable with respect to this basis and

q(e1)=x, q(e2)=y; x, y > 0. The map h from Example 3.4.5 is then conjugate to the map

b h (a + ib)=a + √ e .e . β xy 1 2

The induced cocharacter μhβ is the map

1 z − 1 μ (z)= (z +1− i √ e .e ) hβ 2 xy 1 2 √ √ 1 which is clearly deﬁned over Q( −xy). Conjugating by 1 + e1.e3 in −xq(e3) Q Spin( ) sends μhβ to its conjugate map,

1 z − 1 σ(μ )(z)= (z +1+i √ e .e ) hβ 2 xy 1 2 and therefore the reflex field of (GSpin,X)isQ. By composing with the rational map from GSpin to SO(V ), we conclude that the reflex field for the associated orthogonal group is also Q.

74 The relevance of the reﬂex ﬁeld lies in the existence of so called canonical models for Shimura varieties. A canonical model for a Shimura datum

(G, X) is a projective system of E(G, X)-varieties {ZK } parameterized by the open affine subgroups of G(Af ) such that: (1) after base-changing to C, it is isomorphic to the projective system {ShK (G, X)}, and (2) certain rationality conditions are satisfied on so-called special points of X. We will not be too concerned with the exact definition of canonical models but see [43] Chapter 11 for the details. What is of interest to us is that canonical models exist ([43] Theorem 11.18), they are unique ([43] Corollary 11.11), and they extend uniquely to a model of the projective system of Baily-Borel compactifications ([43] Theorem 12.3). Furthermore, maps of Shimura varieties induce maps of their canonical models in the following sense.

Proposition 3.4.12 (Pink). Let (G, X) and (G,X) be Shimura data and

suppose ϕ : G → G induces a morphism of Shimura varieties. Let K ⊂ G(Af )

be a compact open subgroup and K ⊂ G (Af ) a compact open subgroup that

contains ϕ(K). Then the map ShK (G, X) → ShK (G ,X ) arises from a map → ZK ZK , deﬁned after base-changing to E(G, X). The same result holds if we replace each variety with its Baily-Borel compactiﬁcation.

Proof. [43] Proposition 11.10 and 12.3.

Now that we have established that ShK (G, X) has the structure of an E(G, X)-variety that is natural with respect to morphisms and compactifications, we want to take this one step further and ask about the field of definition of the connected Shimura varieties making up the connected components of

ShK (G, X) as a complex variety. Clearly this reduces to studying the Ga- lois action on the set π0(ShK (G, X)) of connected components of ShK (G, X)

75 over C (or equivalently over E(G, X)). Chapter 5 of [35] produces the following structure result on the set π0(ShK (G, X)) :

Proposition 3.4.13. Let (G, X) be a Shimura datum with Gder simply connected. Let T = G/Gder,Zbe the center of G, and T (R)† be the image of Z(R) in T (R) under the projection map v. Then T (R)† contains T (R)+ and so the ﬁnite group Y = T (R)/T (R)† has a transitive T (R)/T (R)+-action. The map

G(Q)\X × G(Af )/K → T (Q)\Y × T (Af )/v(K)

(x, g) → (1,v(g))

induces a bijection between connected components. Therefore π0(ShK (G, X)) has the structure of a 0-dimensional Shimura variety.

The identiﬁcation of π0(ShK (G, X)) as a 0-dimensional Shimura variety with G(Af )-action allows us to explicitly describe the relevant Gal(C/E(G, X))- action through the Artin map artE(G,X) (recall that one deﬁnes the Artin map

A× → ab artE : E Gal(E /E) as the reciprocal of the reciprocity map recE from Class ﬁeld theory). More precisely, suppose that we are in the situation of

Proposition 3.4.13 above. For any x ∈ X, the map h = v ◦ hx is defined over E(G, X) and independent of choice because T is commutative. Therefore, the cocharacter μh of T is defined over E(G, X) and the homomorphism, r(T,μh)(P )= ρ(μh(P )), ρ:E→Q maps E× into T (Q). The action of σ ∈ Gal(C/E(G, X)) on the point (y, a) ∈ Q \ × A ∈ A× T ( ) Y T ( f )/v(K) can then be described as follows. If s E(G,X) satisfies

76 artE(G,X)(s)=σ|E(G,X)ab then

σ((y, a)) = (r(T,μh)(s)∞ · y, r(T,μh)(s)f · a), ∀y ∈ Y,a ∈ T (Af ),

where (r(T,μh)(s)∞,r(T,μh)(s)f ) ∈ T (R)×T (Af )=T (AQ). All of this follows from the precise deﬁnition of special points and canonical models, and is proven in [35] Chapters 11-13.

Example 3.4.14. Once again we return to the Shimura datum associated to

G = GSpin(V ). Recall from Example 3.4.11 that the reflex field associated to (G, X)isQ. Since Gder is simply connected, Proposition 3.4.13 applies and we can use the Artin map artQ to determine the field of definition of a connected component of ShK (G, X) (and hence also for Shπ(K)(SO,X) since the two varieties are E(G, X)-isomorphic by Example 3.4.9 and Proposition 3.4.12). Recall that the Galois group Gal(Qab/Q) is isomorphic to Z×, with a ∈ Z× Q a ≡ ∈ Z Z × acting on [ζN ] by sending ζN to ζN , where a a ( /N ) . Furthermore, this identification determines artQ : artQ factors through the projection

Q×\A× → Q×\{± }×A× ∼ Z× Q 1 f =

× and the action of artQ(a) for a ∈ Z is equal to that given above. For GSpin, we observe that the quotient group T := GSpin / Spin is isomorphic over Q to Gm via the Cliﬀord norm v, as given in the exact sequence

+ of (3.1). The center Z of G is the rational torus Gm · 1 ∈ C (V ) and so the image of Z(R)inT (R) is just T (R)+ = R+,×. Therefore, for any compact ⊂ open subgroup K GSpin(VAf ), we get

Q×\{± }×A× π0(ShK (GSpin(V ),X)) = 1 f /v(K)

= Z×/v(K)

77 where we use v(K) to mean its image in Z× in the second line. The Galois action is then trivial to deﬁne through this identiﬁcation: If σ ∈ Gal(C/Q) restricts to a ∈ Z× ∼= Gal(Qab/Q) then σ acts through multiplication by a.

3.5 An embedding of orthogonal Shimura varieties

In this section we will show that we can view orthogonal Shimura varieties as subvarieties of symplectic Shimura varieties of much higher dimension. Through this process, we will consider an intermediate Shimura variety associated to the subgroup of the even Cliﬀord algebra that is actually of PEL-type, ie. it provides a moduli space for abelian varieties with non-trivial endomorphism rings. In the next chapter, we will study the real structure of these subgroups and in turn, determine the dimensions of their corresponding moduli spaces. References for this section include van Geemen [16] and Deligne [11].

As we have assumed before, let (V,q) be a rational vector space of signature (2,n). The algebra C+(V ) with canonical anti-involution ι is then of dimension 2n+1 as a vector space, and contains the group GSpin(V ), which gives rise to a Shimura datum (GSpin(V ),X) as we have seen before. Recall that the even Cliﬀord algebra C+(V ) acts on itself, say by right multiplication, and so

+ any map h ∈ X deﬁnes a complex structure on the real vector space C (VR)

+ since the square of h(i) ∈ GSpin(VR) ⊂ GL(C (VR)) is −1. We wish now to construct a symplectic form ψ on C+(V ) that is preserved (up to scalars) by GSpin(V ) and such that ±ψ acts as a polarization for one (and hence for any) of the Hodge structures corresponding to h ∈ X. By Chapter 6 of [35], this is equivalent to checking that the GSp-conjugacy class of maps containing h : S → GSpin(V )(R) ⊂ GSp(ψ)(R) satisﬁes the axioms of a Shimura datum.

78 To this end, ﬁx some diagonal basis of V, {ei}, with q(e1),q(e2) > 0. As we have seen in the previous examples, the map h : a + ib → a · 1+b · √ e1.e2 is q(e1)q(e2) an element of X. We also have the Trace map Tr : C+(V ) → Q induced from right multiplication.

Claim 3.5.1. The map

ψ : C+(V ) × C+(V ) → Q

ψ(x, y)=Tr(e1.e2.ι(x).y) is a polarization of the Hodge structure corresponding to h.

Proof. Proposition 5.9 of [16].

The claim incidently proves that ψ is symplectic, as a polarization of an odd weight Hodge structure is alternating. Furthermore, it is trivial to check that GSpin(V ) preserves ψ up to scalars:

ψ(g.x, g.y)=Tr(e1.e2.ι(g.x).g.y)

=Tr(e1.e2.ι(x).ι(g).g.y)

= v(g) · Tr(e1.e2.ι(x).y), where the last equality comes from the fact that v(g)=ι(g).g is a scalar. It follows that we have constructed an embedding of Shimura datum induced from the map GSpin(V ) → GSp(ψ). This realizes (GSpin(V ),X) as a Shimura datum of Hodge type. In particular, for any Shimura variety ShK (GSpin(V ),X),

there exists K ⊂ GSp(Af ) such that ShK (GSpin(V ),X) is a closed subvariety of the Shimura variety associated to K ([11] Proposition 1.15) and therefore

79 parameterizes a certain subset of the 2n-dimensional abelian varieties with polarization determined by ψ and level structure determined by K.

The abelian varieties that lie in the image of ShK (GSpin(V ),X)havean atypical endomorphism structure. Following §4.9 of [11], we replace the standard involution ι on C+(V ) with the conjugate map

ι : C+(V ) → C+(V ) e .e .ι(x).e .e ι(x)= 1 2 2 1 . q(e1)q(e2)

The vector space C+(V ) has a left action by the algebra C+(V )opp induced from multiplication on the right. The map ψ satisﬁes the following property with respect to this action:

ψ(cx,y)=ψ(x, ι(c) y), which is easily proved from the fact that the trace map satisﬁes Tr(A.B)= Tr(B.A) for linear operators A and B. Deﬁne G ⊂ GSp(ψ) as the subgroup preserving this C+(V )opp-action. Note that any subgroup of C+(V )× that acts on C+(V ) through left multiplication and preserves ψ is contained in G,in particular GSpin(V ) ⊂ G. In fact, when we proved that GSpin(V ) preserves ψ above, we only used the fact that ι(g).g is a scalar. Therefore

G ⊃{g ∈ C+(V )× : ι(g).g ∈ Q×}.

Lemma 3.5.2. G = {g ∈ C+(V )× : ι(g).g ∈ Q×}.

Proof. This follows from Lemma 6.5 of [16].

80 The map GSpin(V ) → GSp(ψ) clearly factors through the group G and we get an intermediate Shimura datum determined by G and the G(R)-conjugacy class of any map in X (call this class X0). The following result characterizes the abelian varieties lying in the image of a Shimura variety corresponding to

(G, X0).

Theorem 3.5.3. Given a compact open subgroup K ⊂ G(Af ), the space

G(Q)\X0 ×G(Af )/K classiﬁes the following quadruples (A, τ, p, ηK)/ ∼ where 1. A is a complex abelian variety, 2. p : C+(V )opp → End(A) ⊗ Q such that the trace of p(b) acting on Lie(A)

equals trX (b),

3. ±τ is a polarization of the Hodge structure H1(A, Q) that induces the involution ι (meaning τ(b.x, y)=τ(x, ι(b).y)),

+ opp 4. ηK is a K-orbit of C (V ) ⊗ Af -linear isomorphisms η : V (Af ) → 1 Q ⊗ A× A H (A, ) f sending ψ to an f multiple of τ,

+ opp 5. there exists a C (V ) -linear isomorphism a : H1(A, Q) → V sending τ

∗ to a Q multiple of ψ and the induced Hodge structure on V lies in X0. We say two such quadruples (A, ψ, p, ηK) ∼ (A,ψ,p,ηK) if there is an isogeny A → A commuting with the action of B, sending ηK to ηK and ψ to a Q× multiples of ψ.

Proof. Special case of [11] Scholie 4.11.

81 CHAPTER 4 The Hermitian Symmetric Spaces of Even Cliﬀord Algebra PEL-type Shimura Varieties

We saw in Section 3.5 that associated to any rational quadratic space (V,q)of

+ signature (2,n), there is an algebraic group G := {x ∈ C (V ):ι(x).x ∈ Gm} that gives rise to the PEL-Shimura datum parameterizing abelian varieties of dimension 2n with C+(V )-multiplication (Theorem 3.5.3). In this chapter, we will calculate the isomorphism class of GR so that we can determine various invariants of this moduli space, such as its dimension.

Over R, the Cliﬀord algebra of a vector space is uniquely determined by the signature, so we will use the notation Cr,s to denote the algebra arising

r 2 − r+s 2 { }r+s from the real quadratic form qr,s := i=1 xi j=r+1 xj . Let Ei i=1 denote the generating elements of Cr,s arising from the orthogonal basis {xi} (so that − 2 ≤ Ei.Ej = Ej.Ei when i = j, Ei =1ifi r and is -1 otherwise). Such a set will be referred to as a set of good generators.

As was stated in the proof of Theorem 2.2.9, there is an isomorphism be-

+ tween the full Clifford algebra Cr,s and the even Clifford algebra Cr,s+1 defined by

Ei → Ei.Er+s+1.

+ Note that the canonical anti-involution ι on Cr,s+1 does not agree with the canonical anti-involution on Cr,s under this identiﬁcation. Instead, ι is identi-

ﬁed with the anti-involution w on Cr,s that is uniquely deﬁned by the property − ∼ + R that w(Ei)= Ei,so(Cr,s,w) = (Cr,s+1,ι). It follows that the -group GR,

82 associated to V of signature (2,n) as above, can be identiﬁed with the algebraic group { ∈ ∈ G× } G2,n := x C2,n−1 : w(x).x m,R .

We will construct a representation of C2,n−1 that will allow us to interpret the group G2,n as the group of similitudes preserving a certain form on our representation. The proof comes from adapting the methods used in the paper Hile-Lounesto [22], where the authors inductively construct explicit bases for matrix representations of Cliﬀord algebras over R. The isomorphism types of the algebras Cr,s are broken up depending on the congruency class of s − r mod 8. Therefore we will proceed in a case by case basis depending on this value. The general line of reasoning for each case will be as follows:

1. Start with a low dimensional Cr,s and a given presentation with a set of { }r+s good generators Ei i=1 .

2. Construct the maps ι and w for the given Cr,s and use w to give a

representation of Gr,s+1.

3. Inductively construct a presentation of Cr+1,s+1 with good generators

from a presentation of Cr,s with good generators. Inductively construct

the two involutions on Cr+1,s+1 and give a representation of Gr+1,s+2 from

the associated data on Cr,s.

4. Use isomorphisms between Cr,s and Cr±4k,s∓4k, as well as between Cr,s

and Cs+1,r−1, in order to deduce the structure of G2,n from Gr,s.

Fixing notation for the rest of the chapter, let m = r + s, and for any

ﬁeld k, k(N):=MN (k), the k-algebra of N by N square matrices.

83 4.1 Case 1: G2,n,n ≡ 1, 3mod8.

In order to ﬁnd a representation of G2,n, we will need to consider the corresponding groups Gr,s for other values of r and s. By our isomorphism between + − ≡ Cr,s and Cr,s+1, the groups Gr,s for s r 1, 7 mod 8 can be determined by looking at the algebras Cr,s for s − r ≡ 0, 6 mod 8. As referenced in [22] (or see [33] Chapter V, §4 for this and all future Cliﬀord algebra isomorphisms),

m such algebras are isomorphic to R(2 2 ). As we shall do for all cases, we start by looking at what we need from [22]. The base example we will be interested

∼ R 01 0 −1 in is C1,1 = (2). Here we identify E1 =(10),E2 =(10) which clearly satisfy the necessary conditions to be a set of good generators. As will be a recurring theme in each case, we wish to ﬁnd two matrices g and h such that

−1 T −1 T − g .Eα .g = Eα and h .Eα .h = Eα so that we can identify ι with the map on matrices X → g.XT .g−1 and likewise w with X → h.XT .h−1. We say that the matrices g and h represent ι and w with respect to the basis {Ei}. It is easy to see that the matrices

01 01 g =(10) and h =(−10) satisfy the required conditions for our given E1 and

E2 and therefore represent ι and w respectively. In particular, we see that

∼ { ∈ 01 −1 T 01 ∈ G× } G1,2 = M M2,R :(−10) .M .( −10).M m

which is GSp1.

Assuming a given matrix representation for Cr,s, pages 59-60 of [22] describe how to construct a basis for Cr+1,s+1:

Eα 0 0 I 0 I Fα =( 0 −Eα ),α=1,...,m; Fm+1 =(I 0 ),Fm+2 =(−I 0 ).

84 It is easy to check that this will give a good set of generators (after rearranging

m+1 the indices) for Cr+1,s+1 = R(2 ) if the Eα are themselves good generators

m for Cr,s = R(2 ). We now show how to go from {Eα}-representatives of ι and w on Cr,s to {Fα}-representatives on Cr+1,s+1. One can check directly that if g represents ι and h represents w as above, then gnew and hnew will represent the corresponding maps on Cr+1,s+1 where 0 h 0 g gnew = ,hnew = . h 0 −g 0

Recall that for C1,1 and our given basis, g is a symmetric matrix of signature (1,1) while h is symplectic. We note this because of the following four observations:

(i) If h is symplectic, gnew will be symplectic.

(ii) If h is symmetric of signature (k, k), gnew will be symmetric of signature (2k, 2k).

(iii) If g is symplectic, hnew will be symmetric of signature (k, k).

(iv) If g is symmetric, hnew will be symplectic.

Using our base example G1,2 as a starting point, it follows from these observations that we can describe Gr,r+1 (r ≥ 1) in terms of well-known matrix groups:1

≡ ∼ (1) If m =2r +1 3, 5 mod 8 then Gr,r+1 = GSp2r−1 . (4.1) ∼ r−1 r−1 (2) If m =2r +1≡ 1, 7 mod 8 then Gr,r+1 = GO(2 , 2 ).

1 The notation GO(2r−1, 2r−1) indicates the R-group of orthogonal similitudes preserving the form q2r−1,2r−1 .

85 On page 58 of [22], the authors show how to construct a good generating set of

Cr−4,s+4 from that of Cr,s (assuming r ≥ 4), by deﬁning the following matrices:

Fα = Eα for α =1,...,r− 4 and α = r +1,...,m,

Fα = Er−3Er−2Er−1ErEα for α = r − 3,r− 2,r− 1,r.

Although not used in their paper, note that we can also go in the opposite direction (assuming s ≥ 4) by replacing the generating element Em−i with the new element Fm−i = Em−3Em−2Em−1EmEm−i for i ∈{0,...,3} as above. In either case, note that if ι and w are the usual maps on Cr,s, then

ι(Fα)=Fα when Fα = Eα,ι(Fα)=−Fα otherwise,

w(Fα)=−Fα when Fα = Eα,w(Fα)=Fα otherwise.

Thus we need to modify the ι and w to get the appropriate maps on the algebra

Cr±4,s∓4. We can do this by modifying the construction of the matrices gnew and hnew starting from the step Cr−4,s−4 → Cr−3,s−3. Recall that in each step we are looking at

Eα 0 0 I 0 I Fα =( 0 −Eα ),α=1,...,m; Fm+1 =(I 0 ),Fm+2 =(−I 0 ).

It will suﬃce for our purposes to assume r − 4,s− 4 > 1, which will allow us to use the matrices g and h already calculated for Cr−4,s−4. Now consider matrices g 0 h 0 gnew = ,hnew = , 0 −g 0 h

g 0 h 0 gnew = ,hnew = . 0 g 0 −h Deﬁning ι ,w,ι ,w in the obvious manner, we note that ι(Fα)=ι (Fα) except

at Fm+1 where ι (Fm+1)=−Fm+1 = −ι(Fm+1). A similar observation holds for each of the other three maps. We can now apply this modiﬁed procedure

three more times to get maps on Cr,s (which we again call ι ,...,w ) that diﬀer

86 from our usual maps only at either Er−i or Em−i for i =0,...,3. In any case

it is easy to check that ι (Er−3Er−2Er−1ErEα)=Er−3Er−2Er−1ErEα and so ι is the desired map for Cr−4,s+4. Likewise one can check similar statements for the other maps. Finally, from the construction above it is easy to see that the matrices representing the maps ι,...,w will be orthogonal (or symplectic) if the matrices representing the maps ι and w on Cr−4,s−4 are orthogonal (or symplectic). Since one could easily apply the operations above multiple times

(ie. Cr,s → Cr+4,s−4 → ...→ Cr+4k,s−4k), the corollary to all this is that if we assume r ± 4k ≥ 1 for some k ∈ Z then by (4.1):

≡ ∼ (1) If m =2r +1 3, 5 mod 8 then Gr±4k,r+1∓4k = GSp2r−1 .

∼ r−1 r−1 (2) If m =2r +1≡ 1, 7 mod 8 then Gr±4k,r+1∓4k = GO(2 , 2 ). (4.2) We use one ﬁnal result from [22] before completing this case. The authors show

(pg. 58) that one can go from the matrix representation of Cr,s to Cs+1,r−1 by the following map of matrices

Fα = E1.Er+α for α =1,...,s

Fs+α = E1.Eα for α =2,...,r

Fs+1 = E1.

It is useful to note that the map w on Cr,s gets sent to the corresponding map ∼ on Cs+1,r−1 (ie. w(Fα)=−Fi for all α). It follows that Gr,s = Gs,r.

Now we can conclude our ﬁrst case. If n ≡ 3 mod 8, then n − 3=8.k and

n +1 n +1 (2,n)=( − 4k, +1+4k). 2 2 ∼ ∼ Therefore G2,n = G n+1 − n+1 = GSp n−1 by (4.2). 2 4k, 2 +1+4k 2 2

87 ∼ Alternatively, if n ≡ 1 mod 8, then G2,n = Gn,2 and 2 − n ≡ 1 mod 8. Using the same argument as above and using 1 − n =8k, we see

∼ ∼ ∼ G2,n = Gn,2 = G n+1 − n+1 = GSp n−1 . 2 4k, 2 +1+4k 2 2

4.2 Case 2: G2,n,n≡ 0, 4mod8. We proceed very similarly to the previous case. Again we will be interested in ∼ m−1 Cliﬀord algebras Cr,s with s − r ≡ 1, 5mod8, in which case Cr,s = C(2 2 ). ∼ We start with the base example C0,1 = C, E1 = i. In the previous case we were able to view ι and w as arising from conjugation by matrices plus transposition. Clearly this will not be possible for w since we are in the 1-dimensional case. However we can replace transposition by conjugate transposition which will work: ι : C → C,x→ (1).xT .(1)−1 w : C → C,x→ (1).xT .(1)−1.

× Thus we see that G0,2 is the real algebraic group {z ∈ C : z.z ∈ R } = GU(1).

Now suppose we have a presentation with good generators for Cr,r+1 =

r C(2 ). As in Case 1, we get a presentation for Cr+1,r+2:

Eα 0 0 I 0 I Fα =( 0 −Eα ),α=1,...,m; Fm+1 =(I 0 ),Fm+2 =(−I 0 ).

Again we wish to construct ιnew and wnew. Suppose (as in the case C0,1 above) that we have representative matrices g and h such that

T −1 ι : Cr,r+1 → Cr,r+1,x→ g.x .g

T −1 w : Cr,r+1 → Cr,r+1,x→ h.x .h .

88 Then as in Case 1, we have 0 h 0 g gnew = ,hnew = , h 0 −g 0 and it is not hard to see that we can deﬁne our new maps via

→ → T −1 ιnew : Cr+1,r+2 Cr+1,r+2,x gnew.x .gnew → → T −1 wnew : Cr+1,r+2 Cr+1,r+2,x hnew.x .hnew.

On the other hand, if it is ι that involves conjugate transposition while w only requires transposition as in

T −1 ι : Cr,r+1 → Cr,r+1,x→ g.x .g

T −1 w : Cr,r+1 → Cr,r+1,x→ h.x .h , then we can deﬁne

→ → T −1 ιnew : Cr+1,r+2 Cr+1,r+2,x gnew.x .gnew → → T −1 wnew : Cr+1,r+2 Cr+1,r+2,x hnew.x .hnew.

We see that the matrices h and g will once again oscillate between being symplectic and symmetric in a 4-cycle as in Case 1, with both matrices starting as symmetric for C0,1. This time however, we also have to consider which maps involve transposition versus conjugate transposition. As this pattern repeats every 2 steps, we see that we can partition the groups Gr,r+2 as follows:

∼ (0) G0,2 = GU(1) ≡ ∼ (1) If m =2r +2 4 mod 8 then Gr,r+2 = GResC/R(Sp2r−1 ).

∼ r−1 r−1 (2) If m =2r +2≡ 2, 6mod8(r>0) then Gr,r+2 = GU(2 , 2 ).

∼ r (3) If m =2r +2≡ 0 mod 8 then Gr,r+2 = GResC/R(O(2 )). (4.3)

r Here the notation GResC/R(Sp2r−1 ) and GResC/R(O(2 )) indicates the (real) subgroup of complex similitudes of the appropriate form that scale by a factor

89 in R. The R-group GU(2r−1, 2r−1) is the group of complex similitudes preserving a skew-Hermitian form of dimensional 2r. Note that the two families T for m ≡ 2 or 6 coincide because the group {M ∈ C(2r): M .h.M = h} does not depend on whether h is real and orthogonal of type (k, k) or real and symplectic, as we have the base change identity I −i · I 0 I Ii· I 2i · I 0 . . = . −i · II −I 0 i · II 0 −2i · I

The rest of this case proceeds identically to that of the previous case.

Namely one can go from Cr,r+1 to Cr−4k,r+1+4k (k ∈ Z) and from Cr,s−1 to

Gs−1,r, assuming that r −|4k|≥0. As in Case 1, the map w does not need to be altered when switching between Cr,s−1 and Cs,r−1. When switching between

Cr,s and Cr−4,s+4, the map needs to be altered in the exact same way as before. In particular, the symmetric or symplectic type of the matrices h defining w is the same for Cr,s and Cr−4,s+4 (assuming r −|4k| > 0, otherwise h switches from being symmetric and indefinite to symmetric and definite). The result is ∼ ∼ Gr,r+2 = Gr−4k,r+4k+2 = Gr+4k+2,r−4k assuming r −|4k| > 0.

We conclude this case with the desired results. If n ≡ 4mod8, then n − 4=8k and n n (2,n)=( − 4k, +4k +2). 2 2

n −1 n −1 Therefore G2,n = GU(2 2 , 2 2 ) by (4.3). If n ≡ 0 mod 8 then n =8k and

n n (n, 2)=( +4k, − 4k +2). 2 2

∼ n −1 n −1 Therefore G2,n = Gn,2 is also isomorphic to GU(2 2 , 2 2 ) by (4.3).

90 4.3 Case 3: G2,n,n≡ 5, 7mod8. ∼ m−2 Again we will be looking at Cr,s with s − r ≡ 2, 4, so that Cr,s = H(2 2 ). We ∼ start with the base example C0,2 = H with E1 = i and E2 = j. In order to realize the maps ι and w in terms of conjugation and transposition, note that because H is not commutative, we will have to replace transposition of matrices

T T T by conjugate transposition at all times (because (M1.M2) = M2 .M1 for matrices with values in the division ﬁeld H). For our base example we have

ι : H → H,x→ (ij).xT .(ij)−1 w : H → H,x→ (1).xT .(1)−1.

Therefore g is the 1 by 1 matrix (k)=(ij) and likewise h = (1). We conclude ∼ that G0,3 = {z ∈ H : z.z ∈ Gm} =: GSp(1).

The matrix representation of Cr,r+2 is used to build Cr+1,r+3 in the exact same way as the previous two cases, and likewise, the same deﬁnitions of gnew and hnew will give us appropriate representative matrices for ιnew and wnew. This causes the matrices to oscillate between being conjugate symmetric and conjugate skew-symmetric. The result is as follows:2

∼ (0) G0,3 = GSp(1).

∼ ∗ r (1) If m =2r +3≡ 5, 7 mod 8 then Gr,r+3 = GO (2.2 ).

∼ r−1 r−1 (2) If m =2r +3≡ 1, 3mod8(r>0) then Gr,r+2 = GSp(2 , 2 ). (4.4) Again, the same observations made in the ﬁrst two cases allow one to construct Gr+4k,r+3−4k from Gr,r+3 assuming r ≥|4k| and these groups will

2 The notation GO∗(2.2r) represents the R-group of similitudes scaling a skew-symmetric H-Hermitian form of dimension 2r. Likewise, GSp(2r−1, 2r−1) is used for a symmetric H- Hermitian form of signature (2r−1, 2r−1).

91 be isomorphic if r>|4k|. Unfortunately that will not be good enough for us as we wish to use this case to determine G2,n for n ≡ 7 mod 8 by ﬁrst determining Gn,2. We can still do this by modifying the procedure outlined ∼ in Case 1. First let us return to G0,3, and look at C0,2 = H with E1 = i and

E2 = j. Suppose that 4k = r + 1 and we want to consider a basis of Cr+4k,1 arising from a basis of Cr,r+2 as before. Then the map on Cr,r+2 that is sent to w on Cr+4k,1 needs to be positive on all the generators with negative squares except one. We have seen in the earlier cases that such maps can be “built up” from the lower dimensional algebras. Indeed this is the case here but we need to start with a map on the base example C0,2 that is negative on E1 = i but positive on E2 = j. Such a map is

w : H → H,x→ (i).xT .(i)−1.

Calling h =(i) to match our notation from Case 1, we can build up inductively h 0 starting at C0,2 by deﬁning hnew = 0 −h as in the ﬁrst two cases. Once one gets to Cr,r+2, the map corresponding to h on this algebra will correspond to

w on Cr+4k,1 under the isomorphism described in Case 1. Since the matrix h T satisﬁes h = −h, it follows that the group of matrices preserving it up to an R× scalar will be GO∗.

∼ ∗ q−1 This concludes the third case: If n ≡ 5, 7 mod 8 then G2,n = GO (2.2 2 ).

∗ n−1 − ≡ ∼ ∼ 2 n 1 Indeed when n 5, then G2,n = Gr,r+3 = GO (2.2 ) for r = 2 by (4.4). ∼ ∼ ∗ n−1 When n ≡ 7, then G2,n = Gn,2 = GO (2.2 2 ) by the paragraph above.

4.4 Case 4: G2,n,n≡ 2mod8. The last two cases need to be handled slightly differently than the first three because the method for constructing presentations for larger Clifford algebras

92 from smaller ones given in [22] is diﬀerent for the semi-simple algebras compared with the simple ones of the previous three cases.

For this case we will look at Cr,s with s − r ≡ 7 mod 8 which means ∼ m−1 m−1 m−1 m−1 Cr,s = R(2 2 ) × R(2 2 ). We always view R(2 2 ) × R(2 2 ) as lying

m+1 m+1 R 2 { A 0 ∈ R 2 } diagonally inside (2 ), ie. Cr,s = ( 0 B ) M2( (2 )) . Starting with

10 the base example C1,0, we have E1 =(0 −1 ). The construction of a basis for

Cr+1,s+1 from Cr,s goes as follows: First, every Eα ∈ Cr,s can be written in the form Gα 0 Eα = ,α=1,...,m. 0 Hα

Then generators for Cr+1,s+1 will be deﬁned by ⎛ ⎞

Gα 000 ⎜ ⎟ ⎜ 0 −Gα 00⎟ Fα = ⎜ ⎟ α =1,...,n. ⎝ 00Hα 0 ⎠

000−Hα ⎛ ⎞ ⎛ ⎞ 0 I 00 0 I 00 ⎜ ⎟ ⎜ ⎟ ⎜I 000⎟ ⎜−I 000⎟ Fm+1 = ⎜ ⎟,Fm+2 = ⎜ ⎟. ⎝000I⎠ ⎝ 000I⎠ 00I 0 00−I 0

Before describing how to construct the representing matrix h for the map w we want to make a few observations.

(1) From the paragraph above and the base example C1,0, it is clear that if Gα 0 we write Eα = 0 Hα , then Gα = Hα for all α EXCEPT one, which we

shall always refer to as E1. In this case G1 = −H1.

m+1 h0 0 ∈ R 2 (2) Consider a matrix of the form h =(0 h0 ) (2 ). Then by (1), h T −1 T −1 − represents the map w (ie. w(M)=h.M .h )iﬀh0.Gj .h0 = Gα for all α.

93 m+1 0 h0 ∈ R 2 (3) Consider a matrix of the form h =(h0 0 ) (2 ). Then by (1), h T −1 T −1 − represents the map w iﬀ (i) h0.G1 .h0 = G1, and (ii) h0.Gα .h0 = Gα for all α>1.

(4) If we are not in the base example of C1,0, we may order the Eα such that

Gα 0 G = ,α=1,...m− 2, α 0 −Gα 0 I 0 I Gm−1 =(I 0 ) ,Gm = −I 0 .

where the Gα are the Gα occurring in Cr−1,s−1. −1 (5) The map G → 0 h1 .GT . 0 h1 ﬁxes ( 0 I ) , 0 I and it sends α h1 0 α h1 0 I 0 −I 0 − T −1 Gα 0 h1.Gα .h1 0 ≤ − − to T −1 for α m 2. 0 Gα 0 h1.Gα .h1 −1 (6) The map G → 0 h1 .GT . 0 h1 sends ( 0 I ) , 0 I to their neg- α −h 1 0 α −h1 0 I 0 −I 0 − T −1 Gα 0 h1.Gα .h1 0 ≤ − atives and − to T −1 for all α m 2. 0 Gα 0 h1.Gα .h1

h0 0 Observations (2) and (3) give us conditions for a matrix of the form ( 0 h0 )or

0 h0 ( h0 0 ) to represent w. In the base example, G1 = 1 and so only the properties of (3) could possibly be satisﬁed. In particular, we see that the group G1,1 is equal to

{ a 0 ∈ a 0 T 01 a 0 G · 01 } ∼ G 2 ( 0 b ) M2,R :(0 b ) .( 10).( 0 b )= m ( 10) = ( m) .

If we are not in the base example, then using observations (4) and (5), we see 0 h1 that h0 cannot be of the form h1 0 . However, if we assume h0 is of the 0 h1 form −h1 0 , then by (6) we have reduced the conditions of (2) and (3) to a smaller set of conditions on h1 : T −1 ≤ − (2) holds if h1.Gj .h1 = Gj for all j m 2, T −1 − T −1 ≤ (3) holds if (i) h1.G1 .h1 = G1, and (ii) h1.Gj .h1 = Gj for 1

94 For example, in C2,1 there is only one Gi, namely G1 = 1 and so only condition

(2) could possibly hold under our assumption on h0 by choosing h1 =1.In particular, h0 is a symplectic 2 by 2 matrix and

{ A 0 ∈ A 0 T h0 0 A 0 G · h0 0 } G2,2 = ( 0 B ) M2,R :(0 B ) .( 0 h0 ).( 0 B )= m ( 0 h0 )

{ ∈ 2 T 01 T 01 } = (M1,M2) (GSp1) : M1 .( −10).M1 = M2 .( −10).M2

2 =: G(Sp1).

We observe that this argument can be generalized inductively: Suppose the

m−1 −x conditions on h have been reduced to conditions on a matrix hx in R(2 2 ) of the form T −1 − x+1 ≤ ≤ − (2) holds if hx.Gα .hx =( 1) Gα for all 1 α m 2x, T −1 − x T −1 − x+1 (3) holds if (i) hx.G1 .hx =( 1) G1, and (ii) hx.Gα .hx =( 1) Gα for all 1 <α≤ m − 2x. If (r − x, s − x)=(1, 0), then (2) will hold if x is odd and (3) will hold if x is

0 hx+1 x even. Otherwise, using observations (5) and (6), writing hx =((−1) hx+1 0 ) will reduce the conditions on h to the analogous conditions on hx+1 as above.

When condition (3) holds, the matrix type (symmetric or symplectic) of h does not matter. However when condition (2) holds, the type does matter, and can be determined by checking the type of hx at each step. The results are as follows.

∼ (1) If m =2r +2≡ 2, 6 mod 8 then Gr+1,r+1 = Gm × GL2r . ≡ ∼ 2 (4.5) (2) If m =2r +2 4 mod 8 then Gr+1,r+1 = G(Sp2r−1 ).

∼ r−1 r−1 2 (3) If m =2r +2≡ 0 mod 8 then Gr+1,r+1 = G(O(2 , 2 ) ).

95 r−1 r−1 2 2 2 The deﬁnitions of G(O(2 , 2 ) ) and G(Sp2r−1 ) are analogous to G(Sp1) above, namely elements consist of pairs of matrices that scale the appropriate form but share the same similitude factor.

As in the previous three cases, we must also consider how to go from

Gr+1,r+1 to Gr+1−4k,r+1+4k (we can assume k is positive and r − 4k ≥ 0). In this case, we modify h by replacing h0,...,h4k−1 with the matrices

hx+1 0 h := . x 0 hx+1

To check this, simply note that transposition plus conjugation by h sends ( 0 I ) x I 0 T −1 0 I Gα 0 hx+1.Gα .hx+1 0 to itself, −I 0 to its negative and 0 −Gα to T −1 . 0 hx+1.Gα .hx+1 ≥ The remaining hx for x 4k will have the same form as hx and the form of h in terms of condition (2) or (3) does not change because 4k is even. The type (symmetric or symplectic) of h also does not change. Therefore ∼ Gr+1,r+1 = Gr+1−4k,r+1+4k assuming r − 4k ≥ 0.

∼ Since G2,n = Gn,2 does not yield any new information, the result is as ∼ ∼ follows: If n ≡ 2 mod 8 then G2,n = G2+4k,n−4k = G(Sp n−2 × Sp n−2 )by 2 2 2 2 (4.5), where n − 2=8k.

4.5 Case 5: G2,n,n≡ 6mod8.

Our ﬁnal case is very similar to Case 4. This time we are looking at Cr,s

m−3 m−3 for s − r ≡ 3 mod 8. Such algebras are isomorphic to H(2 2 ) × H(2 2 )

m−1 which we again view as living diagonally inside H(2 2 ). The base example is ∼ C0,3 = H × H with generators

i 0 j 0 k 0 E1 =(0 −i ),E2 =(0 −j ),E3 =(0 −k ).

96 The construction of generators for Cr+1,s+1 from Cr,s proceeds identically from that of the previous case, namely ⎛ ⎞

Gα 000 ⎜ ⎟ ⎜ 0 −Gα 00⎟ Fα = ⎜ ⎟,α=1,...,m. ⎝ 00Hα 0 ⎠

000−Hα ⎛ ⎞ ⎛ ⎞ 0 I 00 0 I 00 ⎜ ⎟ ⎜ ⎟ ⎜I 000⎟ ⎜−I 000⎟ Fm+1 = ⎜ ⎟,Fm+1 = ⎜ ⎟. ⎝000I⎠ ⎝ 000I⎠ 00I 0 00−I 0

We start with the following observations, similar to the previous case.

(1) From the paragraph above and the base case C0,3, it is clear that writing Gα 0 Eα = 0 Hα ,Gα = Hα for all α EXCEPT three, which we shall always

refer to as E1,E2,E3. For these three cases, Gα = −Hα.

m−3 m−3 h0 0 ∈ R 2 ⊂ H 2 (2) Write h =(0 h0 ), with h0 (2 ) (2 ). Then the map w is → T −1 T −1 − represented by M h.M .h iﬀ h0.Gα .h0 = Gα for all α. m−3 m−3 0 h0 ∈ R 2 ⊂ H 2 (3) Write h =(h0 0 ), with h0 (2 ) (2 ). Then the map w is → T −1 T −1 represented by M h.M .h iﬀ (i) h0.Gα .h0 = Gα, α =1, 2, 3 and T −1 − (ii) h0.Gα .h0 = Gα for all α>3.

(4) If we are not in the base case of C0,3, we may order the Eα such that

Gα 0 G = ,α=1,...m− 2 α 0 −Gα 0 I 0 I Gm−1 =(I 0 ) ,Gm = −I 0 .

T −1 (5) The map M → 0 h1 .M . 0 h1 ﬁxes ( 0 I ) , 0 I and it sends h1 0 h1 0 I 0 −I 0 T −1 Gα 0 −h1.Gα .h1 0 to T . 0 −Gα −1 0 h1.Gα .h1 T −1 (6) The map M → 0 h1 .M . 0 h1 sends ( 0 I ) , 0 I to their neg- −h1 0 −h1 0 I 0 −I 0 T −1 Gα 0 −h1.Gα .h1 0 atives and to T . 0 −Gα −1 0 h1.Gα .h1

97 We observe that for the base example C0,3, the conditions in (3) could not possibly hold because if x.i.x−1 = −i and x.j.x−1 = −j, then x commutes with k. On the other hand, the conditions of (2) hold if we choose h0 equal T ∼ { a 0 ∈ a 0 10 a 0 ∈ G× } to 1, and doing so gives us G0,4 = ( 0 b ) M2,H : ( 0 b ) .( 01).( 0 b ) m,R ,

2 which is isomorphic to the group of Gm ×SU(2) /{±1}. Given arbitrary Cr,r+3, the argument for constructing h is identical to that in Case 4. The resulting groups are as follows.

∼ 2 0) G0,4 = Gm × SU(2) /{±1} ∼ 1) If m =2r +4≡ 2, 6 mod 8 then Gr,r+4 = Gm × ResH/R(GL2r ).

∼ ∗ r 2 2) If m =2r +4≡ 0 mod 8 then Gr,r+4 = G(O (2.2 ) ).

∼ r−1 r−1 2 3) If m =2r +4≡ 4mod8(r>0) then Gr,r+4 = G(Sp(2 , 2 ) ). (4.6)

The deﬁnitions of G(O∗(2.2r)2) and G(Sp(2r−1, 2r−1)2) are analogous to those from Case 4, namely pairs of elements that scale the appropriate H-form by the same similitude factor.

∼ As in the previous case, Gr,r+4 = Gr−4k,r+4+4k as long as r>4k, which we show by modifying the hx for the ﬁrst 4k steps in the exact same way as Case 4, ie.

hx+1 0 h := . x 0 hx+1

If r − 4k>0, this will not change the symmetry type of h and so we get the desired isomorphism. If r =4k then in fact h is a positive deﬁnite matrix and

× r 2 so G0,r+4+4k is R · UH(2 ) , similar to the base example G0,4. The conclusion

∼ ∗ n −1 2 regarding G2,n is as follows: If n ≡ 6 mod 8 then G2,n = G(O (2.2 2 ) )by setting n − 6 ≡ 8k and applying (4.6).

98 The following table summarizes the information we have discussed throughout this chapter. The second column lists the isomorphism types of the corresponding Cliﬀord algebras, as given in [33]. The third column gives the isomorphism classes of G2,n as we have determined above and ﬁnally the fourth column includes the dimension of the corresponding symmetric space Gder(R)/K (see for example, [21]) where K is a maximal compact subgroup of Gder(R).

Table 4–1: Classiﬁcation of G2,n over R

+ der n (8) C (VR) G2,n dimR(G (R)/K)

n n − n − − 0 C(2 2 ) GU(2 2 1, 2 2 1) 2n 1

n+1 n−1 n−1 1 R(2 2 ) GSp n−1 2 2 (2 2 +1) 2 2 n n 2 n n −1 2 R(2 2 ) × R(2 2 ) G(Sp n−2 ) 2 2 (2 2 +1) 2 2 n+1 n−1 n−1 3 R(2 2 ) GSp n−1 2 2 (2 2 +1) 2 2 n n − n − − 4 C(2 2 ) GU(2 2 1, 2 2 1) 2n 1

n−1 ∗ n−1 n−1 n−1 5 H(2 2 ) GO (2.2 2 ) 2 2 (2 2 − 1)

n − n − ∗ n − n n − 6 H(2 2 1) × H(2 2 1) G(O (2.2 2 1)2) 2 2 (2 2 1 − 1)

n−1 ∗ n−1 n−1 n−1 7 H(2 2 ) GO (2.2 2 ) 2 2 (2 2 − 1)

We remark from the table that given a rational quadratic space (V,q) of signature (2,n), if n =1, 2, 3, then the group G must be equal to GSpin(V ), since they are equal over R. When n =4,Gder = GSpin(V )der = Spin by dimension considerations, but the center of G is strictly larger than the center of GSpin(V ), as G contains the entire 2-dimensional center of C+(V ), while Z(GSpin(V ))0 is 1-dimensional.

99 Remark 4.5.1. Observe that for n ≡ 1, 5mod8, the dimension of a Shimura variety associated to G2,n is precisely half the dimension of the variety associated to G2,n+1. Suppose that (V,q) is a rational quadratic space of signature (2,n+ 1) and U ⊂ V is a subspace of signature (2,n). The inclusion of the Q-groups G(U) → G(V ) induces an embedding of PEL-Shimura varieties that gives a closed cycle of precisely half the dimension of the larger space. In the case when n = 1, this has been studied in great detail in the context of special divisors (see Chapter 7) on Hilbert modular surfaces, the so-called Hirzebruch-Zagier divisors. In particular, the intersection pairings of these cycles have deep arithmetic meaning, including representing special zeta function values as well as the coeﬃcients of modular forms [23]. The commutative square / G(OU) G(VO )

GSpin(U) /GSpin(V ) of Q-groups induces a commutative diagram of Shimura datum, which for appropriate choices of level structure consists entirely of embeddings of closed subvarieties. In the case when n =1, the diagram degenerates since GSpin(V )= G(V ) in this case. However, in higher dimensions, it is interesting to note that the intersection of two cycles corresponding to G(U) and G(U ) ⊂ G(V ) will never be proper, since it will contain the intersection of the special divisors corresponding to GSpin(U) and GSpin(U ) inside GSpin(V ). Nevertheless, the intersection number of these two cycles must reﬂect data derived entirely from information pertaining to the two quadratic space U and V , and is a natural generalization of the intersection pairing in the Hilbert modular surfaces case.

100 CHAPTER 5 Counting Cusps for Maximal Lattices

In this chapter we will narrow our focus to the main topic of the thesis, which is discussing the geometry of the boundary of orthogonal Shimura varieties in the Baily-Borel compactiﬁcation. Using results discussed in Chapters 2 and 3, we will be able to completely determine the composition of the boundary for Shimura varieties that arise from maximal lattices that split two hyperbolic planes. We then consider some examples that our theory applies to, including the special case of unimodular lattices which was proved in Kemp [26]. Many of these results are contained in the author’s previous publication [2]. The chapter concludes with a discussion on scaled lattices.

5.1 Counting the orbits of isotropic lines

For this section, we always assume that (L, q)isaZ-lattice of signature (2,n) that splits 2 hyperbolic planes. This assumption is not particularly restrictive; see Remark 5.2.13. Thus we may write L ∼= H ⊥ H ⊥ A for a negative deﬁnite sublattice A. As deﬁned in Remark 2.1.9, ΓL will denote the discriminant

+ kernel of the group SO (L). Fix such an A, and bases e1,e2,f1,f2 for the two hyperbolic planes such that q(ei)=q(fj)=B(ei,fj) = 0. By Example 3.3.4, we know that the ΓL-orbits of isotropic lines in L ⊗ Q correspond bijectively to the 0-dimensional cusps of the ΓL-Shimura variety. Therefore we will study these orbits to draw our desired conclusion.

For given u, v ∈ L with q(u)=B(u, v) = 0, we call the automorphism of L corresponding to 1 + uv ∈ Spin(L) the Eichler transformation E(u, v). This

101 can be written explicitly as

E(u, v)(a)=a − B(a, u)v + B(a, v)u − q(v)B(a, u)u.

By Corollary 2.5.3, E(u, v) lies in ΓL. Let EO(L) be the subgroup of ΓL generated by Eichler transformations.

Lemma 5.1.1. Two primitive vectors a, b ∈ L∨ such that q(a)=q(b) are contained in the same orbit of EO(L) if and only if a = b ∈ Δ(L).

Proof. [15] Lemma 4.4.

Since EO(L) ⊆ ΓL and ΓL acts trivially on Δ(L), Lemma 5.1.1 gives the best possible result for ΓL-orbits of isotropic vectors, and hence also the best result for the orbits of the lines they generate. Since every isotropic line Q · v is generated by a primitive vector v∗ ∈ L∨ and v∗ is unique up to sign, we conclude the following.

Corollary 5.1.2. Let (L, q) be as above. Then the number of 0-dimensional cusps in the associated ΓL-Shimura variety is equal to the cardinality of the set

{x ∈ Δ(L): q(x)=0}/{x ∼−x}.

Proof. It only remains to show that given any x ∈ Δ(L) with q(x)=0,x lifts to a primitive vector in L∨ of norm 0. Since Δ(L) ∼= Δ(A), we can ﬁrst lift x

∨ ∨ to an element y of A ⊂ L . Then q(y) ∈ Z,soy := e1 − q(y)e2 + y is also in L∨. But now y is isotropic, primitive and lifts x.

Corollary 5.1.3. Let (L, q) be as above and assume L is maximal. Then there is exactly one 0-dimensional cusp.

102 Proof. When L is maximal, the set {x ∈ Δ(L): q(x)=0} is {0}.

Corollary 5.1.4. Let (L, q) be as above and assume Δ(L) is cyclic of order N.IfN is odd, consider the decomposition N = S2 · M for M squarefree. If N is even, consider the decomposition N =2S2 · M for M squarefree. Then the number of 0-dimensional cusps is

S S +1 +1or 2 2 if S is even or odd respectively.

Proof. Let ϕ denote Euler’s totient function. We ﬁrst claim that if N is even, then the number of 0-dimensional cusps is ϕ(s) , 2 2s2|N and if N is odd, then the number of 0-dimensional cusps is ϕ(s) . 2 s2|N

Suppose the claim holds. Then in either case, the formula can be rewritten as ϕ(s) . 2 s|S

It is a well-known formula that s|S ϕ(s)=S. On the other hand, ϕ(s)isodd if and only if s = 1 or 2. Therefore ϕ(s) S = +1 2 2 s|S if S is even and ϕ(s) S +1 = 2 2 s|S if S is odd. Therefore it suﬃces to prove the claim.

103 To this end, we observe that since Δ(L) is cyclic of order N, it is therefore generated by some x ∈ L∨. Since B(x, N · x)=2N · q(x) ∈ Z, it follows that a ∈ Z | · q(x)= 2N for some a such that 2 a N and GCD(a, N) = 1 (the condition on the GCD comes from the fact that |Δ(L)| = N). If s is a divisor of N such that 2s2|aN, then N aN 2 q( · x)= ∈ Z s 2s2N N · ∈ and therefore s x Δ(L) corresponds to a 0-dimensional cusp. In fact, for each such s, we can consider the sets

Xs = {1 ≤ r ≤ s − 1 : GCD(r, s)=1}/{r ∼ s − r} (X1 = {1}).

Then each element of Xs corresponds to a distinct 0-dimensional cusp under Nr· ∈ the identiﬁcation of Corollary 5.1.2 by sending r to s x Δ(L). Moreover, all cusps are realized in this way. Indeed, if q(k · x) ∈ Z for some natural number

2 2 N aN a GCD(k,N) 0

Remark 5.1.5. The two preceding corollaries combine to show that when L is maximal and Δ(L) is cyclic, |Δ(L)| must be free of odd squares and not divisible by 8. In fact even when Δ(L) is not cyclic, the discriminant group of a maximal lattice L is quite restricted, see Remark 5.2.14.

Remark 5.1.6. We observed in the proof of Corollary 5.1.4 that if Δ(L)is ∈ ∨ a cyclic of order N, then for some generator x L , we have q(x)= 2N for

104 GCD(a, N) = 1. Identifying Δ(L) with Z/N Z, we can identify q with the map

Z/N Z → Z/N Z t → a · t2

By replacing x with a scalar multiple (relatively prime to N), we can see that a is uniquely determined up to a square unit in Z/N Z. We conclude that when Δ(L) is cyclic of order N, the number of non-isomorphic quadratic spaces (Δ(L), q) is equal to the size of the group (Z/N Z)× modulo the subgroup of squares. By the CRT, such a group has size 2a, where a is equal to the number of odd primes dividing N if 4 N, the number of primes dividing a if 4 N, and one more than the number of primes dividing N otherwise. Curiously, Corollary 5.1.4 shows that the number of 0-dimensional cusps does not depend on q once a cyclic Δ(L) is ﬁxed.

5.2 Counting the orbits of isotropic planes

As in the previous section, we will restrict our attention to Z-lattices (L, q) that split two hyperbolic planes. Additionally, we will suppose that L is maximal, so that we can assume L ∼= H ⊥ H ⊥ A for a maximal negative deﬁnite lattice A. Our goal will be to determine the number of 1-dimensional cusps in the compactiﬁcation of the Shimura variety associated to ΓL, the discriminant

+ kernel of SO (L). We will again do this through investigating the action of ΓL on the isotropic planes of L ⊗ Q. Assume throughout this section that W denotes such an isotropic plane.

Lemma 5.2.1. There exists a basis {u1,u2} of W ∩L that can be extended to a { }∪{ }n+2 basis u1,v1,u2,v2 ei i=5 of L such that u1,v1 and u2,v2 are hyperbolic ⊥ ⊥ { }n+2 planes and L can be written as L = u1,v1 u2,v2 ei i=5 .

105 Proof. We start with an arbitrary basis {u1,u2} of W ∩L. Following the proof ∈ 01 H of Lemma 2.1.14, there exists a v1 L such that u1,v1 =(10)= and L ⊥ is of the form L = u1,v1⊥ u1,v1 . Replace u2 with u2 − B(u2,v1)u1 so that u2 ⊥ v1. The result now follows from again applying the proof of Lemma

⊥ 2.1.14 to u2 in the lattice u1,v1 .

Thus for a given isotropic plane W , we get the negative deﬁnite sublattice spanned by the vectors ei in the previous lemma. Call this lattice AW . Clearly

AW is not uniquely determined because a diﬀerent choice of basis will lead to a diﬀerent sublattice. However, we have the following result.

Lemma 5.2.2. The isomorphism class of AW is uniquely determined by W .

{ } { } Proof. Suppose we have two diﬀerent bases of W , u1,u2 and u1,u2 leading to the decompositions

⊥ ⊥ ⊥ ⊥ u1,v1 u2,v2 AW = L = u1,v1 u2,v2 AW .

The identity map on L restricted to (W ∩ L)⊥ induces an isomorphism,

∼ ⊥ −→= ⊥ φ : u1,u2 AW u1,u2 AW .

⊥ → Since W is isotropic, the projection π : u1,u2 AW AW preserves the ◦ ⊥ ◦ quadratic form q (ie. q = q π on u1,u2 AW ) and hence so does π φ. But then π ◦ φ is surjective with kernel W ∩ L = u1,u2 (because AW is negative

deﬁnite), so its restriction to AW is an isomorphism with AW .

The association of AW to W completely determines the O(L)-orbit of W .

106 ∼ Corollary 5.2.3. An isotropic plane W lies in the O(L)-orbit of W iﬀ AW =

AW .

Proof. Immediate.

To determine the orbit for the discriminant kernel ΓL, we ﬁrst look at the decomposition of the subgroup PW ⊂ O(L) stabilizing W. As discussed in Section 3.3, the connected component of the real Lie group PW (R) has a decomposition as a semidirect product

+ R · · PW ( )=(Gh(W ) Gl(W ) M(W )) U(W )

where Gh(W ) is semi-simple with no compact factors, Gl(W ) is reductive with no compact factors, M(W ) is compact and U(W ) is the unipotent radical. In our case, writing 1 ⊥ 1 ⊥ L = u1,u2,v2,v1 AW = 1 AW , (5.1) 1

+ R we see that PW ( ) is exactly of the form discussed in Example 3.3.4. Fur- thermore, we can extend this decomposition to all of PW (R) by adding in appropriate coset representatives of O(L ⊗ R)/SO+(L ⊗ R) that ﬁx W . The resulting decomposition is:

× PW (R)=Z/2Z · ((SL2(R) · R · O(AW ⊗ R)) U) where 01 a −b 10 −cd Z/2Z = 01 ,SL2(R)= ab , 10 cd I I t 1 t × −1 1 R = t ,O(AW ⊗ R)= 1 t−1 1 I M

107 and U is the unipotent radical ⎧⎛ ⎞⎫ ⎪ 10 m1,3 m1,4 m1,5 ... m1,n+2 ⎪ ⎪ ⎪ ⎪⎜ 01 m z m ... m ⎟⎪ ⎪⎜ 2,3 2,5 2,n+2 ⎟⎪ ⎨⎜ 00 1 0 0 ... 0 ⎟⎬ ⎜ 00 0 1 0 ... 0 ⎟ U = ⎪⎜ ⎟⎪, ⎪⎜ 00 x1 y1 ⎟⎪ ⎪ . . . . ⎪ ⎪⎝ . . . . ⎠⎪ ⎩⎪ . . . . I ⎭⎪ 00xn−2 yn−2 where the matrix coeﬃcients satisfy the equations (3.4).

We check from this description that the subgroup PW of O(L) therefore decomposes as

Z/2Z · ((SL2(Z) · O(AW )) U(Z)). (5.2)

Indeed, using the representation of an arbitrary element z ∈ PW (R) given in equation (3.3), it is clear that the factor matrix from O(AW ⊗ R) and the

× product of the two factor matrices from R and SL2(R) must have integer coefficients if z does. Since the product of an element in R× with an element in SL2(R) has integer matrix coefficients only if the product of any two (not necessarily distinct) elements in the set S = {a, b, c, d} lie in Z, it follows √ that each element of S must be an integer multiplied by y, for some fixed squarefree y ∈ N. But ad − bc =1soy must be 1 and therefore if z has integer coefficients, the matrix factor from SL2(R) must lie in SL2(Z). It only remains to consider that the factor matrix from the unipotent radical must be integral, but this follows from the fact that the factor matrix from O(AW ⊗ R) is invertible over Z.

Corollary 5.2.4. The number of SO+(L)-orbits in the set O(L) · W is equal to 2 . [O(AW ):SO(AW )]

108 Proof. The matrix 01 10 01 10 I + represents the non-trivial element in SO(L)/SO (L) and lies in PW , so the orbit of W with respect to either SO(L)orSO+(L) is the same. On the other hand, from the decomposition of PW above, it is clear that we have the equality [PW : PW ∩ SO(L)]=[O(AW ):SO(AW )]. Hence the O(L)-orbit of W equals that of SO(L) exactly when O(AW ) contains automorphisms of negative determinant, and otherwise splits into the SO(L)-orbit of W and the

SO(L)-orbit of u1,v2.

Remark 5.2.5. If the signature of L is (2,n), n ≥ 2 odd, then the index

[O(AW ):SO(AW )] = 2 because multiplication by −1 will be an automorphism of negative determinant.

+ Finally, to decompose the SO (L)-orbit of W into ΓL-orbits, we use the following result relating the restriction of automorphisms of L to the discriminant group.

Theorem 5.2.6. Let (L, q) be a maximal lattice splitting two hyperbolic planes. Then gen(L) contains only one class and the projection map ρ : O(L) → O(Δ(L)) is surjective. The map remains surjective when restricted to SO+(L).

Proof. [40] Theorem 1.14.2 proves that gen(L) contains only one class and the map from O(L)toO(Δ(L)) is surjective. Hence it remains to show the restriction is surjective. Since [O(L):SO+(L)] = 4, it is equivalent to show that the kernel of the map restricted to SO+(L) has index 4 in the kernel of the full map. To that end, observe that O(H2) lies in the kernel of the map

109 from O(L). Since [O(H2):O(H2) ∩ SO+(L)]=[O(H2):SO+(H2)] = 4 the result follows.

+ Corollary 5.2.7. The number of ΓL-orbits in the set SO (L) · W is equal to

|O(Δ(L))| , |ρ(SO(AW ))| where ρ : O(AW ) → O(Δ(AW )) is the projection map.

+ Proof. This follows from (5.2) and observing that [SO (L):ΓL]=|O(Δ(L))|

+ and [PW ∩ SO (L):PW ∩ ΓL]=|ρ(SO(AW ))|.

We have seen that for every isotropic plane W in V , we get a well-defined isomorphism class of negative definite lattices AW . The above analysis determined the number of orbits for all planes W sharing the same isomorphism class of negative definite lattices. It remains to consider which isomorphism classes appear as AW for some isotropic plane W .

Theorem 5.2.8. Suppose L ∼= H ⊥ H ⊥ A for a maximal, negative deﬁnite lattice A. Then L ∼= H ⊥ H ⊥ A if and only if gen(A) = gen(A). Conse- quently, the lattices appearing in the association W → AW are precisely those in the genus of A.

Proof. By Theorem 5.2.6, the lattice L is uniquely determined by its genus class. Therefore it suﬃces to show that

gen(A) = gen(A) if and only if gen(H ⊥ H ⊥ A)=gen(H ⊥ H ⊥ A).

110 One direction is clear: If A and A are isomorphic after completion at a prime, then obviously they remain isomorphic after completion after attaching hyperbolic planes to both. Thus the content lies in the opposite direction, which is immediate from Theorem 2.1.19 and Theorem 2.1.21.

This concludes our analysis of the orbits of isotropic planes and hence of 1-dimensional cusps for the associated Shimura varieties. We summarize our results in the following.

Theorem 5.2.9. Let (L, q) be a maximal lattice of signature (2,n) that splits two hyperbolic planes and choose a decomposition

L = H ⊥ H ⊥ A.

Let Y denote the connected Hermitian space associated to SO+(L) and let

+ ΓL be the discriminant kernel of SO (L). Then the number of 1-dimensional cusps in the Baily-Borel compactiﬁcation of SO+(L)\Y is equal to

2 . [O(A):SO(A)] A∈gen(A)

In the Baily-Borel compactiﬁcation of ΓL\Y , the number of 1-dimensional cusps is equal to

2 |O(Δ(A))| · , [O(A):SO(A)] |ρ(SO(A))| A∈gen(A) where ρ : O(A) → O(Δ(A)) is the projection map.

111 Remark 5.2.10. In order to simplify notation in the sequel, we will make use of the following abbreviation:

2 |O(Δ(A))| Ω(A):= · . [O(A):SO(A)] |ρ(SO(A))|

Theorem 5.2.11. Assuming the notation of Theorem 5.2.9, the boundary of

ΓL\Y in the Bailey-Borel compactiﬁcation can be described as a collection of projective lines, all intersecting at a single common point.

Proof. The single point of intersection is the unique 0-dimensional cusp (Corol- lary 5.1.3). As described in Claim 3.3.6 and the discussion immediately following, the boundary component F corresponding to the isotropic plane W is isomorphic to the upper half plane H and can be viewed inside of P(L ⊗ C)as the set of vectors v =[v1 + iv2] where {v1,v2} form an oriented basis of W ⊗ R for a ﬁxed choice of orientation. Using the basis of L given by (5.1), the set F can be written as the space

{z =[−ω :1:0:0...0] : ω ∈ H}, as given at the end of Example 3.3.4. Using this representation, we see that the action of SL2 ⊂ PW on F is the familiar map ab aω + b .ω = . cd cω + d

In particular we conclude that each 1-dimensional cusp of ΓL\Y is the aﬃne line and its closure is a projective line P1.

Remark 5.2.12. Theorem 5.2.9 actually generalizes to the slightly larger arithmetic subgroups of O(L) that ﬁx Y . Indeed, replacing SO+(L) with the group

+ O (L) (the index 2 subgroup of O(L) ﬁxing Y ) and ΓL with the discriminant

112 + 2 |O(Δ(A ))| kernel of O (L) has the effect of replacing the factors [O(A):SO(A)] and |ρ(SO(A))| |O(Δ(A))| in Theorem 5.2.9 with 1 and |ρ(O(A))| respectively. The first assertion becomes clear when viewing O+(L) as the subgroup of O(L) that preserves the orientation on the positive definite part of any decomposition of L ⊗ R. Then the

+ Z/2Z part of PW does not lie O (L) and the result follows. The argument for

+ the discriminant kernel of O (L) is identical to that for ΓL found in Theorem 5.2.6 and Corollary 5.2.7 above.

Remark 5.2.13. It is a well-known result (Meyer’s Theorem) that over Q,any non-degenerate indeﬁnite quadratic form of rank at least 5 splits a hyperbolic plane. Thus by Lemma 2.1.14, the same holds for an indeﬁnite maximal lattice.

In particular, if n ≥ 5 above, then rk(L) ≥ 7 and we can apply Meyer’s The- orem twice, thus making the assumption that L splits two hyperbolic planes redundant.

Remark 5.2.14. The maximality of L imposes restrictions on the structure of the abelian group Δ(L). Note that because B(x, r2x)=B(rx, rx)=2q(rx), the order of any x ∈ Δ(L) is free of odd squares. Similarly, B(x, 8x)= 1 | | 2 B(4x, 4x)=q(4x), so the order of x is not divisible by 8. Suppose Δ(L) = pa · m, where p is prime and (p, m)=1.Ifp is odd, it follows from these observations that the Sylow p-subgroup of Δ(L)is(Z/pZ)a. In fact a ≤ 2, because any quadratic form of 3 or more variables has a non-trivial solution in

Z/pZ. When p =2, the Sylow 2-subgroup can be slightly more complicated. Analyzing the possible Q/Z-forms shows that the only possible factorizations are Z/4Z × (Z/2Z)a or (Z/2Z)b with a ≤ 1 and b ≤ 3. If one is interested in calculating |O(Δ(L))| for a given L, then proceeding forward from this point makes the problem tractable in general because |O(Δ(L))| is equal to

113 the product of the size of the automorphism groups of each Sylow subgroup

(with restricted form q) by the Chinese remainder theorem. In particular, for every odd prime p, the Sylow p-subgroup with restricted form q can be viewed as a quadratic form over the ﬁnite ﬁeld Z/pZ by replacing the image space 1 Z Z Z Z p / with /p and the automorphism groups of these spaces are well-known ([29] Theorem 1.3.2). See, for example, the case of a cyclic discriminant group below.

5.3 Examples

We conclude our discussion of Theorem 5.2.9 by discussing some examples that may already be of interest to those studying orthogonal Shimura varieties. Our

ﬁnal example describes a source of lattices L that suit our conditions and arise from CM number ﬁelds.

5.3.1 The unimodular case

If (L, q) is an even unimodular lattice of signature (2,n+ 2), then n ≡ 0 mod 8 and L is uniquely determined by Theorem 2.4.4. The lattice L can be written as

L ∼= H ⊥ H ⊥ E for an even, negative deﬁnite unimodular lattice E of rank n, and any such lattice will do because they all lie in the same genus class (Corollary 2.4.5).

Since E is unimodular, the index [O(E):SO(E)]=2ifE has roots (vectors of q-norm 1). Therefore by Theorem 5.2.9, the number of distinct 1-dimensional

+ cusps for the associated connected SO -Shimura variety (equivalently, ΓL- Shimura variety) is bounded from below by the number of E of dimension n and from above by

114 ⎧ ⎫ ⎧ ⎫ ⎨⎪ ⎬⎪ ⎨⎪ ⎬⎪ Even, n-dimensional neg. deﬁnite Even, n-dimensional neg. deﬁnite · ⎪ ⎪+2 ⎪ ⎪ . ⎩ unimodular lattices with roots ⎭ ⎩ unimodular lattices without roots ⎭

For n<32, the number of cusps attains the maximum bound because the only even, negative deﬁnite unimodular lattices of dimensions less than 32 without roots are the 0 lattice and the Leech lattice, neither of which contain automorphisms of negative determinant (the quotient of the automorphism group of the Leech lattice by its center is simple). The table below gives the relevant data. Table 5–1: Cusps for Unimodular Lattices

Lattices w/ # of 1-dim’l n +2 Lattices of dim n no roots cusps

2 1 (0-lattice) 1 2

10 1(E8) 0 1 ⊕ + 18 2(E8 E8,D16) 0 2 26 24 (Niemeier) 1 (Leech) 25

Remark 5.3.1. Note that our result for n = 24 differs slightly from the result stated in [20] Lemma 1.1, where the boundary is described as containing only 24 1-dimensional components intersecting at a common point. This discrep- ancy arises from the different arithmetic groups used in defining the Shimura

+ + variety. In [20], the author uses the group O (II2,26) instead of SO (II2,26) and our results agree in that case (see Remark 5.2.12 above).

115 Remark 5.3.2. For n ≥ 32, the number of negative deﬁnite unimodular lattices of dimensional n grows astronomically, both with or without roots [28]. It is unclear whether it is possible for higher dimensional unimodular lattices without roots to have automorphisms of negative determinant.

5.3.2 L with cyclic discriminant groups

Suppose the abelian group Δ(L) is cyclic of order k. By the maximality a ≤ ≤ of L, k =2 i=1 pi where 0 a 2 and the pi are distinct odd primes. 1 Z Z 1 Z Z Furthermore, q maps Δ(L)into 2k / (or simply k / if k odd). If x is a generator of Δ(L), any automorphism sends x to λ · x, where λ2 ≡ 1modk (note that if k is even, λ2 will also be well deﬁned mod 2k and congruent to 1) and conversely such a λ uniquely determines an automorphism. In this case, |O(Δ(L))| will be equal to either 2 if 4 k or 2 +1 otherwise, by the Chinese remainder theorem.

The prototypical example of such lattices is the rank 5 lattice given by

− 2 H ⊥ H ⊥ − q(x)=x1x2 + x3x4 tx5 = 2t where k =2t is of the form above. The lattice will be maximal if and only if t is square-free. The ΓL-Shimura variety associated with such a lattice parameterizes abelian surfaces of polarization type (1,t) (this arises from the isomorphism of real lie algebras sp4 and o(2, 3), see [19, 24]). The negative deﬁnite lattice A := −tx2 is uniquely determined by its genus and Δ(A)= Z/2tZ. Since SO(A) is trivial, the number of 1-dimensional cusps in this case is either 2 if t oddor2 +1 if t even. In other words, there is a 1-dimensional cusps for every positive divisor of t. This agrees with Theorem 1.2 of [42] regarding the cusps of paramodular groups.

116 5.3.3 Lattices of signature (2,n) from CM-ﬁelds

In the paper [13], there is signiﬁcant discussion of the invariants of certain quadratic forms arising from CM-algebras F . The construction of these forms is very accessible and of interest to those studying orthogonal Shimura varieties, so we describe now a method for extracting a maximal lattice on these spaces when F is an algebraic number ﬁeld with complex multiplication. The following can be done in much more generality, see the above reference.

Fix a CM-ﬁeld F with complex conjugation σ and [F : Q]=2k. Let ΔF/Q

σ and δF/Q denote the discriminant and different of F respectively. Let K = F be the totally real number field of index 2 in F and assume that F/K is a tamely ramified extension. Fix a λ ∈OK , and define the quadratic form qλ on the 2k-dimensional rational vector space F by

qλ(x)=TrK (λxσ(x)) so that the bilinear form is

Bλ(x, y)=TrF (λxσ(y)).

If λ has r negative embeddings and s positive ones, then the signature of qλ is (2s, 2r) ([13] Lemma 5.2). Thus for our purposes, we want to choose λ with precisely one positive embedding. Many such λ exist in K because the

k embedding K→ R sending x to (ϕ(x))ϕ∈Hom(K,R) is dense.

The obvious lattice to take with respect to this form is OF . The discrim-

2 inant of q with respect to this lattice is NK (λ) · ΔF/Q ([13] Lemma 3.1). In general this will not be maximal. However in certain cases we can replace OF

117 with a fractional ideal I on which Bλ remains integral and has square-free determinant. Since the extension from K to F is tamely ramiﬁed, the integrality of Bλ implies the integrality of qλ ([5] Proposition 2.12) and so (I,qλ) will be a maximal lattice of appropriate signature.

More speciﬁcally, observe that the ideal (λ) OF is ﬁxed by σ. If one can write (λ)=J · σ(J) for fractional ideal J,thenBλ is still integral on the

−1 lattice J and has determinant ΔF/Q. Likewise, δF/Q is also ﬁxed by σ. If one

can write δF/Q = J · σ(J ) · S, where S is an integral ideal of square-free norm,

−1 −1 then Bλ restricted to the fractional ideal I = J · (J ) remains integral and has determinant equal to the ideal norm of S. The existence of such a J is equivalent to the statement that any prime of OF that is ﬁxed by σ and exactly divides δF/Q to an odd power has trivial degree of inertia and every such prime lies over a distinct prime of Z. Likewise, the existence of J is equivalent to the statement that primes of OF ﬁxed by σ and dividing (λ) must exactly divide it to an even power.

Example. Let F be the cyclotomic field of 7th-roots of unity, and K its maximal totally real subfield. Let ω denote a primitive root of unity in this field. The unit λ = ω + ω−1 has precisely 1 positive embedding. The discriminant of the field F is −75 and 7 ramifies completely in F . Therefore the ideal (7) = P 6 for prime ideal P in OF (in particular, the ramification is tame) and we must

5 5 have δF/Q = P because the ideal norm of δF/Q is equal to |ΔF/Q| =7. Obviously P is ﬁxed by σ, so in our notation above, J = 1 and J = P 2. By a well-known result, P is the principal ideal generated by (1 − ω). Taking

118 { ωn }5 basis en = (1−ω)2 n=0,wehave ⎛ ⎞ 43−1 −4 −4 −1 ⎜ ⎟ ⎜ 343−1 −4 −4⎟ ⎜ ⎟ ⎜− − − ⎟ ⎜ 13431 4⎟ Bλ(x, y):=⎜ ⎟ ⎜−4 −1343−1⎟ ⎝−4 −4 −1343⎠ −1 −4 −4 −134 which has determinant 7 as expected. Looking at the matrix, we see the vector u1 = e2 + e5 is isotropic and deﬁning v1 := −e1 − 2e2 − e5, we get u1,v1 = H. { }4 Calculating the orthogonal complement, there is a basis ei i=1 such that ⎛ ⎞ −2 −121 ⎜ ⎟ ⎜−14−10⎟ | ⊥ Bλ(x, y) u1,v1 = ⎜ ⎟. ⎝ 2 −1 −20⎠ 100−2

Again we ﬁnd the isotropic vector u2 = e1 + e3 and its pair v2 = e1 + e3 + e4. The orthogonal complement to this is spanned by e3 and 4e1 + e2 +2e4.We conclude that the lattice spanned by P −2 ⊂ F is isomorphic to the form

H ⊥ H ⊥ −27 7 −28 .

There is a unique reduced, negative deﬁnite, even, integral quadratic form of

−2 −1 determinant 7 ([10] Table 15.1) which is A := ( −1 −4 ). Observe that if M is

−1 −1 T the matrix ( 01), then M .A.M = A. Hence [O(A):SO(A)] = 2 and since x →−x lies in SO(A) and generates the automorphism group of O(Δ(L)), we conclude by Theorem 5.2.9 that there is exactly one cusp of dimension 1 for the corresponding connected ΓL-Shimura variety.

5.4 Scaled lattices and level structure

Let L be a maximal lattice that splits two hyperbolic planes and let ΓL denote the discriminant kernel of SO+(L). In the ﬁnal section of this chapter, we look to extend our results on cusps by replacing the lattice L with the scaled lattice

119 N · L. Since all lattices are contained in a maximal lattice and contain some suﬃciently large scaling of said lattice, this eﬀectively bounds the number of cusps for a Shimura variety arising from the discriminant kernel of an arbitrary lattice.

Lemma 5.4.1. Let N>1 be an arbitrary integer. The group ΓN·L is a normal subgroup of ΓL. If GCD(N,det(L))=1, then ΓN·L is the subgroup of ΓL of

2 2 matrices congruent with 1 modulo N (denoted ΓL(N )).

Proof. Observe that the discriminant group Δ(N · L) is equal to the quotient · ∨ · 1 · ∨ · · (N L) /(N L)= N L /N L and the automorphism group O(L)=O(N L). In particular, if α ∈ SO+(L) acts trivially on Δ(N · L), then it acts trivially

∨ 2 on L /N · L and so ΓN·L is a subgroup of ΓL. Furthermore, it is contained

2 2 in the normal subgroup ΓL(N )={α ∈ ΓL : α ≡ 1onL/N · L} since

2 ∨ 2 L/N · L ⊂ L /N · L. Normality of ΓN·L is also clear as ΓL acts on Δ(N · L). Suppose now that N and the discriminant of L are relatively prime. Let v ∈ L∨

2 and r the order of v in Δ(L). Then for any α ∈ ΓL(N ),α(v)=v + , with ∈ L. But r · ∈ N 2 · L and since (r, N )=1, it follows that ∈ N 2 · L.

Theorem 5.4.2. Fix a positive integer N>1.Let(L, q) be a maximal lattice of signature (2,n) that splits two hyperbolic planes and choose a decomposition

L = H ⊥ L = H ⊥ H ⊥ A.

Let Y denote the connected Hermitian space associated to SO+(L). In the

Baily-Borel compactiﬁcation of ΓN·L\Y , the number of 0-dimensional cusps is

[Γ :Γ · ] L N L , 2n 2N · [ΓL :ΓN·L ]

120 and the number of 1-dimensional cusps is equal to

[Γ :Γ · ] Ω(A ) L N L · , 4n −2 N (1 − p ) [ΓA :ΓN·A ] p A∈gen(A) where p runs through the primes dividing N (see Remark 5.2.10 for the definition of Ω(A)). If N is sufficiently large, the number of 1-dimensional cusps simplifies to 2[Γ :Γ · ] |O(Δ(A ))| L N L · . N 4n (1 − p−2) |O(A)| p A∈gen(A) + + Proof. For each N>1, the maps ΓN·L\Y → ΓL\Y extend to their Baily- Borel compactifications (Remark 3.4.8) and so we would like to know how

+ many distinct boundary components of ΓN·L\Y lie over a given component

+ of ΓL\Y . By normality of ΓN·L, the number of 1-dimensional cusps in the compactiﬁcation of ΓN·L\Y lying over the cusp of ΓL\Y corresponding to an isotropic plane W is equal to

[ΓL :ΓN·L]

[PW ∩ ΓL : PW ∩ ΓN·L] and likewise, the number of 0-dimensional cusps of ΓN·L\Y lying over the unique 0-dimensional cusp of ΓL\Y is equal to

[Γ :Γ · ] L N L . [PI ∩ ΓL : PI ∩ ΓN·L]

We will therefore approach this problem in the same way we approached the problem when comparing O(L) with ΓL in the previous section: by looking at the appropriate parabolic subgroups and comparing their indices.

+ We start with the unique 0-dimensional cusp of ΓL\Y . We choose an { }∪{ }n appropriate basis e1,e2 fi i=1 for L so that we get the decomposition

1 ⊥ B =(1 ) B .

121 The parabolic subgroup P e1 of O(L) corresponding to the stabilizer of the

× ﬁrst basis vector has the Levi decomposition over R into (R ·O(B ⊗R))UI (see Example 3.3.4) where:

× t 1 R = t−1 ,O(B ⊗ R)= 1 In M

and U e1 is the unipotent radical ⎧⎛ ⎞⎫ ⎪ 1 m1,2 m1,3 ... m1,n+2 ⎪ ⎪⎜ ⎟⎪ ⎨⎪⎜ 01 0 ... 0 ⎟⎬⎪ ⎜ ⎟ U e1 := ⎜ 0 x1 ⎟ , ⎪⎜ . . ⎟⎪ ⎪⎝ . . In ⎠⎪ ⎩⎪ ⎭⎪ 0 xn where the matrix coeﬃcients satisfy (3.2).

Z ∩ 2Z n We claim that U e1 ( ) ΓN·L is (N ) (here we are making the identiﬁcation Z Zn of U e1 ( ) with using the free xi variables). Indeed it is clear from (3.2)

n 2 that the matrix represented by (xi)i=1 will be congruent with 1 mod N if and

2 · ∈ Z only if xi = N xi,xi . Suppose that vector v = λ1e1 + λ2e2 + ηjfj lies · ∨ 1 ∨ − in the dual lattice (N L) = N L . Then v (zi).v equals n n n · · · · · · N Nλ2 q( xjfj)+B( xifi,N ηjfj) e1 +N Nλ2 xjfj . i=1 j=1 j=1

∨ ∨ Since Nv ∈ L , it follows that Nλ2e2 and N · ηjfj ∈ L . Therefore Nλ2

· ∈ Z − ∈ · and B( xifi,N ηjfj) , so we conclude (I (xi)).v N L as claimed.

As in the case of the parabolic subgroup stabilizing the isotropic plane, an R element of P e1 ( ) lies in P e1 if and only if each matrix factor in the Levi decomposition has integer coeﬃcients (see Equation (5.2) and the following discussion). We conclude that when N>1, the intersection of P e1 with ΓN·L

2 n is the group ΓN·L (N Z) , where (L ,q|L ) denotes the lattice spanned by

122 Z n ∩ the fi. When N =1,ΓL ( ) is a subgroup of index 2 in P e1 ΓL since the matrix −I4 ⊥ In−2 lies in the intersection only in this case. In particular, when N>1 we have the formula

2n ∩ ∩ · [P e1 ΓL : P e1 ΓN·L]=2N [ΓL :ΓN·L ].

Next we move on to the parabolics PW ⊂ O(L) for an isotropic plane W in

L ⊗ Q. Recall the decomposition of PW into factors given by Equation (5.2). As in the 0-dimensional cusp case above, it is easy to check that we have

∼ 2 2n−3 an isomorphism U(Z) ∩ ΓN·L = (N Z) by using the equations in (3.4). Furthermore,

2 SL2(Z) ∩ ΓN·L =Γ(N ), ∩ O(AW ) ΓN·L =ΓN·AW ,

× and Z ⊂ SL2(Z) so we can ignore it. Again, we note that the intersection of

PW with ΓN·L is equal to the product of the intersection of each factor in the Levi decomposition. Hence we have

∩ ∩ 4n−6 Z 2 [PW ΓL : PW ΓN·L]=N [SL2( ):Γ(N )][ΓAW :ΓN·AW ].

2 Additionally, the index [SL2(Z):Γ(N )] is well-known: 2 6 −2 [SL2(Z):Γ(N )] = N (1 − p ), p where p runs through the prime factors of N. Therefore the two formulas for the number of cusps can be written as claimed. Furthermore, since AW is negative definite, O(AW ) is finite and hence for N sufficiently large, the groups

123 is the projection map from O(AW )toO(Δ(AW )). This gives the simpliﬁed formula for N large.

Proposition 5.4.3. In the setting of Theorem 5.4.2, the closure of each 1- dimensional cusp is isomorphic to the compact modular curve of full level N 2- structure, X(N 2).

Proof. Fix some 1-dimensional cusp and let W be an isotropic plane in L associated to this cusp. Following the same reasoning as Theorem 5.2.11, we see that our 1-dimensional boundary component is isomorphic to the noncompact modular curve of full level N 2-structure. It therefore suﬃces to show that 2 isotropic lines in W are ΓN·L equivalent only if they are ΓN·L ∩ PW equivalent. Suppose W = e1,e2 is an isotropic plane in L, and vector e1 =

2 ae1 + ce2 lies in the same ΓN·L-orbit as e1. Then a ≡ 1modN ,c≡ 0 mod N 2 and GCD(a, c)=1. We claim that there exists b, d ∈ Z such that b ≡ 0modN 2, d ≡ 1modN 2 and ad − bc = 1. If such b and d exists, it ∩ follows that the vectors e1 and e1 are equivalent under the action of ΓN·L PW as required. Clearly there exists b and d such that ad − bc =1 and d must be equivalent to 1 mod N 2 by the conditions on a and c. But b can be altered by any integer multiple of a, therefore some choice of b will be congruent with 0, giving us our desired conclusion.

Ideally we would like to simplify Theorem 5.4.2 by giving a formula for indices [ΓL :ΓN·L] and [ΓL :ΓN·L ] in terms of the number of automorphisms of the discriminant groups Δ(N · L) and Δ(N · L), similar to the formula

+ [SO (L):ΓL]=|O(Δ(L))|. Unfortunately, Theorem 1.14.2 of [40] which proves that ρ : O(L) → O(Δ(L)) is surjective when L (or L) is maximal never

124 applies as soon as N>1 and so calculating the size of the image of SO+(L) in O(N · L) (or SO+(L)inO(N · L)) is not so easy. Nevertheless, we can still say something when the following three conditions are satisﬁed:

(i) N and Δ(L) are relatively prime.

+ (ii) ΓL = {ψ ∈ SO (L) : The spinor norm of ψ =1}.

+ (iii) ΓL = {ψ ∈ SO (L ) : The spinor norm of ψ =1} and dim(L ) ≥ 3.

Remark 5.4.4. Unimodular lattices (of appropriate signature) satisfy the conditions above. More generally, Theorem 8.6 of [49] gives suﬃciency conditions on a maximal Zp-lattice Lp to ensure that Spin(Lp) is the maximal subgroup of

Spin(Lp⊗Q) preserving Lp. Clearly this property holding locally for Z-lattice L (or its sublattice L) guarantees conditions (ii)or(iii) being satisﬁed.

Remark 5.4.5. Only condition (ii) is relevant for calculating [ΓL :ΓN·L] and likewise, only condition (iii) is relevant for [ΓL :ΓN·L ], as we will see below.

Lemma 5.4.6. Fix a positive integer N>1 with prime factorization N 2 = r ni i=1 pi .Let(L, q) be a maximal lattice of signature (2,n) that splits two hyperbolic planes and choose a decomposition

L = H ⊥ L.

Deﬁne δ =2if 4|N and δ =1otherwise. Suppose conditions (i) and (ii) given above are satisﬁed. Then

|O(L/(N 2 · L))| [Γ :Γ · ]= L N L 4rδ and if condition (iii) is also satisﬁed then

|O(L/(N 2 · L))| [Γ :Γ · ]= . L N L 4rδ

125 Proof. We start by invoking the following approximation theorem:

Theorem 5.4.7. Let T be a Z-lattice with a non-degenerate quadratic form and assume dim(T ) ≥ 3. Fix a ﬁnite set of primes {p1,...,pr} and positive { }r integers ni i=1.LetTpi denote the completion of T at pi. Suppose that for ∈ ∈ each i, there exists φpi SO(Tpi ) of spinor norm 1. Then there exists φ SO(T ) of spinor norm 1 such that for all i, under the canonical identiﬁcation

ni · ni · between Tpi /pi Tpi and T/pi T , we have

≡ ni · φ φpi on Tpi /pi Tpi .

Proof. This is a special case of Theorem 1.5 in [39].

Suppose we are given an automorphism ϕ ∈ O(L/(N 2 ·L)) (recall that since { ∈ ≡ 2 · } 2 r ni (N,Δ(L)) = 1, ΓN·L = ψ ΓL : ψ 1modN L ). If N = i=1 pi is the prime factorization of N 2, then ϕ is uniquely determined by its restriction

2 to the Sylow pi-subgroups of L/N · L, which are naturally isomorphic to

ni · the quotient spaces L/pi L. Hence by Theorem 5.4.7 above, if we can ﬁnd automorphisms of Lpi of spinor norm 1 having the appropriate action on the quotient group, then we can ﬁnd a global ψ ∈ ΓL (by condition (ii)) such that

ψ ≡ ϕ mod N 2 · L.

By Corollary 1.9.6 of [40], the projection map

ni · → ni · ρpi : O(pi Lpi ) O(Δ(pi Lpi )) is surjective for all i. On the other hand, by condition (i) and Theorem 2.4.6,

the index [O(Lpi ):ΓLpi ] = 4 (or 8 if pi = 2) and ΓLpi is exactly the group of proper automorphisms of Lpi with spinor norm 1 (since Lpi splits a hyperbolic

126 plane, it has automorphisms of determinant −1 and the exact sequence in Theorem 2.4.6 is also right exact). Thus we would like to calculate the index | [Ker(ρ i ) : Ker(ρ i )]. It is easy to see that if ψ i is an automorphism of p p ΓLpi p ≡ ni ≥ determinant -1, then ψpi 1modpi since ni 2. It remains to determine ∈ whether or not an element ψ SO(Lpi ) with non-trivial spinor norm can act

ni trivially modulo pi . Recall from Theorem 2.5.1 that the image of the group · · GSpin(N Lpi )inSO(N Lpi )=SO(Lpi ) is exactly the subgroup acting trivially ∈ · on the discriminant group. But if g GSpin(N Lpi ), then the norm v(g) must

ni be congruent with a square modulo pi . Hence all elements of SO(Lpi ) acting

ni trivially mod pi have trivial spinor norm except in the explicit case when pi = 2 and ni = 2. In this case, it is possible to have spinor norms in the non-

× trivial class represented by 5 ∈ Z/8Z , for example the element 1 + 2e1 · 2e2, where e1,e2 = H. We conclude that

| [Ker(ρ2):Ker(ρ2 ΓL2 )] = 2 if 2 exactly divides N and

| [Ker(ρ i ) : Ker(ρ i )] = 1 p p ΓLpi in all other cases. This completes the proof of the formula

|O(L/(N 2 · L))| [Γ :Γ · ]= . L N L 4rδ

For the proof of the second formula, we simply remark that everything we have said above holds if we replace L with the sublattice L and substitute condition (ii) for condition (iii).

127 We introduce the following map χ to simplify notation. Let L be a Zp- lattice, deﬁne χ(L) as follows ⎧ ⎪ ∼ dim L ⎪ 1ifL H 2 ⎨⎪ = χ(L)= 0 if dim(L)isodd ⎪ ⎪ ⎩ −1 otherwise.

Theorem 5.4.8. Assume the conditions and notation of Lemma 5.4.6. Let maximal lattice L of signature (2,n),n≥ 3, have decomposition

L = H ⊥ L = H ⊥ H ⊥ A.

Let Y denote the connected Hermitian space associated to SO+(L). In the

Baily-Borel compactiﬁcation of ΓN·L\Y , the number of 0-dimensional cusps is

2(n+1) − n −1 N 1 − χ(Lp)p 2 − n+1 · − 2 2 − n (1 p ), 2 1 − χ(L )p 2 p|N p and the number of 1-dimensional cusps is equal to ⎛ ⎞ ⎜ ⎟ n2−n+2 ⎜ ⎟ N − n − − ⎜ 2 1 e ⎟ · ⎜ (1 − χ(Lp)p ) (1 − p )⎟ 2rδ ⎝ ⎠ p|N 3 ≤ e ≤ n +1 e even Ω(A) × , [ΓA :ΓN·A ] A∈gen(A) where p runs through the primes dividing N, Ω(A) as in Remark 5.2.10, and χ is as above. If N is suﬃciently large, the number of 1-dimensional cusps simpliﬁes to ⎛ ⎞ ⎜ ⎟ 2− n n+2 ⎜ n ⎟ | | N ⎜ − −1 −e ⎟ O(Δ(A )) · (1 − χ(L )p 2 ) (1 − p ) · . r−1 ⎜ p ⎟ | | 2 δ ⎝ ⎠ O(A ) p|N 3 ≤ e ≤ n +1 A ∈gen(A) e even

128 Both formulas for the number of 1-dimensional cusps hold even when condition

(iii) does not apply.

Proof. We start by applying Lemma 5.4.6 to Theorem 5.4.2 and it remains to calculate an explicit value for |O(L/(N 2 · L))| (as well as |O(L/(N 2 · L))|, which will be derived by an identical argument). By the Chinese remainder theorem, it suﬃces to calculate locally at each prime pi dividing N. Applying the theory of local densities (see for example, Lemma 5.6.1 of [29]), since Lpi is unimodular (by condition (i)), we have the equality

− dim(L)(dim(L)−1) ni (ni 1) 2 |O(L/(p · L))| = p ·|O(L/(pi · L))|.

On the other hand, the group |O(L/(p · L))| is just the set of isometries over the ﬁnite ﬁeld Z/pZ with quadratic form induced by q. The size of this set is well-known, given in ([29] Theorem 1.3.2):

be simpliﬁed using the equalities dim(L)=n + 2, dim(L )=n and χ(Lp)=

χ(Lp).

Example 5.4.9. The simplest example is using the unimodular lattice L0 = H ⊥ H H , so L0 = . This is the only case when L splits two hyperbolic planes but n<3 and so the ﬁrst part of Theorem 5.4.8 does not apply. However

L0 satisﬁes condition (ii), so the second part of the corollary still applies.

Plugging in the values n = 2 and χ(Lp) = 1 for all p|N, we get that the

129 number of 1-dimensional cusps is

N 4 (1 − p−2). 2r−1δ p|N

Let us compare this answer with the one we get from using the surjective map

∼ + ϕ : SL2 × SL2 = Spin(L0) SO (L0), given at the end of Section 2.2 and deﬁned by sending aa −ba −ab −bb ab a b −→ϕ −ca da cb db ( cd) , c d −ac bc ad bd . −cc dc cd dd

2 Observe that if the matrix on the right is congruent with I4 mod N , then we easily deduce the following:

a ≡ d ≡ a ≡ d mod N 2,

a2 ≡ 1modN 2,

b, c, b,c ≡ 0modN 2.

If N is either 2 or a power of an odd prime, then it follows that the subgroup of × ± · 2 × 2 2 SL2 SL2 mapping to ΓN·L0 is the group 1 (Γ(N ) Γ(N )), where Γ(N )

2 denotes the standard N -congruence subgroup in SL2(Z). If N is a power of 2 and at least 4, then the kernel of ϕ will contain ±1·(Γ(N 2) × Γ(N 2)) as a subgroup of index 2. More generally, the number of prime factors (and the powerof2)inN will determine the index of ±1·(Γ(N 2) × Γ(N 2)) inside

−1 ϕ (ΓN·L0 ) by the CRT. In particular, we have

−1 2 × 2 r [ϕ (ΓN·L0 ):Γ(N ) Γ(N )]=2δ.

Furthermore, if P∞ is the parabolic subgroup

± · 1 x × Z P∞ = 1 ( 01) SL2( )

130 then ∩ −1 ∩ 2 × 2 [P∞ ϕ (ΓN·L0 ):P∞ Γ(N ) Γ(N )] = 2.

−1 2 × 2 Since both ϕ (ΓN·L0 ) and Γ(N ) Γ(N ) are normal subgroups, it follows that ⎧ ⎫ ⎧ ⎫ ⎨⎪ ⎬⎪ ⎨⎪ ⎬⎪ # of 1-dim’l cusps of # of 1-dim’l cusps of 1 = · . ⎪ ⎪ ⎪ ⎪ r−1 ⎩ −1 \ Z × Z ⎭ ⎩ 2 \ Z 2 ⎭ 2 δ ϕ (ΓN·L0 ) SL2( ) SL2( ) (Γ(N ) SL2( ))

2 2 But the number of 1-dimensional cusps of (Γ(N )\SL2(Z)) is just twice the

2 number of 0-dimensional cusps of Γ(N )\SL2(Z). Hence ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ 4 −2 # of 1-dim’l cusps of N | (1 − p ) 1 =2 p N · ⎪ ⎪ r−1 ⎩ −1 \ Z × Z ⎭ 2 2 δ ϕ (ΓN·L0 ) SL2( ) SL2( ) N 4 = (1 − p−2) 2r−1δ p|N which is exactly the value we found above.

To determine the number of 0-dimensional cusps, we can still use Theorem + H 5.4.2. Since SO ( ) is trivial and we can solve for [ΓL0 :ΓN·L0 ] by using Lemma 5.4.6, plugging in to the formula gives us ⎧ ⎫ ⎨⎪ ⎬⎪ # of 0-dim’l cusps of N 8 = · (1 − p−2)2. ⎪ ⎪ r+1 ⎩ −1 · \ Z × Z ⎭ 2 δ ϕ (N ΓL0 ) SL2( ) SL2( ) p|N

One can again check this by considering the number of 0-dimensional cusps 2 \ Z N 4 − −2 in Γ(N ) SL2( ), which is 2 (1 p ), squaring to give the number of 0- 2 2 r−1 dimensional cusps in the space (Γ(N )\SL2(Z)) , and then dividing by 2 δ

−1 2 2 (the ratio of the indices of ϕ (ΓN·L0 ) and (Γ(N )) compared to the intersection with an appropriate parabolic).

131 Remark 5.4.10. In the previous example, note that the number of 1-dimensional N 4 − −2 cusps (i.e. 2r−1δ p|N (1 p )) times the number of 0-dimensional cusps in the N 4 − −2 closure of each 1-dimensional cusp (i.e. 2 (1 p )) is precisely twice the N 8 · − −2 2 number of 0-dimensional cusps, 2r+1δ p|N (1 p ) . This is precisely as one would expect for a quotient of a product of Hermitian symmetric domains, as in our case where D = H × H: a 0-dimensional cusp corresponds to the orbit of an ordered pair of boundary components on H (say (a, b)), while a 1-dimensional cusp corresponds to the orbit of a single boundary component on one of the two copies of H. Therefore the 1-dimensional cusps that contain (a, b) in their closure are exactly a × H and H × b. This is an example of the biregularity of the boundary components, see Theorem 6.3.2.

Example 5.4.11. Consider the lattices of the form L = H ⊥ H ⊥ −2t, where t is positive, odd, and square-free. Since t is relatively prime to 2, L satisﬁes condition (ii) and L satisﬁes condition (iii) by ([49] Theorem 8.6). Thus we can choose N relatively prime to Δ(L)=2t and all of Theorem 5.4.8 will apply. Since n = 3 is odd, χ(Lp) = 0 for all p|N. Therefore the number

+ of 0-dimensional cusps of ΓN·L\Y is

N 6 · (1 − p−4) 2 p|N and the number of 1-dimensional cusps is ⎛ ⎞ N 8 ⎝ (1 − p−4)⎠ · 2s−1 2r−1δ p|N where r is the number of primes dividing N and s is the number of primes dividing t.

132 CHAPTER 6 Algebraic and Topological Properties of the Boundary

In this chapter we will study some algebraic and topological invariants as-

+ sociated to the compactification of the connected Shimura variety ΓN·L\Y , for a scaled maximal lattice N · L as in the previous chapter. In particular, we will look at the field of definition of both the variety itself as well as the components of the boundary. We will also show that the boundary is always connected and discuss the graph representing its Tits building. Throughout this chapter, X (or X+) will denote the (connected component of) Hermitian symmetric space associated to the Shimura datum of GSpin while Y (or Y +) will denote the analogous objects when referring to the orthogonal group SO. Of course, as complex manifolds these two spaces are the same, so this is just to differentiate them as conjugacy classes of certain maps into the respective groups over R. L0 will always denote the Z-lattice H ⊥ H.

6.1 Connected components and their ﬁeld of deﬁnition

In this section we will use the results of Sections 2.5 and 3.4 to deduce the ﬁeld of deﬁnition of the connected Shimura variety associated to a lattice (L, q) and

+ the discriminant kernel ΓL of SO (L) equipped with its canonical model.

Proposition 6.1.1. Let (L, q) be a lattice of signature (2,n) that splits two

+ hyperbolic planes. Then the canonical model of ΓL\Y is deﬁned over Q. Proof. Let K := GSpin(L ⊗ Z) ⊂ GSpin(Af ). Clearly K is a compact open subgroup of GSpin(Af ) because it is exactly the integral points of the natural

133 + representation of GSpin on C (L ⊗ Af ). Recall from Section 3.4 that we have

∼ × π0(ShK (GSpin,X)) = Z /v(K).

Therefore, in order to calculate the number of connected components, it suﬃces to calculate the image of K under v. However, L splits two hyperbolic planes and therefore GSpin(L) ⊃ GSpin(L0), where L0 := H ⊥ H. By Theorem 2.2.9, it follows that K contains the subgroup

∼ 2 GSpin(L0 ⊗ Z) = {(A1,A2) ∈ GL2(Z) : det(A1)=det(A2)}

and v restricted to this subgroup sends (A1,A2) to det(A1). It follows immedi- × ately that v(K)=Z and therefore ShK (GSpin,X) contains only 1 connected component, which is the variety

K ∩ GSpin(Q)+\X+ = Spin(L)\X+.

By [43] 11.10, the rational map GSpin → SO(L⊗Q) induces a map of Shimura

+ + varieties over Q and by Theorem 2.5.3, the image of Spin(L)\X is ΓL\Y .

Remark 6.1.2. Recall from Lemma 2.1.14 that a maximal lattice will split

2 hyperbolic planes if the encompassing vector space does. If n ≥ 5, then this will always be the case (Meyer’s Theorem). Although we are mainly interested in maximal lattices, it is easy to construct non-maximal lattices where Proposition 6.1.1 still applies.

Remark 6.1.3. In [34], the author shows that when L is a maximal lattice, the associated Shimura variety ShK (GSpin,X) admits a regular canonical model Z 1 over [ 2 ].

134 We would like to extend Proposition 6.1.1 to the scaled lattices N ·L of the previous chapter, assuming again that L splits two hyperbolic planes (although not necessarily maximal).

Proposition 6.1.4. Let (L, q) be a lattice of signature (2,n) that splits two hyperbolic planes and N a positive integer. Let KN = GSpin(N · L ⊗ Z).

Then the canonical model of the Shimura variety ShKN (GSpin,X) induces the

+ structure of an FN -variety on ΓN·L\Y , where FN ⊂ Q(ωN 2 ) is the subfield of the N 2-cyclotomic field fixed by the subgroup of squares in (Z/N 2Z)× ∼=

Gal(Q(ωN 2 )/Q).

Proof. Proceeding similarly to the proof of Proposition 6.1.1, we once again look at the 0-dimensional Shimura variety of connected components

∼ Z× π0(ShKN (GSpin,X)) = /v(KN ).

We know that N ·L ⊃ N ·L0. We again recall the description of GSpin(L0 ⊗Z) as a matrix group:

∼ 2 GSpin(L0 ⊗ Z) = {(A1,A2) ∈ GL2(Z) : det(A1)=det(A2)}

and the norm v(A1,A2) = det(A1). Furthermore, we know that the map from GSpin to SO is deﬁned as aa −ba −ab −bb ab a b ϕ 1 −ca da cb db ( ) , −→ . cd c d − −ac bc ad bd ad bc −cc dc cd dd

2 As in Example 5.4.9, since ϕ(GSpin(N · L0 ⊗ Z)) equals the full level N subgroup of the discriminant kernel of SO(L0 ⊗ Z) (Theorem 2.5.1), we can

135 make the following observations about the matrix coeﬃcients:

a ≡ d ≡ a ≡ d mod N 2,

a2 ≡ ad − bc mod N 2,

b, c, b,c ≡ 0modN 2.

In particular, we see that v(KN ) ⊃ v(GSpin(N · L0 ⊗ Z)) contains the set

× 2 2 BN := {x ∈ Z : x ≡ s mod N }.

On the other hand, it is easy to see that BN ⊃ v(KN ) because any element

+ 2 + z ∈ C (N · L ⊗ Af ) is of the form z = a · 1+N · y, where y ∈ C (L ⊗ Af ) and therefore v(z)=z.ι(z) ∈ BN . Z 2Z × Thus π0(ShKN (GSpin,X)) is in bijection with the quotient of ( /N ) by the subgroup of quadratic residues mod N 2. By the Galois action described in Example 3.4.14, the canonical model induces the structure of an FN -variety

· \ + + × ⊂ on the connected component Spin(N L) X = X 1 ShKN (GSpin,X). Applying the rational map of Shimura varieties between Sh(GSpin,X) and Sh(SO,Y ) and using Theorem 2.5.3 completes the proof.

Remark 6.1.5. The field FN can be described as follows: It is the smallest field √ containing all the fields Q( p) where p is an odd prime dividing N congruent √ with 1 mod 4, Q( −p) where p is an odd prime dividing N congruent with 3 mod 4, Q(ω4)ifN is even, and Q(ω8) if 4 divides N.

Corollary 6.1.6. Let L be a lattice such that L ⊃ N · L, where L is a lattice splitting two hyperbolic planes as above and L⊗Q = L⊗Q. Then the canonical model of Sh(SO,Y ) induces an F -structure on the connected Shimura variety

+ ΓL \Y , where F is a subfield of the field FN defined in Proposition 6.1.4.

136 Proof. Since GSpin(L) ⊃ GSpin(N · L), there exists a quotient map from Q \ × A ⊗ Z ShKN (GSpin,X) onto GSpin( ) X GSpin( f )/ GSpin(L ) that sends Spin(N · L)\X+ onto Spin(L)\X+ and therefore the latter component must be deﬁned over a subﬁeld of FN . Composing with the map of Shimura varieties

+ + that sends Spin(L )\X isomorphically to ΓL \Y completes the proof.

6.2 Boundary components and their ﬁeld of deﬁnition

The purpose of this section is to determine the ﬁeld of deﬁnition for the indi-

+ vidual cusps of ΓN·L\Y , where L is maximal and splits two hyperbolic planes. In Chapter 5 we showed that over C, the boundary consists of a collection of projective lines, all intersecting at a common point and the exact number of such lines depends only on the genus of a maximal negative deﬁnite sublattice of L. In this section, we show that each boundary component is already de- ﬁned over Q and then follow that up with a similar result for lattices of the form N · L.

Theorem 6.2.1. Let L = H ⊥ H ⊥ A be a maximal lattice of signature (2,n)

+ and ΓL the discriminant kernel of SO (L). Then each boundary component

+ in the Bailey-Borel compactiﬁcation of ΓL\Y is deﬁned over Q.

Proof. By Proposition 6.1.1 and the discussion of canonical models in Section

+ + + 3.4, the boundary ∂S := ΓL\Y − ΓL\Y is Q-rational because ΓL\Y is. As we have seen in Section 5.2, choosing one projective line in the boundary is tantamount to choosing a ΓL-orbit of totally isotropic planes in L ⊗ Q.

Fixing such a plane W , then by Lemma 5.2.1, there exists a basis {e1,e2} for W such that the ei lie in L and can be extended to an ordered basis

{e1,e2,e3,e4,f1,...,fn−2} of L such that the bilinear form with respect to this

137 basis is represented by the matrix 1 1 ⊥ 1 B . 1

We consider the map i : SO(L0 ⊗ Q) → SO(L ⊗ Q) induced from the identiﬁ- cation of L0 with {e1,...,e4}. This induces a morphism between the Shimura ⊗Q ⊗Q ⊗Z data (SO(L0 ),YL0 ) and (SO(L ),Y). In particular, if KL0 = SO(L0 ) ⊗ Z ⊂ and KL is the discriminant kernel of SO(L ) then i(KL0 ) KL and we get a Q-map

⊗ Q → ⊗ Q τ :ShKL0 (SO(L0 ),YL0 ) ShKL (SO(L ),Y), which extends to a Q-map of the Bailey-Borel compactiﬁcations of the respective varieties ([43] Theorem 12.3 (b)). We know from Proposition 6.1.1 that + \ + ⊆ ⊗ Q SO (L0) YL0 ShKL0 (SO(L0 ),YL0 ) is a rational variety and it maps to \ + ⊆ ⊗ Q rational variety ΓL Y ShKL (SO(L ),Y). Therefore the restriction of τ extends to a map between the minimal compactiﬁcation of these two spaces,

+ \ + → \ + SO (L0) YL0 ΓL Y , which is a map between projective Q-varieties (by Baily-Borel) and hence proper. We also use τ to denote this extension. The map τ is induced by

+ + a map φi from YL0 to Y which extends to a continuous map between the respective compact dual symmetric spaces. Using the basis {e1,...,e4} of L0 and the extended basis including the fi for L above, the two symmetric spaces can be viewed as Siegel domains of the third kind relative to W as in Example

138 3.3.4

+ { − ∈ P ⊗ C } YL0 = [ ω1ω2 : ω2 : ω1 :1] (L0 ): (ω1), (ω2) > 0

+ t→− →− →− Y = {[−ω1ω2 − x.B. x : ω2 : ω1 :1: x ] ∈ P(L ⊗ C):

t →− →− (ω1).(ω2) > − ( x ).B .( x ), (ω1), (ω2) > 0}

and then φi is the map

→− t t [−ω1ω2 : ω2 : ω1 :1]→ [−ω1ω2 : ω2 : ω1 :1: 0].

+ There are three SO (L0)-orbits of proper rational boundary components of

+ YL0 . In the projective coordinates above they are represented by the spaces

t t t [ω1 : −1:0:0], [ω2 :0:−1:0], and [1:0:0:0]

((ω1), (ω2) > 0), and these get mapped into the ΓL-orbit of

→− →− →− t t t [ω1 : −1:0:0: 0], [ω2 :0:−1:0: 0], and [1:0:0:0: 0]

respectively ((ω1), (ω2) > 0). Note that the ﬁrst component in the image is exactly the boundary component corresponding to isotropic plane W (the span of e1 and e2), the second component corresponds to the boundary component of e1,e3 and the third to the orbit of e1, which is the unique 0-dimensional

ΓL-orbit of rational boundary components. We can therefore recover the rationality of the cusp corresponding to ΓL · W by showing the rationality of + \ + · the SO (L0) YL0 -cusp corresponding to ΓL0 W . We do this by considering modular forms as follows.

+ Suppose σ ∈ SO (L ⊗ Q) is represented by matrix C =(cij) under our given basis for L. Following Chapter 3, §3.3 of [9], the factor of automorphy j(σ, z) for modular forms of weight 1 on Y + is the map from SO+(L⊗Q)×Y +

139 to C× deﬁned by

t t→− →− →− (σ, [−ω1ω2 − x.B. x : ω2 : ω1 :1: x ])

−→ − − t→− →− n+2 c4,1( ω1ω2 x.B. x )+c4,2ω2 + c4,3ω1 + c4,4 + j=5 c4,jxj−4.

Composing j with the map i × φi therefore gives the factor of automorphy for + ⊗ Q × + weight 1 modular forms on SO (L0 ) YL0 (which we will identify as the Q × Q {± }× + pair (SL2( ) SL2( ))/ 1 YL0 ):

ab a b t − − t→− →− → ((( cd), ( c d )), [ ω1ω2 x .A. x : ω2 : ω1 : 1]) (cω1 + d)(c ω2 + d ).

+ \ + Therefore weight k modular forms on SO (L0) YL0 are identified with finite sums of ordered pairs of weight k modular forms on the usual upper half plane H. But using the discriminant modular form Δ(ω1) and the weight 12 + \ + Eisenstein series E12(ω2), one can define a modular form on SO (L0) YL0 that vanishes on only 1 of the two 1-dimensional components at infinity. Since the coefficients in the q-expansion of these modular form are rational, their vanishing cycles are rational ([43] §12.20) and hence so are their intersections with the + \ + boundary. Therefore the closures of each 1-dimensional cusp in SO (L0) YL0 are defined over Q. Hence so are their intersections and complements. This + \ + Q shows that all 3 distinct cusps of SO (L0) YL0 are defined over . But then + \ + the map τ sends each component onto a component of SO (L) YL .Asτ is a proper Q-map, in particular the closure of the cusp corresponding to

ΓL · W is therefore deﬁned over Q, as is the unique 0-dimensional cusp. But the 1-dimensional cusp is equal to the complement of its closure and the 0- dimensional cusp, hence it too is deﬁned over Q. Since W was arbitrary, this completes the proof.

140 Similar to Proposition 6.1.1 and Proposition 6.1.4, we would like to provide a result analogous to Theorem 6.2.1 for the boundary components of the connected Shimura variety associated to the discriminant kernel of N · L. The following gives our result.

+ Theorem 6.2.2. The boundary of the connected Shimura variety ΓN·L\Y (with L as above) consists of points and 1-dimensional curves isomorphic to

2 Γ(N )\H. Each component is deﬁned over a subﬁeld of Q(ωN 2 ) containing the

ﬁeld FN deﬁned in Proposition 6.1.4.

Proof. The structure of each individual cusp was already discussed in Propo- sition 5.4.3 so nothing needs to be proved for the ﬁrst part of the theorem.

As in the proof of Theorem 6.2.1 above, we start with an isotropic plane W of L ⊗ V and a decomposition of L corresponding to W, 1 1 ⊥ L = 1 A 1 where W ∩L is spanned by the ﬁrst two vectors of this basis. We again look at the inclusion map i of orthogonal spaces SO(L0 ⊗ Q) → SO(L ⊗ Q) induced Z · ⊗ Z from this decomposition. The discriminant kernel ΓN·L0 ( )ofSO(N L0 ) maps into the discriminant kernel ΓN·L(Z)ofSO(N · L ⊗ Z) and so induces a map on the corresponding Shimura varieties that is deﬁned over Q:

⊗ Q \ × ⊗ A Z SO(L0 ) (YL0 SO(L0 f ))/ΓN·L0 ( ) ↓ SO(L ⊗ Q)\(Y × SO(L ⊗ Af ))/ΓN·L(Z) .

By Proposition 6.1.4 above, this induces an FN -map on the connected components \ + → \ + ΓN·L0 YL0 ΓN·L Y .

141 \ + From Example 5.4.9, we know that ΓN·L0 YL0 is isomorphic to a finite quotient of the space (Γ(N 2) × Γ(N 2))\H × H, with equality if N =2oris the power of an odd prime. We also know that the space of weight k modular forms on (Γ(N 2)×Γ(N 2))\H×H is identified with finite sums of ordered pairs of weight k modular forms on Γ(N 2)\H. But the algebra of modular forms

2 2 for Γ(N ) is deﬁned over Q(ωN 2 ) and separates the cusps of Γ(N )\H (ie. for any cusp, one can ﬁnd a modular form that vanishes on it but at none of the other cusps). Therefore, as in the proof of Theorem 6.2.1, each 0 and

2 2 1-dimensional cusp of (Γ(N ) × Γ(N ))\H × H is deﬁned over Q(ωN 2 ) and is

+ mapped surjectively onto a subset of the cusps of ΓN·L\Y that includes both the 1-dimensional cusp corresponding to ΓN·L ·W as well as each 0-dimensional cusp in its closure. It follows that W and the 0-dimensional cusps in its closure are deﬁned over Q(ωN 2 ). Since W is arbitrary, we are done.

6.3 The geometry of the boundary

Recall that for a maximal lattice L, we can describe the boundary as a collection of projective lines intersecting at a common point. We conclude this chapter by describing some of the geometrical details of the Baily-Borel com-

+ pactiﬁcation of ΓN·L\Y for N>1.

Theorem 6.3.1. The boundary of any arithmetic quotient of Y + (in particular

+ ΓN·L\Y ) is connected.

Proof. We will prove this claim by considering the structure of the Tits building associated to SO+(L ⊗ Q). The Tits building T is the simplicial complex whose vertices correspond to the (proper) rational boundary components of

+ Y , and the vertices {x1,...,xk} form a simplex if (possibly after reordering) the closure of x1 contains x2, the closure of x2 contains x3, and so on (by

142 + + closure we mean in the Harish-Chandra embedding of Y ⊂ Y ⊂ p+, see Section 3.3). Since we only have boundary components of dimensions 0 and

1, T will be a bipartite graph, with lines connecting vertex x1 (corresponding to a 1-dimensional boundary component) to x2 (corresponding to a 0-dimensional component) if x2 lies in the closure of x1. Note that any arithmetic subgroup of SO+(L ⊗ Q) acts on T naturally, and the quotient of T by such a group gives the simplicial complex of the boundary for the connected Shimura variety associated to that group. In particular, if T is connected then every quotient of T is connected and so the boundary in the Bailey-Borel compactiﬁcation of any arithmetic quotient of Y + will be connected. Thus it remains to show T is a connected graph. First, recall that each proper rational boundary component corresponds to either a 1 or 2-dimensional totally isotropic subspace of L. We note that V1 V2 is a chain of isotropic subspaces if and only if we have xV1 xV2 as boundary components (this can be checked by observing that it holds for one particular chain of subspaces, then apply [1] Chapter III, Proposition 3.4 (ii) along with Witt’s theorem).

Suppose that V and W are two distinct, totally isotropic sublattices of rank 2.

We say V and W are connected if the corresponding vertices xV and xW are path connected in T .IfV ∩ W is not trivial, then there is an isotropic line in their intersection. Letting x denote the vertex of T corresponding to this line, it follows that {xV ,x} and {xW ,x} both form lines in T . Hence V and W are connected in this case. If V ∩ W = {0}, then take arbitrary non-zero vector v ∈ V and choose non-zero w ∈ v⊥ ∩ W. Then v, w is connected to both V and W by the ﬁrst case. Hence all vertices corresponding to isotropic planes are connected, and since every isotropic line is connected to some isotropic plane, it follows T is connected.

143 The Tits building T from the previous proof can be used to tell us more about the geometry of the boundary of our Shimura variety. Let T (ΓN·L) denote the graph realized as the quotient of T by ΓN·L, ie. the simplicial

+ complex of the boundary of ΓN·L\Y .

∼ Theorem 6.3.2. Let L = H ⊥ L . The graph T (ΓN·L) is a biregular graph of bidegrees ⎛ ⎞ 4 − −2 N | (1 p ) [Γ :Γ · ] Ω(A ) p N and L N L · ⎝ ⎠ , 2n−4 2 N [ΓA :ΓN·A ] A∈gen(A) for the vertices corresponding to 1 and 0-dimensional cusps respectively.

Proof. For simplicity, we refer to vertices from the two disjoint sets of T as either 0 or 1-dimensional vertices, with obvious meanings. We start the proof

with the following observation: If x1 is a 1-dimensional vertex and x2,x2 are ∼ ∈ 0-dimensional vertices that lie in the closure of x1, then x2 x2 T (ΓN·L)if ∈ and only if there exists γ ΓN·L such that γ(x1)=x1 and γ(x2)=x2 (this was explained in the proof of Proposition 5.4.3). It follows that the degree of x1 in T (ΓN·L) is exactly equal to the number of cusps of the modular curve Γ(N 2), which is 1 if N = 1 and otherwise is equal to 4 −2 N | (1 − p ) p N . 2

To show that the degree of 0-dimensional vertices in T (ΓN·L) is constant, it is enough to recall that ΓL acts transitively on isotropic lines and contains

ΓN·L as a normal subgroup. Therefore the automorphism group of T (ΓN·L)

(which contains ΓL/ΓN·L) acts transitively on the 0-dimensional vertices and so their degrees must be constant.

144 We can determine the degree of the 0-dimensional vertices through the formula

0-dimensional vertex degree {# 1-dimensional vertices} = . 1-dimensional vertex degree {# 0-dimensional vertices}

Using the formulas from Theorem 5.4.2, we conclude that the degree of a

0-dimensional vertex in T (ΓN·L)is ⎛ ⎞ [Γ :Γ · ] Ω(A ) L N L · ⎝ ⎠ . 2n−4 N [ΓA :ΓN·A ] A∈gen(A)

It is interesting to note that for a scaled lattice N · L with N>1, the graph T (ΓN·L) has cycles and its girth is bounded by 8, which is the girth of T . Furthermore, the graph T is highly connected, having a diameter of 4. This means that the quotient T (ΓN·L) will have a diameter bounded by 4 as well.

Theorem 6.3.3. The diameter and girth of T is 4 and 8 respectively. When

N>1, T (ΓN·L) has cycles of length 8.

Proof. We start with proving the statements about cycles. Suppose we de- 1 1 ⊥ { }{ } compose L = 1 A with the usual basis vectors e1,...,e4 fi . 1 The following set of vertices then gives a cycle of T :

o o o e1,e2 e2 e2,e4 eO4

/ / / e1 e1,e3 e3 e3,e4.

When N>1, the projection of this cycle into T (ΓN·L) is injective. It remains to show that T does not have cycles of shorter length. Since the graph is bipartite, all cycles must be of even length. Suppose there existed a cycle

145 of length 6 in T . Let {v1,v2,v3} be the three 0-dimensional cusps appearing in the cycle. Then { v1,v2, v1,v3, v2,v3} are the three isotropic planes in the cycle. It follows that v1,v2,v3 is an isotropic space. But this is a contradiction since the maximal rank of an isotropic sublattice in L is 2. It remains to show that the distance between any 2 vertices of T is at most 4. Suppose x, y and u, v are two isotropic planes corresponding to vertices of T . The intersection of u⊥ and x, y must be non-trivial by dimension considerations. Suppose z lies in this intersection. Then we have a path

u, v / u / u, z / z / x, y / x.

We conclude

(i) d( u, v, x, y) ≤ 4, (ii) d( u, x, y) ≤ 3, (iii) d( u, x) ≤ 4. Since the vertices were arbitrary, we are done.

146 CHAPTER 7 Special Divisors and their Compactiﬁcations

The purpose of this chapter is to discuss the interplay between special divisors (see Definition 7.1.1 below) and the boundary of the connected orthogonal Shimura varieties already discussed in the early chapters. The importance of these special divisors arises from the fact that they appear as the vanishing locus of certain modular forms on our varieties, called Borcherds products. Understanding their compactification at infinity, which is the main goal of this chapter, will therefore help to determine the behavior of these forms at the cusps. This chapter lays some groundwork for future studies on this topic, as well as explore the possibility of an intersection pairing between special divisors and the 1-dimensional boundary. A good reference for the theory of special divisors and Borcherds products is Bruinier [9].

For this chapter, we again assume that (L, q)isaZ-lattice of signature (2,n) splitting two hyperbolic planes, although this is not necessary for deﬁning special divisors in general.

7.1 Special divisors at the boundary

Let λ ∈ L∨ be an arbitrary vector of negative norm. Recall from Example 3.2.12 that the Hermitian symmetric domain D associated to the connected Shimura variety arising from L is isomorphic as a real SO+(L ⊗ R)-manifold to the Grassmannian of positive definite planes, denoted Gr(L ⊗ R). Under this identification, we define λ⊥ ⊂ D as the closed subset of planes that are

147 perpendicular to the vector λ. Note that if Lλ denotes the sublattice of L

⊥ perpendicular to λ, then λ is just the image of Gr(Lλ ⊗ R) in Gr(L ⊗ R).

Deﬁnition 7.1.1. Let β ∈ L∨/L and m<0. The special divisor 1 H(β,m) is the set H(β,m):= λ⊥. λ=β q(λ)=m

We observe that the set H(β,m) is invariant under the action of ΓL and

+ so is the preimage of a closed set in the connected Shimura variety ΓL\Y . In fact we claim that the image of H(β,m) is a ﬁnite union of sub-Shimura varieties of codimension one, hence the use of the term divisor.

Lemma 7.1.2. Let S be the set

m {(n, α) ∈ (Z\{0}) × Δ(L): ∃ primitive λ ∈ L∨,q(λ)= , λ = α, n · α = β}, n2 and deﬁne the relation (n, α) ∼ (−n, −α). Then the image of H(β,m) in

+ ΓL\Y consists of M closed, irreducible components of codimension 1, where M = |S/ ∼|.

∨ Proof. Recall from Lemma 5.1.1 that two primitive vectors in L are ΓL- conjugate if and only if they share the same norm and the same projection into Δ(L). It follows that H(β,m) is comprised of ﬁnitely many ΓL-orbits of

⊥ the form ΓL · (n · λ) , with each λ arising from a pair (n, α) as in the deﬁnition

⊥ ⊥ of S. Note that since λ =(−λ) , the ΓL-orbits corresponding to the pairs

1 The term Heegner divisor is also commonly used in the literature.

148 + (n, α) and (−n, −α) have the same image in ΓL\Y . This shows that the image of H(β,m) is equal to the image of λ⊥ for precisely M distinct λ.

⊥ It remains to show that ΓL · λ corresponds to an irreducible divisor in

+ ΓL\Y . This follows from the map of Shimura data arising from the inclusion

+ GSpin(Lλ) → GSpin(L) : Note that the image of Spin(Lλ)inSO (L)isthe subgroup of ΓL that ﬁxes Lλ, acts trivially on λ, and projects trivially to

∨ Lλ /Lλ. Therefore we get a map of connected Shimura varieties

\ ⊗ R → \ + ΓLλ Gr(Lλ ) ΓL Y . (7.1)

By passing to an appropriate level subgroup of each arithmetic group, the corresponding induced map between the varieties becomes a closed immersion

⊥ + ([11] Proposition 1.15). It follows that the image of λ in ΓL\Y is a closed subvariety of codimension 1 as claimed (in fact the given map (7.1) is proper as it is the restriction of a map between projective varieties to the preimage of an open subset by Remark 3.4.8).

Example 7.1.3. We again explore the case where (L, q)=H ⊥ H ⊥ −2t for some positive, square-free integer t. Suppose λ ∈ L∨ is primitive with norm − m ∈ N 4t for some non-zero m . By Lemma 5.1.1, there exists some λ in the ΓL-orbit of λ such that

b λ = a.e + e + .e , 1 2 2t 5

− b2 − m b · ≡ 2 ≡ with a 4t = 4t and 2t e5 λ in Δ(L). In particular, we see that b m mod 4t. From this description of λ, we can write

⊥ − H ⊥ −2a −b Lλ = e3,e4 a.e1 e2,be1 + e5 = −b −2t .

149 −2a −b − We observe that the determinant of the integral matrix −b −2t is m and √ therefore it represents the bilinear form arising from the norm of Q( m), restricted to a lattice inside the ring of integers of this ﬁeld. In the case √ when m is a square, we remark that Q( m) degenerates into Q × Q with the quadratic form x2 − y2 and maximal order Z × Z. √ The appearance of the ﬁeld Q( m) is not an accident. As we have stated ⊗ Q Q before, through the accidental isomorphism of Spin(L ) and Sp4( ), the

+ space ΓL\Y can be viewed as a moduli space of abelian surfaces with polarization type (1,t). In this setting, the union of the image of λ⊥ in the moduli − m space of abelian surfaces, for all primitive λ of norm 4t , is known as a Hum- bert surface of invariant m. The importance of Humbert surfaces arises from the realization of the embedding

\ ⊗ R → \ + ΓLλ Gr(Lλ ) ΓL Y as the inclusion of a Hilbert modular surface parameterizing abelian surfaces √ with Q( m)-multiplication. This is stated precisely in [18] and can be summarized as follows.

Proposition 7.1.4. Let Om be the order of discriminant m in the real quadratic √ Q − ﬁeld ( m). Then the image of β∈Δ(L) H(β, m) in the moduli space of abelian surfaces consists of all abelian surfaces whose endomorphism rings contain Om.

Proof. [18] Proposition 2.3.

In the case when m is a square, this statement degenerates into a statement about abelian surfaces that are isogenous to a pair of elliptic curves.

150 Theorem 2.4 of [18] goes on to state that the number of irreducible components of a Humbert surface of invariant m is equal to the cardinality of the set

{b mod 2t : b2 ≡ m mod 4t}.

In terms of our interpretation of Humbert surfaces as the union of components ∈ ∨ − m of special divisors, this makes sense: every primitive λ L of norm 4t has b ∼ Z Z − m − b2 an image 2t .e5 in Δ(L) = /2t . Then 4t = q(λ) is congruent to 4t mod Z, and since every possible b arises in this way for some λ, we arrive at the same conclusion. We will return to this example in Section 7.1.3.

The term special divisor, as well as the notation H(β,m), can be used to

+ denote the subset of Gr(L ⊗ R), its image in ΓL\Y , or its divisor class in

+ + Div(ΓL\Y ). As an element of Div(ΓL\Y ), we consider every irreducible subdivisor of H(β,m) to have multiplicity 2 if 2β =0∈ Δ(L), due to the fact that λ⊥ and (−λ)⊥ both appear in the deﬁnition of H(β,m) above. If 2β = 0 then each factor appears with multiplicity 1, but note that H(β,m)= H(−β,m). The argument in Lemma 7.1.2 shows that as a divisor, H(β,m) contains |S| irreducible components (counted with multiplicity).

As the image of connected Shimura varieties, the closure of H(β,m) in the

+ Baily-Borel compactiﬁcation of ΓL\Y is just the image of the corresponding projective varieties under the extended proper maps (7.1). In terms of the larger space D consisting of D ∼= Gr(L ⊗ R) and its rational boundary components (see §3.3), the closure of λ⊥ also has a nice interpretation.

151 Lemma 7.1.5. The closure of λ⊥ in D contains the boundary components corresponding to isotropic lines and planes that are perpendicular to λ. The

∗ image of this closure under the projection map πL : D → D(ΓL) is equal to

⊥ πL(λ ).

Proof. The ﬁrst statement is easy because as a subset of D∨,λ⊥ is the intersection of a closed set with D :

λ⊥ = {z ∈ P(V ⊗ C):B(z,λ)=0}∩D.

Since B(z,λ)=0iﬀB((z),λ)=0=B((z),λ), the result follows from our association of boundary components with isotropic lines and planes given in Example 3.3.4. The second statement follows from the description of the cylindrical sets relative to the rational boundary components given in Example 3.3.4. For example, suppose that e1 =[1:0:...: 0] is an isotropic line in V . Using the notation of Example 3.3.4, a typical cylindrical set in D relative to e1 is

D v = {[−B ((x), (x)):1:x1 : ...: xn] ∈ P(VC):

B ((x) − v, (x) − v) > 0, (x1) >v1} ,

∈ where v C(F e1 ) is an arbitrary vector in the light cone. Since λ has negative norm, if it is also orthogonal to e1 then we may choose our basis so that

λ =(a :0:0:b :0:...:0)

⊥ and the set λ ∩ D v is equal to

a {x ∈ Cn : B((x) − v, (x) − v) > 0, (x ) >v,x = }. 1 1 2 b

∈ Clearly this set is non-empty for any v C(F e1 ), and therefore [1 : 0 : ...:0] lies in the closure of λ⊥ as claimed.

152 On the other hand, suppose λ is not perpendicular to e1. Then we can choose our basis so that

λ =[a :1:0:...:0].

⊥ Then λ ∩ D v is equal to

{x ∈ D v : B ((x), (x)) = a}, which is clearly empty for v of suﬃciently large norm.

The proof for isotropic planes follows the same reasoning.

We see from the lemma that in low dimension, it is possible for the components of special divisors to not contain 1-dimensional boundary components in their closures, even though Y + does. When n ≥ 6 however, λ⊥ always contains 1-dimensional boundary components because there will always be isotropic planes in the quadratic space λ⊥ of signature (2, 5) by Meyer’s theorem.

7.1.1 0-dimensional Cusps

We can use Lemma 7.1.5 along with the results of the previous chapters to describe which cusps lie in the image of the extended maps (7.1). Recall that each 0-dimensional cusp corresponds to an element of the set

Φ={x ∈ Δ(L): q(x)=0}/{x ∼−x}.

By the previous lemma, a 0-dimensional cusp corresponding to the ΓL-orbit of an isotropic line lies in the closure of the image of λ⊥ iﬀ some representative of that orbit class is perpendicular to λ. Equivalently, given any isotropic line

153 in that ΓL-orbit, there must exist a λ in the ΓL-orbit of λ orthogonal to this line. We can use this interpretation to prove the following result.

Proposition 7.1.6. Let (L, q) be a lattice of signature (2,n) splitting two hyperbolic planes. Let λ ∈ L∨ be a primitive vector of negative norm, and let α = λ ∈ Δ(L). The closure of the image of λ⊥ in the Baily-Borel compactiﬁ-

+ cation of ΓL\Y contains all cusps corresponding to points in the set

{x ∈ Φ: q(x + α)=q(α)}.

Proof. First we check that the statement makes sense, since a priori x ∈ Φ is only deﬁned up to multiplication by ±1. If q(x + α)=q(α), then this is equivalent to saying that B(x, α)=0, since q(x)=0. But then B(−x, α)=0, so q(−x + α)=q(α). One direction of the proof is straightforward: if the cusp corresponding to

⊥ + some x ∈ Φ lies in the closure of the image of λ in ΓL\Y , then there exists

∨ an isotropic lift τ of a representative of x to L and some γ ∈ ΓL such that B(γ · λ, τ) = 0. Since γ · λ = λ = α this proves that x lies in the given subset of Φ.

Conversely, suppose x satisfies q(x + α)=q(α) and fix a representative of x in Δ(L), which we also denote x. By assumption on L, there exists a decomposition L = H ⊥ L = H ⊥ H ⊥ A, for some negative definite sublattice A. We claim that there exists a vector λ ∈ L that lies in the

∨ ΓL-orbit of λ. Indeed, let z be any lift of α to A , which exists since the inclusion of A into L induces an isomorphism of discriminant groups. Then λ − z ∈ L and so q(λ) − q(z)=q(λ − z)+B(λ − z,z) ∈ Z. Therefore the

vector λ =(q(λ) − q(z))e3 + e4 + z lies in the same orbit as λ by Lemma 5.1.1

154 (here we use e1,...,e4 for the usual basis pairing of H ⊥ H) and this proves the claim.

Now let w be any lift of x to A∨. By assumption, the pairing B(λ,w) ∈ Z. Since λ is primitive in L∨ (hence also primitive in (L)∨ ⊂ L∨), there exists v ∈ L such that B(λ,v)=B(λ,w). The vector w = w − v ∈ L is now a lift of x (in particular it has integral norm) that is orthogonal to λ. It follows

that the vector w = −q(w )e1 + e2 + w ∈ L is an isotropic lift of x that is orthogonal to λ.

Remark 7.1.7. The proposition immediately implies that every component of every special divisor contains the cusp corresponding to 0 ∈ Φ in its closure.

+ Hence if L splits two hyperbolic planes, none of the special divisors on ΓL\Y are compact.

Corollary 7.1.8. Let S be the subset of Z × Δ(L) deﬁned in Lemma 7.1.2. The number of 0-dimensional cusps in the closure of H(β,m) is equal to the size of the set

{x ∈ Φ: ∃ (n, α) ∈ S st. q(x + α)=q(α)}.

Proof. Apply Proposition 7.1.6 to each component given in Lemma 7.1.2.

7.1.2 1-dimensional Cusps

We now move on to discussing the 1-dimensional cusps in the closure of the components of a special divisor. In contrast to the previous section, we now restrict to maximal L so that we can use the results of Chapter 5. As used previously, for such an L we have a decomposition L ∼= H ⊥ H ⊥ A, for a maximal negative deﬁnite sublattice A that is unique up to its genus class.

155 For each such A in the genus class of A, ﬁx a rational boundary component

WA and an associated decomposition of L into H ⊥ H ⊥ A and consider the

induced isomorphism τA of Δ(L) with Δ(A ). Then the discussion culminating

+ in Theorem 5.2.9 says that every 1-dimensional cusp of ΓL\Y is associated to a pair (A, {ϕ}), where A is the element of the genus class determined by our cusp and {ϕ} is the O(A)-orbit of automorphisms of Δ(A) consisting of maps

−1 ◦ ◦ ∈ of the form τA γ τA , where γ O(L) is any automorphism sending a rational boundary component in the given cusp to WA (note that at most 2 distinct cusps can be associated to any given pair). The question of determining which cusps lie in the closure of the image of some primitive λ⊥ can now be framed in terms of these pairs.

+ Proposition 7.1.9. With notation as above, a 1-dimensional cusp of ΓL\Y corresponding to the pair (A, {ϕ}) is contained in the closure of the image of

⊥ ∨ λ iﬀ there exists v ∈ (A ) such that q(v)=q(λ) and v = ϕ ◦ τA (λ) (note that the existence of v is independent of choice of ϕ).

Proof. Suppose ﬁrst that such a v exists. Let W be a rational boundary com- ∼ ponent representing our cusp and let L = H ⊥ H ⊥ AW be a decomposition of L with respect to W , so that e1 and e3 span W as usual and AW is isomorphic to A. By assumption on v, there exists some γ ∈ O(L) that sends W

−1 to WA such that γ (v) has the same image in Δ(L)asλ. But if we replace

−1 −1 γ (v) with γ (v)+a · e1 for some appropriate choice of a ∈ Z,wenowhave a primitive vector with the same norm and image in Δ(L)asλ. The result follows from Lemma 5.1.1.

Now suppose that there exists λ ∈ ΓL · λ that is orthogonal to W . Using

⊥ the decomposition of L above we have W = e1,e3⊥AW . Let v ∈ AW be

156 the projection of λ. Then q(v)=q(λ) and v = λ ∈ Δ(L). Let γ be any

automorphism of L sending W to WA and AW to A . Then v = γ(v ) satisﬁes the required equality.

Remark 7.1.10. Note that a compactiﬁed special divisor will either contain a 1-dimensional cusp or be completely disjoint from it. However, such a divisor will always intersect the closure of the 1-dimensional cusp because it always contains the unique 0-dimensional cusp.

Example 7.1.11. We return to our earlier example of the maximal lattice

(L, q)=H ⊥ H ⊥ −2t for a positive, squarefree t ∈ N. As discussed in Ex- ample 5.3.2, there is a 1-dimensional cusp for every positive divisor of t. More precisely, the negative deﬁnite sublattice A = −2t is uniquely determined by its genus, and the group Δ(L)=Z/2tZ decomposes into the product Z/2a+1Z × Z/pZ, p where p runs over the odd primes dividing t and 0 ≤ a ≤ 1 is the power of 2 dividing t. With respect to this decomposition, the automorphisms of Δ(L) that preserve q can be represented by multiplication by ±1ineachof the individual factors (in the case when a =0, clearly multiplication by ±1in Z/2Z is redundant and therefore can be ignored). Since the image of O(A)in O(Δ(A)) is central, the O(A)-orbits of Δ(A)-automorphisms reduce to single points. Therefore each 1-dimensional cusp is represented by a unique α-tuple in {(±1modp1,...,±1modpα)} (or {(±1mod4,...,±1modpα)} if t is even), where α is the number of prime divisors of t (except in the case when t = 1, where there is then a single automorphism of the group Δ(A) ∼= Z/2Z).

157 We next observe that the set of norms of elements of A∨ is equal to {k ∈ Q : −4tk = s2 ∈ N}. In particular, if m is not a square, then no component of the Humbert surface of invariant m will contain a 0-dimensional cusp in its closure. This is precisely as we would expect, since such components are the images of Hilbert modular surfaces, which do not contain 1-dimensional cusps. Conversely, suppose m = s2. Recall from Example 7.1.3 that the Humbert surface of invariant m has an irreducible component for every element of the set {b mod 2t : b2 ≡ s2 mod 4t}. From our assumption that t is squarefree, it is not hard to see that the size of this set is exactly equal to the number of positive divisors of t that are relatively prime s. If we use the CRT to decompose Δ(A) into its Sylow subgroups as above, then each component will correspond to a unique α-tuple in the set {(±s mod p1,...,±s mod pα)}

(or {(±s mod 4,...,±s mod pα)}). Note that for the coordinates indexed by primes p dividing both t and m, s ≡−s mod p (or mod 4 in the case when p = 2). This set of α-tuples is equipped with the obvious homogeneous O(Δ(A))-action deﬁned by pointwise multiplication of the coordinates indexed by the same prime. Clearly this action is free precisely when s and t are relatively prime.

Let β be the number of primes dividing both s and t. There are precisely ∨ − m ± ± s two elements of A of norm 4t , namely v = 2t e5. Hence the image v under the CRT as above is one of {(s,...,s), (−s,...,−s)}.Ift = 1 and t |s, it follows that the closure of each component will contain precisely 2β+1 1-dimensional cusps, namely the cusps corresponding to the automorphisms of

Δ(A) that send the α-tuple (±s,...,±s)to±(s,...,s). If t =1, Δ(A) and O(Δ(A)) are both trivial and the closure of the unique component contains the unique cusp. If t|s, then every 1-dimensional cusp lies in the closure of the unique component.

158 Example 7.1.12. Consider the unique unimodular lattice L of signature (2, 26). Recall from Example 5.3.1 that the associated connected Shimura

+ variety ΓL\Y has 25 1-dimensional cusps intersecting at a single point. The closure of the special divisor H(0, −1) contains the unique 0-dimensional cusp, as well as the 23 1-dimensional cusps corresponding to Niemeier lattices that contain roots. Since the Leech lattice does not contain vectors of norm 1, the two cusps corresponding to the Leech lattice are not in the closure of H(0, −1).

7.2 Intersections with the cusps

It is often the case that one is interested in a certain linear combination of special divisors that represents the divisor of a line bundle of modular forms. That such exists is a result of the theory of Borcherds products which can be brieﬂy described as follows. Given a lattice L of signature (2,n) (here L need not split 2 hyperbolic planes), one deﬁnes vector-valued holomorphic modular → C Z forms f : H [Δ(L)] through the Weil representation of Mp2( ) on the group algebra C[Δ(L)] (see [9] pg. 16-18). Using eβ(τ) to denote the map

τ → e2πiτ · β from H into the β-coordinate of C[Δ(L)] (under the standard basis), every holomorphic modular form has a Fourier expansion around ∞ of the form f = c(β,m).eβ(mτ). β∈Δ(L) m∈Z+q(β) m≥0

We say the map f : H → C[Δ(L)] is a weakly holomorphic2 modular form if it satisﬁes the same conditions as a holomorphic modular form, except that the

2 This is referred to as nearly holomorphic in [9] (Deﬁnition 1.11).

159 Fourier expansion around ∞ may have a ﬁnite principal part in each component.

Assume L ⊗ Q splits two hyperbolic planes and suppose we are given a − n weakly holomorphic modular form f of weight k =1 2 with ﬁnite principal part c(β,m).eβ(mτ) β∈Δ(L) m∈Z+q(β) m<0 such that the c(β,m) are all integral for m<0. Then f lifts to a meromorphic

+ 3 modular form Ψ of weight c(0, 0)/2 with multiplier system χ on ΓL\Y . The divisor of Ψ is given by

1 c(β,m).H(β,m) 2 β∈Δ(L) m∈Z+q(β) m<0 and Ψ has a normally convergent product expansion in a suﬃciently small cylindrical neighbourhood of any 0-dimensional rational boundary component

([9] Theorem 3.22). Moreover, when L splits two hyperbolic planes, any meromorphic modular form whose divisor is a linear combination of special divisors must arise in this way ([9] Theorem 5.12).

One may wonder which possible combinations of special divisors can appear as divisors of such modular forms. The answer to this is given by a certain obstruction space of cusps forms:

3 Any multiplier system of ΓL has ﬁnite order and χ is a character when c(0, 0)/2is integral. See [9] pg. 85-86 for details.

160 Theorem 7.2.1. There exists a weakly holomorphic C[Δ(L)]-modular form of weight k on H with prescribed principal part c(β,m).eβ(mτ) β∈Δ(L) m∈Z+q(β) m<0

(with c(β,m)=c(−β,m)) iﬀ the functional c(β,m).aβ,−m β∈Δ(L) m∈Z+q(β) m<0 C n is identically 0 on the [Δ(L)]-cusp forms of weight κ =1+2 with respect to the dual of the Weil representation (here aβ,−m represents the coeﬃcients in the q-expansion of such cusp forms).

Proof. Theorem 1.17 of [9].

Suppose that one is given a linear combination D of special divisors as a

+ divisor of ΓL\Y . Since the boundary of the connected Shimura variety is a cycle of dimension 1, it is natural to hope to assign an integer representing the intersection number of these cycles of complementary dimension. Unfortu- nately, the boundary is part of the singular locus of our variety and therefore we cannot necessarily expect a well-deﬁned pairing between cycles as in the nonsingular case. However, if D (or some integer multiple of D) represents the Cartier divisor Div(f) of a line bundle, then we can still consider the restriction of this line bundle to the closure of a 1-dimensional cusp and consider the divisor of this restriction. On the other hand, line bundles of modular

+ forms of high weight on ΓL\Y extend to line bundles over the Baily-Borel compactiﬁcation because if Mk(ΓL,D) denotes the global sections of such a sheaf, then ProjC(⊕nMnk(ΓL,D)) is isomorphic to the Baily-Borel compactiﬁ-

+ cation and the restriction of O(1) to ΓL\Y is clearly Mk(ΓL,D) (this is just

161 an example of Theorem 3.2.15. See also [14] Theorem 2.3.2). Therefore, we can consider the restriction of the Cartier divisor representing some multiple of D to the boundary components. Since each such component is isomorphic to a projective line, the restricted line bundle will be isomorphic to OP1 (r) for some r ∈ Z. We now describe how to explicitly calculate r when L is maximal and none of the special divisors contain the given 1-dimensional cusp in their closure.

Theorem 7.2.2. Let (L, q) be a maximal lattice of signature (2,n) splitting 2

1 hyperbolic planes and let D = 2 aβ,mH(β,m) be the divisor of a Borcherds + product ϕ on X =ΓL\Y arising from the lift of a modular form f (we assume aβ,m = a−β,m as in the notation of Theorem 3.22 (ii) in [9]). Suppose there exists a 1-dimensional cusp X of X that does not lie in the closure of the support of D and let A ⊂ L be the negative deﬁnite sublattice associated to this cusp as in Theorem 5.2.9. For λ ∈ L∨ and v ∈ A∨ such that 0 ≥ q(v) >q(λ) and v = λ, let tλ,v be the number of SL2(Z)-orbits in the following set

Sλ,v := {M ∈ M2(Z) : det(M)=q(v) − q(λ),M≡ 0modp, ∀ p | B(v, A)}.

Then the weight of ϕ (and hence half the constant term of f) is equal to 12aβ,m · tλ,v. ⊥ ∨ H(±β,m)∈supp(D) ΓL·λ ∈H(β,m) v∈A λ primitive v=λ q(v)>q(λ)

Proof. For W an isotropic plane corresponding to a rational boundary com- 1 ⊥ 1 ⊥ ponent representing X , we can write L = L0 A = 1 A with W 1 spanned by the ﬁrst two basis vector. As in Section 6.2, the inclusion of L0 into L induced from the ﬁrst 4 basis vectors leads to a map between con- \ + \ + nected Shimura varieties ΓL0 YL0 and ΓL Y that is described on the relative

162 Hermitian symmetric domains H × H ⊂ Y + by

→− t t [−ω1ω2 : ω2 : ω1 :1]→ [−ω1ω2 : ω2 : ω1 :1: 0].

We know that that this map extends to the compactiﬁed varieties, which in

1 1 the case of L0 is just the space P × P , and the restriction of this map to each cusp is an isomorphism onto its image (although the map may send the \ + \ + two 1-dimensional cusps of ΓL0 YL0 to the same cusp of ΓL Y in general). It therefore suﬃces to consider the pullback of ϕ to P1 × P1 and calculate its restriction to P1 × {∞}, which we identify as the closure of X. This has the advantage that the space P1 × P1 is a nonsingular variety and therefore we can calculate intersections at algebraically equivalent cycles, for example

P1 ×{ 0} 0 ∈ ω2 , for any ω2 H. We remark that since the closure of D does not contain the cusp corresponding to the image P1 × {∞}, it also does not contain the cusp corresponding to the image of {∞} × P1 as these two cusps correspond to the same “pairing” described in Proposition 7.1.9 (and may in fact be the same cusp). Therefore intersections can be determined completely by looking at the coordinates of the Hermitian symmetric domain H × H given above. Suppose that H(β,m) is one of the divisors appearing in D and suppose λ ∈ L∨ is a primitive vector such that q(n · λ)=m, n · λ = β for some integer n.

⊥ ∩ × Lemma 7.2.3. λ H H = φ iﬀ q(πL0 (λ)) < 0, where πL0 is the projection

∨ ∨ ∨ map from L to L0 induced from the decomposition L = L0 ⊥ A as above.

In the case when the intersection is non-empty, we have 0 ≥ q(πA∨ (λ)) >q(λ).

163

4 ∈ ∨ ⊥ ∩ × Proof. Observe that if λ =( i=1 aiei)+v, with v A , then λ H H equals

→− a3ω2 + a1 {z =[−ω1ω2 : ω2 : ω1 :1: 0]∈ P(V ⊗ C): ω1,ω2 ∈ H,ω1 = }. a4ω2 − a2

If ω ∈ , then a3ω2+a1 lies in iff a a + a a < 0. This proves the first 2 H a4ω2−a2 H 1 4 3 2 statement. The second statement follows from the first and the fact that q(πL0 (λ)) + q(v)=q(λ).

4 We return to the proof of Theorem 7.2.2. Writing λ =( i=1 aiei)+v as above, we observe that because A is negative deﬁnite, there are only a ﬁnite number of possible v ∈ A∨ that satisfy 0 ≥ q(v) >q(λ) and λ = v.Asv runs over all the possible vectors satisfying these two conditions, the ΓL-orbit of λ is partitioned into the following sets ⊥ Aλ,v {η ∈ ΓL · λ : η ∩ H × H = φ} v where

Aλ,v := {η ∈ ΓL · λ : η − v ∈ L0}.

Note that this partition is invariant under the action of ΓL0 . ×{ 0}⊂ × Suppose we fix a line := H w2 H H and consider its intersection ⊥ ⊥ with η for all η in some fixed Aλ,v. By the lemma, each η intersects properly at exactly the point b ω0 + b 3 2 1 ,ω0 , 0 − 2 b4ω2 b2 where η = biei + v. On the other hand, two such points of intersection are \ + ∈ Z identified in the quotient space ΓL0 YL0 iff there exists matrix M SL2( )

164 such that b b b3 b1 3 1 ∈ SL (Z) · . 2 b4 −b2 b4 −b2 Observe that the determinant of the above matrices must be equal to the negative of the norm of the vector (b1,b2,b3,b4) ∈ L0, which is precisely the diﬀerence q(v) − q(λ), and so is independent of η ∈ Av. Furthermore, the sets

4 Aλ,v can be identiﬁed with the sets Sλ,v by the map ⊥ b3 b1 η → (η = biei + v). b4 −b2

⊥ We can conclude that the union of η ∩ over all η in Aλ,v contains exactly tλ,v ± distinct ΓL0 -orbits. By summing over all appropriate H( β,m) in the support

⊥ of D, irreducible components ΓL · λ of each H(β,m), and appropriate v for each λ, this shows that the divisor of the pull-back of ϕ to P1 × P1 intersects the image of at exactly aβ,m · tλ,v ⊥ ∨ H(±β,m)∈supp(D) ΓL·λ ∈H(β,m) v∈A λ primitive v=λ q(v)>q(λ) points, counted with multiplicity. Therefore, the restriction of ϕ to the closure of the 1-dimensional cusp X vanishes with multiplicity equal to this value. On the other hand, ϕ is a modular form with character, so the same holds for its

restriction to X . But X is isomorphic to SL2(Z)\H and the restricted map has a zero or pole only at inﬁnity, hence this restriction must be a multiple of the appropriate power of the holomorphic cusp form Δ. Since the weight of the restricted map is equal to the weight of the original map, the result follows.

4 v ∨ η ∨ Note that if p|bi for all i and p lies in L , then p must also lie in L , which is impossible since λ is primitive.

165 Proposition 7.2.4. In the notation of the previous theorem, suppose that q(λ) − q(v)=N 2 · s for squarefree s.If(N,B(v, A)) = 1, then 2 tλ,v = σ1(N · s)= d. d|N 2·s

Proof. When (N,B(v, A))=1, the set Sλ,v is equal to the set of 2x2 integral matrices of determinant n := q(λ) − q(v).

Recall the action of the Hecke operator Tn on the free abelian group generated by lattices of C (here by lattice we just mean free Z-modules of rank

2). In this setting, Tn sends a lattice to the sum of its sublattices of index n. The degree of this operator is σ1(n), see for example Proposition 32 of

ab [31]. On the other hand, an integer matrix ( cd) of determinant n represents a sublattice of index n of the lattice Z[i]= 1,i via the association

ab → ( cd) a + ib, c + id .

Two such matrices map to the same sublattice if and only if there is a change of basis map xy · a+ib a+ib ( zw) c+id = c+id .

xy Since the map must be orientation preserving, we conclude ( zw) ∈ SL2(Z). Since every sublattice of Z[i] of index n can be realized in this way, we conclude that the degree of Tn is equal to tλ,v as claimed.

7.3 Discussion and examples

Given a modular form ϕ on a Shimura variety X =Γ\D and a cusp X in the Baily-Borel compactiﬁcation of X, the restriction map ϕ → ϕ|X is known as the Siegel operator and sends the vector space of modular forms on

166 X of a given weight k (ie. holomorphic functions from Γ × D to C satisfying

k f(γ·x)=jD(γ,x) f(x) for a certain factor of automorphy jD(γ,x)) to modular forms on X of the same weight. Clearly the kernel of this map is precisely the set of modular forms that vanish along X. In the case of orthogonal Shimura varieties, the image of Borcherds products under the Siegel operator corresponding to a 1-dimensional cusp will either be trivial or will be a modular unit, as special divisors either contain a 1-dimensional cusp or are disjoint from it. We have seen that using an arithmetic group ΓL associated to a maximal lattice L results in all the 1-dimensional cusps being isomorphic to the modular curve Y (1) parameterizing abelian curves without level structure. Therefore, up to a scalar, the only possible modular units are powers of Δ, which is the reasoning that we used in the previous theorem. If one does not restrict to maximal lattices, then it is possible to have 1-dimensional cusps isomorphic to modular curves with level structure (for example Y (N 2) as we have seen from scaled lattices). In this case, the Siegel operator on Borcherds products may possibly have other modular units besides Δ in its image.

Example 7.3.1. We consider the case when (L, q)=H ⊥ H ⊥ −2, which is the special case of Example 7.1.3 where t = 1 and so parameterizes principally ∼ Z Z 1 · polarized abelian surfaces. Since Δ(L) = /2 is generated by 2 e5, we see ∈ ∨ − m ≡ that the norm of an element v L must be of the form 4 for m 0, 1 mod 4 and m ≡ 0iﬀv ≡ 0 ∈ Δ(L). Recall from Example 7.1.3 that for t = 1, every Humbert surface contains exactly 1 irreducible component. It − m follows that the irreducible components of the special divisor H(β, 4 ) are the m 2 Humbert surfaces of invariant s2 , where s runs through the square divisors of m ≡ m such that s2 0, 1 mod 4. By Example 7.1.11, we know that the Humbert surface of index m contains the unique 1-dimensional cusp iﬀ m is a square.

167 Therefore the same can be said about the special divisor H(β,−m/4), since

m s2 will be a square only when m is. In the context of Borcherd’s lifts, we will be interested in studying the space of vector-valued weakly holomorphic modular forms of weight −1/2 with respect to the Weil representation of q. The space of such modular forms can be identiﬁed with weight 0, index 1 weak Jacobi forms (see for example pg. 57-59 of [12] or Example 1.3 of [9]). In a similar way, the obstruction space from Theorem 7.2.1 consisting of C[Δ(L)]-cusp forms of weight 5/2 can be identiﬁed with a subspace of weight 3, index 1 skew holomorphic Jacobi cusp forms. But the space of such cusp forms is trivial (by Main Theorem of [50], it maps injectively into the cusp forms of weight 4 for the modular curve Y (1), which is a trivial vector space). Since β = −β for all β ∈ Δ(L), we can conclude from Theorem 7.2.1 that every special divisor is twice the divisor of a unique Borcherd’s lift.

We shall now look more closely at H(β,−m/4) for some non-square values of m and apply Theorem 7.2.2 and Proposition 7.2.4.

≡ e5 − A. The ﬁrst non-square value for m is 5. Then β 2 mod L and H(β, m/4) ⊥ contains only one irreducible divisor, the image mod ΓL of λ , where λ =

− e5 ∨ e5 e1 e2 + 2 . There are only 2 vectors in A congruent with 2 and having norm − ± e5 −1 greater than 5/4, namely 2 , each of which has norm 4 . For either choice, we are looking at the orbits of integral matrices of determinant 5/4 − 1/4=1, acted on by SL2(Z) from the left. Clearly there is only 1 such orbit. By 1 − Theorem 7.2.2, the weight of the Borcherds lift whose divisor is 2 H(β, 5/4) 2· − · 2· · · E4 E4,1 E6 E6,1 88E4 E4,1+56E6 E6,1 is therefore 12 (1+1) = 24. Let T1 = 144Δ and T4 := 144Δ be Jacobi forms corresponding to the notation set on pg. 17 of [12]. Then the

1 e5 − weak Jacobi form lifting to the Borcherds product with divisor 2 H( 2 , 5/4)

168 is E5 · E − E3 · E T := 4 4,1 6 6,1 − 22T + 376T 5 144Δ2 4 1 − 5 3 and the q-expansion of T5 is q 4 +48+O(q 4 ).

B. The next possible value for m is 8, which is non-square and therefore gives rise to a special divisor that does not contain the 1-dimensional cusp.

Necessarily β ≡ 0modL and again there is a unique ΓL-orbit in H(β,−m/4)

8 because the only non-trivial square divisor of 8 is 4, and 4 = 2 is not congruent with 0 or 1 mod 4. This time there are 3 vectors in A∨ congruent with 0 mod

A and having norm greater than −2, namely 0 and ±e5. In the ﬁrst case we are looking at the orbits of integral matrices with determinant 2 − 0=2. By applying Proposition 7.2.4, we see that there are 3 such orbits. In the latter two cases, we again have determinant 1 and so we have 1 orbit in each 1 − case. Thus the weight of the Borcherds lift whose divisor is 2 H(β, 8/4) is 12 · (3 + 1 + 1) = 60. The associated weak Jacobi form is

88E5 · E +56E3 · E T := 4 4,1 6 6,1 − 142T − 91520T 8 144Δ2 1 4

− 3 with q-expansion q 2 + 120 + O(q 4 ). C. Since 9 is a square, the associated special divisor H(β,−9/4) will contain our 1-dimensional cusp. Therefore we skip it and move on to consider the case when m =12. Again β ≡ 0modL and by the same reasoning as above, H(β,−m/4) consists of a single component. The same 3 choices for vectors in A∨ appear as in the previous case. For 0, we must consider determinant 3 matrices (4 orbits by the proposition) while for ±e5, we must consider matrices of determinant 3 − 1 = 2 (3 orbits each). Note that Proposition 7.2.4 still applies, despite B(e5,A)=2, because the determinant is not divisible by 1 − 4. The weight of the Borcherds lift with divisor 2 H(β, 12/4) is therefore

169 12 · (4 + 3 + 3) = 120. The associated weak Jacobi form is

88E8 · E +56E5 · E T := 4 4,1 6 6,1 +12647424T −2433360T −150656T −214T 12 144Δ3 1 4 5 8

− 3 with q-expansion q 3 + 240 + O(q 4 ).

≡ e5 D. We conclude with the case when m =13,β 2 mod L. Since m is prime, H(β,−m/4) is irreducible. Now we have four possibilities for vectors in A∨ :

± e5 ± 3e5 2 , 2 . The ﬁrst two cases correspond to matrices of determinant 3 (4 orbits each by Proposition 7.2.4), while the second pair of cases corresponds to determinant 1 matrices (1 orbit each). The weight of the Borcherds lift with 1 − · divisor 2 H(β, 13/4) is therefore 12 (4+4+1+1)=120. Let T9 be the weak Jacobi form

E8 · E − E5 · E T := 4 4,1 6 6,1 − 2017383T + 7332T + 688T − 34T . 9 144Δ3 1 4 5 8

The associated weak Jacobi form is then

11· − 7· E4 E4,1 E6 E6,1 − T13 := 144Δ4 + 1037987416T1 57813856T4

−4236795T5 + 15372T8 + 1000T9 − 46T12

− 13 3 with q-expansion q 4 + 240 + O(q 4 ). All these values were checked explicitly using the MAGMA algebra system by constructing the Jacobi form with prescribed principal components and checking its constant term.

Remark 7.3.2. Comparing with §8 of [17], we see that the values of the weights calculated in the previous example can be realized as Fourier coeﬃcients of a certain modular form of weight 5/2 on Γ0(4) introduced by Cohen. Addi- tionally, these values are proportional to the volume of the special divisors

H(β,−m/4), relative to the product of the Poincar`e metrics on H × H (the Hermitian symmetric domain associated to each irreducible component). This

170 suggests that in general, an intersection pairing between Heegner divisors and the 1-dimensional cusps should in some way reﬂect the volume of these divisors.

171 CHAPTER 8 Conclusion

The motivation of this thesis was to explore orthogonal Shimura varieties (particularly their boundary) and try to derive results analogous to those already known for symplectic or unitary Shimura varieties. We were able to obtain algebraic and geometric results for the very general class of varieties arising from quadratic forms that split two hyperbolic planes, which encompasses all possible orthogonal Shimura varieties of dimension at least 5. Specific results include counting cusps from arithmetic groups arising from maximal sublattices as well as determining the fields of definition for connected components of these varieties and their boundary components. Although these results were as general as possible, we further explored some special examples where our rational form was of a specific type, including the unimodular case.

Many of the results presented in this thesis suggest additional questions for future study. One problem that could be pursued is the question of the number of 1-dimensional cusps for arithmetic groups arising from non-maximal lattices. In the case of a lattice L with cyclic discriminant group, such a lattice is contained in a unique maximal lattice (say M), and there is a linear ordering on lattices between L and M deﬁned through inclusion. Given any isotropic plane W in L⊗Q, there is a minimal lattice between L and M that decomposes with respect to W as in Lemma 5.2.1, and this same lattice holds for all planes in the ΓL-orbit of W . It is quite possible that a result similar to Theorem 5.2.9 holds for such lattices once one takes into account this extra invariant.

172 There is also the question of studying the properties of Shimura varieties for rational spaces that split only a single hyperbolic plane. Although some of the motivating examples for this thesis are special cases of forms of this type (modular curves and Hilbert modular surfaces for example), in the end, many of our proofs relied heavily on the fact that the automorphism structure of H ⊥ H is well-known and can be used to provide structural results on the orthogonal groups of lattices splitting two hyperbolic planes. The structure imposed by only a single copy of H is much less useful, as the group O(H) has only 4 elements. It is possible that in order to get concrete results in this case, one would have to restrict to speciﬁc interesting families of rational quadratic forms of this type, and combine the general theory of orthogonal Shimura varieties discussed in this thesis along with additional structure imposed by these families (for example comparing 0-dimensional cusps in Hilbert modular surfaces to the ideal class group of the associated real number ﬁeld).

Another potential topic to pursue is to further study the PEL-Shimura variety describing abelian varieties with even Clifford algebra endomorphism structure discussed in Chapters 3 and 4. Since PEL-type Shimura varieties have much more structure than a general Shimura variety, there are more tools available to be used to answer various questions than in the generic case, and even if one is only interested in orthogonal Shimura varieties, it is quite possible that studying these larger spaces will produce results for the orthogonal sub-Shimura varieties embedded within them. One specific example that immediately comes to mind is further progress in the Kudla program, whose goal can be summarized as attempting to find modular forms with values in arithmetic Chow groups associated to integral models of Shimura

173 varieties. This program has seen great success for low dimensional orthogonal Shimura varieties, due in large part to the fact that such spaces have the

PEL-interpretation aﬀorded to them by their isomorphism with the group G from Chapter 4. It would be of interest to try and adapt the methods for the low dimensional orthogonal Shimura varieties to the larger PEL-space G in higher dimensions. One could even hope to bootstrap some results down to the embedded orthogonal sub-Shimura variety.

Finally, it would be remiss if we did not mention the further study and development of the theory of special divisors and their interaction with the boundary discussed in Chapter 7.1. Despite the boundary lying in the singular locus of our variety, we have seen that we can sometimes still attach an algebraic meaning to the intersection of the boundary with certain linear combinations of special divisors. It would be interesting to see if this theory can be made more robust and what it says about the Borcherd’s products lying on these spaces. We saw in Chapter 7.1 that the compactification of such divisors all intersect non-trivially at the boundary. It is an interesting question to consider the common locus shared by these divisors and compare it to the negative definite sublattices that can appear within a given L. The study of this locus or a similar space could reveal certain structural restrictions on Borcherd’s products that vanish along these divisors. Another avenue to pursue would be to consider generalizing special divisors to the encompassing PEL-Shimura variety as we discussed previously in Remark 4.5.1. Recall that the irreducible components of special divisors arise from certain decompositions of the vector space V into Vλ ⊥ λ and the inclusion of GSpin(Vλ)into GSpin(V ). This is known to generalize to cycles of arbitrary codimension by replacing λ with a negative definite subspace U of arbitrary dimension, ie.

174 V = VU ⊥ U and GSpin(VU ) → GSpin(V ). It is interesting to note that this same decomposition leads to an embedding of Shimura varieties associated to the groups G(VU ) → G(U) (where G is our group from Chapter 4) and G(VU ) ∩ GSpin(V ) = GSpin(VU ). The codimensions of the cycles generated in this way can be determined by looking at Table 4–1. In particular, if the signature of V is (2,n), then when n ≡ 2 or 6 mod 8, the analogues of the special divisors (ie. the cycles corresponding to G(V λ )) have codimension exactly half the dimension of the encompassing space. This suggests that their intersection pairing may provide interesting arithmetic or algebraic data, generalizing the study of the intersection pairing of special divisors on Hilbert modular surfaces (see for example [23]), which is the case when n = 2. Note however that the intersection of any two of these cycles is not proper as it contains the intersection of the two corresponding special divisors in the embedded orthogonal Shimura variety. In any case, the intersection number of these two cycles must reﬂect data derived entirely from the lattice L and a choice of 2 vectors of negative norm in L. It would be very interesting to see precisely what this data is.

175 Index of Notation

L ⊥ L,8 SO(L), 9 (G, X), 67 SO+(V ), 25 (G, X+), 41 U(F ), 55 (L, q), 6 V (F ), 55 Aadj,21 W (F ), 54 AW , 106 Δ(L), 8 B(x, y), 7 GSpn,69 C(F ), 55 GSpin(L), 21 C(V ), 15 GSpin(V ), 18 + C(V ) ,16 ΓL,9 Cr,s,82 Gr(V ⊗ R), 49 D,45 Ω(A), 112 D(Γ), 51 ShK (G, X), 68 ∗ D(Γ) ,52 Spn,43 D∨,53 Spin(L), 21 Dc,53 Spin(V ), 18 Dr(O), 56 det(L), 8 E(G, X), 74 gen(L), 27 G(O∗(2.2r)2), 98 ι,16 2 H G(Sp2r−1 ), 96 ,10 G(Q)+,68 S,40 G(R)+,68 D,55 GO∗(2.2r), 91 H,59 r−1 r−1 GU(2 , 2 ), 90 Hn,47 G0,35 g,38 GDer,36 D,54 Gad,37 q,8 Gh(F ), 55 ∂D,52 Gl(F ), 55 k(N), 83 G2,n,83 nL,12 H(β,m), 148 qr,s,25 L∨,8 sL,12 L0,21 v(x), 19 M(F ), 55 M ⊥ N,44 O(L), 9 P (F ), 53 PW , 107 R,7

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