An example-based introduction to Shimura varietiesjavier.fresan.perso.math.cnrs.fr/lan.pdf · An...

An example-based introduction to Shimuravarieties

Kai-Wen Lan

Abstract. In this introductory article, we will explain what Shimuravarieties are and why they are useful, with an emphasis on examples. Wewill also describe various compactifications of Shimura varieties, againwith an emphasis on examples, and summarize what we know about theintegral models of Shimura varieties and their compactifications.

Contents

1. Introduction 3

2. Double coset spaces 5

2.1. Algebraic groups 5

2.2. Manifolds 7

2.3. Shimura data 10

2.4. Shimura varieties and their canonical models 12

3. Hermitian symmetric domains 15

3.1. The case of Sp2n(R) 15

3.1.1. Siegel upper half-spaces 15

3.1.2. Transitivity of action 17

3.1.3. Bounded realization 17

3.1.4. Interpretation as conjugacy classes 18

3.1.5. Moduli of polarized abelian varieties 19

3.2. The case of Ua,b 20


The author was partially supported by the National Science Foundation under agreementNo. DMS-1352216, and by an Alfred P. Sloan Research Fellowship. Any opinions, findings,

and conclusions or recommendations expressed in this article are those of the author anddo not necessarily reflect the views of these organizations.

2 Kai-Wen Lan

3.2.2. Unbounded realization 21

3.2.3. The special case where a = b: Hermitian upper half-spaces 22

3.2.4. Generalized Cayley transformations and transitivity of actions 23


3.3. The case of SO∗2n 25

3.3.1. Unbounded realization, and interpretation as a conjugacy class 25

3.3.2. The special case where n = 2k: quaternion upper half-spaces 26

3.4. The case of SOa,2(R) 26

3.4.1. Projective coordinates 26

3.4.2. The case where a ≥ 1 and b = 2 27

3.4.3. Connected components and spin groups 28


3.5. The case of E7 29

3.5.1. Cayley numbers 30

3.5.2. An exceptional Jordan algebra over R 30

3.5.3. A real Lie group with Lie algebra e7(−25) 31

3.5.4. An octonion upper half-space 32


3.6. The case of E6 34

3.6.1. Bounded and unbounded realizations 34

3.6.2. A real Lie group of Lie algebra e6(−14) 35


3.7. A brief summary 36

4. Rational boundary components and compactifications 37

4.1. Examples of rational boundary components 38

4.1.1. Cases with D = H 38

4.1.2. Cases with D = H×H 38

4.1.3. Cases with D = Hn 39

4.1.4. Cases with D = Ha,b, with a ≥ b 43

4.1.5. Cases with D = HSO∗2n46

4.1.6. Cases with D+ = H+SOa,2

, with a ≥ 2 48

4.1.7. Cases with D = HE6 or D = HE7 51

4.2. Compactifications and algebraicity 52

4.2.1. Overview 52

4.2.2. (Γ\D)min

= Γ\D∗ 52

4.2.3. (Γ\D)torΣ 52

4.2.4. (Γ\D)BS 53

An example-based introduction to Shimura varieties 3

4.2.5. (Γ\D)RBS 54

4.2.6. Models over number fields or their rings of integers 54

4.2.7. Algebro-geometric definition of modular forms 54

4.2.8. Mixed Shimura varieties 55

5. Integral models 56

5.1. PEL-type cases 56

5.1.1. PEL datum 56

5.1.2. Smooth PEL moduli problems 59

5.1.3. PEL-type Shimura varieties 59

5.1.4. Toroidal and minimal compactifications of PEL moduli 62

5.1.5. Integral models not defined by smooth moduli 63

5.1.6. Langlands–Rapoport conjecture 65

5.2. Hodge-type and abelian-type cases 65

5.2.1. Hodge-type Shimura varieties 65

5.2.2. Abelian-type Shimura varieties 66

5.2.3. Integral models of these Shimura varieties 69

5.2.4. Integral models of toroidal and minimal compactifications 70

5.3. Beyond abelian-type cases 70

5.4. Beyond Shimura varieties? 72

Acknowledgements 72

References 72

Index 82

1. Introduction

Shimura varieties are generalizations of modular curves, which have playedan important role in many recent developments of number theory. Just tomention a few examples with which this author is more familiar, Shimuravarieties were crucially used in the proof of the local Langlands conjecturefor GLn by Harris and Taylor (see [64]); in the proof of the Iwasawa mainconjecture for GL2 by Skinner and Urban (see [173]); in the construction ofp-adic L-functions for unitary groups by Eischen, Harris, Li, and Skinner (see[63] and [45]); in the construction of Galois representations (in the context ofthe global Langlands correspondence) for all cohomological automorphic rep-resentations of GLn by Harris, Taylor, Thorne, and this author (see [62]); andin the analogous but deeper construction for torsion cohomological classesof GLn by Scholze (see [158]).

4 Kai-Wen Lan

One of the reasons that Shimura varieties are so useful is because theycarry two important kinds of symmetries—the Hecke symmetry and the Ga-lois symmetry. Roughly speaking, the Hecke symmetry is useful for studyingautomorphic representations, the Galois symmetry is useful for studying Ga-lois representations, and the compatibility between the two kinds of symme-tries is useful for studying the relations between the two kinds of representa-tions, especially in the context of the Langlands program.

However, the theory of Shimura varieties does not have a reputationof being easy to learn. There are important foundational topics such as thenotion of Shimura data and the existence of canonical models, and coursesor seminars on Shimura varieties often spend a substantial amount of thetime on such topics. But learning these topics might shed little light on howShimura varieties are actually used—in some of the important applicationsmentioned above, the “Shimura varieties” there were not even defined usingany Shimura data or canonical models. In reality, it is still a daunting taskto introduce people to this rich and multifaceted subject.

In this introductory article, we will experiment with a somewhat differ-ent approach. We still urge the readers to read the excellent texts such as [36],[39], and [119]. But we will not try to repeat most of the standard materials,such as variation of Hodge structures, the theory of canonical models, etc,which have already been explained quite well by many authors. We simplyassume that people will learn them from some other sources. Instead, we willtry to explain the scope of the theory by presenting many examples, andmany special facts which are only true for these examples. Our experienceis that such materials are rather scattered in the literature, but can be veryhelpful for digesting the abstract concepts, and for learning how to apply thegeneral results in specific circumstances. We hope that they will benefit thereaders in a similar way.

We will keep the technical level as low as possible, and present mainlysimple-minded arguments that should be understandable by readers who arewilling to see matrices larger than 2×2 ones. This is partly because this arti-cle is based on our notes for several introductory lectures targeting audiencesof very varied backgrounds. The readers will still see a large number of tech-nical phrases, often without detailed explanations, but we believe that suchphrases can be kept as keywords in mind and learned later. We will providereferences when necessary, but such references do not have to be consultedimmediately—doing that will likely disrupt the flow of reading and obscurethe main points. (It is probably unrealistic and unhelpful to try to consultall of the references, especially the historical ones.) We hope that the readerswill find these consistent with our goal of explaining the scope of the theory.There will inevitably be some undesirable omissions even with our limitedgoal, but we hope that our overall coverage will still be helpful.


Here is an outline of the article. In Section 2, we define certain doublecoset spaces associated with pairs of algebraic groups and manifolds, and in-troduce the notion of Shimura data. We also summarize the main results con-cerning the quasi-projectivity of Shimura varieties and the existence of theircanonical models. In Section 3, we give many examples of Hermitian sym-metric domains, which provide the connected components of the manifoldsneeded in the definition of Shimura data. Our examples actually exhaust allpossibilities of them. In Section 4, we give many examples of rational bound-ary components and describe the corresponding minimal compactifications.We also summarize the properties of several kinds of useful compactifications.In Section 5, we explain how the so-called integral models of Shimura varietiesare constructed. This is very sensitive to the types of Shimura varieties weconsider, and we also take this opportunity to summarize many special factsand many recent developments for each type of them. For the convenience ofthe readers, we have also included an index at the end of the article.

2. Double coset spaces

2.1. Algebraic groups

Let Z := lim←−N (Z/NZ), A∞ := Z ⊗Z Q, and A := R×A∞. Suppose G is

a reductive algebraic group over Q. Then there is a natural way to definethe topological groups G(R), G(A∞), and G(A) = G(R)×G(A∞), with thetopology given either as a restricted product (following Weil) (see [138, Sec-tion 5.1]), or as an affine scheme (of finite presentation) over the topologicalring A (following Grothendieck), and the two approaches coincide in this case(see [33] for a detailed discussion on this matter). We will not try to defineand summarize all the needed properties of reductive linear algebraic groups(from textbooks such as [15] and [175]). Instead, we will just provide manyexamples of such groups.

Example 2.1.1 (general and special linear groups). For each ring R and eachinteger n ≥ 0, we define GLn(R) to be the group of invertible n×n matriceswith entries in R, and define SLn(R) to be the subgroup of GLn(R) formed bymatrices of determinant one. When n = 1, we have GL1(R) = Gm(R) := R×.The assignments of GLn(R) and SLn(R) are functorial in R, and in fact theyare the R-points of the affine group schemes GLn and SLn over Z, respec-tively. The pullbacks of these group schemes to Q are affine algebraic varietiesserving as prototypical examples of reductive algebraic groups. Moreover, thepullback of SLn to Q is a prototypical example of a semisimple algebraicgroup.

For simplicity of exposition, we will often introduce a linear algebraicgroup G by describing it as the pullback to Q of some explicitly given closedsubgroup scheme of GLm, for some m, over Z. By abuse of notation, we willoften denote this group scheme over Z by the same symbol G, which is then

6 Kai-Wen Lan

equipped with a fixed choice of a faithful matrix representation G ↪→ GLm.The readers should not be too worried about these technical terminologiesconcerning group schemes over Z. Concretely, they just mean we will explic-itly represent the elements of G(R) as invertible m × m matrices, for eachring R, whose entries satisfy certain defining conditions given by algebraicequations that are compatible with all base ring extensions R→ R′.

Example 2.1.2 (symplectic groups). Let n ≥ 0 be any integer, and let 0nand 1n denote the zero and identity matrices of size n. Consider the skew-symmetric matrix

Jn :=

(0n 1n−1n 0n

)(2.1.3)

of size 2n. Then the assignments

Sp2n(R) := {g ∈ GL2n(R) : tgJng = Jn} (2.1.4)

and

GSp2n(R) := {(g, r) ∈ GL2n(R)×R× = (GL2n×Gm)(R) : tgJng = rJn}(2.1.5)

are functorial in R, and define closed subgroup schemes Sp2n and GSp2n ofGL2n and GL2n+1, respectively. Here we view GL2n×Gm as a closed sub-group scheme of GL2n+1 by block-diagonally embedding the matrix entries.To better understand the meaning of these definitions, consider the alternat-ing pairing

〈 · , · 〉 : R2n×R2n → R (2.1.6)

defined by setting

〈x, y〉 = txJny, (2.1.7)

for all x, y ∈ R2n (written as vertical vectors). For each g ∈ GL2n(R), wehave

〈gx, gy〉 = tx tgJngy. (2.1.8)

Therefore, g satisfies the condition tgJng = Jn exactly when

〈gx, gy〉 = 〈x, y〉, (2.1.9)

for all x, y ∈ R2n. That is, g preserves the above pairing 〈 · , · 〉. Note that wehave not used exactly what Jn is, except for its skew-symmetry. Similarly,(g, r) ∈ GL2n(R)×R× satisfies the condition tgJng = rJn exactly when

〈gx, gy〉 = r〈x, y〉, (2.1.10)

for all x, y ∈ R2n. That is, g preserves the above pairing 〈 · , · 〉 up to thescalar factor r. The assignment of r to (g, r) defines a homomorphism

ν : GSp2n(R)→ Gm(R), (2.1.11)

called the similitude character of GSp2n(R), whose kernel is exactly Sp2n(R),for each ring R. We can define similar groups and homomorphisms by replac-ing Jn with any other matrices (with entries over Z, if we still want to definea group scheme over Z). In fact, many people would prefer to define Sp2n and


GSp2n using anti-diagonal matrices (just not a “block anti-diagonal matrix”

like the above Jn =

(0n 1n−1n 0n

)).

Example 2.1.12 (orthogonal groups and special orthogonal groups). Considerany integers n, a, b ≥ 0, and consider the symmetric matrices 1n and

1a,b :=

(1a−1b

), (2.1.13)

with empty slots being filled up by zero matrices of suitable sizes. Then theassignments

On(R) := {g ∈ GLn(R) : tgg = 1n}, (2.1.14)

SOn(R) := On(R)∩SLn(R), (2.1.15)

Oa,b(R) := {g ∈ GLn(R) : tg1a,bg = 1a,b}, (2.1.16)

SOa,b(R) := Oa,b(R)∩SLa+b(R) (2.1.17)

are functorial in R, and define closed subgroup schemes On, SOn, Oa,b,and SOa,b of GLn, GLn, GLa+b, and GLa+b, respectively. As in the caseof Sp2n(R) above, On(R) and Oa+b(R) are the group of matrices preservingthe symmetric bilinear pairings defined by 1n and 1a,b, which have signatures(n, 0) and (a, b), respectively.

2.2. Manifolds

Let us fix a choice of G, and consider any manifold D with a smooth tran-sitive action of G(R). For technical reasons, we shall consider only algebraicgroups G over Q each of whose connected components in the Zariski topol-ogy contains at least one rational point. This requirement is satisfied by everyconnected algebraic group, because the identity element is always rational,and also by all the examples in Section 2.1. Then G(Q) is dense in G(R) inthe real analytic topology, by applying a special case of the weak approxima-tion theorem as in [138, Section 7.3, Theorem 7.7] to the identity componentof G (namely, the connected component containing the identity element) inthe Zariski topology, and by translations between this identity componentand other connected components.

Example 2.2.1. For each integer n ≥ 0, consider the manifold

Sn := {X ∈ Symn(R) : X > 0, det(X) = 1}= {X ∈ Mn(R) : tX = X > 0, det(X) = 1},

(2.2.2)

where Mn (resp. Symn) denotes the space of n×n matrices (resp. symmetricmatrices), and where X > 0 means being positive definite. Consider theaction of g ∈ SLn(R) on Sn defined by

X 7→ tgXg, (2.2.3)

for each X ∈ Sn. This action is transitive because each positive definitematrix X is of the form

X = Y 2 = tY Y (2.2.4)

8 Kai-Wen Lan

for some positive definite matrix Y , and if det(X) = 1 then det(Y ) = 1 aswell. The stabilizer of X = 1n ∈ Sn is, by definition, SOn(R), and so we have

Sn = SLn(R) · 1n ∼= SLn(R)/SOn(R). (2.2.5)

Note that

dimR(Sn) = 12n(n+ 1)− 1. (2.2.6)

In the classification in [65, Chapter X, Section 6], Sn is a noncompact Rie-mannian symmetric space of type A I.

Example 2.2.7. When n = 2, we also have the familiar example of

g =

(a bc d

)∈ SL2(R) (2.2.8)

acting on the Poincare upper half-plane

H := {z ∈ C : Im(z) > 0} (2.2.9)

by the Mobius transformation

z 7→ gz =az + b

cz + d, (2.2.10)

for each z ∈ H. Note that this is actually induced by the natural SL2(C)

action on P1(C): If we identify z ∈ C with

(z1

)∈ P1(C), then(

a bc d

)(z1

)=

(az + bcz + d

)∼(gz1

). (2.2.11)

For readers who have studied modular forms, the factor (cz + d) involved inthe above identification between projective coordinates is exactly the sameautomorphy factor (cz + d) in the definition of holomorphic modular forms.(This is not just a coincidence.) It is well known that this action of SL2(R)on H is transitive (which will be generalized in Section 3.1.2 below), and thestabilizer of i ∈ H is{(

a −bb a

)∈ SL2(R)

}= SO2(R) =

{(cos θ − sin θsin θ cos θ

): θ ∈ R

}, (2.2.12)

so that we have

H = SL2(R) · i ∼= SL2(R)/SO2(R). (2.2.13)

Example 2.2.14. The isomorphisms H ∼= SL2(R)/SO2(R) ∼= S2 show that Hcan be viewed as a special case of Sn with n = 2. But this is the onlyexample of Sn, with n ≥ 1, such that Sn has a complex structure. SincedimR(Sn) = 1

2n(n+ 1)− 1 is odd when n = 3, for example, there is no hopefor S3 to be a complex manifold.

For any (G,D) as above, and for any open compact subgroup U ofG(A∞), we can define the double coset space

XU := G(Q)\(D×G(A∞))/U , (2.2.15)


where G(Q) acts diagonally on D×G(A∞) from the left-hand side, andwhere U only acts on G(A∞) from the right-hand side. The reader mightnaturally wonder why we need to consider a complicated double quotientas in (2.2.15). One important justification is that the group G(A∞) has anatural right action on the collection {XU}U , induced by

D×G(A∞)∼→ D×G(A∞) : (x, h) 7→ (x, hg), (2.2.16)

for each g ∈ G(A∞), which maps XgUg−1 to XU because h(gug−1)g = hgu,for all h ∈ G(A∞) and u ∈ U . Such an action provides natural Hecke actionson, for example, the limit of cohomology groups lim−→U H

∗(XU ,C). Such a

symmetry of {XU}U is what we meant by Hecke symmetry in the introduction(see Section 1). This is crucial for relating the geometry of such double cosetspaces to the theory of automorphic representations.

Let D+ be a connected component of D, which admits a transitiveaction of G(R)+, the identity component (namely, the connected componentcontaining the identity element) of G(R) in the real analytic topology. Let

G(Q)+ := G(Q)∩G(R)+, (2.2.17)

which is the subgroup of G(Q) stabilizing D+. Then

G(Q)+\(D+×G(A∞))/U → G(Q)\(D×G(A∞))/U (2.2.18)

is surjective because G(Q) is dense in G(R) (see the beginning of this Sec-tion 2.2), and is injective by definition. It is known (see [12, Theorem 5.1]) that

#(G(Q)+\G(A∞)/U) <∞, (2.2.19)

which means there exists a subset {gi}i∈I of G(A∞) indexed by a finite set Isuch that we have a disjoint union

G(A∞) =∐i∈I

G(Q)+ gi U . (2.2.20)

Then

XU ∼= G(Q)+\(D+×G(A∞))/U

=∐i∈I

G(Q)+\(D+×G(Q)+ gi U)/U

=∐i∈I

Γi\D+,

(2.2.21)

whereΓi := (giUg−1

i )∩G(Q)+, (2.2.22)

because γgiu = giu exactly when γ ∈ giUg−1i , for any γ ∈ G(Q)+, i ∈ I,

and u ∈ U .

Each such Γi is an arithmetic subgroup of G(Q); namely, a subgroupcommensurable with G(Z) in the sense that Γi ∩G(Z) has finite indices inboth Γi and G(Z). In fact, since U is an open compact subgroup of G(A∞),each such Γi is a congruence subgroup of G(Z); namely, a subgroup containingthe principal congruence subgroup ker(G(Z) → G(Z/NZ)) for some integer

10 Kai-Wen Lan

N ≥ 1. By definition, congruences subgroups are arithmetic subgroups. Notethat, although the definition of principal congruence subgroups depends onthe choice of some faithful matrix representation G ↪→ GLm over Z (seeSection 2.1), the definition of congruence subgroups does not, because con-gruence subgroups of G(Q) can be characterized alternatively as the intersec-tions of G(Q) with open compact subgroups of G(A∞). More generally, thedefinition of arithmetic subgroups of G(Q) does not depend on the choice ofthe faithful matrix representation either (see [13, 7.13] or [138, Section 4.1]).

If each Γi acts freely on D+, then XU is a manifold because D+ is.This is the case when each Γi is a neat arithmetic subgroup of G(Q) (asin [13, (17.1)]), which is in turn the case when U is a neat open compactsubgroup of G (as in [137, 0.6]). For reading the remainder of this article, itsuffices to know that, when people say such groups are neat (or, much lessprecisely, sufficiently small), they just want to ensure that the stabilizers ofgeometric actions are all trivial. For simplicity, we shall tacitly assume thatthe arithmetic subgroups or open compact subgroups we will encounter areall neat, unless otherwise stated.

Now the question is what each Γi\D+ is, after knowing that it is amanifold because Γi is neat. What additional structures can we expect fromsuch quotients?

Example 2.2.23. When G = SL2 and D = H as in Example 2.2.7, thegroups Γi are congruence subgroups of SL2(Z), each of which contains theprincipal congruence subgroup

Γ(N) := ker(SL2(Z)→ SL2(Z/NZ)) (2.2.24)

for some integer N ≥ 1. The quotients Γi\H are familiar objects called mod-ular curves, each of which admits a good compactification into a compactRiemann surface (or, in other words, complete algebraic curves over C) byadding a finite number of points, which are called cusps. (We will revisit thisexample in more detail in Section 4.1.1 below.)

Remark 2.2.25. We may replace the manifold D above with other geometricobjects over D that still have a smooth (but not necessarily transitive) actionof G(R). For example, when G = SL2 and D = H, we may consider the space

H× Symk(C2), and obtain natural vector bundles or local systems over XU .

2.3. Shimura data

Shimura varieties are not just any double coset spaces XU as above. Forarithmetic applications, it is desirable that each XU is an algebraic varietyover C with a model over some canonically determined number field.

This led to the notion of a Shimura datum (G,D) (see [39, 2.1.1] and[119, Definition 5.5]). Consider the Deligne torus S, which is the restrictionof scalars ResC/R Gm,C (as an algebraic group over R), so that S(R) = C×.Then we require G to be a connected reductive algebraic group, and require D


to be a G(R)-conjugacy class of homomorphisms

h : S→ GR, (2.3.1)

where the subscript “R” means “base change to R” as usual, satisfying thefollowing conditions:

(1) The representation defined by h and the adjoint representation of G(R)on g = Lie G(C) induces a decomposition

g = k⊕ p+⊕ p− (2.3.2)

such thatz ∈ S(R) = C× (2.3.3)

acts by 1, z/z, and z/z, respectively, on the three summands.

(2) h(i) induces a Cartan involution on Gad(R). (Here Gad denotes theadjoint quotient of G; namely, the quotient of G by its center.)

(3) GadR has no nontrivial Q-simple factor H such that H(R) is compact (or,

equivalently, such that the composition of h with the projection to H istrivial, because of the previous condition (2)).

In this case, any finite dimensional R-representation of G(R) defines a varia-tion of (real) Hodge structures, and D is a finite union of Hermitian symmet-ric domains. Without going into details, let us just emphasize that D andits quotients XU , for neat open compact subgroups U of G(A∞), are thencomplex manifolds, not just real ones.

For studying only the connected components D+ (of some D whichappeared in the last paragraph) and their quotients, it is easier to work withwhat is called a connected Shimura datum (G,D+) (see [119, Definition 4.22]),which requires G to be a connected semisimple algebraic group, and requiresD+ to be a Gad(R)+-conjugacy class of homomorphisms

h : S→ GadR , (2.3.4)

where Gad(R)+ denotes (as before) the identity component of Gad(R) in thereal analytic topology, satisfying the same three conditions (1), (2), and (3)as above.

A homomorphism as in (2.3.4) determines and is determined by a ho-momorphism

h : U1 → GadR , (2.3.5)

whereU1 := {z ∈ C : |z| = 1} (2.3.6)

is viewed as an algebraic group over R here, via the canonical homomorphism

S→ U1 (2.3.7)

induced by z 7→ z/z. In many cases we will study, the restriction of thehomomorphism (2.3.4) to U1 (now viewed as a subgroup of S) lifts to ahomomorphism

h : U1 → GR, (2.3.8)

12 Kai-Wen Lan

whose composition with GR → GadR then provides a square-root of the h

in (2.3.5). This is nonstandard, but sometimes more convenient, when weprefer to describe GR instead of Gad

R in many practical examples (to be givenin Section 3 below).

2.4. Shimura varieties and their canonical models

Now we are ready to state the following beautiful results, whose assertionscan be understood without knowing anything about their proofs:

Theorem 2.4.1 (Satake, Baily–Borel, and Borel; see [9] and [14]). Suppose(G,D) is a Shimura datum as in Section 2.3. Then the whole collection ofcomplex manifolds {XU}U , with U varying among neat open compact sub-groups of G(A∞), is the complex analytification (see [163, §2]) of a canonicalcollection of smooth quasi-projective varieties over C. Moreover, the analyticcovering maps

XU → XU ′ , (2.4.2)

when U ⊂ U ′, are also given by the complex analytifications of canonical finiteetale algebraic morphisms between the corresponding varieties.

Alternatively, suppose (G,D+) is a connected Shimura datum as inSection 2.3. Then analogous assertions hold for the collection {Γ\D+}Γ,with Γ varying among neat arithmetic subgroups of Gad(Q) that are con-tained in Gad(R)+. (If we consider also (G,D), where D is the correspond-ing G(R)-conjugacy class of any h in D+, then the connected componentsof XU , where U is a neat open compact subgroup of G(A∞), are all of theform Γ\D+, as explained in Section 2.2.)

Theorem 2.4.3 (Shimura, Deligne, Milne, Borovoı, and others; see [168], [36],[39], [116], [23], and [24]). Suppose (G,D) is a Shimura datum as in Sec-tion 2.3. Then there exists a number field F0, given as a subfield of C de-pending only on (G,D), called the reflex field of (G,D), such that the wholecollection of complex manifolds {XU}U , with U varying among neat open com-pact subgroups of G(A∞), is the complex analytification of the pullback to Cof a canonical collection of smooth quasi-projective varieties over F0, whichsatisfies certain additional properties qualifying them as the canonical modelsof {XU}U . Moreover, the analytic covering maps

XU → XU ′ , (2.4.4)

when U ⊂ U ′, are also given by the complex analytifications of canonicalfinite etale algebraic morphisms defined over F0 between the correspondingcanonical models.

Remark 2.4.5. Theorem 2.4.1 is proved by constructing the so-called Sa-take–Baily–Borel or minimal compactifications of XU or Γ\D+, which areprojective varieties over C. (See Sections 4.2.2 and 4.2.7 below.) The asser-tion that the quasi-projective varieties XU and Γ\D+ are defined over Q, the


algebraic closure of Q in C, can be shown without using the theory of canon-ical models, as in [46]. But these general results cannot explain why each XUis defined over some particular number field.

Remark 2.4.6. In Theorem 2.4.3, since each XU is algebraic and defined overF0, its arithmetic invariants such as the etale cohomology group H∗et(XU ,Q`),for any prime number `, carry a canonical action of Gal(Q/F0), where thealgebraic closure Q of Q in C contains F0 because F0 is given as a subfieldof C. (Recall that any base change between separably closed fields inducesa canonical isomorphism between the etale cohomology groups—see, for ex-ample, [38, Arcata, V, 3.3].) This is an instance of what we meant by Galoissymmetry in the introduction (see Section 1). Since the maps XU → XU ′ arealso algebraic and defined over F0, we have a canonical action of Gal(Q/F0)on lim−→U H

∗et(XU ,Q`), which is compatible with the Hecke action of G(A∞).

This is an instance of what we meant by the compatibility between Hecke andGalois symmetries in the introduction (see Section 1).

Remark 2.4.7. In Theorem 2.4.3, if we assume instead that (G,D+) is aconnected Shimura datum, and consider the collection {Γ\D+}Γ as in The-orem 2.4.1, then it is only true that each Γ\D+ is defined over a numberfield depending on both (G,D+) and Γ, but in general we cannot expect thewhole collection {Γ\D+}Γ, or just the subcollection formed by those Γ whosepreimage in G(Q) are congruence subgroups, to be defined over a single num-ber field depending only on (G,D). This is a generalization of the followingclassical phenomenon: When (G,D) = (SL2,H), and when interpreting thequotient Γ(N)\H as in Example 2.2.23 as a parameter space for complexelliptic curves Ez = C/(Zz + Z) with level N -structures

αN : (Z/NZ)2 ∼→ Ez[N ] :

(a, b) 7→ ( aN z + bN ) mod (Zz + Z),

(2.4.8)

where z ∈ H and Ez[N ] denotes the N -torsion subgroup of Ez, by identifyingEz with Eγz for all γ ∈ Γ(N), the value of the Weil pairing of αN ((1, 0), (0, 1))is a constant primitive N -th root ζN of unity on Γ(N)\H. (See the introduc-tion of [43].) We can only expect Γ(N)\H to have a model over Q(ζN ), ratherthan Q.

Remark 2.4.9. In the context of Theorem 2.4.3, people may refer to either thevarieties XU over C, or their canonical models over F0, or the projective limitsof such varieties over either C or F0, as the Shimura varieties associated withthe Shimura datum (G,D). For the sake of clarity, people sometimes saythat XU and its canonical model are Shimura varieties at level U . In thecontext of Remark 2.4.7, people often refer to quotients of the form Γ\D+,where Γ are arithmetic subgroups of

Gad(Q)+ := Gad(Q)∩Gad(R)+ (2.4.10)

14 Kai-Wen Lan

whose preimages under the canonical homomorphism G(Q) → Gad(Q) arecongruence subgroups of G(Q), as the connected Shimura varieties associ-ated with the connected Shimura datum (G,D+). We allow all such sub-groups Γ of Gad(Q)+, rather than only the images of the congruence sub-groups of G(Q), so that the connected Shimura varieties thus defined areuseful for studying all connected components of Shimura varieties.

Remark 2.4.11. By a morphism of Shimura data

(G1,D1)→ (G2,D2), (2.4.12)

we mean a group homomorphism G1 → G2 mapping D1 into D2. In caseD1 = G1(R) ·h0 is the conjugacy class of some homomorphism h0 : S→ G1,R,then this means the composition of h0 with G1,R → G2,R lies in the conjugacyclass D2. If U1 and U2 are open compact subgroups of G1(A∞) and G2(A∞),respectively, such that U1 is mapped into U2, then we obtain the corre-sponding morphism XU1 → XU2 between Shimura varieties. This morphismis defined over the subfield of C generated by the reflex fields of (G1,D1)and (G2,D2). There are also analogues of these assertions for connectedShimura data and the corresponding connected Shimura varieties.

Remark 2.4.13. An important special case of a morphism of Shimura dataas in Remark 2.4.11 is when the homomorphism G1 → G2 is injective, andwhen U1 is exactly the pullback of U2 under the induced homomorphismG1(A∞)→ G2(A∞). (It is always possible to find such a U2 for any given U1—see [36, Proposition 1.15]. The neatness of U2 can be arranged when U1 is neat,by taking U2 to be generated by the image of U1 and some suitable neat opencompact subgroup of G2(A∞).) In this case, XU1 is a closed subvariety of XU2 .Subvarieties of this kind are called special subvarieties. The most importantexamples are given by special points or (with some additional assumptions onthe Shimura data) CM points, which are zero-dimensional special subvarietiesdefined by subgroups G1 of G2 that are tori (see [119, Definition 12.5]). Theseare generalizations of the points of modular curves parameterizing CM ellipticcurves with level structures, which are represented by points of the Poincareupper half-plane with coordinates in imaginary quadratic field extensionsof Q. Zero-dimensional Shimura varieties and special points are importantbecause their canonical models can be defined more directly, and hence theyare useful for characterizing the canonical models of Shimura varieties ofpositive dimensions.

We will not try to explain the proofs of Theorems 2.4.1 and 2.4.3 in thisintroductory article. In fact, we will not even try to explain in any depth thebackground notions such as variations of Hodge structures or Hermitian sym-metric domains, or key notions such as zero-dimensional Shimura varieties,the Shimura reciprocity law, and the precise meaning of canonical models,because most existing texts already present them quite well, and there is lit-tle we can add to them. Instead, we will start by presenting many examplesof (G,D) and (G,D+), so that the readers can have some idea to where the


theory applies. We hope this will benefit not only readers who would liketo better understand the existing texts, but also readers who have concretearithmetic or geometric problems in mind.

3. Hermitian symmetric domains

In this section, we will explore many examples of Hermitian symmetric do-mains, which can be interpreted as the generalizations of the Poincare upperhalf-plane H (see Example 2.2.7) in many different ways. Most of these willgive the (G(R),D) or (G(R),D+) for some Shimura data (G,D) or connectedShimura data (G,D+), respectively, as in Section 2.3, although we will notbe using the structure of G(Q) in this section. (Later we will indeed use thestructure of G(Q) in Section 4.)

However, we will not define Hermitian symmetric domains. Our excusefor this omission is that there is a complete classification of all irreducible ones(namely, those that are not isomorphic to nontrivial products of smaller ones),and we will see all of them in our examples. (For a more formal treatment,there are general introductions such as [119, Section 1], with references tomore advanced texts such as [65].)

In our personal experience, especially in number-theoretic applicationsrequiring actual calculations, the abstract theory of Hermitian symmetricdomains cannot easily replace the knowledge of these explicit examples. (Al-ready in the classical theory for SL2 or GL2, it is hard to imagine studyingmodular forms without ever introducing the Poincare upper half-plane.)

We shall begin with the case of Sp2n(R) in Section 3.1, followed bythe case of Ua,b in Section 3.2, the case of SO∗2n in Section 3.3, the caseof SOa,2(R) in Section 3.4, the case of E7 in Section 3.5, the case of E6 inSection 3.6, and a summary of all these cases in Section 3.7.

3.1. The case of Sp2n(R)

3.1.1. Siegel upper half-spaces. Let n ≥ 0 be any integer. Consider the Siegelupper half-space

Hn := {Z ∈ Symn(C) : Im(Z) > 0}= {Z ∈ Mn(C) : tZ − Z = 0, 1

2i (tZ − Z) < 0}

=

Z ∈ Mn(C) :

t

(Z1

)(0 1−1 0

)(Z1

)= 0,

12it

(Z1

)(0 1−1 0

)(Z1

)< 0

,

(3.1.1.1)

where Mn (resp. Symn) denotes the space of n×n matrices (resp. symmetricmatrices) as before, and where the notation > 0 (resp. < 0) means positivedefiniteness (resp. negative definiteness) of matrices, which admits a natural

16 Kai-Wen Lan

(left) action of g = (A BC D ) ∈ Sp2n(R):

Z 7→ gZ := (AZ +B)(CZ +D)−1 (3.1.1.2)

(note that (CZ +D)−1 is multiplied to the right of (AZ +B)). Again, suchan action should be understood in generalized projective coordinates:(

A BC D

)(Z1

)=

(AZ +BCZ +D

)∼(gZ1

)(3.1.1.3)

(by multiplying (CZ +D)−1 to the right).

Remark 3.1.1.4. By generalized projective coordinates given by an m×n ma-trix, we mean anm×nmatrix of rank n that is identified up to the right actionof invertible n× n matrices. Giving such a matrix is equivalent to defining apoint of the Grassmannian of n-dimensional subspaces of an m-dimensionalvector space. Later we will encounter some similar generalized projective co-ordinates when defining other Hermitian symmetric domains, and the readerscan interpret them as defining points on some other Grassmannians. This isclosely related to the interpretation of Hermitian symmetric domains as pa-rameter spaces of variations of Hodge structures, where the Grassmanniansparameterize some Hodge filtrations.

Let us briefly explain why the action is indeed defined. Since

t

(A BC D

)(0 1−1 0

)(A BC D

)=

(0 1−1 0

), (3.1.1.5)

we have

t

(Z1

)(0 1−1 0

)(Z1

)= t

(Z1

)tg

(0 1−1 0

)g

(Z1

)= t(CZ +D) t

(gZ1

)(0 1−1 0

)(gZ1

)(CZ +D) = 0

(3.1.1.6)

if and only if

t

(gZ1

)(0 1−1 0

)(gZ1

)= 0; (3.1.1.7)

and

12it

(Z1

)(0 1−1 0

)(Z1

)= 1

2it

(Z1

)tg

(0 1−1 0

)g

(Z1

)= 1

2it(CZ +D) t

(gZ1

)(0 1−1 0

)(gZ1

)(CZ +D) < 0

(3.1.1.8)

if and only if

12it

(gZ1

)(0 1−1 0

)(gZ1

)< 0. (3.1.1.9)

Thus, Z ∈ Hn if and only if gZ ∈ Hn, and the action is indeed defined.


3.1.2. Transitivity of action. If Z = X + iY ∈ Hn, then(1 −X0 1

)Z = iY, (3.1.2.1)

with Y > 0, and so there exists some A ∈ GLn(R) such that

tAY A = 1n, (3.1.2.2)

which shows that (tA 00 A

)(iY ) = i1n. (3.1.2.3)

These show that

Hn = Sp2n(R) · (i1n). (3.1.2.4)

The stabilizer of i1n is

K :=

{(A B−B A

)∈ Sp2n(R)

}∼= Un :=

{g = A+ iB : tgg = 1n

}(3.1.2.5)

(where the conditions on A and B are the same for(A B−B A

)∈ Sp2n(R) and

for g = A + iB ∈ Un), which is a maximal compact subgroup of Sp2n(R).(Certainly, the precise matrix expressions of these elements depend on thechoice of the point i1n.) Thus, we have shown that

Hn ∼= Sp2n(R)/Un. (3.1.2.6)

When n = 1, this specializes to the isomorphism H = H1∼= SL2(R)/SO2(R)

in Example 2.2.7, because SL2(R) ∼= Sp2(R) and SO2(R) ∼= U1. Note that

dimCHn = dimC(Symn(C)

)= 1

2n(n+ 1), (3.1.2.7)

for all n ≥ 0.

3.1.3. Bounded realization. There is also a bounded realization of Hn. Notethat (

1 −i1 i

)(i1

)=

(02i

)∼(

01

). (3.1.3.1)

Set

U := (Z − i1n)(Z + i1n)−1, (3.1.3.2)

so that (1 −i1 i

)(Z1

)=

(Z − i1nZ + i1n

)∼(U1

)(3.1.3.3)

in generalized projective coordinates (cf. Remark 3.1.1.4). Note that thisassignment maps Z = i10 to U = 0. Since(

1 1−i i

)(0 1−1 0

)(1 −i1 i

)=

(0 2i−2i 0

)= 2i

(0 1−1 0

), (3.1.3.4)

we havetZ = Z ⇐⇒ tU = U, (3.1.3.5)

18 Kai-Wen Lan

and

12i (

tZ − Z) = 12it

(Z1

)(0 1−1 0

)(Z1

)= 1

8it

(Z1

)(1 1i −i

)(1 −i1 i

)(0 1−1 0

)(1 1i −i

)(1 −i1 i

)(Z1

)< 0

⇐⇒ tUU − 1n = t

(U1

)(1 00 −1

)(U1

)< 0,

(3.1.3.6)

and so

Hn ∼= Dn := {U ∈ Symn(C) : tUU − 1n < 0}. (3.1.3.7)

This generalizes the Cayley transforms that move the Poincare upper half-plane H to the open unit disc {u ∈ C : |u| < 1}. (But the formula for theaction of Sp2n(R) has to be conjugated, and is more complicated.)

In the classification in [65, Chapter X, Section 6], Hn ∼= Dn is a Hermit-ian symmetric domain of type C I. In Cartan’s classification, it is a boundedsymmetric domain of type IIIn.

3.1.4. Interpretation as conjugacy classes. Now consider the homomorphism

h0 : U1 → Sp2n(R) : x+ yi 7→(x −yy x

). (3.1.4.1)

For simplicity, in the matrix at the right-hand side, we denoted by x and ythe scalar multiples of the n× n identity matrix 1n by x and y, respectively.(We will adopt similar abuses of notation later, without further explanation.)Then we have (

A BC D

)(0 1−1 0

)=

(0 1−1 0

)(A BC D

)⇐⇒

(−B A−D C

)=

(C D−A −B

)⇐⇒ A = D, B = −C,

(3.1.4.2)

which shows that

Centh0(Sp2n(R)) = Un (3.1.4.3)

as subgroups of Sp2n(R), where Cent denotes the centralizer, and so that

Hn = Sp2n(R) · h0, (3.1.4.4)

with the action given by conjugation. That is, Hn is the Sp2n(R)-conjugacyclass of h0. In this case, (Sp2n,Hn) is a connected Shimura datum as inSection 2.3 (where we use Sp2n to denote the algebraic group over Q, notthe group scheme over Z), taking into account the variant (2.3.8) of (2.3.5)as explained there. (For simplicity, we shall not repeat such an explanationin what follows.)


The above is compatible with the similarly defined action of GSp2n(R)on

H±n := {Z ∈ Mn(C) : tZ = Z, and either Im(Z) > 0 or Im(Z) < 0}.(3.1.4.5)

Then

H±n = GSp2n(R) · h0, (3.1.4.6)

where

h0 : C× → GSp2n(R) : x+ yi 7→(x −yy x

)(3.1.4.7)

also encodes the action of R× ⊂ C×, and is better for the purpose of studyingvariations of Hodge structures (and also for various other reasons). In thiscase, (GSp2n,H±n ) is a Shimura datum as in Section 2.3 (where we use GSp2n

to denote the algebraic group over Q, not the group scheme over Z).

3.1.5. Moduli of polarized abelian varieties. For each Z ∈ Hn, we have alattice

LZ := ZnZ + Zn (3.1.5.1)

in Cn such that

AZ := Cn/LZ (3.1.5.2)

is not only a complex torus, but also a polarized abelian variety . (See [124,Sections 1–3] and [128, Chapter II, Sections 1 and 4].) When n = 1, this isjust an elliptic curve.

The point is that the abelian variety AZ = Cn/LZ (as a complex torus)can be identified with the fixed real torus

R2n/Z2n (3.1.5.3)

with its complex structure induced by the one on R2n defined by

h(i) ∈ Sp2n(R) ⊂ GL2n(R), (3.1.5.4)

where the homomorphism

h = g · h0 ∈ Hn = Sp2n(R) · h0 (3.1.5.5)

corresponds to the point

Z = g · (i1n) ∈ Hn = Sp2n(R) · (i1n). (3.1.5.6)

Accordingly, varying the lattice LZ in Cn with Z = g · (i1n) correspondsto varying the complex structure J = h(i) on R2n with h = g · h0. (Such avariation of complex structures can be interpreted as a variation of Hodgestructures.)

The pairing on Z2n defined by the perfect pairing 〈 · , · 〉 defines (up toa sign convention) a principal polarization λZ on AZ (and the existence ofsuch a polarization was the reason that the complex torus AZ is an abelianvariety). If (AZ , λZ) ∼= (AZ′ , λZ′) for Z and Z ′ corresponding to h and h′,

respectively, then we have an induced isomorphism R2n ∼→ R2n preserving

20 Kai-Wen Lan

Z2n and 〈 · , · 〉, which defines an element g = (A BC D ) ∈ Sp2n(Z) conjugating

h to h′ and mapping Z to Z ′. Conversely, if we have

Z ′ = gZ = (AZ +B)(CZ +D)−1 (3.1.5.7)

for some g = (A BC D ) ∈ Sp2n(Z), then we have a homothety of lattices

LZ = Z2nZ + Z2n = Z2n(AZ +B) + Z2n(CZ +D)

= (Z2nZ ′ + Z2n)(CZ +D) = LZ′(CZ +D) ∼ LZ′ ,(3.1.5.8)

which implies that (AZ , λZ) ∼= (AZ′ , λZ′). If g lies in a congruence subgroup Γof Sp2n(Z), then the above isomorphism (AZ , λZ) ∼= (AZ′ , λZ′) also respectscertain (symplectic) level structures. Therefore, Γ\Hn (or the manifolds XUdefined using G = GSp2n and D = H±n ) are Siegel modular varieties param-eterizing polarized abelian varieties with certain level structures (dependingon Γ or U). The spaces Γ\Hn are, a priori, just complex manifolds. The name“Siegel modular varieties” will be justified in Sections 4.1.3 and 4.2.2 below,where we shall see that they are indeed the complex analytifications of somecanonical algebraic varieties.

The fact that spaces like Γ\Hn parameterize polarized abelian varietieswith level structures is perhaps just a coincidence, but such a coincidenceis extremely important. One can redefine them as the C-points of certainmoduli problems of polarized abelian schemes with level structures; and thentheir constructions can be extended over the integers in number fields, be-cause the moduli schemes of polarized abelian schemes, called Siegel modulischemes, have been constructed over the integers. (See [129, Chapter 7] forthe construction based on geometric invariant theory. Alternatively, geomet-ric invariant theory can be avoided by first constructing algebraic spaces orDeligne–Mumford stacks by verifying Artin’s criterion—see [2] and [90, Sec-tion B.3] for such a criterion, and see [47, Chapter I, Section 4] and [90,Chapter 2] for its verification for the moduli of polarized abelian schemes;and then showing that they are quasi-projective schemes by constructingtheir minimal compactifications as in [47, Chapter V] or [90, Chapter 7]—seeSection 5.1.4 below.)

Such integral models of Γ\Hn are very useful because we can understandtheir reductions in positive characteristics by studying polarized abelian va-rieties over finite fields, for which we have many powerful tools. We willencounter more general cases of Γ\D later, but we have good methods forconstructing integral models for them (namely, methods for which we cansay something useful about the outputs) only in cases that can be related toSiegel modular varieties at all.

3.2. The case of Ua,b

We learned some materials here from [171] and [170].

3.2.1. Bounded realization. Let a ≥ b ≥ 0 be any integers. Recall that wehave introduced the symmetric matrix 1a,b =

(1a−1b

)in (2.1.13), which


we now view as a Hermitian matrix, which defines a Hermitian pairing ofsignature (a, b). Consider the group

Ua,b := {g ∈ GLa+b(C) : tg1a,bg = 1a,b}, (3.2.1.1)

which acts on

Da,b :=

{U ∈ Ma,b(C) : t

(U1

)(1a−1b

)(U1

)= tUU − 1b < 0

},

(3.2.1.2)where Ma,b denotes the space of a× b matrices, by

U 7→ gU = (AU +B)(CU +D)−1 (3.2.1.3)

for each g = (A BC D ) ∈ Ua,b. This action is based on the identity(

A BC D

)(U1

)=

(AU +BCU +D

)∼(

(AU +B)(CU +D)−1

1

)(3.2.1.4)

in generalized projective coordinates (cf. Remark 3.1.1.4). Note that Ua,b isa real Lie group, but not a complex one, despite the use of complex numbersin its coordinates; and that

dimCDa,b = dimC(Ma,b(C)

)= ab. (3.2.1.5)

For any (A BC D ) ∈ Ua,b, as soon as B = 0, we must also have C = 0, by

the defining condition in (3.2.1.1). Hence, the stabilizer of 0 ∈ Da,b is

K :=

{(A

D

)∈ Ua,b

}∼= Ua×Ub :

(A

D

)7→ (A,D). (3.2.1.6)

Remark 3.2.1.7. For the purpose of introducing Hermitian symmetric do-mains, it might be more natural to consider the semisimple Lie group SUa,b—namely, the subgroup of Ua,b of elements of determinant one—or its adjointquotient, but we still consider the reductive Lie group Ua,b, because it iseasier to write down homomorphisms of the form U1 → Ua,b, as in (2.3.8).

However, to show the transitivity of the action of Ua,b on Da,b, and formany other computations for which it is easier to work with block upper-triangular matrices, it will be easier to switch to some unbounded realization.

3.2.2. Unbounded realization. Consider

U′a,b := {g ∈ GLa+b(C) : tgJa,bg = Ja,b}, (3.2.2.1)

where the matrix

Ja,b :=

1bS

−1b

(3.2.2.2)

is skew-Hermitian with −iS > 0, for some choice of a skew-Hermitian ma-trix S (depending on the context), so that there exists some T ∈ GLn(C) such

22 Kai-Wen Lan

that −iS = tTT . We warn the readers that the notation of Ja,b and U′a,b here

is rather nonstandard. Then U′a,b acts on

Ha,b :=

(ZW

)∈ Ma,b(C) ∼= Mb(C)×Ma−b,b(C) :

−i tZW1

Ja,bZW1

= −i( tZ − Z + tWSW ) < 0

(3.2.2.3)

by (ZW

)7→ g

(ZW

)=

((AZ + EW +B)(CZ +HW +D)−1

(FZ +MW +G)(CZ +HW +D)−1

), (3.2.2.4)

for each g =(A E BF M GC H D

)∈ U′a,b, based on the identity

g

ZW1

=

AZ + EW +BFZ +MW +GCZ +HW +D

∼ (AZ + EW +B)(CZ +HW +D)−1

(FZ +MW +G)(CZ +HW +D)−1

1

(3.2.2.5)

in generalized projective coordinates (cf. Remark 3.1.1.4).

If we consider

Hermb(C) := {X ∈ Mb(C) : tX = X}, (3.2.2.6)

where Hermb denotes the space of b×b Hermitian matrices (which are over Chere), then we have a canonical isomorphism

Mb(C)∼→ Hermb(C)⊗

RC : X 7→ Re(X) + i Im(X), (3.2.2.7)

where

Re(X) := 12 (X + tX) (3.2.2.8)

and

Im(X) := 12i (X −

tX) (3.2.2.9)

are the Hermitian real and imaginary parts (abusively denoted by the usualsymbols), and we can rewrite the condition

− i( tZ − Z + tWSW ) < 0 (3.2.2.10)

as

Im(Z) > 12tW (−iS)W. (3.2.2.11)

3.2.3. The special case where a = b: Hermitian upper half-spaces. In thiscase, we can omit the second coordinate W in the above unbounded realiza-tion, and we have

Hb,b = {Z ∈ Mb(C) ∼= Hermb(C)⊗RC : Im(Z) > 0}, (3.2.3.1)

which can be viewed as a generalization of

Hn = {Z ∈ Symn(C) ∼= Symn(R)⊗RC : Im(Z) > 0}. (3.2.3.2)


That is, we can view the symmetric matrices in Symn(R) as being Hermitianwith respect to the trivial involution of R, which then generalizes to the usualcomplex Hermitian matrices in Hermn(C) as being Hermitian with respectto the complex conjugation of C over R. These two examples will be furthergeneralized with R and C replaced with the other two normed (but possiblynonassociative) division algebras H and O over R, the rings of Hamilton andCayley numbers, respectively, in Sections 3.3.2 and 3.5.4 below.

3.2.4. Generalized Cayley transformations and transitivity of actions. Since

−i

1√2

− i√2

tT−1

1√2

i√2

1

S−1

1√2

1√2

T−1

i√2

− i√2

=

11−1

,

(3.2.4.1)

we have an isomorphism

U′a,b∼→ Ua,b : g 7→

1√2

− i√2

T1√2

i√2

g

1√2

1√2

T−1

i√2

− i√2

, (3.2.4.2)

and accordingly an isomorphism

Ha,b∼→ Da,b :

ZW1

7→

1√2

− i√2

T1√2

i√2

ZW1

∼ (U1

)(3.2.4.3)

equivariant with the above isomorphism (3.2.4.2). This is a generalization ofthe classical Cayley transformation and maps

(i1b0

)∈ Ha,b to 0 ∈ Da,b.

Now, given any ( ZW ) ∈ Ha,b, we have1 ∗ ∗1 −W

1

ZW1

=

Z ′01

, (3.2.4.4)

1 −Re(Z ′)1

1

Z ′01

=

i Im(Z ′)01

, (3.2.4.5)

tA−1

1A

i Im(Z ′)01

=

i1b01

, (3.2.4.6)

for some Z ′ ∈ Mb(C) and some A ∈ GLb(C) such that Im(Z ′) = tAA, wherethe first matrix in each equation is some element of U′a,b. These show that

Ha,b = U′a,b ·(i1b0

)∼= Da,b = Ua,b · 0 ∼= Ua,b/(Ua×Ub), (3.2.4.7)

24 Kai-Wen Lan

and hence the actions of Ua,b and U′a,b on Da,b and Ha,b, respectively, aretransitive.

In the classification in [65, Chapter X, Section 6], Ha,b ∼= Da,b is aHermitian symmetric domain of type A III. In Cartan’s classification, it is abounded symmetric domain of type Iba.

3.2.5. Interpretation as conjugacy classes. Consider the homomorphism

h0 : U1 → Ua,b : x+ yi 7→(x− yi

x+ yi

), (3.2.5.1)

with x− yi and x+ yi denoting multiples of 1a and 1b, respectively. Then

Centh0(Ua,b) = Ua×Ub (3.2.5.2)

(as subgroups of Ua,b) and so that

Da,b = Ua,b · h0. (3.2.5.3)

Via the inverse of the isomorphism Ua,b∼→ U′a,b above defined by conjugation,

we obtain

h′0 : U1 → U′a,b : x+ yi 7→

x −yx− yi

y x

, (3.2.5.4)

so that

Ha,b = U′a,b · h′0. (3.2.5.5)

Both of these are compatible with the extensions of the actions of Ua,b

and U′a,b on Da,b and Ha,b, respectively, to some actions of

GUa,b := {(g, r) ∈ GLa+b(C)×R× : tg1a,bg = r1a,b} (3.2.5.6)

and

GU′a,b := {(g, r) ∈ GLa+b(C)×R× : tgJa,bg = rJa,b} (3.2.5.7)

on some

D±a,b = GUa,b · h0 (3.2.5.8)

and

H±a,b = GUa,b · h′0 (3.2.5.9)

(where we omit the explicit descriptions of D±a,b and H±a,b, for simplicity),

where h0 and h′0 are now homomorphisms

h0 : C× → GUa,b (3.2.5.10)

and

h′0 : C× → GU′a,b (3.2.5.11)

defined by the same expressions as in (3.2.5.1) and (3.2.5.4), respectively.(As in Section 3.1.4, the upshot here is that the pairs (GUa,b,D±a,b) and

(GU′a,b,H±a,b) can be of the form (G(R),D) for some Shimura data (G,D).

We shall omit such remarks in later examples.)


Since the real part of any skew-Hermitian pairing is alternating, thereis an injective homomorphism of the form

Ua,b∼= U′a,b ↪→ Sp2n(R), (3.2.5.12)

with n = a + b, whose pre-composition with h0 : U1 → Ua,b lies in theconjugacy class of the analogous homomorphism for Sp2n(R) in (3.1.4.1).

3.3. The case of SO∗2n

3.3.1. Unbounded realization, and interpretation as a conjugacy class. Con-sider the subgroup SO∗2n of

SO2n(C) = {g ∈ SL2n(C) : tgg = 12n} (3.3.1.1)

preserving the skew-Hermitian form defined by(

1n−1n

). That is,

SO∗2n = SO2n(C)∩U′n,n (3.3.1.2)

as subgroups of GL2n(C), where U′n,n is defined as in (3.2.2.1), with Liealgebra

so∗2n =

{(A B−B A

): A,B ∈ Mn(C), tA = −A, tB = B

}(3.3.1.3)

as Lie subalgebras of M2n(C) (see, for example, [79, Chapter I, Section 1]or [80, Chapter I, Section 17]). By explanations as before, SO∗2n acts transi-tively on

HSO∗2n:=

Z ∈ Mn(C) : t

(Z1

)(Z1

)= tZZ + 1 = 0,

−i t(Z1

)(1

−1

)(Z1

)= −( tZ − Z) < 0

, (3.3.1.4)

whose stabilizer at i1n ∈ HSO∗2nis

K :=

{(A B−B A

)∈ SO∗2n : A,B ∈ Mn(R)

}∼= Un, (3.3.1.5)

so that

HSO∗2n= SO∗2n · h0

∼= SO∗2n/Un, (3.3.1.6)

where

h0 : U1 → SO∗2n : x+ yi 7→(x −yy x

). (3.3.1.7)

As before, there is also another version of SO∗2n which acts on some boundedrealization DSO∗2n

, but we shall omit them for simplicity. Since SO∗2n ⊂ U′n,n,by the same explanation as in Section 3.2.5, there are injective homomor-phisms

SO∗2n ↪→ Sp4n(R), (3.3.1.8)

whose pre-composition with h0 : U1 → SO∗2n lies in the conjugacy class ofthe analogous homomorphism for Sp4n(R) as in (3.1.4.1).

26 Kai-Wen Lan

In the classification in [65, Chapter X, Section 6], HSO∗2n∼= DSO∗2n

is aHermitian symmetric domain of type D III. In Cartan’s classification, it is abounded symmetric domain of type IIn.

3.3.2. The special case where n = 2k: quaternion upper half-spaces. Sincewe have the isomorphism

H ∼= C2 : x+ yi+ j(z + wi) 7→(x+ yiz + wi

)(3.3.2.1)

of right C-modules, where H denotes the Hamiltonian quaternion algebraover R, we have a homomorphism

H→ M2(C) : x+ yi+ j(z + wi) 7→(x+ yi −z + wiz + wi x− yi

)(3.3.2.2)

induced by the left action of H on itself. Using this, we can show that

SO∗4k∼={g ∈ GL2k(H) : tg

(1k

−1k

)g =

(1k

−1k

)}, (3.3.2.3)

namely, the subgroup Sp2k(H) of GL2k(H) respecting the skew-Hermitianpairing on Hk defined by

(1k

−1k

)(see, for example, [56, Section 1.1.4], com-

bined with a Gram–Schmidt process as in [90, Section 1.2.3]); and that

HSO∗4k∼= {Z ∈ Hermk(H)⊗

RC : Im(Z) > 0}, (3.3.2.4)

with the action of g = (A BC D ) ∈ GL2k(H) given by

gZ = (AZ +B)(CZ +D)−1, (3.3.2.5)

which is similar to the action of Sp2k(R) on Hk in Section 3.1.1. This isa quaternion upper half-space, further generalizing the examples from Sec-tions 3.1.1 and 3.2.3.

3.4. The case of SOa,2(R)

We learned some materials here from [152, Appendix, Section 6].

3.4.1. Projective coordinates. Let a, b ≥ 0 be any integers. Consider

SOa,b(R) := {g ∈ SLa+b(R) : tg1a,bg = 1a,b}, (3.4.1.1)

where 1a,b =(

1a−1b

)is defined as before. Let us temporarily write V := Ra+b

and VC := V ⊗RC ∼= Ca+b. Then, by definition, SOa,b(R) acts on

HSOa,b :=

{v ∈ VC : tv

(1a−1b

)v = 0, tv

(1a−1b

)v < 0

}/C×.

(3.4.1.2)Note that v 6= 0 in the above definition, and hence the quotient by C× meanswe are actually working with a subset of Pa+b−1(C) ∼= (VC − {0})/C×. (Thisis consistent with our use of generalized projective coordinates in previous ex-amples.) However, HSOa,b is in general not connected, and even its connected


components do not have the structure of Hermitian symmetric domains forarbitrary a, b ≥ 0.

3.4.2. The case where a ≥ 1 and b = 2. In this case, we do get Hermitiansymmetric domains. (This follows from the general theory, but since we havenot defined Hermitian symmetric domains, we are merely stating this asa fact.)

Suppose we have an orthogonal direct sum

V = Ra+2 = Ra⊕R2 (3.4.2.1)

such that the pairings on Ra and R2 are defined by(

1a−1

−1

)and(

1/21/2

), respectively, which have signatures (a− 1, 1) and (1, 1) summing

up to (a, 2) in total. Note that we have(1 1−1 1

)(12

12

)(1 −11 1

)=

(1−1

). (3.4.2.2)

Then we also have

VC ∼= Ca⊕C2, (3.4.2.3)

and we denote the coordinates of Ca and C2 by (z1, . . . , za) and (w1, w2),respectively. With these coordinates, we have

HSOa,2∼={

z21 + · · ·+ z2

a−1 − z2a + w1w2 = 0,

|z1|2 + · · ·+ |za−1|2 − |za|2 + Re(w1w2) < 0

}. (3.4.2.4)

If w2 = 0, then the two conditions in (3.4.2.4) imply that

|z1|2 + · · ·+ |za−1|2 < |za|2 = |z1 + · · ·+ za−1|2, (3.4.2.5)

which is not possible. Hence, we must have w2 6= 0, and so, up to scaling ofthe projective coordinates, we may assume that w2 = 1. Then

w1 = z2a − z2

1 − · · · − z2a−1 (3.4.2.6)

by the first condition, and the second condition becomes

|z1|2 + · · ·+ |za−1|2 − |za|2 + Re(w1) < 0. (3.4.2.7)

If we write zj = xj + iyj , then

|zj |2 − Re(z2j ) = 2y2

j , (3.4.2.8)

for each 1 ≤ j ≤ a. Thus, we have

HSOa,2∼= HSOa,2 :=

z = (zj) ∈ Ca : y = Im(z) > 0 in the sense that

ty

(1a−1

−1

)y < 0, i.e., y2

a > y21 + · · · y2

a−1

.

(3.4.2.9)

The condition

y2a > y2

1 + · · · y2a−1 (3.4.2.10)

28 Kai-Wen Lan

defines a familiar light cone with two connected components, one being de-fined by

ya >√y2

1 + · · · y2a−1, (3.4.2.11)

the other being defined by

ya < −√y2

1 + · · · y2a−1. (3.4.2.12)

These two connected components correspond to two connected componentsH+

SOa,2and H−SOa,2

of HSOa,2 .

Note that

dimC HSOa,2 = dimCHSOa,2 = dimCH+SOa,2

= dimC Ca = a. (3.4.2.13)

Also, HSO1,2∼= H±1 and H+

SO1,2

∼= H1 = H, when a = 1. In particular, we

have obtained yet another generalization of the Poincare upper half-plane H.

3.4.3. Connected components and spin groups. Under the assumption thata ≥ 1, the real Lie group SOa,2(R) has two connected components (see, forexample, [65, Chapter X, Section 2, Lemma 2.4] or [80, Proposition 1.145]),and the stabilizer of H+

SOa,2in G(R) = SOa,2(R) is the identity compo-

nent G(R)+ = SOa,2(R)+; namely, the connected component containing theidentity element (in the real analytic topology). The group SOa,2(R) can beviewed as the group of R-points of the algebraic group SOa,2 defined over R,which admits a simply-connected two-fold covering group Spina,2. This is onlya covering of algebraic groups, which means we can only expect a surjectionon C-points. In fact, the induced morphism

Spina,2(R)→ SOa,2(R) (3.4.3.1)

on R-points is not surjective, because Spina,2(R) is connected and factors

through SOa,2(R)+. Here we are keeping the language as elementary as pos-sible, without introducing the spin groups in detail—see, for example, [56,Chapter 6], for a review of their construction based on Clifford algebras. Butthe connectedness of Spina,2(R), nevertheless, is a general feature of the realpoints of any isotropic simply-connected connected simple algebraic groupover Q—see [138, Section 7.3, Theorem 7.6].

To understand this issue better, consider the Gal(C/R)-equivariant shortexact sequence

1→ {±1} → Spina,2(C)→ SOa,2(C)→ 1, (3.4.3.2)

which induces by taking Galois cohomology the long exact sequence

1→ {±1} → Spina,2(R)→ SOa,2(R)→ H1(Gal(C/R), {±1}) ∼= {±1},(3.4.3.3)

where the homomorphism

SOa,2(R)→ {±1} (3.4.3.4)


(induced by the connecting homomorphism) is the spinor norm. The ker-nel of the spinor norm is SOa,2(R)+, which is at the same time the imageof Spina,2(R).

We shall omit the verification that SOa,2(R)+ does act transitivelyon H+

SOa,2, with stabilizers isomorphic to

SOa(R)×SO2(R). (3.4.3.5)

Accordingly, SOa,2(R) acts transitively on HSOa,2 , with stabilizers also iso-morphic to SOa(R)×SO2(R). We shall also omit the description of somebounded realizations D+

SOa,2and DSOa,2 of H+

SOa,2and HSOa,2 , respectively.

In the classification in [65, Chapter X, Section 6],H+SOa,2

∼= D+SOa,2

(with

a ≥ 1) is a Hermitian symmetric domain of type BD I (with (p, q) = (a, 2)in the notation there). In Cartan’s classification, it is a bounded symmetricdomain of type IVa.


h0 : U1 → SOa,2(R) : eiθ 7→

1acos 2θ − sin 2θsin 2θ cos 2θ

. (3.4.4.1)

Note that the image of h0 is connected because U1 is, and hence lies inSOa,2(R)+. Moreover,

K := Centh0(SOa,2(R)+) =

{(A

D

)∈ SOa,2(R)+

}∼= SOa(R)×SO2(R) :(

AD

)7→ (A,D).

(3.4.4.2)

Then K is a maximal compact subgroup of SOa,2(R)+, and we have

H+SOa,2

= SOa,2(R)+ · h0∼= SOa,2(R)+/(SOa(R)×SO2(R)). (3.4.4.3)

Note that there is a coefficient 2 of θ in the above definition of h0.This h0 lifts to a homomorphism

h0 : U1 → Spina,2(R), (3.4.4.4)

where Spina,2(R) is the above double cover of SOa,2(R)+, and there is aninjective homomorphism

Spina,2(R) ↪→ Sp2n(R), (3.4.4.5)

with n = 2a (cf. Examples 5.2.1.3 and 5.2.2.11 below), whose pre-composition

with h0 lies in the conjugacy class of the analogous homomorphism for Sp2n(R)in (3.1.4.1).

3.5. The case of E7

Our main references are [8], [74], [55], and [44, Chapter 12].

30 Kai-Wen Lan

3.5.1. Cayley numbers. Consider the algebra O of Cayley numbers, whichis an octonion (i.e., degree eight) division algebra over R which is normed,distributive, noncommutative, and nonassociative. We have

O = R⊕Re1⊕ · · ·⊕Re7 (3.5.1.1)

as an R-vector space, which satisfies the multiplication rules given by

e2i = −1 (3.5.1.2)

and

(eiei+1)ei+3 = ei(ei+1ei+3) = 1 (3.5.1.3)

(with the indices periodically identified modulo 7) for all 1 ≤ i ≤ 7. We have

x := x0 − x1e1 − · · · − x7e7 (3.5.1.4)

and

N(x) := xx = x20 + x2

1 + ·+ x27 ≥ 0 (3.5.1.5)

for each

x = x0 + x1e1 + · · ·+ x7e7 ∈ O, (3.5.1.6)

where the square-root of N(x) defines the norm of x.

3.5.2. An exceptional Jordan algebra over R. Consider the Albert algebra

A := Herm3(O), (3.5.2.1)

the 3× 3 Hermitian matrices over O, which is a formally real Jordan algebraof exceptional type, with the Jordan algebra multiplication given by

X · Y = 12 (XY + Y X), (3.5.2.2)

where XY and Y X are the usual matrix multiplications. (See, for example,[48, Chapter II–V and VIII] for a general introduction to Jordan algebrasand the classification of formally real ones. See also [5, Chapter II, Remarkat the end of Section 1] or [6, Chapter II, Remark 1.11], and the sectionsfollowing there, for why they matter.)

For

X =

a x yx b zy z c

∈ A (3.5.2.3)

(with a, b, c ∈ R and x, y, z ∈ O), we can define the trace

tr(X) = a+ b+ c (3.5.2.4)

and the determinant

det(X) = abc− aN(z)− bN(y)− cN(x) + tr((xz)y). (3.5.2.5)

Then we also have an inner product

(X,Y ) := tr(X · Y ) (3.5.2.6)

and a trilinear form ( · , · , · ) satisfying the identity

(X,X,X) = det(X). (3.5.2.7)


This trilinear form defines a cross product X × Y such that

(X × Y,Z) = 3(X,Y, Z), (3.5.2.8)

for X,Y, Z ∈ A, which specializes to

X ×X =

bc−N(z) yz − cx xz − byzy − cx ac−N(y) xy − azzx− by yx− az ab−N(x)

(3.5.2.9)

when X = Y .

Since O is nonassociative, it is rather a miracle that the above definitionscan be made for 3× 3 Hermitian matrices over O—nothing similar works forlarger matrices.

3.5.3. A real Lie group with Lie algebra e7(−25). Consider the Lie group

M := {(g, r) ∈ GL(A)×R× : det(gX) = r det(X)}, (3.5.3.1)

so that we have a homomorphism

ν : M → R× : (g, r) 7→ r. (3.5.3.2)

Then we have

ker(ν) = {g ∈ GL(A) : det(gX) = det(X)}. (3.5.3.3)

It is known that ker(ν) is a connected semisimple real Lie group of real rank 2with Lie algebra isomorphic to e6(−26). The number −26 in the notationof e6(−26) means it is the real form of the complex simple Lie algebra oftype E6 such that the Killing form has signature (a, b) with a− b = −26; or,equivalent, that every Cartan involution has trace 26 = −(−26). For complexsimple Lie algebras of exceptional type, such numbers already classify theirreal forms up to isomorphism. (The same explanation applies to other realLie algebras of exceptional type below.) Also, the center of M is

Z(M) ∼= R× (3.5.3.4)

with x ∈ R× acting as multiplication by the scalar x on A, so that ν(x) = x3.In this case, the Lie algebra of M is the direct sum of those of Z(M) andker(ν), and ker(ν) is the derived group of M . The ker(ν)-orbit of 1 in A isisomorphic to the quotient of ker(ν) by a maximal compact subgroup with Liealgebra isomorphic to f4. In the classification in [65, Chapter X, Section 6],this orbit is a noncompact Riemannian symmetric space of type E IV, whichis not Hermitian.

Moreover, g ∈M induces g∗ ∈M by

(gX, g∗Y ) = (X,Y ). (3.5.3.5)

Then M acts on

W := A⊕R⊕A⊕R (3.5.3.6)

by

g(X, ε,X ′, ε′) := (gX, ν(g)−1ε, g∗X ′, ν(g)ε′). (3.5.3.7)

32 Kai-Wen Lan

There is also an additive action

ρ : A ↪→ GL(W) : B 7→ ρB (3.5.3.8)

with ρB defined by

ρB

XεX ′

ε′

:=

X + 2B ×X ′ + εB ×B

εX ′ + εB

ε′ + (B,X) + (B ×B,X ′) + εdet(B)

, (3.5.3.9)

with elements of W represented by column vectors (cf. (3.5.3.6)).

Consider the Lie group

U := image(ρ). (3.5.3.10)

It is known that M normalizes U . Let ι ∈ GL(W) be defined by

ι(X, ε,X ′, ε′) := (−X ′,−ε′, X, ε). (3.5.3.11)

Consider the Lie groupsP := M n U (3.5.3.12)

andG := 〈ι, P 〉 (3.5.3.13)

(i.e., G is generated by ι and P ) as subgroups of GL(W). It is known that G isa connected semisimple Lie group of real rank 3 with Lie algebra isomorphicto e7(−25), and that P is the maximal parabolic subgroup of G stabilizing thesubspace {(0, 0, 0, ε′) : ε′ ∈ R} of W.

These definitions might seem rather complicated and unmotivated, butit is worth drawing the analogy of the element ι and the subgroups U , M ,and P of G with the corresponding element

(1n

−1n

)and the subgroups{(

1n ∗1n

)}, {( ∗ ∗ )}, and {( ∗ ∗∗ )} of Sp2n(R).

A more systematic definition is that G is the Lie subgroup of GL(W)consisting of elements preserving both the quartic form

Q(X, ε,X ′, ε′) :=(X ×X,X ′ ×X ′)− εdet(X)− ε′ det(X ′)− 1

4 ((X,X ′)− εε′)2(3.5.3.14)

and the skew-symmetric bilinear form

〈(X1, ε1, X′1, ε′1), (X2, ε2, X

′2, ε′2)〉 := (X1, X

′2)− (X2, X

′1) + ε1ε

′2 − ε2ε

′1

(3.5.3.15)on W. But, as we shall see, it is useful to know the explicit facts (3.5.3.13)and (3.5.3.12).

3.5.4. An octonion upper half-space. Consider

HE7:= {Z ∈ A⊗

RC : Im(Z) > 0}, (3.5.4.1)

where Im(Z) > 0 means the left multiplication of Im(Z) on A induces apositive definite linear transformation, or equivalently that Im(Z) lies in theinterior of the set {X2 : X ∈ A} with its natural real analytic topology (see


[48, Theorem III.2.1]). (For an analogy, consider the case of symmetric ma-trices over R, where positive definite matrices are exactly those matrices inthe interior of the set of squares of symmetric matrices with its natural realanalytic topology.) This is the most exotic generalization of the various clas-sical upper half-spaces in Sections 3.1.1, 3.2.3, and 3.3.2 defined by Hermitianmatrices over R, C, and H, respectively. Together with the “semi-classical”examples in Section 3.4.2, these are all the irreducible examples (namely,ones that are not isomorphic to nontrivial products of smaller ones) of theso-called tube domains. They are all of the form

{X + iY : X,Y ∈ J, Y > 0} ⊂ J⊗RC, (3.5.4.2)

where J is a formally real Jordan algebra (see [48, Chapters VIII–X]; seealso [5, Chapter III, Section 1] or [6, Chapter III, Section 1]), where thesymbol i denotes the action of 1⊗ i on J⊗R C. In the definition of HE7

, wehave J = A = Herm3(O).

Note that

dimCHE7= dimC(A⊗

RC) = dimR(A) = 27. (3.5.4.3)

The group U acts on HE7 by the translation

Z 7→ Z +B, (3.5.4.4)

for each ρB ∈ U associated with B ∈ A. The group M acts on HE7by

its natural action on A⊗RC (up to a sign convention), which preserves the

subset HE7. The element ι ∈ G acts on HE7

by

ι(Z) := −Z−1, (3.5.4.5)

where Z−1 is defined by inverting the Jordan algebra multiplication in A⊗RC.One can check that these actions are compatible with each other, and definean action of G on HE7

.

Given any Z = X + iY ∈ HE7 , where X,Y ∈ A and where the symbol idenotes the action of 1⊗ i on A⊗R C, we have

ρ−X(Z) = iY (3.5.4.6)

for some Y > 0. By Freudenthal’s theorem (which intriguingly asserts thatmatrices in Herm3(O) can also be diagonalized ; see [48, Theorem V.2.5]),there exists some g ∈M such that

gY = 13, (3.5.4.7)

and so that

g(iY ) = i13. (3.5.4.8)

This shows that

HE7 = G · i13 (3.5.4.9)

and so that the action of G on HE7 is transitive.

34 Kai-Wen Lan

Let K denote the stabilizer of i13 in G. Then

K = Centι(G) (3.5.4.10)

as subgroups of G, which is a maximal compact subgroup of G and is aconnected compact Lie group with Lie algebra e6⊕R. Consider the homo-morphism

h0 : U1 → G (3.5.4.11)

defined in block matrix form by setting

h0(cos θ + i sin θ) =

cos θ − sin θ

cos 3θ sin 3θsin θ cos θ

− sin 3θ cos 3θ

∈ GL(W)

(3.5.4.12)(by left multiplication on column vectors representing elements in W; cf.(3.5.3.6)). In particular, h0(i) = ι, so that we also have

HE7= G · h0

∼= G/K. (3.5.4.13)

There is no homomorphism of the form G → Sp2n(R), for any n ≥ 1,whose pre-composition with h0 : U1 → G lies in the conjugacy class of theanalogous homomorphism for Sp2n(R) in (3.1.4.1) (see [39, 1.3.10(i)] or [120,Section 10, p. 531]; cf. the end of Section 5.2.2 below).

3.5.5. Bounded realization. There is also a bounded realization DE7 of HE7 ,also with coordinates in A⊗R C, with a change of coordinates

HE7

∼→ DE7: Z 7→ (Z − i13)(Z + i13)−1, (3.5.5.1)

where (Z + i13)−1 is (as before) defined by inverting the Jordan algebramultiplication in A⊗R C, which is yet another generalization of the classicalCayley transform (cf. Sections 3.1.3 and 3.2.4).

In the classification in [65, Chapter X, Section 6], HE7∼= DE7

is aHermitian symmetric domain of type E VII. In Cartan’s classification, it is abounded symmetric domain of type VI.

3.6. The case of E6

3.6.1. Bounded and unbounded realizations. There is (up to isomorphism)just one irreducible Hermitian symmetric domain that we have not discussedyet. (Again, being irreducible means not isomorphic to a nontrivial productsof smaller ones.) Let us start with a bounded realization, given by

DE6:= DE7

∩

0 x yx 0 0y 0 0

: x, y ∈ O⊗RC

(3.6.1.1)

in A⊗RC (see Section 3.5.5), where x and y are the conjugations induced bythe ones in O, which can be embedded as an open submanifold of O2⊗RC. Letus also let HE6

denote the preimage of DE6under the isomorphism (3.5.5.1),

which is then an unbounded realization.


Essentially by construction, we have

dimCHE6 = dimCDE6 = 2 dimC(O⊗RC) = 2 dimR O = 16. (3.6.1.2)

By [70, Section 3], DE6and henceHE6

admit transitive actions of a connectedsemisimple Lie group G of rank 2 with Lie algebra isomorphic to e6(−14), withstabilizers given by maximal compact subgroups of G which are connectedLie groups with Lie algebras isomorphic to so10⊕R.

In the classification in [65, Chapter X, Section 6], HE6∼= DE6

is aHermitian symmetric domain of type E III. In Cartan’s classification, it is abounded symmetric domain of type V.

3.6.2. A real Lie group of Lie algebra e6(−14). It is not easy to describe DE6

and HE6, together with the actions of certain G as above, in explicit coordi-

nates. Nevertheless, let us define a real Lie group G with Lie algebra e6(−14),which can be given the structure of the R-points of an algebraic group over Q,and which acts on DE6

and HE6by the abstract theory of Hermitian sym-

metric domains.

Consider the pairing

〈X,Y 〉 := (Xc, Y ), (3.6.2.1)

for X,Y ∈ A ⊗R C, where c denotes the complex conjugation with respectto the tensor factor C; consider the involution σ on A = Herm3(O) (whichextends linearly to A⊗R C) defined bya x y

x b zy z c

σ

=

a −x −y−x b z−y z c

; (3.6.2.2)

and consider

G := {g ∈ GL(A⊗RC) : det(gX) = det(X), 〈gX, gY 〉σ = 〈X,Y 〉σ},

(3.6.2.3)where

〈X,Y 〉σ := 〈Xσ, Y 〉. (3.6.2.4)

We note that

{g ∈ GL(A⊗RC) : det(gX) = det(X)} (3.6.2.5)

is a complex Lie group with Lie algebra e6(C), and so the upshot is that thecondition

〈gX, gY 〉σ = 〈X,Y 〉σ (3.6.2.6)

defines a real form of rank 2 of this complex Lie group with Lie algebraisomorphic to e6(−14), which is not isomorphic to either its compact form(see (3.5.4.10) and the remark following it) or the noncompact real form

{g ∈ GL(A) : det(gX) = det(X)} (3.6.2.7)

36 Kai-Wen Lan

(see (3.5.3.3) and the remark following it) whose Lie algebra is isomorphicto e6(−26). (Thus far, we have seen three kinds of real Lie groups of type E6

in our examples, with one compact and the other two of real rank 2.)


h0 : U1 → G (3.6.3.1)

defined by

h0(α)

a x yx b zy z c

=

α4a αx αyαx α−2b α−2zαy α−2z α−2c

, (3.6.3.2)

with α acting as 1⊗α on O⊗Z C. Then it can be checked directly that

K := Centh0(G) (3.6.3.3)

is a maximal compact subgroup of G, which is a connected Lie group withLie algebra isomorphic to so10⊕R, so that we have

HE6 = G · h0∼= G/K. (3.6.3.4)

There is no homomorphism of the form G→ Sp2n(R), for some n ≥ 1,whose pre-composition with h0 : U1 → G lies in the conjugacy class of theanalogous homomorphism for Sp2n(R) in (3.1.4.1) (see [39, 1.3.10(i)] or [120,Section 10, p. 530]; cf. the end of Section 5.2.2 below).

3.7. A brief summary

Let us summarize the groups and symmetric domains we have consideredas follows. Let us also include the information of the corresponding specialnode on each Dynkin diagram, marked with a unique double circle on thediagram. Such special nodes have not been explained, but should be usefulfor the readers when they consult more advanced texts.

(1) G(R) = Sp2n(R), K ∼= Un, D = Hn:d d · · · d d < da (3.7.1)

(2) G(R) = Ua,b, K ∼= Ua×Ub, D = Ha,b:d d · · · da · · · d d (3.7.2)

Here the special node is the b-th from the left, or alternatively, from theright, because it is only up to automorphisms of the Dynkin diagram.

(3) G(R) = SO∗2n, K ∼= Un, D = HSO∗2n, with n ≥ 2:

d d · · · d · · · d��HH

dda (3.7.3)

Again, the special node is only up to automorphisms of the Dynkindiagram. Therefore, it can be, alternatively, the other right-most node.

(4) G(R)+ = SOa,2(R)+, K ∼= SOa(R)×SO2(R), D+ = H+SOa,2

, with a ≥ 1

but a 6= 2:


(a) a = 2n− 1 is odd, with n ≥ 1:da d · · · d d > d (3.7.4)

(b) a = 2n− 2 is even, with n ≥ 3:

da d · · · d · · · d��HH

dd (3.7.5)

(In the excluded case a = 2, we have Spin2,2(R) ∼= SL2(R)× SL2(R),

and H+SO2,2

∼= H×H is not irreducible.)

(5) LieG ∼= e6(−14), LieK ∼= so10⊕R, D = HE6:

da d dd

d d (3.7.6)

Yet again, the special node is only up to automorphisms of the Dynkindiagram. Therefore, it can be, alternatively, the right-most node.

(6) LieG ∼= e7(−25), LieK ∼= e6⊕R, D = HE7:

da d d dd

d d (3.7.7)

These exhaust all the possibilities of irreducible Hermitian symmetric do-mains.

In the cases (1), (2), and (3), there exist injective homomorphisms ofthe form G(R) → Sp2n′(R), for some n′ ≥ 1, whose pre-composition withh0 : U1 → G(R) lies in the conjugacy class of the analogous homomorphismfor Sp2n′(R) as in (3.1.4.1). In the case (4), the analogous assertion is trueafter replacing SOa,2(R)+ with its double cover Spina,2(R). In the cases (5)and (6), the analogous assertion is simply false. (In Section 5.2.2 below, wewill encounter this issue again in the classification of abelian-type Shimuradata, although the question there is more subtle because we need to alsoconsider the structure of algebraic groups over Q.)

4. Rational boundary components and compactifications

In this section, we will explain how the double coset spaces

XU = G(Q)\(D×G(A∞))/U ∼=∐i∈I

Γi\D+

as in (2.2.15) and (2.2.21) are compactified and shown to be algebraic varietiesin cases where D are Hermitian symmetric domains or their finite unions.

38 Kai-Wen Lan

4.1. Examples of rational boundary components

Let us start by explaining the notion of rational boundary components inmany examples, based on the Hermitian symmetric domains we have seen inSection 3. We will begin with the cases with D = H in Section 4.1.1, followedby the cases with D = H×H in Section 4.1.2; the cases with D = Hn inSection 4.1.3; the cases with D = Ha,b, with a ≥ b, in Section 4.1.4; the caseswith D = HSO∗2n

in Section 4.1.5; the cases with D+ = H+SOa,2

, with a ≥ 2,

in Section 4.1.6; and the cases with D = HE6or D = HE7

in Section 4.1.7.

4.1.1. Cases with D = H. Suppose G = SL2, so that we have D = H as inExamples 2.2.7 and 2.2.23, and we have to consider quotients of the form Γ\D,where Γ is an arithmetic subgroup of SL2(Q). Note that the action of SL2(Q)preserves

H∗ := H∪P1(Q) (4.1.1.1)

as a subset of P1(C).

In the natural topology of P1(C), the boundary ofH is P1(R)=SL2(R)·∞,where ∞ means just ( 1

0 ) in projective coordinates. The reason to considerinstead the rational subset P1(Q) = SL2(Q) · ∞ is the reduction theory : Thefundamental domain ofH under the action of Γ can be compactified by addingonly points in P1(Q) = SL2(Q) · ∞. The irrational points are completely ir-relevant. Therefore, by topologizing H∗ at the points of P1(Q) using opendiscs bounded by the so-called horocycles, namely, circles tangent to P1(R)at the points of P1(Q) (see [169, Section 1.3]), we obtain a compactification

Γ\H ↪→ Γ\H∗ (4.1.1.2)

such that Γ\H∗ has the structure of a compact Riemann surface (or, in otherwords, a complete algebraic curve over C), with finitely many boundary points

Γ\H∗ − Γ\H ∼= Γ\P1(Q) (4.1.1.3)

called cusps. (Here the finiteness of Γ\P1(Q) can be deduced from the el-ementary fact that P1(Q) = SL2(Q) · ∞ = SL2(Z) · ∞.) The points ofP1(Q) = SL2(Q) · ∞ are the so-called rational boundary components of H,which are prototypical examples of their analogues for more general Hermit-ian symmetric domains.

The notions of cusps and rational boundary components depend on thestructure of G = SL2 as an algebraic group over Q in an essential way. Thereexist other algebraic groups G′ over Q such that G′(R) ∼= G(R) = SL2(R)and such that the fundamental domains for the actions of the correspondingarithmetic subgroups Γ′ of G′(Q) on D are already compact, in which casethere is no need to add any cusps (cf. the example of Shimura curves inSection 5.2.2 below).

4.1.2. Cases with D = H×H. There are two important cases of G with G(R)acting on D = H×H.


The first G is simply SL2×SL2. Then we should take

D∗ := H∗×H∗

= (H×H)∪(G(Q) · (H×∞)

)∪(G(Q) · (∞×H)

)∪(G(Q) · (∞×∞)

)(4.1.2.1)

as subsets of P1(C)×P1(C). There are three G(Q)-orbits of rational boundarycomponents (which are those orbits other than H×H).

The other G is the restriction of scalars ResF/Q SL2, for some realquadratic extension field F of Q. It is the group scheme G over Q whoseR-points are

G(R) = SL2(R⊗QF ), (4.1.2.2)

for each Q-algebra R. In particular, we have G(Q) = SL2(F ). This is thecase of Hilbert modular surfaces, which are generalized to higher-dimensionalHilbert modular varieties by replacing F with totally real extension fieldsof Q of higher degrees. Then we should take

D∗ := (H×H)∪(G(Q) · (∞×∞)

), (4.1.2.3)

with only one G(Q)-orbit of rational boundary components (which is theunique orbit other than H×H).

In both cases, we can topologize D∗ such that Γ\D∗ is a compactificationof Γ\D for each arithmetic subgroup Γ of G(Q). When G is simply SL2×SL2,this is just doubling the case where G = SL2. When G is ResF/Q SL2, this isessentially proved in [165], in a different language. This generalizes the casewhere G = SL2 but will be superseded by the introduction of Satake topologyin general.

Here are some general principles, which are valid for not only these twoexamples, but also for the forthcoming ones:

(1) The rational boundary components are in one-one correspondence with(rational) parabolic subgroups P of the algebraic group G over Q whosepullback to every Q-simple (almost) factor G′ of G is either the wholeG′ or a maximal proper parabolic subgroup P′. When the group G isQ-simple, we will just say that such a P is a maximal parabolic subgroup.

(2) For classical groups G defined by nondegenerate pairings over finite-dimensional semisimple algebras over Q, the (rational) parabolic sub-groups of G are always stabilizers of increasing sequences of totallyisotropic subspaces.

4.1.3. Cases with D = Hn. Suppose G = Sp2n, so that G(R) acts on Hn asin Section 3.1.1.

When n = 2, there are three G(Q)-orbits of proper parabolic subgroupsof G(Q), with the following representatives, respectively: (We warn the read-ers that, although we will write down only the groups of Q-points, we willstill be working with algebraic groups over Q. This is a somewhat misleadingabuse of language, which we adopt only for simplicity of exposition.)

40 Kai-Wen Lan

(1) A Borel subgroup

B(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗∗ ∗

(4.1.3.1)

(which is a minimal parabolic subgroup) stabilizing the increasing se-quence

∗000

⊂

∗∗00

(4.1.3.2)

of totally isotropic subspaces of Q4. Note that we have∗000

⊥

=

∗∗0∗

(4.1.3.3)

and ∗∗00

⊥

=

∗∗00

(4.1.3.4)

with respect to the alternating pairing

〈 · , · 〉 : Q4×Q4 → Q (4.1.3.5)

defined by the matrix J2 =(

12−12

), and so we can complete the above

sequence (4.1.3.2) into a flag

0000

⊂

∗000

⊂

∗∗00

⊂

∗∗0∗

⊂

∗∗∗∗

(4.1.3.6)

in Q4. This also explains why the entries of B are not exactly upper-triangular. Sometimes it is convenient to work with upper-triangularmatrices—then one should replace J2 with a matrix that is indeed anti-diagonal, not just block anti-diagonal, as remarked in Example 2.1.2.

(2) The Klingen parabolic subgroup

P(1)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

(4.1.3.7)


(which is a maximal parabolic subgroup) stabilizing the totally isotropicsubspace

∗000

(4.1.3.8)

of Q4.

(3) The Siegel parabolic subgroup

P(2)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗∗ ∗

(4.1.3.9)

(which is a maximal parabolic subgroup) stabilizing the totally isotropicsubspace

∗∗00

(4.1.3.10)

of Q4.

The Borel subgroup B is not maximal, and hence will not be needed in theconsideration of rational boundary components. Nevertheless, we have

B(Q) = P(1)(Q)∩P(2)(Q) (4.1.3.11)

by definition, and P(1)(Q) and P(2)(Q) will both be needed.

Now that we have seen the maximal parabolic subgroups when n = 2,let us consider a more general n. The rational boundary components of Hnare given by elements of the G(Q)-orbits of(

∞r

Hn−r

)∼= Hn−r, (4.1.3.12)

which means

1rZ

0r1n−r

: Z ∈ Hn−r

(4.1.3.13)

in generalized projective coordinates (cf. Remark 3.1.1.4), for r = 1, . . . , n.(When r = 0, we just get Hn itself.)

There is a topology, called the Satake topology (see [147], [9, Theorem4.9], and [19, Section III.3] for systematic treatments not just for this case),on the set

H∗n := Hn ∪(∪

1≤r≤nG(Q) · Hn−r

), (4.1.3.14)

where Hn−r stands for(∞r

Hn−r)

as in (4.1.3.12), such that Hn−r lies in theclosure ofHn−s exactly when r ≥ s, and such that Γ\H∗n is a compactification

42 Kai-Wen Lan

of Γ\Hn as a topological space, for every arithmetic subgroup Γ of G(Q) (bythe reduction theory, as in [18] and [11]). According to [9], Γ\H∗n is a normalprojective variety over C, which is called the Satake–Baily–Borel or minimalcompactification of Γ\Hn. (The references given above are not the historicalones mainly for H∗n and Γ\H∗n, such as [146], [7], [28], and [154]—see [153,Section 2] for a nice review. Rather, they are more recent ones that also workfor our later examples.)

Each

(∞r

Hn−r

), or rather

1rZ

0r1n−r

: Z ∈ Hn−r

in gener-

alized projective coordinates (see (4.1.3.13)), is stabilized by a maximal par-abolic subgroup of G(Q) of the form

P(r)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

(4.1.3.15)

because ∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

1∗

01

=

∗ ∗∗

01

∼

1∗

01

, (4.1.3.16)

where the sizes of the block matrices are given by r, n − r, r, and n − r,both horizontally and vertically. This P(r)(Q) is the stabilizer of the totallyisotropic Q-subspace

x000

: x ∈ Qr

(4.1.3.17)

of

Q2n ∼= Qr ⊕Qn−r ⊕Qr ⊕Qn−r. (4.1.3.18)

We have

P(r)(Q) = L(r)(Q) n U(r)(Q), (4.1.3.19)

where

L(r)(Q) =

X

A BtX−1

C D

:X ∈ GLr(Q),(

A BC D

)∈ Sp2n−2r(Q)

∼= GLr(Q)×Sp2n−2r(Q)

(4.1.3.20)


is a Levi subgroup, and where

U(r)(Q) =

1 ∗ ∗ ∗1 ∗

1∗ 1

(4.1.3.21)

is the unipotent radical . Note that L(r)(Q) acts on the rational boundarycomponent H2n−2r via the second factor Sp2n−2r(Q). This is the so-called

Hermitian part of the Levi subgroup L(r)(Q). The unipotent radical U(r)(Q)is an extension

1→W(r)(Q)→ U(r)(Q)→ V(r)(Q)→ 1, (4.1.3.22)

where

W(r)(Q) =

1 Y1

11

: Y ∈ Symr(Q)

∼= Symr(Q) (4.1.3.23)

is the center of U(r)(Q), and where

V(r)(Q) ∼= Q2n−2r (4.1.3.24)

(by viewing Symr(Q) and Q2n−2r as commutative algebraic groups over Q).

4.1.4. Cases with D = Ha,b, with a ≥ b. Suppose E is some imaginaryquadratic extension field of Q. Suppose G is a group scheme over Q defined by

G(R) := {g ∈ GLa+b(E⊗QR) : tgJa,bg = Ja,b}, (4.1.4.1)

for each Q-algebra R, where Ja,b is some skew-Hermitian matrix as in (3.2.2.2)with S a skew-Hermitian matrix with entries in E. (We do not need to chooseany S when a = b.) Then G(R) is the group U′a,b defined by Ja,b as in (3.2.2.1),which acts on Ha,b as in Section 3.2.2.

In this case, the rational boundary components of Ha,b (with respect tothe choice of G) are given by elements of the G(Q)-orbits of(

∞r

Ha−r,b−r

)∼= Ha−r,b−r, (4.1.4.2)

which means

1r

ZW

0r1b−r

:

(ZW

)∈ Ha−r,b−r

(4.1.4.3)

in generalized projective coordinates (cf. Remark 3.1.1.4), for r = 1, . . . , b.(When r = 0, we just get Ha,b itself.)

44 Kai-Wen Lan

With the Satake topology (see the reference given in Section 4.1.3, whichwe shall not repeat) on the set

H∗a,b := Ha,b ∪(∪

1≤r≤bG(Q) · Ha−r,b−r

), (4.1.4.4)

where Ha−r,b−r stands for(∞r

Ha−r,b−r)

as in (4.1.4.2), Ha−r,b−r lies inthe closure of Ha−s,b−s exactly when r ≥ s, and the quotient Γ\H∗a,b is

a compactification of Γ\Ha,b as a topological space, for every arithmeticsubgroup Γ of G(Q). Again, according to [9], Γ\H∗a,b is a normal projective

variety over C, which is the minimal compactification of Γ\Ha,b.

Each(∞r

Ha−r,b−r), or rather

{( 1rZW

0r1b−r

): ( ZW ) ∈ Ha−r,b−r

}in gen-

eralized projective coordinates (see (4.1.4.3)), is stabilized by a maximal par-abolic subgroup of G(Q) of the form

P(r)(Q) =

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗∗ ∗ ∗ ∗

, (4.1.4.5)

where the sizes of the block matrices are given by r, b− r, a− b, r, and b− r,both horizontally and vertically. This P(r)(Q) is the stabilizer of the totallyisotropic E-subspace

x0000

: x ∈ Er

(4.1.4.6)

of

Ea+b ∼= Er ⊕Eb−r ⊕Ea−b⊕Er ⊕Eb−r. (4.1.4.7)

We have

P(r)(Q) = L(r)(Q) n U(r)(Q), (4.1.4.8)

where

L(r)(Q) =

X

A E BF M G

tX−1

C H D

:

X ∈ GLr(E),A E BF M GC H D

∈ G(r)(Q)

∼= GLr(E)×G(r)(Q)

(4.1.4.9)


is a Levi subgroup, where G(r) is the analogue of G defined by the smallermatrix

Ja−r,b−r =

1b−rS

−1b−r

(4.1.4.10)

with the same S; and where

U(r)(Q) =

1 ∗ ∗ ∗ ∗

1 ∗1 ∗

1∗ 1

(4.1.4.11)

is the unipotent radical. Note that L(r)(Q) acts on the rational boundarycomponentHa−r,b−r via the second factor G(r)(Q). This is the Hermitian part

of the Levi subgroup L(r)(Q). The unipotent radical U(r)(Q) is an extension

1→W(r)(Q)→ U(r)(Q)→ V(r)(Q)→ 1, (4.1.4.12)

where

W(r)(Q) =

1 Y

11

11

: Y ∈ Hermr(E)

∼= Hermr(E)

(4.1.4.13)is the center of U(r)(Q), and where

V(r)(Q) ∼= E(a−r)+(b−r) (4.1.4.14)

(by viewing Hermr(E) and E(a−r)+(b−r) as commutative algebraic groupsover Q). We note that the overall picture is very similar to the case of Hn inSection 4.1.3, although we need to work with coordinates over E and hencewith GLr(E) as an algebraic group over Q (which means we are working withthe restriction of scalars ResE/Q GLr), and with the matrix S when a > b.

Let us mention two important variants which frequently appear in theliterature.

The first variant replaces the imaginary quadratic extension field E ofQ in the above definition of G with a CM extension (namely, a totally imagi-nary quadratic extension of a totally real extension) F of Q, and replaces Ja,bwith a nondegenerate skew-Hermitian matrix J over OF such that −iJ de-fines a Hermitian pairing of signatures (a1, b1), (a2, b2), . . . , (ad, bd) at the dreal places of the maximal totally real subextension F+ of Q in F , whered := [F+ : Q], for some integers aj ≥ bj ≥ 0 and n ≥ 0 such that aj + bj = n,for all j = 1, . . . , d. Then

G(R) ∼= U′a1,b1 ×U′a2,b2 × · · ·×U′ad,bd (4.1.4.15)

acts onD ∼= Ha1,b1 ×Ha2,b2 × · · ·×Had,bd , (4.1.4.16)

46 Kai-Wen Lan

and the rational boundary components of D are given by elements of theG(Q)-orbits of some

D(r) ∼= Ha1−r,b1−r ×Ha2−r,b2−r × · · ·×Had−r,bd−r, (4.1.4.17)

for r = 1, . . . ,min(b1, b2, . . . , bd). Note that there is no r at all if at least oneof b1, b2, . . . , bd is zero.

In the special case where (a1, b1) = (a, b) but (aj , bj) = (n, 0) for allj 6= 1, for some integers a ≥ b ≥ 0 and n = a+ b ≥ 0, we simply have

D ∼= Ha,b. (4.1.4.18)

If d = 1, then this is just as before. If d ≥ 2, then D∗ = D and the quo-tient Γ\D is automatically compact, for every arithmetic subgroup Γ of G(Q).Whether d ≥ 2 or not, when (a, b) = (2, 1) (and when (4.1.4.18) holds), thequotient Γ\D is called a Picard modular surface.

The second variant is a further generalization of the above first variant,by replacing F with a central simple algebra B over a CM extension over Qthat is equipped with a positive involution ? (namely, an anti-automorphismof order 2) such that trB/Q(xx?) > 0 for every x 6= 0 in B, and by defin-ing G using a nondegenerate skew-Hermitian pairing over a finitely generatedB-module. Then we still have an isomorphism as in (4.1.4.16), and the ratio-nal boundary components of D are still of the same form as in (4.1.4.17), butthe values of r allowed in (4.1.4.17) are more restrictive, depending on thepossible Q-dimensions of totally isotropic B-submodules. If B is a divisionalgebra and the pairing is defined over a simple B-module, then no r > 0 isallowed, in which case Γ\D is again automatically compact, for every arith-metic subgroup Γ of G(Q).

4.1.5. Cases with D = HSO∗2n. Starting with this case, we will be less explicit

about the choices of G. There exists a group scheme G over Q such thatG(R) ∼= SO∗2n, such that the rational boundary components of HSO∗2n

are

given by elements of the G(Q)-orbits of some D(r) ∼= HSO∗2n−4r(which we

shall abusively denote as HSO∗2n−4r), with r = 1, . . . , bn2 c, and such that, with

the Satake topology on

H∗SO∗2n:= HSO∗2n

∪(

∪1≤r≤bn2 c

G(Q) · HSO∗2n−4r

), (4.1.5.1)

HSO∗2n−4rlies in the closure of HSO∗2n−4s

exactly when r ≥ s, and the quo-

tient Γ\H∗SO∗2ngives the minimal compactification of Γ\HSO∗2n

, for every

arithmetic subgroup Γ of G(Q).

When n = 2k, an example of G is given by

G(R) := {g ∈ GL2k(O⊗ZR) : tgJkg = Jk}, (4.1.5.2)

for each Q-algebra R, where B is a quaternion algebra over Q such thatB ⊗Q R ∼= H, and where Jk =

(1k

−1k

)is as before. Then we can describe


the rational boundary components HSO∗4k−4rof HSO∗4k

as{(∞r

Z

): Z ∈ Hermk−r(H)⊗

RC, Im(Z) > 0

}, (4.1.5.3)

which means

1rZ

0r1k−r

: Z ∈ Hermk−r(H)⊗RC, Im(Z) > 0

(4.1.5.4)

in generalized projective coordinates (cf. Remark 3.1.1.4), for r = 1, . . . , k.Each HSO∗4k−4r

as above is stabilized by a maximal parabolic subgroup of

G(Q) of the form

P(r)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

(4.1.5.5)

because ∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

1∗

01

=

∗ ∗∗

01

∼

1∗

01

, (4.1.5.6)

where the sizes of the block matrices are given by r, n − r, r, and n − r,both horizontally and vertically. This P(r)(Q) is the stabilizer of the totallyisotropic B-submodule

x000

: x ∈ Br

(4.1.5.7)

of

B2n ∼= Br ⊕Bn−r ⊕Br ⊕Bn−r, (4.1.5.8)

with left B-module structures given by having each b ∈ B act by right mul-tiplication of its conjugate b, so that they do not intervene with left multi-plications of matrix entries. We have

P(r)(Q) = L(r)(Q) n U(r)(Q), (4.1.5.9)

where

L(r)(Q) =

X

A BtX−1

C D

:X ∈ GLr(B),(A BC D

)∈ G(r)(Q)

∼= GLr(B)×G(r)(Q)

(4.1.5.10)

48 Kai-Wen Lan

is a Levi subgroup, where G(r) is the analogue of G defined by the smallermatrix

Jk−r =

(1k−r

−1k−r

); (4.1.5.11)

and where

U(r)(Q) =

1 ∗ ∗ ∗1 ∗

1∗ 1

(4.1.5.12)

is the unipotent radical. Note that L(r)(Q) acts on the rational boundarycomponent HSO∗4n−4r

via the second factor G(r)(Q). This is the Hermitian

part of the Levi subgroup L(r)(Q). The unipotent radical U(r)(Q) is an ex-tension

1→W(r)(Q)→ U(r)(Q)→ V(r)(Q)→ 1, (4.1.5.13)

where

W(r)(Q) =

1 Y1

11

: Y ∈ Hermr(B)

∼= Hermr(B) (4.1.5.14)

is the center of U(r)(Q), and where

V(r)(Q) ∼= B2k−2r (4.1.5.15)

(by viewing Hermr(B) and B2k−2r as commutative algebraic groups over Q).We note that, again, the overall picture is very similar to the one in Sec-tion 4.1.3, although we need to work with coordinates over B and hencewith GLr(B) as an algebraic group over Q.

4.1.6. Cases with D+ = H+SOa,2

, with a ≥ 2. Suppose G is a group scheme

over Q defined by some Q-valued symmetric bilinear form Q on Qa+2 whichhas signature (a, 2) over R, so that G(R) ∼= SOa,2(R) and G(R)+ ∼= SOa,2(R)+.(The precise choice of Q will not be important for our exposition.)

The maximal parabolic subgroups of G(Q) = SO(Qa+2, Q), whereSO(Qa+2, Q) denotes the subgroup of SLa+2(Q) consisting of elements pre-serving the pairing Q (and we shall adopt similar notation in what follows),are in bijection with the nonzero subspaces I of Qa+2 that are totally isotropicwith respect to the bilinear form Q. Then we must have 1 ≤ dimQ(I) ≤ 2because the signature of Q is (a, 2) over R. (But it might happen that nosuch I exists.) For each such I, we have a parabolic subgroup of G(Q) of theform

PI(Q) ∼= LI(Q) n UI(Q), (4.1.6.1)

where

LI(Q) ∼= SO(I⊥/I,Q|I⊥/I)×GLQ(I) (4.1.6.2)


is the Levi quotient (which is also noncanonically a subgroup), and whereUI(Q) is the unipotent radical which fits into a short exact sequence

1→ ∧2I → UI(Q)→ HomQ(I⊥/I, I)→ 1. (4.1.6.3)

When dimQ(I) = 1, we have

LI(R) ∼= SOa−1,1(R)×GL1(R), (4.1.6.4)

and so LI(R)∩G(R)+ acts trivially on some zero-dimensional rational bound-ary component

H+I∼= H0. (4.1.6.5)

When dimQ(I) = 2, we have

LI(R) ∼= SOa−2(R)×GL2(R), (4.1.6.6)

and so LI(R)∩G(R)+ acts via the usual Mobius transformation of

GL2(R)+ = {g ∈ GL2(R) : det(g) > 0} (4.1.6.7)

on some one-dimensional rational boundary component

H+I∼= H1. (4.1.6.8)

For example, suppose (for simplicity) that Q is defined by the symmetricmatrix

1a−2

0 11 0

0 11 0

, (4.1.6.9)

where the sizes of the block matrices are given by a− 2, 1, 1, 1, and 1. If

I =

000∗0

, (4.1.6.10)

which is one-dimensional, then

PI(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗

∼= LI(Q) n UI(Q), (4.1.6.11)

where

LI(Q) =

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

XX−1

:

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

∈ SOa−1,1(Q), X ∈ Q×

(4.1.6.12)

50 Kai-Wen Lan

and

UI(Q) =

1 x

1 y1 z

− tx −z −y 1 − 12txx− yz

1

: x ∈ Qa−2, y, z ∈ Q

;

(4.1.6.13)and PI(Q) stabilizes

H+I =

00010

(4.1.6.14)

because ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗

00010

=

000∗0

∼

00010

(4.1.6.15)

in projective coordinates. Alternatively, if

I =

0∗0∗0

, (4.1.6.16)

which is two-dimensional, then

PI(Q) =

∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗ ∗∗ ∗ ∗ ∗ ∗

∗ ∗

∼= LI(Q) n UI(Q), (4.1.6.17)

where

LI(Q) =

X

a ba′ b′

c dc′ d′

:

X ∈ SOa−2(Q),

t

(a′ b′

c′ d′

)−1

=

(a bc d

)∈ GL2(Q)

(4.1.6.18)


and

UI(Q) =

1 0 x 0 y− tx 1 − 1

2txx 0 w

0 0 1 0 0− ty 0 −w − txy 1 − 1

2tyy

0 0 0 0 1

: w ∈ Q, x, y ∈ Qa−2

;

(4.1.6.19)and PI(Q)∩G(R)+ stabilizes

H+I =

0z010

: z ∈ C, Im z > 0

(4.1.6.20)

because∗ ∗ ∗∗ a b ∗

∗ ∗∗ c ∗ d ∗

∗ ∗

0z010

=

0

az + b0

cz + d0

∼

0(az + b)(cz + d)−1

010

(4.1.6.21)

in projective coordinates.

There is a systematic way to combine these into a union

(H+SOa,2

)∗ = H+SOa,2

∪

∪06=I totally

isotropic

H+I

, (4.1.6.22)

equipped with Satake topology, such that H+I lies in the closure of H+

I′ (fornonzero I and I ′) exactly when I ⊂ I ′ (which is qualitatively quite differ-ent from the previous cases in Sections 4.1.3, 4.1.4, and 4.1.5, where rationalboundary components associated with larger totally isotropic subspaces aresmaller) and such that the quotient Γ\(H+

SOa,2)∗ gives the minimal com-

pactification of Γ\H+SOa,2

, for every arithmetic subgroup Γ of G(Q) that is

contained in G(R)+.

4.1.7. Cases with D = HE6 or D = HE7 . We shall be very brief in these twocases. For HE6 , there exists a group scheme G over Q such that G(R) is asin Section 3.6.2 and such that the rational boundary components of HE6

areisomorphic to either H5,1 or H0. For HE7

, there exists a group scheme Gover Q such that G(R) is as in Section 3.5.3 and such that the rationalboundary components of HE7

are isomorphic to either H+SO10,2

, H1, or H0.

In each of these two cases, by [139, Section 3, Proposition 3], there existssome G such that G(R) is as above and such that rankQ GQ = rankR GR;or, in other words, GQ is as split as GR allows. In both cases, we have theminimal compactifications, as usual, for arithmetic subgroups Γ of G(Q).

52 Kai-Wen Lan

4.2. Compactifications and algebraicity

4.2.1. Overview. In all cases considered so far, we have the following diagramof useful compactifications of Γ\D, with canonical morphisms between themdenoted by solid arrows:

(Γ\D)torΣ

&&

not directly related(Γ\D)BS

��

(Γ\D)RBS

uu(Γ\D)

min

(4.2.1.1)

Let us explain the objects in this diagram.

4.2.2. (Γ\D)min

= Γ\D∗. This is the Satake–Baily–Borel or minimal com-pactification of Γ\D, which is a normal projective variety over C (see [9]).Here D∗ is the union of D with its rational boundary components, equippedwith the Satake topology, as explained in Section 4.1. The transition fromthe analytic quotient Γ\D∗ to an algebraic variety is achieved by first en-dowing Γ\D∗ with the structure of a complex analytic space (see [163, §1]) inwhich the closed complement of the open subset Γ\D of Γ\D∗ is a closed com-plex analytic subspace; and then by constructing a closed immersion of Γ\D∗into a projective space PN (C) for some N , and by applying Chow’s theo-rem (or “GAGA” for closed complex analytic subspaces of PNC , showing thatthey are all complex analytifications of algebraic subvarieties—see [163, §3,

19, Proposition 13]). Then the open subspace Γ\D of (Γ\D)min

= Γ\D∗ isa quasi-projective variety over C, by applying Chow’s theorem again to theclosed complement. (This is the only known general method to show that Γ\Dis an algebraic variety.) Moreover, it is proved in [14] that any holomorphicmap from a quasi-projective variety to Γ\D is the complex analytificationof a morphism of algebraic varieties. In particular, the structure of Γ\D asan algebraic variety is unique up to isomorphism. (These are the key resultsused in the proof of Theorem 2.4.1.)

The compactification (Γ\D)min

is canonical and does not depend onany choices, but is highly singular in general. As we have seen in the ex-

amples in Section 4.1, the boundary (Γ\D)min−(Γ\D) has a high codimen-

sion in (Γ\D)min

in general. When the connected components of D are ir-reducible in the sense explained in the beginning of Section 3 but are notone-dimensional, and when Γ\D is not already compact, this codimension isalways at least two. The intuition is that we are collapsing a lot into a smallneighborhood of each boundary point.

4.2.3. (Γ\D)torΣ . This is the toroidal compactification of Γ\D (see [5] or [6],

and see also [126]), which is an algebraic space depending on the choice of


some combinatorial data Σ called a cone decomposition (or fan). (More pre-cisely, Σ is a compatible collection of cone decompositions, but we prefer notto go into the details here.) This means (Γ\D)

torΣ is not a canonical compacti-

fication of Γ\D in the sense that it does not just depend on Γ\D. Nevertheless,

it depends only on Γ\D and Σ, and there exist choices of Σ such that (Γ\D)torΣ

is a smooth projective variety , and such that the boundary (Γ\D)torΣ − (Γ\D)

is a simple normal crossings divisor . Such choices can always be achieved byreplacing any given Σ with a refinement . In particular, we obtain meaning-ful smooth projective compactifications by paying the price of working withthe noncanonical Σ’s. (As a comparison, we have no precise control on the

smooth compactifications obtained by resolving the singularities of (Γ\D)min

using Hironaka’s general theory in [68, 69].)

As in the case of (Γ\D)min

, the boundary of (Γ\D)torΣ is stratified by

locally closed subsets corresponding to Γ-orbits of the proper parabolic sub-groups P of the algebraic group G over Q whose pullback to every Q-simple(almost) factor G′ of G is either the whole G′ or a maximal parabolic sub-group P′. This stratification is further refined by a stratification in terms ofthe cones in the cone decomposition Σ. The incidence relations of the stratain this finer stratification is dual to the incidence relations of the cones in thecone decomposition Σ. The canonical map (Γ\D)

torΣ → (Γ\D)

minin (4.2.1.1)

respects such stratifications.

One subtlety is that, except in very special cases, no choice of Σ can becompatible with all Hecke correspondences. For arithmetic applications, it isnecessary to systematically work with refinements of Σ.

4.2.4. (Γ\D)BS. This is the Borel–Serre compactification of Γ\D (see [20] and[19, Section III.5]; see also [57, Section 4]), which is a (real analytic) manifoldwith corners. It can be defined more generally when D is a Riemannian sym-metric space that is not Hermitian, but is almost never an algebraic variety.Already when G = SL2 and D = H (and when Γ is neat), the compactifi-cation (Γ\D)BS is achieved by adding circles to D (instead of adding pointsto D as in the case of D∗). In general, its boundary is stratified by locallyclosed subsets of G corresponding to the Γ-orbits of all proper rational par-abolic subgroups P of the algebraic group G over Q, and the constructionof the subset associated with P uses the whole P, instead of just its Levi

quotient as in the case of (Γ\D)min

.

The most important feature of (Γ\D)BS is that the canonical open em-bedding Γ\D ↪→ (Γ\D)BS is a homotopy equivalence, which makes (Γ\D)BS

useful for studying the cohomology of Γ\D, and also that of the arithmeticsubgroup Γ of G(Q), which are closely related to the theory of automorphicrepresentations and Eisenstein series (see, for example, [22], [162], [53], [54],and [106]). But let us remark here, as an addendum to Section 4.2.2 above,that cuspidal automorphic representations are more closely related to theL2-cohomology of Γ\D, and this L2-cohomology is canonical isomorphic to

54 Kai-Wen Lan

the intersection cohomology of (Γ\D)min

. This was conjectured by Zucker (see[178, (6.20)]) and proved by Looijenga (see [108]), Saper–Stern (see [145]),and Looijenga–Rapoport (see [109]).

4.2.5. (Γ\D)RBS. This is the reductive Borel–Serre compactification of Γ\D(see [178, Section 4], [58, Part I], and [19, Section III.6]; see also [57, Section5]), which is similar to (Γ\D)BS in the following aspects: It can also be de-fined more generally when D is a Riemannian symmetric space that is notHermitian, and its boundary is also stratified by locally closed subsets of Gcorresponding to the Γ-orbits of all proper rational parabolic subgroups P ofthe algebraic group G over Q. But there is a key difference—the subset asso-ciated with P is constructed using only the (reductive) Levi quotient L of P.The reductive Borel–Serre compactification (Γ\D)RBS is useful for studying,for example, the weighted cohomology of Γ (see [58, Parts II–IV]; see also [57,Section 6.8]).

4.2.6. Models over number fields or their rings of integers. Although(Γ\D)BS and (Γ\D)RBS are very useful analytic objects, when it comes toapplications requiring algebraic varieties (or at least algebraic spaces or al-gebraic stacks) defined over some number fields, we need to work with Γ\D,

(Γ\D)min

, and (Γ\D)torΣ , the last being perhaps the most complicated. It is

known that (Γ\D)min

and (Γ\D)torΣ also have models over the same number

fields over which Γ\D is defined (see [137]), and they extend to good modelsover the integers in many important special cases (see Sections 5.1.4 and 5.2.4below).

All assertions so far in this Section 4.2 have their counterparts forShimura varieties; i.e., the double coset spaces XU defined by Shimura data.(As we explained in Section 4.2.2, the results there are used in the proof ofTheorem 2.4.1.) Moreover, the minimal and toroidal compactifications alsohave canonical models over the same reflex field F0 determined by (G,D) asin Theorem 2.4.3 (see [137] again).

4.2.7. Algebro-geometric definition of modular forms. The projective coor-

dinates of the minimal compactification (Γ\D)min

= Γ\D∗ are given by (holo-morphic) modular forms, namely, holomorphic functions f : D → C satisfyingthe following two conditions:

(1) Automorphy condition: f(γZ) = j(γ, Z)f(Z), for all γ ∈ Γ and Z ∈ D,for some automorphy factor j : Γ×D → C× defining the weight off , satisfying the cocycle condition j(γ′γ, Z) = j(γ′, Z)j(γ, Z) for allγ, γ′ ∈ Γ and Z ∈ D. (This means f represents a section of an automor-phic line bundle over Γ\D defined by j. The line bundles used in [9] arepowers of the canonical bundle, namely, the top exterior power of thecotangent bundle, of the complex manifold Γ\D.)

(2) Growth condition: f(Z) stays bounded near D∗ − D. (When Γ is neat,this implies that the section represented by f extends to a section of a


line bundle over (Γ\D)min

= Γ\D∗.) If f satisfies the additional condi-tion that f(Z)→ 0 as Z → D∗ −D, then f is called a cusp form.

Ratios of such modular forms f define modular functions which are ingeneral highly transcendental as functions of the coordinates of D. Therefore,

to give models of Γ\D and (Γ\D)min

over number fields or over their rings ofintegers, we cannot just naively use the coordinates of D.

Not all modular forms are defined as above using only automorphyfactors valued in C×. It is possible to consider automorphy factors valuedin Centh0

(G(C)), so that it makes sense to consider modular forms withweights given by all finite-dimensional representations W of Centh0

(G(C)).But there is a subtlety here: For such a W , in general, the associated au-tomorphic vector bundle W over Γ\D does not extend to a vector bundle

over (Γ\D)min

. (This is consistent with the fact that, in general, the bound-

ary (Γ\D)min−(Γ\D) has a high codimension in (Γ\D)

min.) Nevertheless, the

automorphic vector bundle W has canonical extensions (see [127] and [60])

to vector bundles W can over the toroidal compactifications (Γ\D)torΣ , whose

sections are independent of Σ. (When W is the restriction of a representationof G(C), the vector bundle W is equipped with an integrable connection, andits canonical extension W can is equipped with an integrable connection withlog poles along the boundary and coincides with the one given in [34].) Theglobal sections or more generally the cohomology classes of such canonicalextensions W can over (Γ\D)

torΣ then provide an algebro-geometric definition

of modular forms with coefficients in W . There are also the subcanonicalextensions W sub := W can(−∞), where ∞ abusively denotes the boundary

divisor (Γ\D)tor−(Γ\D) with its reduced structure, whose global sections are

(roughly speaking) useful for studying cusp forms. The cohomology of W can

and W sub is often called the coherent cohomology of (Γ\D)tor

. It is this ap-proach that will allow us to define modular forms in mixed or positive char-acteristics and study congruences among them using integral models. Evenin characteristic zero, there have been some results which have only beenproved with such an algebro-geometric definition. (See [61], [117], [88], and[92] for some surveys on this topic.)

4.2.8. Mixed Shimura varieties. Just as Shimura varieties parameterize cer-tain variations of Hodge structures with additional structures, there is also anotion of mixed Shimura varieties parameterizing certain variations of mixedHodge structures with additional structures. Then there is also a theory ofcanonical models for such mixed Shimura varieties, and even for their toroidalcompactifications—see [117] and [137]. Many mixed Shimura varieties natu-rally appear along the boundary of the toroidal compactifications of pureShimura varieties (i.e., the usual ones associated with reductive groups as inSections 2.3 and 2.4).

56 Kai-Wen Lan

The theory of mixed Shimura varieties is even heavier in notation thanthe theory of (pure) Shimura varieties, and is beyond the scope of this in-troductory article. But let us at least mention the following prototypical ex-ample. Consider the Shimura varieties XU associated to (G,D) = (GL2,H±),whereH± := H±1 , which are modular curves. For each integer m ≥ 1, consider

the (non-reductive) semidirect product G = GL2 n (G2a)m, with G = GL2

acting diagonally on (G2a)m, where Ga is the additive group scheme over Z,

so that GL2(R) acts by left multiplication on (G2a)(R) = R2 for every ring R.

Consider the corresponding action of G(R) on D = H± × (R2)m. Then weobtain by considering double coset spaces as in Section 2.2 the mixed Shimura

varieties XU over XU for neat open compact subgroups U with image in U un-

der the canonical homomorphism G(A∞)→ G(A∞), which are torsors underabelian schemes over XU that are isogenous to the m-fold self-fiber product

of the universal elliptic curve over XU . Specifically, there exists some U such

that XU is exactly isomorphic to the m-fold self-fiber product of the univer-sal elliptic curve over XU . The projective smooth toroidal compactificationsof such mixed Shimura varieties then provide generalizations of the classicalKuga–Sato varieties, whose cohomology with trivial coefficients are useful forstudying the cohomology with nontrivial coefficients over the usual modularcurves XU , or even for the construction of related motives—see, for example,[35, Section 5] and [155]. (We can also reduce questions about the cohomol-ogy of Shimura varieties with nontrivial coefficients to questions about thecohomology of mixed Shimura varieties with trivial coefficients in many moregeneral cases—see the end of Section 5.1.4 below.)

5. Integral models

5.1. PEL-type cases

All currently known constructions of integral models of Shimura varieties(which we can describe in some reasonable detail) rely on the importantcoincidence that, as explained in Section 3.1.5, when G = Sp2n and Γ is acongruence subgroup of G(Z), the Siegel modular variety Γ\Hn parameterizespolarized abelian varieties with level structures; and on the fact that we candefine moduli problems over the integers that naturally extend such complexvarieties. More generally, we can define the so-called PEL moduli problems,which parameterize abelian schemes with additional PEL structures. As weshall see below, the abbreviation PEL stands for polarizations, endomorphismstructures, and level structures. This is the main topic of this subsection.

5.1.1. PEL datum. The definition of PEL moduli problems require some lin-ear algebraic data. Suppose we are given an integral PEL datum

(O, ?, L, 〈 · , · 〉, h0), (5.1.1.1)

whose entries can be explained as follows:


(1) O is an order in a finite-dimensional semisimple algebra over Q.

(2) ? is a positive involution of O; namely, an anti-automorphism of order2 such that trO⊗ZR/R(xx?) > 0 for every x 6= 0 in O ⊗Z R.

(3) L is an O-lattice; namely, a finitely generated free Z-module L with thestructure of an O-module.

(4) 〈 · , · 〉 : L×L → Z(1) is an alternating pairing satisfying the conditionthat

〈bx, y〉 = 〈x, b?y〉 (5.1.1.2)

for all x, y ∈ L and b ∈ O. Here

Z(1) := ker(exp : C→ C×) = 2πiZ (5.1.1.3)

is the formal Tate twist of Z.

(5) h0 is an R-algebra homomorphism

h0 : C→ EndO⊗ZR(L⊗

ZR), (5.1.1.4)

satisfying the following conditions:(a) For any z ∈ C and x, y ∈ L⊗ZR, we have 〈h0(z)x, y〉 = 〈x, h0(z)y〉,

where z 7→ z denotes the complex conjugation.

(b) The (symmetric) R-bilinear pairing (2πi)−1〈 · , h0(i)( · )〉 on L⊗ZRis positive definite.

What do these mean? Given any integral PEL datum as above, we canwrite down a real torus

A0 := (L⊗ZR)/L (5.1.1.5)

with a complex structure given by h0 : C→ EndO⊗ZR(L⊗ZR), which can beshown to be an abelian variety (not just a complex torus) with a polarization

λ0 : A0 → A∨0∼= (L⊗

ZR)/L#, (5.1.1.6)

where A∨0 is the dual abelian variety of A0, with

L# := {x ∈ L⊗ZQ : 〈x, y〉 ∈ Z(1), for all y ∈ L} (5.1.1.7)

the dual lattice of L with respect to 〈 · , · 〉, and where the λ0 is induced bythe natural inclusion L ⊂ L# (see [124, Chapter I, Section 2, and ChapterII, Section 9] and [87, Section 2.2]; cf. [90, Section 1.3.2]). The assumptionsthat L is an O-lattice and that the compatibility (5.1.1.2) holds imply thatwe have an endomorphism structure

i0 : O → EndC(A0) (5.1.1.8)

58 Kai-Wen Lan

satisfying the Rosati condition; namely, for each b ∈ O, the diagram

A0λ0 //

i(b?)

��

A∨0

(i(b))∨

��

A0λ0

// A∨0

(5.1.1.9)

is commutative (cf. [90, Section 1.3.3]). Moreover, for each integer n ≥ 1, wehave a principal level-n structure

(α0,n : L/nL∼→ A0[n], ν0,n : (Z/nZ)(1)

∼→ µn), (5.1.1.10)

where A0[n] denotes the n-torsion subgroup of A0, which is canonically iso-morphic to ( 1

nL)/L because of (5.1.1.5), where α0,n is induced by this canon-ical isomorphism, and where ν0,n is induced by the exponential functionexp : C→ C× (with kernel Z(1) by definition), so that the diagram

(L/nL)×(L/nL)

α0,n×α0,n o��

〈 · , · 〉// (Z/nZ)(1)

ν0,no��

A0[n]×A0[n]λ0-Weil pairing

// µn

(5.1.1.11)

is commutative (see [87, Sections 2.2 and 2.3]; cf. [90, Section 1.3.6]), and sothat such a level structure is liftable to all higher levels n′ divisible by n. (Thesecond entry ν0,n in (α0,n, ν0,n) is often omitted in the notation, although ν0,n

is not determined by α0,n when 〈 · , · 〉 is not a perfect pairing modulo n.)Furthermore, the homomorphism h0 : C→ EndO⊗ZR(L⊗ZR) defines a Hodgedecomposition

H1(A0,C) ∼= L⊗ZC = V0⊕V 0, (5.1.1.12)

where

V0 := {x ∈ L⊗ZC : h0(z)x = (1⊗ z)x, for all z ∈ C} (5.1.1.13)

and

V 0 := {x ∈ L⊗ZC : h0(z)x = (1⊗ z)x, for all z ∈ C}. (5.1.1.14)

Then we also haveLieA0

∼= V0 (5.1.1.15)

as O⊗ZC-modules. The reflex field F0 for h0 (which is defined in this contextwithout referring to any Shimura datum) is the field of definition of theisomorphism class of V0 as an O⊗Z C-module, or more precisely the subfield

F0 := Q(tr(b|V0) : b ∈ O) (5.1.1.16)

of C (see [90, Section 1.2.5]). This is the smallest subfield of C over whichwe can formulate a trace or determinantal condition on the Lie algebra ofan abelian variety, to ensure that the pullback to C of its Lie algebra isisomorphic to V0 as an O ⊗Z C-module (see [90, Sections 1.2.5 and 1.3.4]).


The upshot is that, by combining all of these, we obtain a tuple

(A0, λ0, i0, (α0,n, ν0,n)), (5.1.1.17)

which is an abelian variety with a polarization, an endomorphism structure,and a level structure, such that LieA0 satisfies the Lie algebra conditiondefined by h0. These additional structures form the so-called PEL structures.Thus, writing down an integral PEL datum means writing down such a tuple.

5.1.2. Smooth PEL moduli problems. Given any integral PEL datum asin (5.1.1.1), we can define the associated PEL moduli problems (at varyinglevels) over (the category of schemes over) F0, or over suitable localizationsof the ring of integers OF0 . Roughly speaking, they are the smallest mod-uli problems (over the respective base rings) parameterizing tuples of abelianschemes with additional PEL structures as above, including the one we wrotedown in (5.1.1.17). See [90, Section 1.4.1] for more details on defining smoothPEL moduli problems over localizations of OF0 of good residue characteris-tics. (A prime number p is good for (O, ?, L, 〈 · , · 〉, h0) if it satisfies all ofthe following conditions: p is unramified in O, as in [90, Definition 1.1.1.18];p does not divide [L# : L]; p 6= 2 whenever (O ⊗Z R, ?) has a simple factorisomorphic to a matrix algebra over the Hamiltonian numbers H with itscanonical positive involution; and there is no nontrivial level structure at p.See [90, Definition 1.4.1.1].) Note that people often talk about good reduc-tions instead of smoothness when describing integral models, even though itis not always clear what reductions should mean without some propernessassumption.

The moduli problems above are defined using the language of isomor-phism classes. If we use the language of isogeny classes instead, then it ismore natural to work with rational PEL datum

(O⊗ZQ, ?, L⊗

ZQ, 〈 · , · 〉, h0), (5.1.2.1)

which is good for defining PEL moduli problems over F0 (in characteristiczero) up to isogenies of all possible degrees, or with its p-integral version

(O⊗ZZ(p), ?, L⊗

ZZ(p), 〈 · , · 〉, h0), (5.1.2.2)

which is good for defining PEL moduli problems over OF0,(p) only up toisogenies of degrees prime to p. The latter was done in [82, Section 5] andgeneralized in [90, Section 1.4.2], and then compared in [90, Section 1.4.3]with the definition by isomorphism classes in [90, Section 1.4.3]. (While thecomparison was not difficult, the author still found it helpful to have thedetails recorded in the literature.)

5.1.3. PEL-type Shimura varieties. The PEL moduli problems as above arerelated to the double coset spaces in Section 2 as follows. Firstly, the subtuple

60 Kai-Wen Lan

(O, ?, L, 〈 · , · 〉) of (5.1.1.1) defines a group scheme G over Z by

G(R) :=

(g, r) ∈ EndO(L⊗ZR)×R× :

〈gx, gy〉 = r〈x, y〉, for all x, y ∈ L⊗ZR

, (5.1.3.1)

for each ring R. The elaborate conditions for a prime p to be good, in the firstparagraph of Section 5.1.2, then imply that G(Zp) is a so-called hyperspecialopen compact subgroup of G(Qp) when GQp is connected as an algebraicgroup over Qp. Secondly, the remaining entry h0 : C → EndO⊗ZR(L ⊗Z R)defines a homomorphism

h0 : C× → G(R), (5.1.3.2)

whose G(R)-conjugacy class defines a manifold

D := G(R) · h0. (5.1.3.3)

Then (G,D) defines a double coset space XU as in (2.2.15), for each neat opencompact subgroup U of G(A∞). When (G,D) is a Shimura datum (which is,however, not always the case), we say it is a PEL-type Shimura datum, andsay XU is a PEL-type Shimura variety . In this case, the reflex field F0 for h0

coincides with the reflex field of (G,D) (see Theorem 2.4.3).

Example 5.1.3.4. Suppose O = Z; ? is trivial; L = Z2n; 〈 · , · 〉 is defined bythe matrix 2πi

(1n

−1n

); and h0(eiθ) =

(cos θ − sin θsin θ cos θ

), for all θ ∈ R. Then

G ∼= GSp2n (see Example 2.1.2), and D = G(R) · h0∼= H±n (see (3.1.4.6)).

This is the important special case of Siegel moduli problems.

Example 5.1.3.5. Suppose O = OE is an order in an imaginary quadraticextension field E of Q, with a fixed isomorphism E ⊗Q R ∼= C; ? is thecomplex conjugation of E over Q; L = Oa+b for some integers a ≥ b ≥ 0;〈 · , · 〉 is defined by 2πi times the composition of the skew-Hermitian pairing

defined by some matrix

(1b

ε1a−b−1b

)with trO/Z : O → Z, where ε ∈ O

satisfies −iε ∈ R>0; and h0(eiθ) =

(cos θ − sin θ

e−iθ

sin θ cos θ

), for all θ ∈ R. Then

G(R) ∼= GUa,b∼= GU′a,b, and D = G(R) · h0

∼= H±a,b (see (3.2.5.9)).

Example 5.1.3.6. SupposeO is an order in a quaternion algebra B over Q suchthat B ⊗Q R ∼= H and O is stabilized by the canonical positive involution ?;L = O2n; 〈 · , · 〉 is 2πi times the composition of the skew-Hermitian pairingdefined by the matrix

(1n

−1n

)with trO/Z :O → Z; and h0(eiθ)=

(cos θ − sin θsin θ cos θ

)for all θ ∈ R. Then the derived group of G(R) is isomorphic to SO∗2n, andD = G(R) · h0 contains HSO∗2n

as a connected component (see Section 3.3.1).However, G is not connected as an algebraic group over Q.

These three examples are prototypical for integral PEL data and theassociated PEL moduli problems. By the classification of positive involutions(see the review in [90, Propositions 1.2.1.13 and 1.2.1.14] based on Albert’sclassification, or just the classification over R in [82, Section 2]), up to cocen-ter (i.e., up to replacing the group with a subgroup containing the derived


group), G(R) factorizes as a product of groups of the form Sp2n(R), Ua,b,or SO∗2n (with varying values of n ≥ 0 and a ≥ b ≥ 0). Thus, the connectedcomponents of D are products of the Hermitian symmetric domains we haveseen in Sections 3.1, 3.2, and 3.3. However, we never see the Hermitian sym-metric domains in Sections 3.4, 3.6, and 3.5 as factors, except when it isbecause of the accidental (or special) isomorphisms H+

SO1,2

∼= H ∼= H1,1,

H+SO2,2

∼= H×H, H+SO3,2

∼= H2, H+SO4,2

∼= H2,2, and H+SO6,2

∼= HSO∗8in low

dimensions (see [65, Chapter X, Section 6]).

For a PEL moduli problem defined by an integral PEL datum withO ⊗Z Q a simple algebra over Q, the complex Lie group G(C) factorizes upto cocenter (i.e., again, up to replacing the group with a subgroup containingthe derived group) as a product of symplectic, special linear group, or specialorthogonal groups, but only one type of them can occur. Accordingly, wesay the PEL moduli problem is of type C, type A, or type D. People oftenalso say that a PEL moduli problem of type A is a unitary Shimura variety,even when G violates Condition (3) in Section 2.3 and so (G,D) is not aShimura datum. However, by orthogonal Shimura varieties, people might notmean PEL moduli problems of type D. On one hand, this is because thereare useful abelian-type Shimura varieties associated with special orthogonalgroups, as we shall see in Example 5.2.2.11 below. On the other hand, sincethe groups G associated with PEL moduli problems of type D are not evenconnected, the corresponding pairs (G,D) violate the conditions for beingShimura data in a more fundamental way, and hence there is some ambiguityin what PEL-type Shimura varieties of type D should mean.

In all cases, whether (G,D) is a Shimura datum or not, when the cor-responding (smooth) PEL moduli problem MU is defined at level U , there isan open and closed immersion

XU ↪→ MU (C) (5.1.3.7)

(see [87, Sections 2.4 and 2.5]), which is generally not an isomorphism, dueto the so-called failure of Hasse’s principle (see [90, Remark 1.4.3.12] and[82, Sections 7–8]). More concretely, this is because the definition of MUuses only the adelic group G(A)—the level structures use G(A∞), while theLie algebra condition uses G(R). But there can be more than one mutuallynonisomorphic algebraic groups G(1), . . . ,G(m) (including G) over Q suchthat G(j)(A) ∼= G(A), for all j, and such that MU (C) is the disjoint union of

the X(j)U ’s defined using G(j)’s. (When the pairs (G(j),D)’s are Shimura data,

by checking the action of Aut(C/F0) on the dense subset of the so-called CMpoints, it can be shown that this disjoint union descends to a disjoint union

over F0 of the canonical models of the X(j)U ’s.) In cases of types A and C,

such a failure is harmless in practice, because all the X(j)U ’s are isomorphic to

each other, even at the level of canonical models over F0 (see [82, Section 8]).However, in cases of type D, such a failure can be quite serious. It can even

62 Kai-Wen Lan

happen that some of the X(j)U ’s are noncompact, while others are not (see [91,

Example A.7.2]).

With the tacit running assumption that U is neat (see [137, 0.6] again,or see [90, Definition 1.4.1.8]), MU is a (smooth) quasi-projective schemebecause it is finite over the base change of some Siegel moduli scheme (seeSection 3.1.5), and hence an open and closed subscheme of MU serves as anintegral model of XU . (When U is not neat, we still obtain a Deligne–Mumfordstack whose associated coarse moduli space is a quasi-projective scheme.)

Remark 5.1.3.8. The consideration of certain Shimura varieties or connectedShimura varieties as parameter spaces for complex abelian varieties withPEL structures has a long history—see, for example, Shimura’s articles [166]and [167]. The theory over the complex numbers is already quite compli-cated and heavy in notation, which seems inevitable for anything involvingsemisimple algebras and positive involutions, but there is an additional sub-tlety when formulating the PEL structures over the integers. That is, we canno longer talk about the singular homology or cohomology of an abelian va-riety over a general base field, let alone their relative versions for an abelianscheme over a general base scheme. Instead, we have resorted to the (relative)de Rham and etale homology or cohomology. This is the reason that PELmoduli problems suffer from the failure of Hasse’s principle, and that thesemoduli problems can only be easily defined in mixed characteristics (0, p)when there is no ramification or level at p. (In Section 5.1.5 below, we willsummarize the known constructions when there are indeed some ramificationand levels at p.)

5.1.4. Toroidal and minimal compactifications of PEL moduli. The abovePEL moduli MU carry universal or tautological objects, which are abelianschemes with additional PEL structures, essentially by definition.

In the important special case of Siegel moduli problems considered inExample 5.1.3.4, the tautological objects are just principally polarized abelianschemes with level structures (but with no nontrivial endomorphism struc-tures or polarization degrees specified in the moduli problem). By vastly gen-eralizing results due to Mumford (see [125]), Faltings and Chai constructedtoroidal compactifications of the Siegel moduli problems by studying the de-generation of polarized abelian schemes into semi-abelian schemes (see [47,Chapters III and IV]). Over such toroidal compactifications, they can alge-bro-geometrically define modular forms, and construct minimal compactifi-cations of Siegel moduli problems as the projective spectra of some gradedalgebras of modular forms (with inputs of positivity results from [123]; see[47, Chapter V]). The main difficulty in Faltings and Chai’s construction wasthat they needed to glue together boundary strata parameterizing degener-ations of different ranks. This was much more complicated than the earlierboundary construction in [140], based on the original results of [125].

The generalization of the results of [47, Chapter III–V] to all smoothPEL moduli problems was carried out in [90]. Later in [87], it was shown that


the toroidal and minimal compactifications in [90] (and also in [47, ChapterIII–V], both based on the theory of degeneration) are indeed compatible withthe toroidal and minimal compactifications in [60] and [137] (based on theconstructions in [9] and [5] using analytic coordinate charts). In other words,toroidal and minimal compactifications of PEL moduli problems provide inte-gral models of toroidal and minimal compactifications of PEL-type Shimuravarieties.

Over the integral models of PEL-type Shimura varieties and theirtoroidal compactifications, by considering the relative cohomology of certainKuga families (which are abelian scheme torsors which are integral modelsof certain mixed Shimura varieties as in Section 4.2.8) and their toroidalcompactifications, we can also define automorphic vector bundles and theircanonical and subcanonical extensions over these integral models (see [47,Chapter VI], [89, especially Section 6], [100, Sections 1–3], and [101, Sec-tions 4–5]), whose global sections and cohomology classes then provide analgebro-geometric definition of modular forms in mixed characteristics, gen-eralizing the theory over C in Section 4.2.7. See [88] and [92] again for somesurveys on this topic. See also [66] and [67], for example, for some appli-cation to the theory of p-adic modular forms, with the assumptions on theintegral models of toroidal and minimal compactifications justified by theconstructions in [90].

5.1.5. Integral models not defined by smooth moduli. Let p be any primenumber, which is not necessarily good for the integral PEL datum. (Theconditions for being good for the integral PEL datum are the same as theconditions for being good for defining the PEL moduli, except that we ignorethe condition of having no nontrivial level structure at p.)

Let Zp := lim←−p-N (Z/NZ) and A∞,p := Zp⊗ZQ, with the superscript “p”

meaning “away from p”. When the level U ⊂ G(A∞) is of the form U = UpUp,where Up ⊂ G(A∞,p) and Up ⊂ G(Qp) such that Up is a parahoric open com-pact subgroup of G(Qp)—that is, when Up is the identity component of thestabilizer of some multichain of Zp-lattices in L⊗Z Qp—there is also a natu-ral way to define a PEL moduli problem parameterizing not just individualabelian schemes with additional PEL structures, but rather multichains ofisogenies of them. (See [142, Chapters 3 and 6].) This generalizes the ideathat, in the case of modular curves, the moduli problem at level Γ0(p) pa-rameterizes degree-p cyclic isogenies of elliptic curves. (See, for example, [73,Sections 3.4 and 6.6], which gave a moduli interpretation of the correspondingconstruction in [43] that was by merely taking normalizations.)

PEL moduli problems at parahoric levels at p are generally far fromsmooth in mixed characteristics (0, p), in which case we say that the integralmodels have bad reductions. Nevertheless, people have developed a theory oflocal models which describes the singularities of such moduli problems—see,for example, [142], [131], [132], [133], and [134]. (See also [141], [59], and [135]for some surveys.) Beginners should be warned that the forgetful morphisms

64 Kai-Wen Lan

from a higher parahoric level to a lower one, or more precisely from onesuch moduli problem defined by some multichain of Zp-lattices to anotherone defined by a simpler multichain, is usually just projective but not finite.This is very different from what we have seen in characteristic zero (or in theclassical modular curve case in [43] and [73]).

For higher levels at p, the older ideas in [43] by taking normalizations,and in [73] by generalizing the so-called Drinfeld level structures, are bothuseful. But they are useful in rather different contexts—exactly which strat-egy is more useful depends on the intended applications.

The idea of generalizing Drinfeld level structures played important rolesin [64], [113], [114], and [115], and in subsequent works based on them suchas [172], where people needed to compute the cohomology of certain nearbycycles and count points, and hence would prefer to work with moduli prob-lems even when they are not flat . (It was known much earlier—see [29, Ap-pendix A]—that moduli problems defined by Drinfeld level structures aregenerally not even flat over the base rings.) But such moduli problems tendto be very complicated, and might be too cumbersome when it comes to theconstruction of compactifications, because most known techniques of gluingrequire at least some noetherian normal base schemes. (The results in [64],[113], [114], [115], and [172] were all for compact PEL-type Shimura varieties.)

On the contrary, the idea of taking normalizations produces, essentiallyby definition, noetherian normal schemes flat over the base rings. It had beenoverlooked for a long time because of its lack of moduli interpretations, andbecause no theory of local models was available for them. While these twoproblems still remain, recent developments beyond the PEL-type cases (seethe discussion on the Hodge-type cases below in Section 5.2.3), and also de-velopments in the construction of p-adic modular forms (as in [62], whichwas based on [?] and did not require the integral models to be constructedas moduli problems) motivated people to systematically reconsider the con-structions by taking normalizations. In the PEL-type cases, we can constructgood compactifications for all such integral models (and their variants)—see[93], [95], and [96]. Such compactifications are good in the sense that the com-plications in describing its local structures and its boundary stratificationsare completely independent of each other. This is useful in, for example, [99,Section 6.3], which generalized the results of [114] and [115] to all noncom-pact cases, without having to compactify any moduli problems defined byDrinfeld level structures.

In [?], it is shown that many familiar subsets of the reductions of the inte-gral models of Shimura varieties in characteristic p > 0, such as p-rank strata,Newton strata, Oort central leaves, Ekedahl–Oort strata, and Kottwitz–Rapoportstrata, and their pullbacks to the integral models defined by taking normal-izations at arbitrarily higher levels at p, admit partial toroidal and minimalcompactifications with many properties similar to the whole toroidal andminimal compactifications.


5.1.6. Langlands–Rapoport conjecture. In [102] and [103], where the termi-nology Shimura varieties first appeared in the literature, Langlands proposedto study Hasse–Weil zeta functions of Shimura varieties and relate them tothe automorphic L-functions, generalizing earlier works for GL2 and its innerforms by Eichler, Shimura, Kuga, Sato, and Ihara. This is very ambitious—see the table of contents of [104] for the types of difficulties that alreadyshowed up in the case of Picard modular surfaces. This motivated the Lang-lands–Rapoport conjecture, which gives a detailed description of the pointsof the reduction modulo p of a general Shimura variety at a hyperspeciallevel at p—see [105], [136], [143], and [141]. See also [81] for a variant of theconjecture due to Kottwitz, which was proved by himself for all PEL-typeShimura varieties of types A and C in [82].

5.2. Hodge-type and abelian-type cases

5.2.1. Hodge-type Shimura varieties.

Definition 5.2.1.1. A Hodge-type Shimura datum (G,D) is a Shimura datumsuch that there exists a Siegel embedding

(G,D) ↪→ (GSp2n,H±n ); (5.2.1.2)

namely, an injective homomorphism G ↪→ GSp2n of algebraic groups over Qinducing an (automatically closed) embedding D ↪→ H±n , for some n ≥ 0 (cf.Remarks 2.4.11 and 2.4.13). A Hodge-type Shimura variety is a Shimuravariety associated with a Hodge-type Shimura datum.

When D = G(R) · h0 is the conjugacy class of some homomorphismh0 : C× → G(R) as before, the above means the composition of h0 withG(R) ↪→ GSp2n(R) lies in the conjugacy class of an analogous homomor-phism for GSp2n(R) as in (3.1.4.7). For each neat open compact subgroup Uof G(A∞), which we may assume to be the pullback of a neat open compactsubgroup U ′ of GSp2n(A∞), the Shimura variety XU associated with (G,D)at level U is a special subvariety of the Siegel modular variety at level U ′.Then the points of XU are equipped with polarized abelian varieties with levelstructures, which are pulled back from the Siegel modular varieties (and hencedepend on the choice of the Siegel embedding above).

Example 5.2.1.3. All the PEL-type Shimura data are, essentially by defini-tion, of Hodge type, because of the definition of G in (5.1.3.1), and because ofthe condition on h0 in Section 5.1.1. But there are many Hodge-type Shimuradata which are not of PEL type. Important examples of this kind are givenby general spin groups (with associated Hermitian symmetric domains as inSection 3.4.3), which admit Siegel embeddings thanks to the so-called Ku-ga–Satake construction (see [149], [151], and [84]; see also the reviews in [37,Sections 3–4] and [112, Sections 1 and 3]).

The injectivity of the homomorphism G ↪→ GSp2n means we have afaithful 2n-dimensional symplectic representation of G. By [42, Chapter I,

66 Kai-Wen Lan

Proposition 3.1], the faithfulness of this representation implies that G(Q)is the subgroup of GSp2n(Q) fixing some tensors (si) in the tensor algebra

generated by Q2n and its dual. But since C× h0−→ G(R)→ GSp2n(R) defines aHodge structure on R2n, by [42, Chapter I, Proposition 3.4], these tensors (si)are Hodge tensors, namely tensors of weight 0 and Hodge type (0, 0) withrespect to the induced Hodge structure on the tensor algebra of R2n.

Then the Shimura varieties XU associated with (G,D) can be inter-preted as parameter spaces of polarized abelian varieties (over C) equippedwith some Hodge cycles and level structures. According to Deligne (see [42,Chapter I, Main Theorem 2.11]), Hodge cycles on complex abelian varietiesare absolute Hodge cycles. Roughly speaking, this means that the Hodge cy-cles can be conjugated, as if they were known to be defined by algebraic cycles(as if the Hodge conjecture were true for them), when the underlying abelianvarieties are conjugated by an automorphism of C. This defines a canonicalaction of Aut(C/F0) on the points of XU by conjugating the Hodge cycles ofthe corresponding abelian varieties, which coincides with the action given bythe structure of XU as the C-points of its canonical model over F0. Becauseof this, people can formulate some moduli interpretation for the points of thecanonical model of XU over F0.

Such a pointwise moduli interpretation has not really helped peoplewrite down any useful moduli problem, but they are still important for study-ing the less direct construction by taking normalizations, to be introduced inSection 5.2.3 below.

5.2.2. Abelian-type Shimura varieties.

Definition 5.2.2.1. An abelian-type Shimura datum (G,D) is a Shimura da-tum such that there exist a Hodge-type Shimura datum (G1,D1) and a centralisogeny

Gder1 → Gder (5.2.2.2)

between the derived groups which induces an isomorphism

(Gad1 ,Dad

1 )∼→ (Gad,Dad) (5.2.2.3)

between the adjoint quotients. An abelian-type Shimura variety is a Shimuravariety associated with an abelian-type Shimura datum.

By definition, every Hodge-type Shimura datum (resp. Shimura variety)is of abelian type. Essentially by definition, the connected components of anabelian-type Shimura variety are finite quotients of those of some Hodge-typeShimura variety. Thus, results for abelian-type Shimura varieties are oftenproved by reducing to the case of Hodge-type Shimura variety.

Example 5.2.2.4. The simplest and perhaps the most prominent examples ofabelian-type Shimura varieties that are not of Hodge type are the Shimuracurves, which are associated with Shimura data (G,D) with G(Q) = B×,


where B is a quaternion algebra over a totally real extension F of Q ofdegree d > 1 such that

B⊗QR ∼= M2(R)×Hd−1, (5.2.2.5)

in which case

D ∼= H± (5.2.2.6)

is one-dimensional. More generally, there are abelian-type Shimura data(G,D) with G(Q) = B×, where B is a quaternion algebra over F such that

B⊗QR ∼= M2(R)a×Hb (5.2.2.7)

for some a, b ≥ 1, in which case

D ∼= (H±)a (5.2.2.8)

(cf. [169, Section 9.2]). Even more generally, as explained in [36, Section 6],with the same B and F (and a, b ≥ 1), for each integer n ≥ 1, there areabelian-type Shimura data (G,D) such that G(Q) is the group of similitudesover F of a symmetric bilinear pairing over Bn, and such that

G(R) ∼= GSp2n(R)a×Hb, (5.2.2.9)

where H is a real form of GSp2n(C) with compact derived group (which is H×when n = 1), in which case

D ∼= (H±n )a. (5.2.2.10)

In all these cases, the related Hodge-type Shimura data (as in Definition5.2.2.1) and their Siegel embeddings (as in Definition 5.2.1.1) can be con-structed with some auxiliary choice of imaginary quadratic field extensionsof Q (see [36, Section 6], and see also [26, Section 2] and [83, Section 12]).However, none of these are of Hodge type (let alone of PEL type) becausethe Hodge structures defined by rational representations of G are not rationalHodge structures (see [119, Section 9, part (c) of the last example]).

Example 5.2.2.11. There are also the abelian-type Shimura varieties associ-ated with Shimura data (G,D) such that the derived group of G as an alge-braic group over Q is the special orthogonal group defined by some symmetricbilinear form over Q of signature (a, 2) over R, for some a ≥ 1, so that G(R)+

is up to cocenter the group SOa,2(R)+ studied in Section 3.4.3. (See also thelast paragraph of Section 3.4.4.) In this case, the related Hodge-type Shimuradata (as in Definition 5.2.2.1) are general spin groups, with their Siegel em-beddings (as in Definition 5.2.1.1) given by the Kuga–Satake construction,as in Example 5.2.1.3, mapping G into GSp2n with n = 2a (see the samereferences in Example 5.2.1.3). Such Shimura varieties associated with spe-cial orthogonal groups are important because they are related to the moduliof interesting algebraic varieties. For example, when a = 19 in the above,the corresponding Shimura varieties contain as open subspaces the modulispaces of polarized K3 surfaces with level structures (see, for example, [144,Section 6], [110, Sections 2 and 4], and [164, Section 4.2]).

68 Kai-Wen Lan

In [39, 2.3], based on earlier works due to Satake (see [148]; and seealso [150] and [152, Chapter IV]), Deligne analyzed all abelian-type Shimuradata (G,D) that are adjoint and simple in the sense that G is adjoint andsimple as an algebraic group over Q. (See also [120, Section 10].) When Gis simple over Q, there is up to isomorphism a unique irreducible Dynkindiagram associated with all the simple factors of GC, or rather Lie G(C). (Infact, in this case, Gad ∼= Resk/Q H for some number field k and some connectedsemisimple algebraic group algebraic group H over k such that HQ remains

simple over the algebraic closure Q of Q in C; see, for example, [21, 6.21(ii)].)We say that the group is of type An, Bn, Cn, Dn, E6, or E7 accordinglyto the type of such a Dynkin diagram (cf. the summary in Section 3.7). Incases of type Dn, for n ≥ 5, we further single out the special case of type DR

n

(resp. DHn) if all the noncompact simple factors of Lie G(R) are isomorphic

to so2n−2,2 (resp. so∗2n). In cases of type D4, the situation is more complicatedbecause so6,2

∼= so∗8, but we can still single out some special cases of type DR4

and DH4 . For simplicity, we shall drop the subscript n when it is not used in the

discussion. Deligne showed that an adjoint and simple Shimura datum (G,D)is of abelian type exactly when it is of types A, B, C, DR, or DH. In casesof types A, B, C, and DR, the related Hodge-type Shimura datum (G1,D1)(as in Definition 5.2.1.1) can be chosen such that the derived group Gder

1 issimply-connected as an algebraic group over Q. However, in the remainingcases of DH

n , where n ≥ 4, the related Hodge-type Shimura datum can only bechosen such that the noncompact simple factors of Gder

1 (R) are all isomorphicto SO∗2n. (Although this could have been a more precise statement in termsof algebraic groups, we are content with the simpler statement here in termsof real Lie groups.)

More generally, suppose that (G,D) is a Shimura datum that is simplein the sense that the adjoint quotient Gad is simple. (But we no longer as-sume that G itself is adjoint.) Then we can still classify the type of (G,D)by classifying the type of (Gad,Dad) as above, and we have the followingpossibilities:

(1) All cases of types A, B, C, and DR are of abelian type.

(2) Cases of type DH can be of abelian type only when Gder is the quotientof Gder

1 for the Hodge-type datum (G1,D1) mentioned in the last para-graph. In particular, Gder

1 (R) cannot have factors isomorphic to the spingroup cover Spin∗2n of SO∗2n.

(3) Cases of mixed type D, namely those that are neither of type DR nor oftype DH, are not of abelian type. Such cases exist by a general resultdue to Borel and Harder (see [17, Theorem B]).

(4) Cases of type E6 and E7 are never of abelian type.

Thus, roughly speaking, all Shimura varieties associated with symplectic andunitary groups are of abelian type, and none of the Shimura varieties asso-ciated with exceptional groups can be of abelian type, while there are some


complications for Shimura varieties associated with spin groups or specialorthogonal groups.

5.2.3. Integral models of these Shimura varieties. The strategy for construct-ing integral models of Hodge-type Shimura varieties goes as follows. Sup-pose (G,D) is a Hodge-type Shimura datum with Siegel embedding

(G,D) ↪→ (GSp2n,H±n ), (5.2.3.1)

and suppose U is an open compact subgroup of G(A∞) that is the pullbackof some open compact subgroup U ′ of GSp2n(A∞) such that U ′ = U ′,pU ′p,where U ′,p is some neat open compact subgroup of GSp2n(A∞,p), and whereU ′p = GSp2n(Zp). (In the literature, people have considered the more generalsetting where U ′p is the stabilizer in GSp2n(Qp) of some Zp-lattice, which cannevertheless be reduced to our setting here by modifying the Siegel embed-ding (5.2.3.1), at the expense of replacing n with 8n, using “Zarhin’s trick”as in [93, Lemma 4.9] or [?, Lemma 2.1.1.9].) Let v|p be a place of F0, andlet OF0,(v) demote the localization of F0 at v. Then we have a canonical openimmersion

XU,F0↪→ XU ′,OF0,(v)

, (5.2.3.2)

where XU,F0abusively denotes the canonical model of XU over F0, and where

XU ′,OF0,(v)denotes the base change to OF0,(v) of the Siegel moduli scheme at

level U ′. Then we define the integral model

XU,OF0,(v)(5.2.3.3)

of XU,F0over OF0,(v) by taking the normalization of XU ′,OF0,(v)

in XU,F0under

the above canonical open immersion.

A priori, this is a very general construction, which also makes sensewhen U is just a subgroup of the pullback of U ′. This is what we did in thePEL-type cases in [?] and [93], as reviewed in Section 5.1.5, and similar ideascan be traced back to [43].

But when U is of the form UpUp such that Up is a neat open compactsubgroup of G(A∞,p) and such that Up is a hyperspecial maximal open com-pact subgroup of G(Qp), Vasiu, Kisin, and others (see [118, Remark 2.15],[177], [122], [76], and [75]) proved that XU,OF0,(v)

is smooth and satisfies cer-

tain extension properties making it an integral canonical model . (The for-mulations of the extension properties vary from author to author. See, inparticular, [118, Section 2], [122, Section 3], and [76, (2.3.7)]. An integralcanonical model then just means an integral model satisfying some formula-tion of the extension property. This notion is not expected to be useful whenthe levels at p are not hyperspecial.) They also showed that, roughly speak-ing, by working with connected components, by taking suitable quotients ofintegral models of Hodge-type ones by finite groups, and by descent, we alsoobtain integral models for all abelian-type Shimura varieties at hyperspeciallevels at p, which are smooth and satisfy the same extension properties.

70 Kai-Wen Lan

When p > 2, and when G splits over a tamely ramified extension of Qp,Kisin and Pappas proved in [?] that all abelian-type Shimura varieties at para-horic levels at p have integral models which have local models as predictedby the group-theoretic construction in [134], and which satisfy a weaker ex-tension property.

In [77], Kisin proved a variant of the Langlands–Rapoport conjecture(partly following Kottwitz’s variant; cf. Section 5.1.6) for all abelian-typeShimura varieties at hyperspecial levels at p, when p > 2.

5.2.4. Integral models of toroidal and minimal compactifications. In [?], itis proved that, by taking normalizations over the toroidal and minimal com-pactifications of Siegel moduli schemes constructed in [47] and [90], we alsohave good integral models of toroidal and minimal compactifications of theintegral models of Hodge-type Shimura varieties constructed in the previoussubsection.

The cone decompositions for the toroidal compactifications thus ob-tained are pullbacks of certain smooth ones for the Siegel moduli, which arerather inexplicit. But it is believed that, by considering normalizations ofblowups of the minimal compactifications as in [95], we can obtain toroidalcompactifications associated with more general projective and smooth conedecompositions as well.

As in the PEL-type cases in Section 5.1.5, such compactifications aregood in the sense that the complications in describing its local structures andits boundary stratifications are completely independent of each other. Usingthe toroidal compactifications thus constructed, we showed in [99] and [?]that, for each integral model of Hodge-type or abelian-type Shimura varietiesconstructed in [76], [75], and [?], and for any prime ` different from the residuecharacteristics, the `-adic etale nearby cycle cohomology over the special fiberis canonically isomorphic to the `-adic etale cohomology over the (geometric)generic fiber, without resorting to the proper base change theorem (cf. [3,XII, 5.1] and [40, XIII, (2.1.7.1) and (2.1.7.3)]). (In [?], the proof in thecase of abelian-type Shimura varieties is achieved by reduction to the caseof Hodge-type Shimura varieties.) Intuitively speaking, the special fibers ofthese integral models have as many points as there should be, at least forstudying the `-adic etale cohomology of Shimura varieties.

5.3. Beyond abelian-type cases

Although abelian-type Shimura varieties are already very useful and general,as we have seen in Section 5.2.2, they do not cover all possibilities of Shimuravarieties. Among those associated with simple Shimura data, we are not justmissing those of types E6 and E7—we are also missing the majority (depend-ing on one’s viewpoint) of type D cases. But these missing cases can still bequite interesting, both for geometric and arithmetic reasons.

For example, in [121], Milne and Suh showed that there are many ex-amples of geometrically connected quasi-projective varieties X over number


fields F such that the complex analytifications of X⊗F,σC and X⊗F,τ C havenonisomorphic fundamental groups for different embeddings σ and τ of Finto C, by considering connected Shimura varieties of mixed type D (whichexist by [17, Theorem B]).

As another example, we have the following result of R. Liu and X. Zhu’s:

Theorem 5.3.1 (see [107, Theorem 1.2]). Suppose XU is a Shimura variety as-sociated with a general Shimura datum (G,D) (which can certainly be simpleof types E6, E7, or mixed D) at some level U . Suppose V is an etale Qp-localsystem associated with some finite-dimensional representation of G, whose re-striction to the largest anisotropic subtorus in the center of G that is R-split istrivial. Suppose x : Spec(F )→ XU is a closed point of XU , together with a geo-metric point x : Spec(F )→ XU lifting x. Then the stalk Vx, when regarded asa representation of Gal(F/F ), is geometric in the sense of Fontaine–Mazur(see [52, Part I, Section 1]). That is, it is unramified at all but finitely manyplaces of F , and is de Rham at the places of F above p.

Recall that Fontaine and Mazur conjectured that every irreducible geo-metric Galois representation comes from geometry in the sense that it ap-pears as a subquotient of the etale cohomology of an algebraic variety (see[52, Part I, Section 1] again). Theorem 5.3.1 would have been less surprisingif it only considered abelian-type Shimura varieties with certain assumptionssuch that the stalks as in the theorem are just the p-adic realizations of someabelian motives. But the theorem does consider non-abelian-type Shimuravarieties as well, and we do not know any families of motives parameterizedby these general Shimura varieties. We hope that there are such families, butthis is a difficult problem that is still wide open.

Both results mentioned above are about Shimura varieties or connectedShimura varieties in characteristic zero. But what can we say about their in-tegral models? Certainly, any quasi-projective variety over the reflex field F0

has some integral model over the rings of integers OF0 in F0, but we arenot interested in such a general construction. It would be desirable to havea construction such that the pullback to OF0,(p) of the integral model atlevel U is smooth and satisfies some reasonable extension property, when Uis of the form UpUp, where Up is a neat open compact subgroup of G(A∞,p)and where Up is a hyperspecial maximal open compact subgroup of G(Qp).Moreover, it would be desirable that the reduction modulo p of such integralmodels can be described by some variant of the Langlands–Rapoport conjec-ture (see Section 5.1.6 and the end of Section 5.2.3). These are also difficultand wide open problems, but their nature might be very different from thatof the problem (in the previous paragraph) of finding motives parameterizedby such Shimura varieties.

72 Kai-Wen Lan

5.4. Beyond Shimura varieties?

There are some exciting recent developments which are not about integralmodels of Shimura varieties, or not even about Shimura varieties, but it wouldhave been a shame not to mention them at all.

In [158], Scholze constructed perfectoid Shimura varieties in all Hodge-type cases, which are limits at the infinite level at p (in a subtle sense) ofShimura varieties at finite levels. Base on these, in [164], Shen also constructedperfectoid Shimura varieties in all abelian-type cases. Although the construc-tions in [158] used Siegel moduli schemes over the integers, these perfectoidShimura varieties are objects (called adic spaces) defined over the character-istic zero base field Cp (the completion of an algebraic closure Qp of Qp). Butthe property of being perfectoid (see [157]) makes them much more powerfulthan integral models at finite levels for studying many questions about torsionin the cohomology of Shimura varieties and related geometric objects—see,for example, [158], [130], and [25].

Even the concept of Shimura varieties itself has been challenged andgeneralized. In his Berkeley–MSRI lectures in the Fall of 2014 (see [161]),Scholze defined certain moduli spaces of mixed-characteristic shtukas, whichcan be viewed as a local analogue of Shimura varieties, but is much more gen-eral than the previously studied local geometric objects such as the so-calledRapoport–Zink spaces (see [142]) and can be defined for all reductive lin-ear algebraic groups over Qp. He proposed to use them to construct localLanglands parameters for all reductive linear algebraic groups over Qp, alongthe lines of Vincent Lafforgues’s construction of global Langlands parametersfor global fields of positive characteristics (see [?]), but in the much fanciercategory of diamonds (see [161] and [159]).

Acknowledgements

Some materials in this article are based on introductory lectures given atthe National Center for Theoretical Sciences (NCTS) in Taipei in 2014; atthe Mathematical Sciences Research Institute (MSRI) in Berkeley in 2014;at the University of Minnesota, Twin Cities, in 2015; and at the summerschool sponsored by ETH Zurich in Ascona, Switzerland, in 2016. We heartilythank the organizers for their invitations, and thank the audiences for theirvaluable feedbacks. We also thank Brian Conrad and Sug Woo Shin, and theanonymous referees, for their helpful comments and suggestions.

References

[1] J. Arthur, D. Ellwood, and R. Kottwitz, editors. Harmonic Analysis, TheTrace Formula, and Shimura Varieties, volume 4 of Clay Mathematics Pro-ceedings. Proceedings of the Clay Mathematics Institute 2003 Summer School,


The Fields Institute, Toronto, Canada, June 2–27, 2003, American Mathemat-ical Society, Providence, Rhode Island, Clay Mathematics Institute, Cam-bridge, Massachusetts, 2005.

[2] M. Artin. Algebraization of formal moduli: I. In Spencer and Iyanaga [174],pages 21–71.

[3] M. Artin, A. Grothendieck, and J.-L. Verdier, editors. Theorie des topos etcohomologie etale des schemas (SGA 4), Tome 3, volume 305 of Lecture Notesin Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1973.

[4] M. Artin and J. Tate, editors. Arithmetic and Geometry, Vol. I. Arithmetic.Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday,volume 35 of Progress in Mathematics. Birkhauser, Boston, 1983.

[5] A. Ash, D. Mumford, M. Rapoport, and Y. Tai. Smooth Compactification ofLocally Symmetric Varieties, volume 4 of Lie Groups: History Frontiers andApplications. Math Sci Press, Brookline, Massachusetts, 1975.

[6] A. Ash, D. Mumford, M. Rapoport, and Y. Tai. Smooth Compactification ofLocally Symmetric Varieties. Cambridge Mathematical Library. CambridgeUniversity Press, Cambridge, New York, 2nd edition, 2010.

[7] W. L. Baily, Jr. Satake’s compactification of Vn. Amer. J. Math., 80(2):348–364, April 1958.

[8] W. L. Baily, Jr. An exceptional arithmetic group and its eisenstein series.Ann. Math. (2), 91(3):512–549, May 1970.

[9] W. L. Baily, Jr. and A. Borel. Compactification of arithmetic quotients ofbounded symmetric domains. Ann. Math. (2), 84(3):442–528, November 1966.

[10] M.-J. Bertin and C. Goldstein, editors. Seminaire de Theorie des Nombres,Paris 1982–1983, volume 51 of Progress in Mathematics. Birkhauser, Boston,Basel, Stuttgart, 1984.

[11] A. Borel. Ensembles fondamentaux pour les groupes arithmetiques. In Colloq.Theorie des Groupes Algebriques (Bruxelles, 1962), pages 23–40. LibrairieUniversitaire, Louvain; GauthierVillars, Paris, 1962.

[12] A. Borel. Some finiteness properties of adele groups over number fields. Publ.

Math. Inst. Hautes Etud. Sci., 16:5–30, 1963.

[13] A. Borel. Introduction aux groupes arithmetiques, volume 1341 of Publicationsde l’Institut de Mathematique de l’Universite de Strasbourg XV, Actualitesscientifiques et industrielles. Hermann, Paris, 1969.

[14] A. Borel. Some metric properties of arithmetic quotients of symmetric spacesand an extension theorem. J. Differential Geometry, 6:543–560, 1972.

[15] A. Borel. Linear Algebraic Groups, volume 126 of Graduate Texts in Mathe-matics. Springer-Verlag, Berlin, Heidelberg, New York, 2nd edition, 1991.

[16] A. Borel and W. Casselman, editors. Automorphic Forms, Representationsand L-Functions, volume 33, Part 2 of Proceedings of Symposia in Pure Math-ematics. held at Oregon State University, Corvallis, Oregon, July 11–August5, 1977, American Mathematical Society, Providence, Rhode Island, 1979.

[17] A. Borel and G. Harder. Existence of discrete cocompact subgroups of reduc-tive groups over local fields. J. Reine Angew. Math., 298:53–64, 1978.

[18] A. Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann.Math. (2), 75(3):483–535, May 1962.

74 Kai-Wen Lan

[19] A. Borel and L. Ji. Compactifications of symmetric and locally symmetricspaces. Mathematics: Theory & Applications. Birkhauser, Boston, 2006.

[20] A. Borel and J.-P. Serre. Corners and arithmetic groups. Comment. Math.Helv., 48:436–491, 1973. with an appendix by A. Douady et L. Herault.

[21] A. Borel and J. Tits. Groupes reductifs. Publ. Math. Inst. Hautes Etud. Sci.,27:55–151, 1965.

[22] A. Borel and N. Wallach. Continuous Cohomology, Discrete Subgroups, andRepresentations of Reductive Groups, volume 67 of Mathematical Surveys andMonographs. American Mathematical Society, Providence, Rhode Island, 2ndedition, 2000.

[23] M. V. Borovoı. Langlands’ conjecture concerning conjugation of connectedShimura varieties. Selecta Math. Soviet., 3(1):3–39, 1983/84.

[24] M. V. Borovoı. The group of points of a semisimple group over a totally real-closed field. In Problems in group theory and homological algebra (Russian),Mathematika, pages 142–149. Yaroslav. Gos. Univ., Yaroslavl′, 1987.

[25] A. Caraiani and P. Scholze. On the generic part of the cohomology of compactunitary Shimura varieties. Ann. Math. (2), 186(3):649–766, 2017.

[26] H. Carayol. Sur la mauvaise reduction des courbes de Shimura. CompositioMath., 59(2):151–230, 1986.

[27] J. A. Carlson, C. H. Clemens, and D. R. Morrison, editors. Complex Geometryand Lie Theory, volume 53 of Proceedings of Symposia in Pure Mathematics.Proceedings of a Symposium on Complex Geometry and Lie Theory held atSundance, Utah, May 26–30, 1989, Springer-Verlag, Berlin, Heidelberg, NewYork, 1991.

[28] H. Cartan. Prolongement des espaces analytiques normaux. Math. Ann., pages97–110, 1958.

[29] C.-L. Chai and P. Norman. Bad reduction of the Siegel moduli scheme of genustwo with Γo(p)-level structure. Amer. J. Math., 112(6):1003–1071, December1990.

[30] L. Clozel and J. S. Milne, editors. Automorphic Forms, Shimura Varieties,and L-Functions. Volume I, volume 10 of Perspectives in Mathematics. Pro-ceedings of a Conference held at the University of Michigan, Ann Arbor, July6–16, 1988, Academic Press Inc., Boston, 1990.

[31] L. Clozel and J. S. Milne, editors. Automorphic Forms, Shimura Varieties,and L-Functions. Volume II, volume 11 of Perspectives in Mathematics. Pro-ceedings of a Conference held at the University of Michigan, Ann Arbor, July6–16, 1988, Academic Press Inc., Boston, 1990.

[32] J. Coates and S. T. Yau, editors. Elliptic Curves, Modular Forms & Fermat’sLast Theorem. Proceedings of a conference held in the Institute of Mathemat-ics of the Chinese Univerisity of Hong Kong, International Press, Cambridge,Massachusetts, 1997.

[33] B. Conrad. Weil and Grothendieck approaches to adelic points. Enseign.Math. (2), 58(1–2):61–97, 2012.

[34] P. Deligne. Equations differentielles a points singuliers reguliers, volume 163of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, NewYork, 1970.


[35] P. Deligne. Formes modulaires et representations `-adiques. In SeminaireBourbaki, expose n◦ 355 (fevrier 1969), volume 179 of Lecture Notes in Math-ematics, pages 139–172. Springer-Verlag, Berlin, Heidelberg, New York, 1971.

[36] P. Deligne. Travaux de Shimura. In Seminaire Bourbaki, expose n◦ 389 (fevrier1971), volume 244 of Lecture Notes in Mathematics, pages 123–165. Springer-Verlag, Berlin, Heidelberg, New York, 1971.

[37] P. Deligne. La conjecture de Weil pour les surfaces K3. Invent. Math., 15:206–226, 1972.

[38] P. Deligne, editor. Cohomologie etale (SGA 4 12), volume 569 of Lecture Notes

in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1977.

[39] P. Deligne. Varietes de Shimura: Interpretation modulaire, et techniques deconstruction de modeles canoniques. In Borel and Casselman [16], pages 247–290.

[40] P. Deligne and N. Katz, editors. Groupes de monodromie en geometriealgebrique (SGA 7 II), volume 340 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1973.

[41] P. Deligne and W. Kuyk, editors. Modular Functions of One Variable II, vol-ume 349 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg,New York, 1973.

[42] P. Deligne, J. S. Milne, A. Ogus, and K. Shih. Hodge Cycles, Motives, andShimura Varieties, volume 900 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1982.

[43] P. Deligne and M. Rapoport. Les schemas de modules de courbes elliptiques.In Deligne and Kuyk [41], pages 143–316.

[44] M. Eie. Topics in Number Theory, volume 2 of Monographs in Number Theory.World Scientific, Singapore, 2009.

[45] E. Eischen, M. Harris, J.-S. Li, and C. Skinner. p-adic L-functions for unitarygroups. preprint, 2016.

[46] G. Faltings. Arithmetic varieties and rigidity. In Bertin and Goldstein [10],pages 64–77.

[47] G. Faltings and C.-L. Chai. Degeneration of Abelian Varieties, volume 22 ofErgebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Springer-Verlag,Berlin, Heidelberg, New York, 1990.

[48] J. Faraut and A. Koranyi. Analysis on Symmetric Cones. Oxford Mathemat-ical Monographs. Clarendon Press, Oxford University Press, Oxford, 1995.

[49] L. Fargues and E. Mantovan. Varietes de Shimura, espaces de Rapoport–Zink,et correspondances de Langlands locales. Number 291 in Asterisque. SocieteMathematique de France, Paris, 2004.

[50] Gavril Farkas and Ian Morrison, editors. Handbook of Moduli: Volume II, vol-ume 25 of Advanced Lectures in Mathematics. International Press, Somerville,Massachusetts, Higher Education Press, Beijing, 2013.

[51] Gavril Farkas and Ian Morrison, editors. Handbook of Moduli: Volume III, vol-ume 26 of Advanced Lectures in Mathematics. International Press, Somerville,Massachusetts, Higher Education Press, Beijing, 2013.

[52] J.-M. Fontaine and B. Mazur. Geometric Galois representations. In Coatesand Yau [32], pages 190–227.

76 Kai-Wen Lan

[53] J. Franke. Harmonic analysis in weighted L2-spaces. Ann. Sci. Ecole Norm.Sup. (4), 31:181–279, 1998.

[54] J. Franke and J. Schwermer. A decomposition of spaces of automorphic forms,and the Eisenstein cohomology of arithmetic groups. Math. Ann., 311:765–790, 1998.

[55] W. T. Gan and H. Y. Loke. Modular forms of level p on the exceptional tubedomain. J. Ramanujan Math. Soc., 12(2):162–202, 1997.

[56] R. Goodman and N. R. Wallach. Symmetry, Representations, and Invariants,volume 255 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, Hei-delberg, New York, 2009.

[57] M. Goresky. Compactifications and cohomology of modular varieties. InArthur et al. [1], pages 551–582.

[58] M. Goresky, G. Harder, and R. MacPherson. Weighted cohomology. Invent.Math., 116:139–213, 1994.

[59] T. J. Haines. Introduction to Shimura varieties with bad reductions of para-horic type. In Arthur et al. [1], pages 583–658.

[60] M. Harris. Functorial properties of toroidal compactifications of locally sym-metric varieties. Proc. London Math. Soc. (3), 59:1–22, 1989.

[61] M. Harris. Automorphic forms and the cohomology of vector bundles onShimura varieties. In Clozel and Milne [31], pages 41–91.

[62] M. Harris, K.-W. Lan, R. Taylor, and J. Thorne. On the rigid cohomology ofcertain Shimura varieties. Res. Math. Sci., 3:article no. 37, 308 pp., 2016.

[63] M. Harris, J.-S. Li, and C. M. Skinner. p-adic L-functions for unitary Shimuravarieties. I. Construction of the Eisenstein measure. Doc. Math., (ExtraVol.):393–464, 2006.

[64] M. Harris and R. Taylor. The Geometry and Cohomology of Some SimpleShimura Varieties. Number 151 in Annals of Mathematics Studies. PrincetonUniversity Press, Princeton, 2001.

[65] S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces, vol-ume 34 of Graduate Studies in Mathematics. American Mathematical Society,Providence, Rhode Island, 2001.

[66] H. Hida. Control theorems of coherent sheaves on Shimura varieties of PELtype. J. Inst. Math. Jussieu, 1(1):1–76, 2002.

[67] H. Hida. p-Adic Automorphic Forms on Shimura Varieties. Springer Mono-graphs in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2004.

[68] H. Hironaka. Resolution of singularities of an algebraic variety over a field ofcharacteristic zero: I. Ann. Math. (2), 79(1):109–203, January 1964.

[69] H. Hironaka. Resolution of singularities of an algebraic variety over a field ofcharacteristic zero: II. Ann. Math. (2), 79(2):205–326, March 1964.

[70] U. Hirzebruch. Uber Jordan-Algebren und beschrankte symmetrische Gebiete.Math. Z., 94:387–390, 1966.

[71] International Colloquium on Discrete Subgroups of Lie Groups and Appli-cations to Moduli, Bombay, 8–15 January 1973. Discrete Subgroups of LieGroups and Applications to Moduli: Papers presented at the Bombay Collo-quium, 1973, volume 7 of Tata Institute of Fundamental Research Studies inMathematics. Oxford University Press, Oxford, 1975.


[72] L. Ji, Y. S. Poon, L. Yang, and S.-T. Yau, editors. Fifth International Con-gress of Chinese Mathematicians, volume 51, part 1 of AMS/IP Studies inAdvanced Mathematics. Proceedings of the Fifth International Congress ofChinese Mathematicians held at Tsinghua University, Beijing, in December2010, American Mathematical Society, Providence, Rhode Island, and Inter-national Press, Cambridge, MA, 2012.

[73] N. M. Katz and B. Mazur. Arithmetic Moduli of Elliptic Curves. Number 108in Annals of Mathematics Studies. Princeton University Press, Princeton,1985.

[74] H. H. Kim. Exceptional modular form of weight 4 on an exceptional domaincontained in C27. Rev. Mat. Iberoamericana, 9(1):139–200, 1993.

[75] W. Kim and K. Madapusi Pera. 2-adic integral canonical models. ForumMath. Sigma, 4:e28, 34 pp., 2016.

[76] M. Kisin. Integral models for Shimura varieties of abelian type. J. Amer.Math. Soc., 23(4):967–1012, October 2010.

[77] M. Kisin. Mod p points on Shimura varieties of abelian type. J. Amer. Math.Soc., 30(3):819–914, July 2017.

[78] M. Kisin and G. Pappas. Integral models of Shimura varieties with parahoriclevel structure. preprint, 2015.

[79] A. W. Knapp. Representation Theory of Semisimple Groups: An OverviewBased on Examples. Princeton Landmarks in Mathematics. Princeton Uni-versity Press, Princeton, 2001.

[80] A. W. Knapp. Lie Groups Beyond an Introduction, volume 140 of Progress inMathematics. Birkhauser, Boston, 2nd edition, 2002.

[81] R. E. Kottwitz. Shimura varieties and λ-adic representations. In Clozel andMilne [30], pages 161–209.

[82] R. E. Kottwitz. Points on some Shimura varieties over finite fields. J. Amer.Math. Soc., 5(2):373–444, April 1992.

[83] A. Kret and S. W. Shin. Galois representations for general symplectic groups.preprint, 2016.

[84] M. Kuga and I. Satake. Abelian varieties attached to polarized K3-surfaces.Math. Ann., 169:239–242, 1967.

[85] J.-P. Labesse and J. Schwermer, editors. Cohomology of Arithmetic Groupsand Automorphic Forms, volume 1447 of Lecture Notes in Mathematics. Pro-ceedings of a Conference held in Luminy/Marseille, France, May 22–27, 1989,Springer-Verlag, Berlin, Heidelberg, New York, 1990.

[86] V. Lafforgue. Chtoucas pour les groupes reductifs et parametrisation de Lang-lands globale. preprint, 2012.

[87] K.-W. Lan. Comparison between analytic and algebraic constructions oftoroidal compactifications of PEL-type Shimura varieties. J. Reine Angew.Math., 664:163–228, March 2012.

[88] K.-W. Lan. Geometric modular forms and the cohomology of torsion auto-morphic sheaves. In Ji et al. [72], pages 183–208.

[89] K.-W. Lan. Toroidal compactifications of PEL-type Kuga families. AlgebraNumber Theory, 6(5):885–966, 2012.

78 Kai-Wen Lan

[90] K.-W. Lan. Arithmetic compactification of PEL-type Shimura varieties, vol-ume 36 of London Mathematical Society Monographs. Princeton UniversityPress, Princeton, 2013. errata available on the author’s website.

[91] K.-W. Lan. Boundary strata of connected components in positive characteris-tics. Algebra Number Theory, 9(9):1955–2054, 2015. an appendix to the article“Families of nearly ordinary Eisenstein series on unitary groups” by Xin Wan.

[92] K.-W. Lan. Cohomology of automorphic bundles. preprint, 2016. to appearin the proceedings of the Seventh International Congress of Chinese Mathe-maticians.

[93] K.-W. Lan. Compactifications of PEL-type Shimura varieties in ramified char-acteristics. Forum Math. Sigma, 4:e1, 98 pp., 2016.

[94] K.-W. Lan. Compactifications of PEL-type Shimura varieties and Kuga fam-ilies with ordinary loci. World Scientific, Singapore, 2017.

[95] K.-W. Lan. Integral models of toroidal compactifications with projective conedecompositions. Int. Math. Res. Not. IMRN, 2017(11):3237–3280, 2017.

[96] K.-W. Lan. Compactifications of splitting models of PEL-type Shimura vari-eties. Trans. Amer. Math. Soc., 370(4):2463–2515, 2018.

[97] K.-W. Lan and B. Stroh. Compactifications of subschemes of integral modelsof Shimura varieties. preprint, 2015.

[98] K.-W. Lan and B. Stroh. Nearby cycles of automorphic etale sheaves, II.preprint, 2017.

[99] K.-W. Lan and B. Stroh. Nearby cycles of automorphic etale sheaves. Compos.Math., 154(1):80–119, 2018.

[100] K.-W. Lan and J. Suh. Vanishing theorems for torsion automorphic sheaveson compact PEL-type Shimura varieties. Duke Math. J., 161(6):1113–1170,2012.

[101] K.-W. Lan and J. Suh. Vanishing theorems for torsion automorphic sheaveson general PEL-type Shimura varieties. Adv. Math., 242:228–286, 2013.

[102] R. P. Langlands. Some contemporary problems with origins in the Jugend-traum. In Mathematical developments arising from Hilbert problems (Proc.Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill.,1974), pages 401–418. American Mathematical Society, Providence, RhodeIsland, 1976.

[103] R. P. Langlands. Shimura varieties and the Selberg trace formula. Canad. J.Math., 29(6):1292–1299, 1977.

[104] R. P. Langlands and D. Ramakrishnan, editors. The Zeta Functions of PicardModular Surfaces. based on lectures delivered at a CRM Workshop in thespring of 1988, Les Publications CRM, Montreal, 1992.

[105] R. P. Langlands and M. Rapoport. Shimuravarietaten und Gerben. J. ReineAngew. Math., 378:113–220, 1987.

[106] J.-S. Li and J. Schwermer. On the Eisenstein cohomology of arithmetic groups.Duke Math. J., 123(1):141–169, 2004.

[107] R. Liu and X. Zhu. Rigidity and a Riemann–Hilbert correspondence for p-adiclocal systems. Invent. Math., 207:291–343, 2017.

[108] E. Looijenga. L2-cohomology of locally symmetric varieties. CompositioMath., 67:3–20, 1988.


[109] E. Looijenga and M. Rapoport. Weights in the local cohomology of a Baily–Borel compactification. In Carlson et al. [27], pages 223–260.

[110] K. Madapusi Pera. The Tate conjecture for K3 surfaces in odd characteristic.Invent. Math., 201(2):625–668, 2015.

[111] K. Madapusi Pera. Toroidal compactifications of integral models of Shimuravarieties of Hodge type. preprint, 2015.

[112] K. Madapusi Pera. Integral canonical models for spin Shimura varieties. Com-pos. Math., 152(4):769–824, 2016.

[113] E. Mantovan. On certain unitary group Shimura varieties. In Varietes deShimura, espaces de Rapoport–Zink, et correspondances de Langlands locales[49], pages 201–331.

[114] E. Mantovan. On the cohomology of certain PEL-type Shimura varieties. DukeMath. J., 129(3):573–610, 2005.

[115] E. Mantovan. `-adic etale cohomology of PEL type Shimura varieties withnon-trivial coefficients. In WIN–Woman in Numbers, volume 60 of Fields In-stitute Communications, pages 61–83. American Mathematical Society, Prov-idence, Rhode Island, 2011.

[116] J. S. Milne. The action of an automorphism of C on a Shimura variety andits special points. In Artin and Tate [4], pages 239–265.

[117] J. S. Milne. Canonical models of (mixed) Shimura varieties and automorphicvector bundles. In Clozel and Milne [30], pages 283–414.

[118] J. S. Milne. The points on a Shimura variety modulo a prime of good reduc-tion. In Langlands and Ramakrishnan [104], pages 151–253.

[119] J. S. Milne. Introduction to Shimura varieties. In Arthur et al. [1], pages265–378.

[120] J. S. Milne. Shimura varieties and moduli. In Farkas and Morrison [50], pages467–548.

[121] J. S. Milne and J. Suh. Nonhomeomorphic conjugates of connected Shimuravarieties. Amer. J. Math., 132(3):731–750, 2010.

[122] B. Moonen. Models of Shimura varieties in mixed characteristic. In GaloisRepresentations in Arithmetic Algebraic Geometry [156], pages 267–350.

[123] L. Moret-Bailly. Pinceaux de varietes abeliennes, volume 129 of Asterisque.Societe Mathematique de France, Paris, 1985.

[124] D. Mumford. Abelian Varieties, volume 5 of Tata Institute of FundamentalResearch Studies in Mathematics. Oxford University Press, Oxford, 1970. withappendices by C. P. Ramanujam and Yuri Manin.

[125] D. Mumford. An analytic construction of degenerating abelian varieties overcomplete rings. Compositio Math., 24(3):239–272, 1972.

[126] D. Mumford. A new approach to compactifying locally symmetric varieties.In Discrete Subgroups of Lie Groups and Applications to Moduli: Papers pre-sented at the Bombay Colloquium, 1973 [71], pages 211–224.

[127] D. Mumford. Hirzebruch’s proportionality theorem in the non-compact case.Invent. Math., 42:239–272, 1977.

[128] D. Mumford. Tata Lectures on Theta I, volume 28 of Progress in Mathematics.Birkhauser, Boston, 1983.

80 Kai-Wen Lan

[129] D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, vol-ume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag,Berlin, Heidelberg, New York, 3rd edition, 1994.

[130] J. Newton and J. A. Thorne. Torsion Galois representations over CM fieldsand Hecke algebras in the derived category. Forum Math. Sigma, 4:e21, 88pp., 2016.

[131] G. Pappas and M. Rapoport. Local models in the ramified case, I. The EL-case. J. Algebraic Geom., 12(1):107–145, 2003.

[132] G. Pappas and M. Rapoport. Local models in the ramified case, II. Splittingmodels. Duke Math. J., 127(2):193–250, 2005.

[133] G. Pappas and M. Rapoport. Local models in the ramified case, III. Unitarygroups. J. Inst. Math. Jussieu, 8(3):507–564, 2009.

[134] G. Pappas and X. Zhu. Local models of Shimura varieties and a conjecture ofKottwitz. Invent. Math., 194:147–254, 2013.

[135] Georgios Pappas, Michael Rapoport, and Brian Smithling. Local models ofShimura varieties, I. Geometry and combinatorics. In Farkas and Morrison[51], pages 135–217.

[136] M. Pfau. The conjecture of Langlands and Rapoport for certain Shimuravarieties of non-rational weight. J. Reine Angew. Math., 471:165–199, 1996.

[137] R. Pink. Arithmetic compactification of mixed Shimura varieties. PhD thesis,Rheinischen Friedrich-Wilhelms-Universitat, Bonn, 1989.

[138] V. Platonov and A. Rapinchuk. Algebraic Groups and Number Theory, volume139 of Pure and Applied Mathematics. Academic Press Inc., New York andLondon, 1994. translated by Rachel Rowen.

[139] G. Prasad and A. S. Rapinchuk. On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior.Adv. Math., 207(2):646–660, 2006.

[140] M. Rapoport. Compactifications de l’espace de modules de Hilbert–Blumenthal. Compositio Math., 36(3):255–335, 1978.

[141] M. Rapoport. A guide to the reduction modulo p of Shimura varieties. InTilouine et al. [176], pages 271–318.

[142] M. Rapoport and T. Zink. Period Spaces for p-divisible Groups. Number 141in Annals of Mathematics Studies. Princeton University Press, Princeton,1996.

[143] H. Reimann. The semi-simple zeta function of quaternionic Shimura vari-eties, volume 1657 of Lecture Notes in Mathematics. Springer-Verlag, Berlin,Heidelberg, New York, 1997.

[144] J. Rizov. Moduli stacks of polarized K3 surfaces in mixed characteristic.Serdica Math. J., 32(2–3):131–178, 2006.

[145] L. Saper and M. Stern. L2-cohomology of arithmetic varieties. Ann. Math.(2), 132(1):1–69, July 1990.

[146] I. Satake. On the compactification of the Siegel space. J. Indian Math. Soc.(N.S.), 20:259–281, 1956.

[147] I. Satake. On compactifications of the quotient spaces for arithmetically de-fined discontinuous groups. Ann. Math. (2), 72(3):555–580, 1960.


[148] I. Satake. Holomorphic imbeddings of symmetric domains into a Siegel space.Amer. J. Math., 87(2):425–461, April 1965.

[149] I. Satake. Clifford algebras and families of abelian varieties. Nagoya Math. J.,27:435–446, 1966.

[150] I. Satake. Symplectic representations of algebraic groups satisfying a certainanalyticity condition. Acta Math., 117:215–279, 1967.

[151] I. Satake. Corrections to “Clifford algebras and families of abelian varieties”.Nagoya Math. J., 31:295–296, 1968.

[152] I. Satake. Algebraic Structures of Symmetric Domains, volume 14 of Pub-lications of the Mathematical Society of Japan. Princeton University Press,Princeton, 1980.

[153] I. Satake. Compactifications, old and new. Sugaku Expositions, 14(2):175–189,2001.

[154] I. Satake and H. Cartan. Demonstration du theoreme fondamental. InSeminaire Henri Cartan 10e annee: 1957/1958, expose 15 (14 avril 1958).Secretariat mathematique, 11 rue Pierre Curie, Paris 5e, 1958.

[155] A. J. Scholl. Motives for modular forms. Invent. Math., 100(2):419–430, 1990.

[156] A. J. Scholl and R. L. Taylor. Galois Representations in Arithmetic AlgebraicGeometry, volume 254 of London Mathematical Society Lecture Note Series.Cambridge University Press, Cambridge, New York, 1998.

[157] P. Scholze. Perfectoid spaces. Publ. Math. Inst. Hautes Etud. Sci., 116:245–313, 2012.

[158] P. Scholze. On torsion in the cohomology of locally symmetric varieties. Ann.Math. (2), 182(3):946–1066, 2015.

[159] P. Scholze. Etale cohomology of diamonds. preprint, 2017.

[160] P. Scholze. p-adic geometry. preprint, 2017.

[161] P. Scholze and J. Weinstein. Berkeley lectures on p-adic geometry. preprint,2017.

[162] J. Schwermer. Cohomology of arithmetic groups, automorphic forms and L-functions. In Labesse and Schwermer [85], pages 1–29.

[163] J.-P. Serre. Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier.Grenoble, 6:1–42, 1955–1956.

[164] X. Shen. Perfectoid Shimura varieties of abelian type. Int. Math. Res. Not.IMRN, 2017(21):6599–6653, 2017.

[165] H. Shimizu. On discontinuous groups operating on the product of the upperhalf planes. Ann. Math. (2), 77(1):33–71, 1963.

[166] G. Shimura. On analytic families of polarized abelian varities and automorphicfunctions. Ann. Math. (2), 78(1):149–192, July 1963.

[167] G. Shimura. Moduli and fibre systems of abelian varieties. Ann. Math. (2),83(2):294–338, March 1966.

[168] G. Shimura. On canonical models of arithmetic quotients of bounded sym-metric domains. Ann. Math. (2), 91(1):144–222, January 1970.

[169] G. Shimura. Introduction to the Arithmetic Theory of Automorphic Functions,volume 11 of Publications of the Mathematical Society of Japan. PrincetonUniversity Press, Princeton, 1971.

[170] G. Shimura. The arithmetic of automorphic forms with respect to a unitarygroup. Ann. Math. (2), 107(3):569–605, May 1978.

[171] G. Shimura. Abelian Varieties with Complex Multiplication and Modular Func-tions. Princeton University Press, Princeton, 1998.

[172] S. W. Shin. Galois representations arising from some compact Shimura vari-eties. Ann. Math. (2), 173(3):1645–1741, 2011.

[173] C. Skinner and E. Urban. The main conjecture for GL(2). Invent. Math.,195(1):1–277, January 2014.

[174] D. C. Spencer and S. Iyanaga, editors. Global Analysis. Papers in honor ofK. Kodaira. Princeton University Press, Princeton, 1969.

[175] T. A. Springer. Linear Algebraic Groups, volume 9 of Progress in Mathematics.Birkhauser, Boston, 2nd edition, 1998.

[176] J. Tilouine, H. Carayol, M. Harris, and M.-F. Vigneras, editors. Formes au-

tomorphes (I): Actes du semestre du Centre Emile Borel, printemps 2000.Number 298 in Asterisque. Societe Mathematique de France, Paris, 2005.

[177] A. Vasiu. Integral canonical models of Shimura varieties of preabelian type.Asian J. Mathematics, 3:401–518, 1999.

[178] S. Zucker. L2 cohomology of warpped products and arithmetic groups. Invent.Math., 70:169–218, 1982.

Index

〈 · , · 〉, 6, 19, 35, 40, 56, 59, 60〈 · , · 〉σ, 350n, 61a,b, 7, 20, 261n, 6, 7

(1) (Tate twist), 56

A, 30A, 5A0, 57, 58A∨0 , 57A0[n], 57A∞, 5A∞,p, 63abelian motive, 71abelian scheme, 20, 55, 56, 58

univeral, 62abelian variety, 19, 57, 61, 65

dual, 57abelian-type

Shimura datum, 37, 66–68Shimura variety, 66, 69–71

absolute Hodge cycle, 65

adelic topology, 5

adic space, 71

adjoint quotient, 11, 21, 66, 68

adjoint representation, 11

adjoint Shimura datum, 66, 67

affine group scheme, 5

Albert algebra, 30

Albert’s classification, 60

algebraic curve, 10, 38

algebraic cycle, 65

algebraic group, 4, 5, 10, 11

simply-connected, 28, 68

algebraic space, 20, 52

algebraic stack, 20, 61

algebraicity, 10, 12, 20, 37, 38, 52,54, 70

α0,n, 57, 58

αN , 13

analytification, 12, 20, 52, 70

approximation, 782

INDEX 83

arithmetic subgroup, 9, 12, 13,41, 43, 46, 51, 53

neat, 10Artin’s criterion, 20Aut(C/F0), 61, 65automatic compactness, 38, 45, 46automorphic L-function, 64automorphic line bundle, 54automorphic representation, 3, 4,

9, 53automorphic vector bundle, 54, 62automorphy condition, 54automorphy factor, 8, 54

B, 46, 48, 60, 66bad reduction, 63blowup, 69Borel subgroup, 39Borel–Serre compactification, 51,

53–54boundary, 38, 52, 55

codimension, 52, 55stratification, 53, 64, 70

boundary component, 5, 37–39,41, 43, 45, 46, 48, 49, 51, 52

bounded realization, 17, 20, 25,28, 34

Cp, 71canonical bundle, 54canonical extension, 55, 62canonical model, 4, 5, 12, 14, 54,

55, 60, 61, 69integral, 69

Cartan involution, 11Cartan’s classification of

Hermitian symmetric domaintype Iba, 23type IIn, 25type IIIn, 18type IVa, 28type V, 34type VI, 34

Cayley numbers, 22, 29Cayley transform, 17, 23, 34

Cent (centralizer), 18, 23, 29, 33,36, 54

center of unipotent radical, 43,45, 48

central leaves, 64Chow’s theorem, 52classical group, 39Clifford algebra, 28closed complex analytic subspace,

52CM elliptic curve, 14CM field, 45, 46CM point, 14, 61coarse moduli space, 61cocenter, 60, 67coherent cohomology, 55, 62cohomology, 9, 53, 56

coherent, 55, 62etale, 12, 70intersection, 53L2, 53nearby cycle, 63, 70relative, 62weighted, 54

commensurable, 9compact factor, 11compact Riemann surface, 10, 38compactification

Borel–Serre, 51, 53–54reductive Borel–Serre, 51,

53–54Satake–Baily–Borel or minimal,

5, 12, 41, 43, 46, 51, 51, 52,54integral model, 20, 62, 64, 69

toroidal, 51–55integral model, 62, 64, 69

complete algebraic curve, 10, 38complex analytic space, 52complex analytic subspace, 52complex analytification, 12, 20,

52, 70complex manifold, 8, 11complex structure, 8, 19, 57complex torus, 19, 57

84 INDEX

condition on Lie algebra, 58, 61cone decomposition, 52, 69

refinement, 52, 53congruence subgroup, 9, 10, 13,

19principal, 9, 10, 13

conjugacy class, 10, 11, 14, 18,23–25, 29, 33–37, 59, 65

connected, 7, 10, 11, 60, 61connected component, 5, 7, 9,

11–13, 26–28, 60, 66, 69connected Shimura datum, 11,

12, 13, 18connected Shimura variety, 13,

61, 70connection, 55cotangent bundle, 54counting point, 63cusp, 10, 38cusp form, 54, 55cyclic isogeny, 63

D, 7, 10, 11, 59Da,b, 20, 23

D±a,b, 24

Dad, 66DE6

, 34DE7

, 34Dn, 18D+, 9, 11D(r), 45, 46DSO∗2n

, 25DSOa,2 , 28

D+SOa,2

, 28

D∗, 38, 39, 52de Rham, 70degeneration, 62Deligne torus, 10Deligne–Mumford stack, 20, 61derived group, 31, 60, 66–68determinantal condition, 58diamond, 72division algebra, 46

normed, 22octonion, 29

double coset space, 4, 5, 8, 9, 37,59

Drinfeld level structure, 63, 64dual abelian variety, 57dual lattice, 57Dynkin diagram, 36, 37, 67

E, 43, 60e6, 33, 37e6(−14), 34, 35, 37e6(−26), 31, 35e6(C), 35e7(−25), 32, 37Ez, 13Eisenstein series, 53Ekedahl–Oort stratification, 64elliptic curve, 13, 19, 63

CM, 14universal, 55

endomorphism structure, 56–58eq-SO-2n-C, 24eq-SO-2n-star, 24eq-so-2n-star, 25etale cohomology, 12, 70etale local system, 70exceptional group, 31–33, 35, 37,

51, 67, 68, 70extension property, 69, 71

F , 45, 66F0, 12, 54, 58, 60, 71f4, 31F+, 45failure of Hasse’s principle, 61, 62faithful representation, 5, 9, 65fan, 52flag, 40flat, 63, 64Fontaine–Mazur conjecture, 70formally real Jordan algebra, 30,

33free action, 10Freudenthal’s theorem, 33fundamental domain, 38fundamental group, 70

INDEX 85

G, 32–35, 37G, 5, 7, 10, 11, 36, 43, 45, 46, 48,

51, 59g, 11Gad, 11, 66Gad(Q)+, 13Gad(R)+, 11(G,D), 8, 10, 12, 13, 59, 65(G,D+), 11–13Gder, 66, 68G(j), 61Gm, 5G(Q)+, 9

G(r), 44, 47G(R)+, 9, 28, 36, 67g∗, 31

G, 55GAGA, 52Gal(Q/F0), 12, 13Galois cohomology, 28Galois representation, 3, 4, 12, 13

geometric, 70Galois symmetry, 3, 4, 13Γ, 12, 13, 41, 43, 46, 51, 53Γ0(p), 63Γ\D, 51, 54(Γ\D)BS, 51, 53, 54

(Γ\D)min

, 51–55Γ\D+, 12, 13(Γ\D)RBS, 51, 53, 54

(Γ\D)torΣ , 51, 52, 54, 55

Γ\D∗, 52, 54Γi, 9Γ(N), 10, 13general linear group, 3, 5, 55, 64general spin group, 65, 67generalized projective

coordinates, 15, 17, 21, 22,26, 38, 41–44, 46

geometric Galois representation,70

geometric invariant theory, 20GL1, 5GL2, 3, 55, 64GL2k(H), 26

GLn, 3, 5GLr(B), 48global Langlands correspondence,

3global Langlands parameter, 72good prime, 59, 62, 63good reduction, 59Gram–Schmidt process, 26Grassmannian, 16group scheme, 5growth condition, 54GSp2n, 6, 60, 65, 67GUa,b, 24, 60GU′a,b, 24, 60

H, 8, 10, 13, 14, 17, 27, 38, 53, 60H, 22, 25, 59, 66, 67h, 10, 11, 19H0, 48, 51h0, 18, 19, 23–25, 29, 33, 35,

56–60, 65h′0, 24

h0, 29H1, 17, 27, 49, 51H1,1, 60

H±1 , 27, 55H2, 60H2,2, 60H5,1, 51Ha,b, 21, 23, 36, 43

H±a,b, 24, 60H∗a,b, 43Ha−r,b−r, 43

h, 11HE6

, 34, 37, 51HE7 , 32, 34, 37, 51H+I , 48, 49Hn, 15, 18, 36, 39H±n , 18, 60, 67H∗n, 41Hn−r, 41H±, 55, 66HSO1,2

, 27

H+SO1,2

, 27, 60

H+SO10,2

, 51

86 INDEX

H+SO2,2

, 36, 60

HSO∗2n, 25, 36, 46, 60

H∗SO∗2n, 46

HSO∗2n−4r, 46

H+SO3,2

, 60

H+SO4,2

, 60

HSO∗4k, 26

H+SO6,2

, 60

HSO∗8, 60

HSOa,2 , 27

H−SOa,2, 27

H+SOa,2

, 27, 28, 36, 65

(H+SOa,2

)∗, 51

HSOa,2 , 27

HSOa,b , 26H∗, 38H∗×H∗, 38H×H, 38, 60Hamiltonian numbers, 22, 25, 59,

66, 67Hasse–Weil zeta function, 64Hecke action, 9Hecke symmetry, 3, 9, 13Herm3(O), 30Hermb(C), 22Hermk(H), 26Hermitian

imaginary part, 22real part, 22

Hermitian part of Levi subgroup,42, 45, 48

Hermitian symmetric domain, 5,11, 14, 26, 36–37

irreducible, 15, 34, 36, 37of type A III, 23of type BD I, 28of type C I, 18of type D III, 25of type E VII, 34

Hermitian upper half-space, 22Hilbert modular surface, 39Hilbert modular variety, 39Hodge conjecture, 65Hodge cycle, 65

absolute, 65Hodge decomposition, 58Hodge filtration, 16Hodge structure

rational, 67variation, 4, 11, 14, 18, 19, 55

Hodge tensor, 65Hodge-type

Shimura datum, 65–68Shimura variety, 65, 66, 68–71

homothety of lattices, 19homotopy equivalence, 53horocycle, 38hyperspecial, 59, 69, 71

i0, 57, 58identity component, 7, 9, 11, 28imaginary quadratic field, 14, 43,

60, 67infinite level, 71∞, 38∞r, 41, 43, 46integrable connection, 55integral canonical model, 69integral model, 5, 20, 54–56, 61,

62, 68, 69, 71minimal compactification, 62,

64, 69toroidal compactification, 62,

64, 69integral PEL datum, 56, 58intersection cohomology, 53ι, 32, 33isogeny class, 59isomorphism class, 59Iwasawa main conjecture, 3

J, 33J2, 40Ja,b, 21, 43Jn, 6Jordan algebra, 30, 33, 34

K, 17, 21, 25, 29, 33, 36, 37k, 11K3 surface, 67

INDEX 87

Klingen parabolic subgroup, 40Kottwitz conjecture, 64, 69Kottwitz–Rapoport stratification,

64Kuga family, 62Kuga–Satake construction, 65, 67Kuga–Sato variety, 56

L, 56, 59, 60L2-cohomology, 53LI(Q), 48L#, 57L(r)(Q), 42, 44, 47λ0, 57, 58Langlands parameter, 71, 72Langlands program, 3, 4Langlands–Rapoport conjecture,

64, 69, 71lattice, 19, 56

dual, 57homothety, 19

level, 13level structure, 19, 20, 56–58, 61,

65, 67Drinfeld, 63, 64

Levi quotient, 48, 53Levi subgroup, 42, 44, 47, 48

Hermitian part, 42, 45, 48LieA0, 58Lie algebra condition, 58, 61light cone, 27line bundle, 54linear algebraic group, 5local analogue of Shimura variety,

71local Langlands conjecture, 3local Langlands parameter, 71local model, 63, 64, 69local system, 10

etale, 70

M , 31, 33Ma,b, 20Mn, 7, 15MU , 61manifold, 4, 7–10

complex, 8, 11with corners, 53

maximal compact subgroup, 17,29, 31, 33, 34, 36

maximal parabolic subgroup, 32,39–42, 44, 46, 48

minimal compactification, 5, 12,41, 43, 46, 51, 51, 52, 54

integral model, 20, 62, 64, 69minimal parabolic subgroup, 40mixed Hodge structure, 55mixed Shimura variety, 55–56

integral model, 62mixed type D, 68, 70mixed-characteristic shtuka, 71Mobius transformation, 8, 49modular curve, 10, 13, 14, 55, 63modular form, 8, 54, 55, 62p-adic, 63, 64

modular function, 54moduli interpretation, 64, 66moduli problem, 20, 56, 67, 71

PEL, 56, 58, 61morphism of Shimura data, 13, 14morphism of Shimura varieties, 14motive, 71multichain of isogenies, 63multichain of lattices, 63

N(X), 29nearby cycle, 63, 70neat, 10, 14, 61Newton stratification, 64normal, 41, 43, 52, 64normal crossings divisor, 52normalization, 63, 64, 66, 69ν, 6, 31ν0,n, 57, 58number field, 10, 12, 13, 20, 54, 70

O, 56, 59, 60O, 22, 29

norm, 30Oa,b, 7OE , 60OF , 45

88 INDEX

OF0, 58, 71

OF0,(p), 71OF0,(v), 68O-lattice, 56On, 7octonion division algebra, 29octonion upper half-space, 32Oort central leaves, 64open compact subgroup, 8, 10,

12, 14, 55, 68hyperspecial, 59, 69, 71neat, 10, 14, 61parahoric, 63, 69

orthogonal group, 7orthogonal Shimura variety, 60,

61, 68

P , 32PI(Q), 48p−, 11p+, 11P(r)(Q), 42, 44, 47p-adic L-function, 3p-adic modular form, 63, 64p-adic realization, 71p-rank stratification, 64parabolic subgroup, 39, 53

Klingen, 40maximal, 39–42, 44, 46, 48minimal, 40Siegel, 41

parahoric, 63, 69parameter space, 13, 20, 56, 61,

65partial minimal compactification,

64partial toroidal compactification,

64PEL datum

integral, 56, 58rational, 59

PEL moduli problem, 56, 58, 61,63

of type A, 60, 61of type C, 60, 61of type D, 60, 61

PEL structure, 56, 58, 61PEL-type

Shimura datum, 59, 65Shimura variety, 60, 61

perfectoid Shimura variety, 71Picard modular surface, 46Poincare upper half-plane, 8, 14,

17, 27polarization, 19, 56–58polarized abelian scheme, 20polarized abelian variety, 19, 65polarized K3 surface, 67positive involution, 46, 56, 60, 61principal congruence subgroup, 9,

10, 13principal level structure, 57principal polarization, 19projective coordinates, 8, 15, 17,

21, 22, 26, 38, 41–44, 46, 54projective spectrum, 62projective variety, 12, 41, 43, 52proper base change theorem, 70

Q, 48Q, 12Qp, 71quasi-projective scheme, 61quasi-projective variety, 4, 12, 52,

70, 71quaternion algebra, 46, 66quaternion upper half-space, 26

Rapoport–Zink space, 71rational boundary component, 5,

37–39, 41, 43, 45, 46, 48, 49,51, 52

rational Hodge structure, 67rational PEL datum, 59real analytic topology, 7, 9, 11, 28real quadratic extension, 38real torus, 19, 57realization

bounded, 17, 20, 25, 28, 34unbounded, 15, 21, 24, 26, 27,

32, 34reduction, 64

INDEX 89

bad, 63good, 59

reduction theory, 38, 41reductive algebraic group, 5, 10,

71reductive Borel–Serre

compactification, 51, 53–54reductive Lie group, 21refinement of cone

decompositions, 52, 53reflex field, 12, 54, 58, 60, 71relative cohomology, 62representation

automorphic, 3, 4, 9, 53Galois, 3, 4, 12, 13

geometric, 70resolution of singularity, 52restricted product, 5restriction of scalars, 10, 38, 45ρ, 31, 33ρB , 31, 33Riemann surface, 10, 38Riemannian symmetric space, 53

noncompact of type A I, 8noncompact of type E IV, 31

Rosati condition, 57

S, 21, 43, 44S, 10S2, 8S3, 8Sn, 7, 8Satake topology, 39, 41, 43, 46,

51, 52Satake–Baily–Borel

compactification, 5, 12, 41,43, 46, 51, 51, 52, 54

integral model, 20, 62, 64, 69semi-abelian scheme, 62semisimple algebra, 39, 56, 61semisimple algebraic group, 5, 11semisimple Lie group, 21Shimura curve, 38, 66Shimura datum, 4, 10, 12, 13, 18,

24, 54, 59, 61adjoint, 66, 67

connected, 11, 12, 13, 18morphism, 13, 14of abelian type, 37, 66–68of Hodge type, 65–68of PEL type, 59, 65simple, 67, 68

Shimura reciprocity law, 14Shimura variety, 3, 4, 10, 13, 54,

55, 64CM point, 14, 61connected, 13, 70integral canonical model, 69integral model, 5, 54–56, 61,

68, 69, 71level, 13local analogue, 71morphism, 14of abelian type, 66, 69–71of Hodge type, 65, 68–71of PEL type, 60, 61perfectoid, 71special point, 14special subvariety, 14zero-dimensional, 14

shtuka, 71Siegel embedding, 24, 25, 29, 34,

36, 37, 65, 65, 67, 68Siegel modular variety, 19, 20, 56,

65Siegel moduli scheme, 20, 56,

60–62, 71Siegel parabolic subgroup, 41Siegel upper half-space, 15Σ, 52σ, 35signature, 7, 20, 26, 45, 48similitude character, 6simple algebra, 60simple normal crossings divisor,

52simple Shimura datum, 67simply-connected algebraic group,

28, 68SL2, 8, 10, 13, 17, 38, 53SL2×SL2, 38

90 INDEX

SLn, 5, 7smooth, 12, 52, 59, 69, 71so10, 34, 36, 37SO2, 8, 17, 28SO∗2n, 24, 36, 46, 60, 68so∗2n, 25, 67so2n−2,2, 67SO∗4k, 25so6,2, 67so∗8, 67SOa,2(R)+, 28, 36, 37, 48, 67SOa,b, 7, 26, 28, 48SOn, 7, 24, 28SO(Qa+2, Q), 48Sp2, 17Sp2k(H), 26Sp2n, 6, 15, 36, 39, 56, 60special linear group, 5, 7, 8, 10,

13, 17, 38, 53, 60special node, 36special orthogonal group, 7, 7, 8,

17, 24–26, 28, 36, 37, 46, 48,60, 67, 68

special point, 14special subvariety, 14special unitary group, 21Spin2,2, 36

Spin∗2n, 68Spina,2, 28, 29, 37spin group, 28, 36, 37, 65, 67, 68spinor norm, 28?, 56, 59, 60stratification

boundary, 53, 64, 70Ekedahl–Oort, 64Kottwitz–Rapoport, 64Newton, 64p-rank, 64

SUa,b, 21subcanonical extension, 55, 62sufficiently small, 10Symn, 7, 15symmetric space

Hermitian, 5, 11, 14, 26, 36–37noncompact of type A III, 23

noncompact of type BD I, 28noncompact of type C I, 18noncompact of type D III, 25noncompact of type E III, 34noncompact of type E VII,

34irreducible, 15, 32, 34, 36, 37Riemannian, 53

noncompact of type A I, 8noncompact of type E IV, 31

symmetryGalois, 3, 4, 13Hecke, 3, 9, 13

symplectic group, 6, 15, 17, 36,39, 56, 60, 65, 67, 68

symplectic Shimura variety, 60,67, 68

tamely ramified, 69Tate twist, 56topology

adelic, 5of H∗, 38real analytic, 7, 9, 11, 28Satake, 39, 41, 43, 46, 51Zariski, 7

toroidal compactification, 51–55integral model, 62, 64, 69

torsion cohomology class, 3, 71torsion subgroup, 13, 57torus, 14totally imaginary extension, 45totally isotropic subspace, 39–42,

44, 47, 48totally real field, 39, 45, 66trace condition, 58transitivity, 7–9, 16, 23, 25, 28, 33tube domain, 15, 22, 25–27, 32–33type A, 20–24, 36, 43–46, 60, 61,

64, 67, 68type B, 26–29, 36, 48–51, 67, 68type C, 15–20, 36, 39–43, 60, 61,

64, 67, 68type D, 24–29, 36, 46–51, 60, 61,

67mixed, 68, 70

INDEX 91

type DH, 67, 68type DR, 67, 68

type E6, 34–37, 51, 67, 68, 70type E7, 29–34, 37, 51, 67, 68, 70

U , 31, 33U , 8, 12, 13, 59, 65, 68U1, 11, 17Ua,b, 20, 21, 36, 45, 60U′a,b, 21, 24, 43

UI(Q), 48Un, 17, 36Up, 63, 69, 71Up, 63, 69, 71U ′, 65, 68U ′p, 68U ′,p, 68U(r)(Q), 42, 44, 47

U , 55unbounded realization, 15, 21, 24,

26, 27, 32, 34unipotent radical, 42, 43, 45, 47,

48unit disc, 17unitary group, 3, 11, 17, 20, 21,

24, 36, 43, 45–46, 60, 68unitary Shimura variety, 60, 68universal abelian scheme, 62universal elliptic curve, 55unramified, 59, 70upper half-plane

Poincare, 8, 14, 17, 27upper half-space

Hermitian, 22octonion, 32quaternion, 26Siegel, 15

V , 26

V0, 58V 0, 58VC, 26V(r)(Q), 42, 45, 48variation of Hodge structures, 4,

11, 14, 18, 19, 55variation of mixed Hodge

structures, 55vector bundle, 10, 54, 62

W , 54W , 54W, 31W can, 55W(r)(Q), 42, 45, 48

W sub, 55weak approximation, 7weight, 54weighted cohomology, 54Weil pairing, 13, 58

XU , 55XU , 8, 9, 10, 12, 13, 37, 54, 55,

59, 65, 69XU,F0

, 68XU,OF0,(v)

, 69

X(j)U , 61

XU ′,OF0,(v), 68

Z, 5

Zp, 63Z(M), 31Zarhin’s trick, 68Zariski topology, 7zero-dimensional Shimura

varieties, 14ζN , 13Zucker’s conjecture, 53

Kai-Wen LanUniversity of Minnesota, Minneapolis, MN 55455, USA

e-mail: [email protected]

An example-based introduction to Shimura varietiesjavier.fresan.perso.math.cnrs.fr/lan.pdf · An...

Documents

Transcript of An example-based introduction to Shimura varietiesjavier.fresan.perso.math.cnrs.fr/lan.pdf · An...