AN EXAMPLE-BASED INTRODUCTION TO SHIMURA ...kwlan/articles/intro-sh-ex.pdfAN EXAMPLE-BASED...

AN EXAMPLE-BASED INTRODUCTION TO SHIMURA

VARIETIES

KAI-WEN LAN

Abstract. In this introductory article, we will explain what Shimura vari-

eties are and why they are useful, with an emphasis on examples. We will alsodescribe various compactifications of Shimura varieties, again with an empha-

sis on examples, and summarize what we know about the integral models of

Shimura varieties and their compactifications.

Contents

1. Introduction 32. Double coset spaces 42.1. Algebraic groups 42.2. Manifolds 62.3. Shimura data 92.4. Shimura varieties and their canonical models 103. Hermitian symmetric domains 133.1. The case of Sp2n(R) 133.1.1. Siegel upper half-spaces 133.1.2. Transitivity of action 143.1.3. Bounded realization 153.1.4. Interpretation as conjugacy classes 163.1.5. Moduli of polarized abelian varieties 163.2. The case of Ua,b 183.2.1. Bounded realization 183.2.2. Unbounded realization 193.2.3. The special case where a = b: Hermitian upper half-spaces 203.2.4. Generalized Cayley transformations and transitivity of actions 203.2.5. Interpretation as conjugacy classes 213.3. The case of SO∗2n 223.3.1. Unbounded realization, and interpretation as a conjugacy class 223.3.2. The special case where n = 2k: quaternion upper half-spaces 233.4. The case of SOa,2(R) 233.4.1. Projective coordinates 23

2010 Mathematics Subject Classification. Primary 14G35; Secondary 11G18.

Key words and phrases. locally symmetric varieties, Shimura varieties, examples.The author was partially supported by the National Science Foundation under agreement

No. 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 and do not

necessarily reflect the views of these organizations.This article will be published in the proceedings of the ETHZ Summer School on Motives and

Complex Multiplication.1

2 KAI-WEN LAN

3.4.2. The case where a ≥ 1 and b = 2 243.4.3. Connected components and spin groups 253.4.4. Interpretation as conjugacy classes 263.5. The case of E7 263.5.1. Cayley numbers 263.5.2. An exceptional Jordan algebra over R 273.5.3. A real Lie group with Lie algebra e7(−25) 273.5.4. An octonion upper half-space 293.5.5. Bounded realization 303.6. The case of E6 313.6.1. Bounded and unbounded realizations 313.6.2. A real Lie group of Lie algebra e6(−14) 313.6.3. Interpretation as conjugacy classes 323.7. A brief summary 324. Rational boundary components and compactifications 334.1. Examples of rational boundary components 344.1.1. Cases with D = H 344.1.2. Cases with D = H×H 344.1.3. Cases with D = Hn 354.1.4. Cases with D = Ha,b, with a ≥ b 394.1.5. Cases with D = HSO∗2n

42

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

, with a ≥ 2 44

4.1.7. Cases with D = HE6 or D = HE7 464.2. Compactifications and algebraicity 474.2.1. Overview 474.2.2. (Γ\D)

min= Γ\D∗ 47

4.2.3. (Γ\D)torΣ 47

4.2.4. (Γ\D)BS 484.2.5. (Γ\D)RBS 484.2.6. Models over number fields or their rings of integers 494.2.7. Algebro-geometric definition of modular forms 494.2.8. Mixed Shimura varieties 505. Integral models 515.1. PEL-type cases 515.1.1. PEL datum 515.1.2. Smooth PEL moduli problems 535.1.3. PEL-type Shimura varieties 545.1.4. Toroidal and minimal compactifications of PEL moduli 565.1.5. Integral models not defined by smooth moduli 575.1.6. Langlands–Rapoport conjecture 585.2. Hodge-type and abelian-type cases 585.2.1. Hodge-type Shimura varieties 585.2.2. Abelian-type Shimura varieties 595.2.3. Integral models of these Shimura varieties 625.2.4. Integral models of toroidal and minimal compactifications 635.3. Beyond abelian-type cases 635.4. Beyond Shimura varieties? 64Acknowledgements 65

AN EXAMPLE-BASED INTRODUCTION TO SHIMURA VARIETIES 3

References 65Index 72

1. Introduction

Shimura varieties are generalizations of modular curves, which have played animportant role in many recent developments of number theory. Just to mentiona few examples with which this author is more familiar, Shimura varieties werecrucially used in the proof of the local Langlands conjecture for GLn by Harris andTaylor (see [HT01]); in the proof of the Iwasawa main conjecture for GL2 by Skinnerand Urban (see [SU14]); in the construction of p-adic L-functions for unitary groupsby Eischen, Harris, Li, and Skinner (see [HLS06] and [EHLS16]); in the constructionof Galois representations (in the context of the global Langlands correspondence) forall cohomological automorphic representations of GLn by Harris, Taylor, Thorne,and this author (see [HLTT16]); and in the analogous but deeper construction fortorsion cohomological classes of GLn by Scholze (see [Sch15]).

One of the reasons that Shimura varieties are so useful is because they carry twoimportant kinds of symmetries—the Hecke symmetry and the Galois symmetry.Roughly speaking, the Hecke symmetry is useful for studying automorphic repre-sentations, the Galois symmetry is useful for studying Galois representations, andthe compatibility between the two kinds of symmetries is useful for studying therelations between the two kinds of representations, especially in the context of theLanglands program.

However, the theory of Shimura varieties does not have a reputation of being easyto learn. There are important foundational topics such as the notion of Shimuradata and the existence of canonical models, and courses or seminars on Shimuravarieties often spend a substantial amount of the time on such topics. But learningthese topics might shed little light on how Shimura varieties are actually used—insome of the important applications mentioned above, the “Shimura varieties” therewere not even defined using any Shimura data or canonical models. In reality, it isstill a daunting task to introduce people to this rich and multifaceted subject.

In this introductory article, we will experiment with a somewhat different ap-proach. We still urge the readers to read the excellent texts such as [Del71b],[Del79], and [Mil05]. But we will not try to repeat most of the standard materials,such as variation of Hodge structures, the theory of canonical models, etc, whichhave already been explained quite well by many authors. We simply assume thatpeople will learn them from some other sources. Instead, we will try to explain thescope of the theory by presenting many examples, and many special facts whichare only true for these examples. Our experience is that such materials are ratherscattered in the literature, but can be very helpful for digesting the abstract con-cepts, and for learning how to apply the general results in specific circumstances.We hope that they will benefit the readers in a similar way.

We will keep the technical level as low as possible, and present mainly simple-minded arguments that should be understandable by readers who are willing tosee matrices larger than 2 × 2 ones. This is partly because this article is basedon our notes for several introductory lectures targeting audiences of very variedbackgrounds. The readers will still see a large number of technical phrases, often

4 KAI-WEN LAN

without detailed explanations, but we believe that such phrases can be kept askeywords in mind and learned later. We will provide references when necessary,but such references do not have to be consulted immediately—doing that will likelydisrupt the flow of reading and obscure the main points. (It is probably unrealisticand unhelpful to try to consult all of the references, especially the historical ones.)We hope that the readers will find these consistent with our goal of explaining thescope of the theory. There will inevitably be some undesirable omissions even withour limited goal, but we hope that our overall coverage will still be helpful.

Here is an outline of the article. In Section 2, we define certain double cosetspaces associated with pairs of algebraic groups and manifolds, and introduce thenotion of Shimura data. We also summarize the main results concerning the quasi-projectivity of Shimura varieties and the existence of their canonical models. InSection 3, we give many examples of Hermitian symmetric domains, which providethe connected components of the manifolds needed in the definition of Shimuradata. Our examples actually exhaust all possibilities of them. In Section 4, we givemany examples of rational boundary components and describe the correspondingminimal compactifications. We also summarize the properties of several kinds ofuseful compactifications. In Section 5, we explain how the so-called integral modelsof Shimura varieties are constructed. This is very sensitive to the types of Shimuravarieties we consider, and we also take this opportunity to summarize many specialfacts and 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⊗ZQ, and A := R×A∞.

Suppose G is a reductive algebraic group over Q. Then there is a natural way todefine the topological groups G(R), G(A∞), and G(A) = G(R)×G(A∞), with thetopology given either as a restricted product (following Weil) (see [PR94, Sec. 5.1]),or as an affine scheme (of finite presentation) over the topological ring A (followingGrothendieck), and the two approaches coincide in this case (see [Con12] for adetailed discussion on this matter). We will not try to define and summarize allthe needed properties of reductive linear algebraic groups (from textbooks such as[Bor91] and [Spr98]). Instead, we will just provide many examples of such groups.

Example 2.1.1 (general and special linear groups). For each ring R and each integern ≥ 0, we define GLn(R) to be the group of invertible n× n matrices with entriesin R, and define SLn(R) to be the subgroup of GLn(R) formed by matrices ofdeterminant one. When n = 1, we have GL1(R) = Gm(R) := R×. The assignmentsof GLn(R) and SLn(R) are functorial in R, and in fact they are the R-points ofthe affine group schemes GLn and SLn over Z, respectively. The pullbacks of thesegroup schemes to Q are affine algebraic varieties serving as prototypical examplesof reductive algebraic groups. Moreover, the pullback of SLn to Q is a prototypicalexample of a semisimple algebraic group.

For simplicity of exposition, we will often introduce a linear algebraic group G bydescribing it as the pullback to Q of some explicitly given closed subgroup schemeof GLm, for some m, over Z. By abuse of notation, we will often denote this groupscheme over Z by the same symbol G, which is then equipped with a fixed choice ofa faithful matrix representation G ↪→ GLm. The readers should not be too worried


about these technical terminologies concerning group schemes over Z. Concretely,they just mean we will explicitly represent the elements of G(R) as invertible m×mmatrices, for each ring R, whose entries satisfy certain defining conditions given byalgebraic equations that are compatible with all base ring extensions R→ R′.

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

(2.1.3) Jn :=

(0n 1n−1n 0n

)of size 2n. Then the assignments

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

and

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

are functorial in R, and define closed subgroup schemes Sp2n and GSp2n of GL2n

and GL2n+1, respectively. Here we view GL2n×Gm as a closed subgroup schemeof GL2n+1 by block-diagonally embedding the matrix entries. To better understandthe meaning of these definitions, consider the alternating pairing

(2.1.6) 〈 · , · 〉 : R2n×R2n → R

defined by setting

(2.1.7) 〈x, y〉 = txJny,

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

(2.1.8) 〈gx, gy〉 = tx tgJngy.

Therefore, g satisfies the condition tgJng = Jn exactly when

(2.1.9) 〈gx, gy〉 = 〈x, y〉,

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

(2.1.10) 〈gx, gy〉 = r〈x, y〉,

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

(2.1.11) ν : GSp2n(R)→ Gm(R),

called the similitude character of GSp2n(R), whose kernel is exactly Sp2n(R), foreach ring R. We can define similar groups and homomorphisms by replacing Jnwith any other matrices (with entries over Z, if we still want to define a groupscheme 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

)).

6 KAI-WEN LAN

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

(2.1.13) 1a,b :=

(1a−1b

),

with empty slots being filled up by zero matrices of suitable sizes. Then the assign-ments

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 case of Sp2n(R) above,On(R) and Oa+b(R) are the group of matrices preserving the symmetric bilinearpairings 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 asmooth transitive action of G(R). For technical reasons, we shall consider onlyalgebraic groups G over Q each of whose connected components in the Zariskitopology contains at least one rational point. This requirement is satisfied by everyconnected algebraic group, because the identity element is always rational, andalso by all the examples in Section 2.1. Then G(Q) is dense in G(R) in the realanalytic topology, by applying a special case of the weak approximation theoremas in [PR94, Sec. 7.3, Thm. 7.7] to the identity component of G (namely, theconnected component containing the identity element) in the Zariski topology, andby translations between this identity component and 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. symmetric ma-trices), and where X > 0 means being positive definite. Consider the action ofg ∈ SLn(R) on Sn defined by

(2.2.3) X 7→ tgXg,

for each X ∈ Sn. This action is transitive because each positive definite matrix Xis of the form

(2.2.4) X = Y 2 = tY Y

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

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

Note that

(2.2.6) dimR(Sn) = 12n(n+ 1)− 1.

In the classification in [Hel01, Ch. X, Sec. 6], Sn is a noncompact Riemanniansymmetric space of type A I.


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

(2.2.8) g =

(a bc d

)∈ SL2(R)

acting on the Poincare upper half-plane

(2.2.9) H := {z ∈ C : Im(z) > 0}by the Mobius transformation

(2.2.10) z 7→ gz =az + b

cz + d,

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

(2.2.11)

(a bc d

)(z1

)=

(az + bcz + d

)∼(gz1

).

For readers who have studied modular forms, the factor (cz + d) involved in theabove identification between projective coordinates is exactly the same automorphyfactor (cz + d) in the definition of holomorphic modular forms. (This is not just acoincidence.) It is well known that this action of SL2(R) on H is transitive (whichwill be generalized in Section 3.1.2 below), and the stabilizer 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

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

Example 2.2.14. The isomorphisms H ∼= SL2(R)/SO2(R) ∼= S2 show that H can beviewed as a special case of Sn with n = 2. But this is the only example of Sn, withn ≥ 1, such that Sn has a complex structure. Since dimR(Sn) = 1

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

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

(2.2.15) XU := G(Q)\(D×G(A∞))/U ,where G(Q) acts diagonally on D×G(A∞) from the left-hand side, and where Uonly acts on G(A∞) from the right-hand side. The reader might naturally won-der why we need to consider a complicated double quotient as in (2.2.15). Oneimportant justification is that the group G(A∞) has a natural right action on thecollection {XU}U , induced by

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

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

UH∗(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 coset spaces to the theoryof automorphic representations.

8 KAI-WEN LAN

Let D+ be a connected component of D, which admits a transitive action ofG(R)+, the identity component (namely, the connected component containing theidentity element) of G(R) in the real analytic topology. Let

(2.2.17) G(Q)+ := G(Q)∩G(R)+,

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

(2.2.18) G(Q)+\(D+×G(A∞))/U → G(Q)\(D×G(A∞))/Uis surjective because G(Q) is dense in G(R) (see the beginning of this Section 2.2),and is injective by definition. It is known (see [Bor63, Thm. 5.1]) that

(2.2.19) #(G(Q)+\G(A∞)/U) <∞,which means there exists a subset {gi}i∈I of G(A∞) indexed by a finite set I suchthat we have a disjoint union

(2.2.20) G(A∞) =∐i∈I

G(Q)+ gi U .

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)

where

(2.2.22) Γi := (giUg−1i )∩G(Q)+,

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 subgroup commensu-rable with G(Z) in the sense that Γi ∩G(Z) has finite indices in both Γi and G(Z).In fact, since U is an open compact subgroup of G(A∞), each such Γi is a congruencesubgroup of G(Z); namely, a subgroup containing the principal congruence subgroupker(G(Z) → G(Z/NZ)) for some integer N ≥ 1. By definition, congruences sub-groups are arithmetic subgroups. Note that, although the definition of principalcongruence subgroups depends on the choice of some faithful matrix representationG ↪→ GLm over Z (see Section 2.1), the definition of congruence subgroups doesnot, because congruence subgroups of G(Q) can be characterized alternatively asthe intersections of G(Q) with open compact subgroups of G(A∞). More generally,the definition of arithmetic subgroups of G(Q) does not depend on the choice ofthe faithful matrix representation either (see [Bor69, 7.13] or [PR94, Sec. 4.1]).

If each Γi acts freely on D+, then XU is a manifold because D+ is. This is thecase when each Γi is a neat arithmetic subgroup of G(Q) (as in [Bor69, (17.1)]),which is in turn the case when U is a neat open compact subgroup of G (as in[Pin89, 0.6]). For reading the remainder of this article, it suffices to know that,when people say such groups are neat (or, much less precisely, sufficiently small),they just want to ensure that the stabilizers of geometric actions are all trivial. Forsimplicity, we shall tacitly assume that the arithmetic subgroups or open compactsubgroups we will encounter are all neat, unless otherwise stated.

Now the question is what each Γi\D+ is, after knowing that it is a manifoldbecause Γi is neat. What additional structures can we expect from such quotients?


Example 2.2.23. When G = SL2 and D = H as in Example 2.2.7, the groups Γi arecongruence subgroups of SL2(Z), each of which contains the principal congruencesubgroup

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

for some integer N ≥ 1. The quotients Γi\H are familiar objects called modularcurves, each of which admits a good compactification into a compact Riemannsurface (or, in other words, complete algebraic curves over C) by adding a finitenumber of points, which are called cusps. (We will revisit this example in moredetail in Section 4.1.1 below.)

Remark 2.2.25. We may replace the manifold D above with other geometric objectsover D that still have a smooth (but not necessarily transitive) action of 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 XUas above. For arithmetic applications, it is desirable that each XU is an algebraicvariety over C with a model over some canonically determined number field.

This led to the notion of a Shimura datum (G,D) (see [Del79, 2.1.1] and[Mil05, Def. 5.5]). Consider the Deligne torus S, which is the restriction ofscalars ResC/R Gm,C (as an algebraic group over R), so that S(R) = C×. Thenwe require G to be a connected reductive algebraic group, and require D to be aG(R)-conjugacy class of homomorphisms

(2.3.1) h : S→ GR,

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

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

(2.3.2) g = k⊕ p+⊕ p−

such that

(2.3.3) z ∈ S(R) = C×

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 the adjoint

quotient of G; namely, the quotient of G by its center.)(3) Gad

R 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 variation of(real) Hodge structures, and D is a finite union of Hermitian symmetric domains.Without going into details, let us just emphasize that D and its quotients XU , forneat open compact subgroups U of G(A∞), are then complex manifolds, not justreal ones.

For studying only the connected components D+ (of some D which appeared inthe last paragraph) and their quotients, it is easier to work with what is called aconnected Shimura datum (G,D+) (see [Mil05, Def. 4.22]), which requires G to be a

10 KAI-WEN LAN

connected semisimple algebraic group, and requires D+ to be a Gad(R)+-conjugacyclass of homomorphisms

(2.3.4) h : S→ GadR ,

where Gad(R)+ denotes (as before) the identity component of Gad(R) in the realanalytic 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 homomorphism

(2.3.5) h : U1 → GadR ,

where

(2.3.6) U1 := {z ∈ C : |z| = 1}

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

(2.3.7) S→ U1

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

(2.3.8) h : U1 → GR,

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 we prefer to describeGR instead of Gad

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

2.4. Shimura varieties and their canonical models. Now we are ready tostate the following beautiful results, whose assertions can be understood withoutknowing anything about their proofs:

Theorem 2.4.1 (Satake, Baily–Borel, and Borel; see [BB66] and [Bor72]). Suppose(G,D) is a Shimura datum as in Section 2.3. Then the whole collection of complexmanifolds {XU}U , with U varying among neat open compact subgroups of G(A∞),is the complex analytification (see [Ser56, §2]) of a canonical collection of smoothquasi-projective varieties over C. Moreover, the analytic covering maps

(2.4.2) XU → XU ′ ,

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

Alternatively, suppose (G,D+) is a connected Shimura datum as in Section 2.3.Then analogous assertions hold for the collection {Γ\D+}Γ, with Γ varying amongneat arithmetic subgroups of Gad(Q) that are contained in Gad(R)+. (If we consideralso (G,D), where D is the corresponding G(R)-conjugacy class of any h in D+,then the connected components of XU , where U is a neat open compact subgroup ofG(A∞), are all of the form Γ\D+, as explained in Section 2.2.)

Theorem 2.4.3 (Shimura, Deligne, Milne, Borovoı, and others; see [Shi70],[Del71b], [Del79], [Mil83], [Bor84], and [Bor87]). Suppose (G,D) is a Shimuradatum as in Section 2.3. Then there exists a number field F0, given as a subfieldof C depending only on (G,D), called the reflex field of (G,D), such that thewhole collection of complex manifolds {XU}U , with U varying among neat opencompact subgroups of G(A∞), is the complex analytification of the pullback to C ofa canonical collection of smooth quasi-projective varieties over F0, which satisfies


certain additional properties qualifying them as the canonical models of {XU}U .Moreover, the analytic covering maps

(2.4.4) XU → XU ′ ,

when U ⊂ U ′, are also given by the complex analytifications of canonical finite etalealgebraic morphisms defined over F0 between the corresponding canonical models.

Remark 2.4.5. Theorem 2.4.1 is proved by constructing the so-called Satake–Baily–Borel or minimal compactifications of XU or Γ\D+, which are projective varietiesover C. (See Sections 4.2.2 and 4.2.7 below.) The assertion that the quasi-projectivevarieties XU and Γ\D+ are defined over Q, the algebraic closure of Q in C, can beshown without using the theory of canonical models, as in [Fal84]. But these generalresults cannot explain why each XU is defined over some particular number field.

Remark 2.4.6. In Theorem 2.4.3, since each XU is algebraic and defined over F0, itsarithmetic invariants such as the etale cohomology group H∗et(XU ,Q`), for any prime

number `, carries a canonical action of Gal(Q/F0), where the algebraic closure Qof Q in C contains F0 because F0 is given as a subfield of C. (Recall that any basechange between separably closed fields induce a canonical isomorphism betweenthe etale cohomology groups—see, for example, [Del77, Arcata, V, 3.3].) This isan instance of what we meant by Galois symmetry in the introduction (see Section1). Since the maps XU → XU ′ are also algebraic and defined over F0, we have acanonical action of Gal(Q/F0) on lim−→

UH∗et(XU ,Q`), which is compatible with the

Hecke action of G(A∞). This is an instance of what we meant by the compatibilitybetween Hecke and Galois symmetries in the introduction (see Section 1).

Remark 2.4.7. In Theorem 2.4.3, if we assume instead that (G,D+) is a connectedShimura datum, and consider the collection {Γ\D+}Γ as in Theorem 2.4.1, thenit is only true that each Γ\D+ is defined over a number field depending on both(G,D+) and Γ, but in general we cannot expect the whole collection {Γ\D+}Γ, orjust the subcollection formed by those Γ whose preimage in G(Q) are congruencesubgroups, to be defined over a single number field depending only on (G,D). Thisis a generalization of the following classical phenomenon: When (G,D) = (SL2,H),and when interpreting the quotient Γ(N)\H as in Example 2.2.23 as a parameterspace for complex elliptic 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 identifying Ezwith Eγz for all γ ∈ Γ(N), the value of the Weil pairing of αN ((1, 0), (0, 1)) is aconstant primitive N -th root ζN of unity on Γ(N)\H. (See the introduction of[DR73].) We can only expect Γ(N)\H to have a model over Q(ζN ), rather than 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 limits ofsuch varieties over either C or F0, as the Shimura varieties associated with theShimura datum (G,D). For the sake of clarity, people sometimes say that XU andits canonical model are Shimura varieties at level U . In the context of Remark2.4.7, people often refer to quotients of the form Γ\D+, where Γ are arithmetic

12 KAI-WEN LAN

subgroups of

(2.4.10) Gad(Q)+ := Gad(Q)∩Gad(R)+

whose preimages under the canonical homomorphism G(Q)→ Gad(Q) are congru-ence subgroups of G(Q), as the connected Shimura varieties associated with theconnected Shimura datum (G,D+). We allow all such subgroups Γ of Gad(Q)+,rather than only the images of the congruence subgroups of G(Q), so that theconnected Shimura varieties thus defined are useful for studying all connected com-ponents of Shimura varieties.

Remark 2.4.11. By a morphism of Shimura data

(2.4.12) (G1,D1)→ (G2,D2),

we mean a group homomorphism G1 → G2 mapping D1 into D2. If D = G1(R) ·h0

is the conjugacy class of some homomorphism h0 : S → G1,R, then this means thecomposition of h0 with G1,R → G2,R lies in the conjugacy class D2. If U1 and U2

are open compact subgroups of G1(A∞) and G2(A∞), respectively, such that U1 ismapped into U2, then we obtain the corresponding morphism XU1 → XU2 betweenShimura varieties. This morphism is defined over the subfield of C generated by thereflex fields of (G1,D1) and (G2,D2). There are also analogues of these assertionsfor connected Shimura data and the corresponding connected Shimura varieties.

Remark 2.4.13. An important special case of a morphism of Shimura data as inRemark 2.4.11 is when the homomorphism G1 → G2 is injective, and when U1 isexactly the pullback of U2 under the induced homomorphism G1(A∞)→ G2(A∞).(It is always possible to find such a U2 for any given U1—see [Del71b, Prop. 1.15].The neatness of U2 can be arranged when U1 is neat, by taking U2 to be generatedby the image of U1 and some suitable neat open compact subgroup of G2(A∞).)In this case, XU1 is a closed subvariety of XU2 . Subvarieties of this kind are calledspecial subvarieties. The most important examples are given by special points or(with some additional assumptions on the Shimura data) CM points, which arezero-dimensional special subvarieties defined by subgroups G1 of G2 that are tori(see [Mil05, Def. 12.5]). These are generalizations of the points of modular curvesparameterizing CM elliptic curves with level structures, which are represented bypoints of the Poincare upper half-plane with coordinates in imaginary quadraticfield extensions of Q. Zero-dimensional Shimura varieties and special points areimportant because their canonical models can be defined more directly, and hencethey are useful for characterizing the canonical models of Shimura varieties of pos-itive dimensions.

We will not try to explain the proofs of Theorems 2.4.1 and 2.4.3 in this intro-ductory article. In fact, we will not even try to explain in any depth the backgroundnotions such as variations of Hodge structures or Hermitian symmetric domains,or key notions such as zero-dimensional Shimura varieties, the Shimura reciprocitylaw, and the precise meaning of canonical models, because most existing texts al-ready present them quite well, and there is little we can add to them. Instead, wewill start by presenting many examples of (G,D) and (G,D+), so that the readerscan have some idea to where the theory applies. We hope this will benefit not onlyreaders who would like to better understand the existing texts, but also readerswho have concrete arithmetic or geometric problems in mind.


3. Hermitian symmetric domains

In this section, we will explore many examples of Hermitian symmetric domains,which can be interpreted as the generalizations of the Poincare upper half-plane H(see Example 2.2.7) in many different ways. Most of these will give the (G(R),D)or (G(R),D+) for some Shimura data (G,D) or connected Shimura data (G,D+),respectively, as in Section 2.3, although we will not be using the structure of G(Q)in this section. (Later we will indeed use the structure of G(Q) in Section 4.)

However, we will not define Hermitian symmetric domains. Our excuse for thisomission 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 willsee all of them in our examples. (For a more formal treatment, there are generalintroductions such as [Mil05, Sec. 1], with references to more advanced texts suchas [Hel01].)

In our personal experience, especially in number-theoretic applications requiringactual calculations, the abstract theory of Hermitian symmetric domains cannoteasily replace the knowledge of these explicit examples. (Already in the classicaltheory for SL2 or GL2, it is hard to imagine studying modular forms without everintroducing the Poincare upper half-plane.)

We shall begin with the case of Sp2n(R) in Section 3.1, followed by the case ofUa,b in Section 3.2, the case of SO∗2n in Section 3.3, the case of SOa,2(R) in Section3.4, the case of E7 in Section 3.5, the case of E6 in Section 3.6, and a summary ofall 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 Siegel upperhalf-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, 1

2it

(Z1

)(0 1−1 0

)(Z1

)< 0

},

(3.1.1.1)

where Mn (resp. Symn) denotes the space of n×n matrices (resp. symmetric matri-ces) as before, and where the notation > 0 (resp. < 0) means positive definiteness(resp. negative definiteness) of matrices, which admits a natural (left) action of

g =

(A BC D

)∈ Sp2n(R):

(3.1.1.2) Z 7→ gZ := (AZ +B)(CZ +D)−1

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

(3.1.1.3)

(A BC D

)(Z1

)=

(AZ +BCZ +D

)∼(gZ1

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

Remark 3.1.1.4. By generalized projective coordinates given by an m×n matrix, wemean an m×n matrix of rank n that is identified up to the right action of invertible

14 KAI-WEN LAN

n×n matrices. Giving such a matrix is equivalent to defining a point of the Grass-mannian of n-dimensional subspaces of an m-dimensional vector space. Later wewill encounter some similar generalized projective coordinates when defining otherHermitian symmetric domains, and the readers can interpret them as defining pointson some other Grassmannians. This is closely related to the interpretation of Her-mitian symmetric domains as parameter spaces of variations of Hodge structures,where the Grassmannians parameterize some Hodge filtrations.

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

(3.1.1.5) t

(A BC D

)(0 1−1 0

)(A BC D

)=

(0 1−1 0

),

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

(3.1.1.7) t

(gZ1

)(0 1−1 0

)(gZ1

)= 0;

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

(3.1.1.9) 12it

(gZ1

)(0 1−1 0

)(gZ1

)< 0.

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

(3.1.2.1)

(1 −X0 1

)Z = iY,

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

(3.1.2.2) tAY A = 1n,

which shows that

(3.1.2.3)

(tA 00 A

)(iY ) = i1n.

These show that

(3.1.2.4) Hn = Sp2n(R) · (i1n).

The stabilizer of i1n is

(3.1.2.5) K :=

{(A B−B A

)∈ Sp2n(R)

}∼= Un :=

{g = A+ iB : tgg = 1n

}


(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 the choice of the pointi1n.) Thus, we have shown that

(3.1.2.6) Hn ∼= Sp2n(R)/Un.

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

(3.1.2.7) dimCHn = dimC(Symn(C)

)= 1

2n(n+ 1),

for all n ≥ 0.

3.1.3. Bounded realization. There is also a bounded realization of Hn. Note that

(3.1.3.1)

(1 −i1 i

)(i1

)=

(02i

)∼(

01

).

Set

(3.1.3.2) U := (Z − i1n)(Z + i1n)−1,

so that

(3.1.3.3)

(1 −i1 i

)(Z1

)=

(Z − i1nZ + i1n

)∼(U1

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

(3.1.3.4)

(1 1−i i

)(0 1−1 0

)(1 −i1 i

)=

(0 2i−2i 0

)= 2i

(0 1−1 0

),

we have

(3.1.3.5) tZ = Z ⇐⇒ tU = U,

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

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

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

In the classification in [Hel01, Ch. X, Sec. 6], Hn ∼= Dn is a Hermitian symmetricdomain of type C I. In Cartan’s classification, it is a bounded symmetric domainof type IIIn.

16 KAI-WEN LAN

3.1.4. Interpretation as conjugacy classes. Now consider the homomorphism

(3.1.4.1) h0 : U1 → Sp2n(R) : x+ yi 7→(x −yy x

).

For simplicity, in the matrix at the right-hand side, we denoted by x and y thescalar multiples of the n× n identity matrix 1n by x and y, respectively. (We willadopt 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

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

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

(3.1.4.4) Hn = Sp2n(R) · h0,

with the action given by conjugation. That is, Hn is the Sp2n(R)-conjugacy class ofh0. In this case, (Sp2n,Hn) is a connected Shimura datum as in Section 2.3 (wherewe use Sp2n to denote the algebraic group over Q, not the 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 explanation in what follows.)

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

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

(3.1.4.6) H±n = GSp2n(R) · h0,

where

(3.1.4.7) h0 : C× → GSp2n(R) : x+ yi 7→(x −yy x

)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 this case,(GSp2n,H±n ) is a Shimura datum as in Section 2.3 (where we use GSp2n to denotethe algebraic group over Q, not the group scheme over Z).

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

(3.1.5.1) LZ := ZnZ + Zn

in Cn such that

(3.1.5.2) AZ := Cn/LZis not only a complex torus, but also a polarized abelian variety . (See [Mum70, Sec.1–3] and [Mum83, Ch. II, Sec. 1 and 4].) When n = 1, this is just an elliptic curve.

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

(3.1.5.3) R2n/Z2n


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

(3.1.5.4) h(i) ∈ Sp2n(R) ⊂ GL2n(R),

where the homomorphism

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

corresponds to the point

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

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

The pairing on Z2n defined by the perfect pairing 〈 · , · 〉 defines (up to a signconvention) a principal polarization λZ on AZ (and the existence of such a po-larization was the reason that the complex torus AZ is an abelian variety). If(AZ , λZ) ∼= (AZ′ , λZ′) for Z and Z ′ corresponding to h and h′, respectively, then

we have an induced isomorphism R2n ∼→ R2n preserving Z2n and 〈 · , · 〉, which de-

fines an element g =

(A BC D

)∈ Sp2n(Z) conjugating h to h′ and mapping Z to

Z ′. Conversely, if we have

(3.1.5.7) Z ′ = gZ = (AZ +B)(CZ +D)−1

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 Γ ofSp2n(Z), then the above isomorphism (AZ , λZ) ∼= (AZ′ , λZ′) also respect certain(symplectic) level structures. Therefore, Γ\Hn (or the manifolds XU defined usingG = GSp2n and D = H±n ) are Siegel modular varieties parameterizing polarizedabelian varieties with certain level structures (depending on Γ or U). The spacesΓ\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 areindeed the complex analytifications of some canonical algebraic varieties.

The fact that spaces like Γ\Hn parameterize polarized abelian varieties withlevel structures is perhaps just a coincidence, but such a coincidence is extremelyimportant. One can redefine them as the C-points of certain moduli problems ofpolarized abelian schemes with level structures; and then their constructions can beextended over the integers in number fields, because the moduli schemes of polar-ized abelian schemes, called Siegel moduli schemes, have been constructed over theintegers. (See [MFK94, Ch. 7] for the construction based on geometric invarianttheory. Alternatively, geometric invariant theory can be avoided by first construct-ing algebraic spaces or Deligne–Mumford stacks by verifying Artin’s criterion—see[Art69] and [Lan13, Sec. B.3] for such a criterion, and see [FC90, Ch. I, Sec. 4] and[Lan13, Ch. 2] for its verification for the moduli of polarized abelian schemes; andthen showing that they are quasi-projective schemes by constructing their minimalcompactifications as in [FC90, Ch. V] or [Lan13, Ch. 7]—see Section 5.1.4 below.)

18 KAI-WEN LAN

Such integral models of Γ\Hn are very useful because we can understand theirreductions in positive characteristics by studying polarized abelian varieties overfinite fields, for which we have many powerful tools. We will encounter more generalcases of Γ\D later, but we have good methods for constructing integral models forthem (namely, methods for which we can say something useful about the outputs)only in cases that can be related to Siegel modular varieties at all.

3.2. The case of Ua,b. We learned some materials here from [Shi98] and [Shi78].

3.2.1. Bounded realization. Let a ≥ b ≥ 0 be any integers. Recall that we have

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 of signature (a, b).Consider the group

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

which acts on

(3.2.1.2) Da,b :=

{U ∈ Ma,b(C) : t

(U1

)(1a−1b

)(U1

)= tUU − 1b < 0

},

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

(3.2.1.3) U 7→ gU = (AU +B)(CU +D)−1

for each g =

(A BC D

)∈ Ua,b. This action is based on the identity

(3.2.1.4)

(A BC D

)(U1

)=

(AU +BCU +D

)∼(

(AU +B)(CU +D)−1

1

)in generalized projective coordinates (cf. Remark 3.1.1.4). Note that Ua,b is areal Lie group, but not a complex one, despite the use of complex numbers in itscoordinates; and that

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

)= ab.

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

(3.2.1.6) K :=

{(A

D

)∈ Ua,b

}∼= Ua×Ub :

(A

D

)7→ (A,D).

Remark 3.2.1.7. For the purpose of introducing Hermitian symmetric domains,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 adjoint quotient, butwe still consider the reductive Lie group Ua,b, because it is easier to write downhomomorphisms of the form U1 → Ua,b, as in (2.3.8).

However, to show the transitivity of the action of Ua,b onDa,b, and for many othercomputations 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

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

(3.2.2.2) Ja,b :=

1bS

−1b

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

(3.2.2.3) Ha,b :=

(ZW

)∈ Ma,b(C) ∼= Mb(C)×Ma−b,b(C) :

−i tZW1

Ja,bZW1

= −i( tZ − Z + tWSW ) < 0

by

(3.2.2.4)

(ZW

)7→ g

(ZW

)=

((AZ + EW +B)(CZ +HW +D)−1

(FZ +MW +G)(CZ +HW +D)−1

),

for each g =

A E BF M GC H D

∈ U′a,b, based on the identity

(3.2.2.5)

g

ZW1

=

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

∼ (AZ + EW +B)(CZ +HW +D)−1

(FZ +MW +G)(CZ +HW +D)−1

1

in generalized projective coordinates (cf. Remark 3.1.1.4).

If we consider

(3.2.2.6) Hermb(C) := {X ∈ Mb(C) : tX = X},where Hermb denotes the space of b×b Hermitian matrices (which are over C here),then we have a canonical isomorphism

(3.2.2.7) Mb(C)∼→ Hermb(C)⊗

RC : X 7→ Re(X) + i Im(X),

where

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

and

(3.2.2.9) Im(X) := 12i (X −

tX)

are the Hermitian real and imaginary parts (abusively denoted by the usual sym-bols), and we can rewrite the condition

(3.2.2.10) − i( tZ − Z + tWSW ) < 0

as

(3.2.2.11) Im(Z) > 12tW (−iS)W.

20 KAI-WEN LAN

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

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

which can be viewed as a generalization of

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

That is, we can view the symmetric matrices in Symn(R) as being Hermitian withrespect to the trivial involution of R, which then generalizes to the usual complexHermitian matrices in Hermn(C) as being Hermitian with respect to the complexconjugation of C over R. These two examples will be further generalized with Rand C replaced with the other two normed (but possibly nonassociative) divisionalgebras H and O over R, the rings of Hamilton and Cayley 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

(3.2.4.2) 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

,

and accordingly an isomorphism

(3.2.4.3) Ha,b∼→ Da,b :

ZW1

7→

1√2

− i√2

T1√2

i√2

ZW1

∼ (U1

)

equivariant with the above isomorphism (3.2.4.2). This is a generalization of the

classical Cayley transformation and maps

(i1b0

)∈ Ha,b to 0 ∈ Da,b.

Now, given any

(ZW

)∈ Ha,b, we have1 ∗ ∗

1 −W1

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, where thefirst matrix in each equation is some element of U′a,b. These show that

(3.2.4.7) Ha,b = U′a,b ·(i1b0

)∼= Da,b = Ua,b · 0 ∼= Ua,b/(Ua×Ub),

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

In the classification in [Hel01, Ch. X, Sec. 6], Ha,b ∼= Da,b is a Hermitian sym-metric domain of type A III. In Cartan’s classification, it is a bounded symmetricdomain of type Iba.

3.2.5. Interpretation as conjugacy classes. Consider the homomorphism

(3.2.5.1) h0 : U1 → Ua,b : x+ yi 7→(x− yi

x+ yi

),

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

(3.2.5.2) Centh0(Ua,b) = Ua×Ub

(as subgroups of Ua,b) and so that

(3.2.5.3) Da,b = Ua,b · h0.

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

obtain

(3.2.5.4) h′0 : U1 → U′a,b : x+ yi 7→

x −yx− yi

y x

,

so that

(3.2.5.5) Ha,b = U′a,b · h′0.

Both of these are compatible with the extensions of the actions of Ua,b and U′a,bon Da,b and Ha,b, respectively, to some actions of

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

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

(3.2.5.8) D±a,b = GUa,b · h0

and

(3.2.5.9) H±a,b = GUa,b · h′0(where we omit the explicit descriptions of D±a,b and H±a,b, for simplicity), where h0

and h′0 are now homomorphisms

(3.2.5.10) h0 : C× → GUa,b

and

(3.2.5.11) h′0 : C× → GU′a,b

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

22 KAI-WEN LAN

be of the form (G(R),D) for some Shimura data (G,D). We shall omit such remarksin later examples.)

Since the real part of any skew-Hermitian pairing is alternating, there is aninjective homomorphism of the form

(3.2.5.12) Ua,b∼= U′a,b ↪→ Sp2n(R),

with n = a + b, whose pre-composition with h0 : U1 → Ua,b lies in the conjugacyclass 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. Consider thesubgroup SO∗2n of

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

preserving the skew-Hermitian form defined by

(1n

−1n

). That is,

(3.3.1.2) SO∗2n = SO2n(C)∩U′n,n

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

(3.3.1.3) so∗2n =

{(A B−B A

): A,B ∈ Mn(C), tA = −A, tB = B

}as Lie subalgebras of M2n(C) (see, for example, [Kna01, Ch. I, Sec. 1] or [Kna02,Ch. I, Sec. 17]). By explanations as before, SO∗2n acts transitively on

(3.3.1.4) HSO∗2n:=

Z ∈ Mn(C) : t

(Z1

)(Z1

)= tZZ + 1 = 0,

−i t(Z1

)(1

−1

)(Z1

)= −( tZ − Z) < 0

,

whose stabilizer at i1n ∈ HSO∗2nis

(3.3.1.5) K :=

{(A B−B A

)∈ SO∗2n : A,B ∈ Mn(R)

}∼= Un,

so that

(3.3.1.6) HSO∗2n= SO∗2n · h0

∼= SO∗2n/Un,

where

(3.3.1.7) h0 : U1 → SO∗2n : x+ yi 7→(x −yy x

).

As before, there is also another version of SO∗2n which acts on some bounded real-ization DSO∗2n

, but we shall omit them for simplicity. Since SO∗2n ⊂ U′n,n, by thesame explanation as in Section 3.2.5, there are injective homomorphisms

(3.3.1.8) SO∗2n ↪→ Sp4n(R),

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

In the classification in [Hel01, Ch. X, Sec. 6], HSO∗2n∼= DSO∗2n

is a Hermitian sym-metric domain of type D III. In Cartan’s classification, it is a bounded symmetricdomain of type IIn.


3.3.2. The special case where n = 2k: quaternion upper half-spaces. Since we havethe isomorphism

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

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

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

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

(3.3.2.3) SO∗4k∼={g ∈ GL2k(H) : tg

(1k

−1k

)g =

(1k

−1k

)},

namely, the subgroup Sp2k(H) of GL2k(H) respecting the skew-Hermitian pairing

on Hk defined by

(1k

−1k

)(see, for example, [GW09, Sec. 1.1.4], combined with

a Gram–Schmidt process as in [Lan13, Sec. 1.2.3]); and that

(3.3.2.4) HSO∗4k∼= {Z ∈ Hermk(H)⊗

RC : Im(Z) > 0},

with the action of g =

(A BC D

)∈ GL2k(H) given by

(3.3.2.5) gZ = (AZ +B)(CZ +D)−1,

which is similar to the action of Sp2k(R) onHk in Section 3.1.1. This is a quaternionupper half-space, further generalizing the examples in Sections 3.1.1 and 3.2.3.

3.4. The case of SOa,2(R). We learned some materials here from [Sat80, Appen-dix, Sec. 6].

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

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

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

(3.4.1.2) HSOa,b:=

{v ∈ VC : tv

(1a−1b

)v = 0, tv

(1a−1b

)v < 0

}/C×.

Note that v 6= 0 in the above definition, and hence the quotient by C× means we areactually working with a subset of Pa+b−1(C) ∼= (VC − {0})/C×. (This is consistentwith our use of generalized projective coordinates in previous examples.) However,

HSOa,bis in general not connected, and even its connected components do not have

the structure of Hermitian symmetric domains for arbitrary a, b ≥ 0.

24 KAI-WEN LAN

3.4.2. The case where a ≥ 1 and b = 2. In this case, we do get Hermitian symmetricdomains. (This follows from the general theory, but since we have not definedHermitian symmetric domains, we are merely stating this as a fact.)

Suppose we have an orthogonal direct sum

(3.4.2.1) V = Ra+2 = Ra⊕R2

such that the pairings on Ra and R2 are defined by

(1a−1

−1

)and

(12

12

),

respectively, which have signatures (a − 1, 1) and (1, 1) summing up to (a, 2) intotal. Note that we have

(3.4.2.2)

(1 1−1 1

)(12

12

)(1 −11 1

)=

(1−1

).

Then we also have

(3.4.2.3) VC ∼= Ca⊕C2,

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

(3.4.2.4) HSOa,2∼={

z21 + · · ·+ z2

a−1 − z2a + w1w2 = 0,

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

}.

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

(3.4.2.5) |z1|2 + · · ·+ |za−1|2 < |za|2 = |z1 + · · ·+ za−1|2,which is not possible. Hence, we must have w2 6= 0, and so, up to scaling of theprojective coordinates, we may assume that w2 = 1. Then

(3.4.2.6) w1 = z2a − z2

1 − · · · − z2a−1

by the first condition, and the second condition becomes

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

If we write zj = xj + iyj , then

(3.4.2.8) |zj |2 − Re(z2j ) = 2y2

j ,

for each 1 ≤ j ≤ a. Thus, we have(3.4.2.9)

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

.

The condition

(3.4.2.10) y2a > y2

1 + · · · y2a−1

defines a familiar light cone with two connected components, one being defined by

(3.4.2.11) ya >√y2

1 + · · · y2a−1,

the other being defined by

(3.4.2.12) ya < −√y2

1 + · · · y2a−1.

These two connected components correspond to two connected components H+SOa,2

and H−SOa,2of HSOa,2

.


Note that

(3.4.2.13) dimC HSOa,2 = dimCHSOa,2 = dimCH+SOa,2

= dimC Ca = a.

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 that a ≥ 1,the real Lie group SOa,2(R) has two connected components (see, for example,[Hel01, Ch. X, Sec. 2, Lem. 2.4] or [Kna02, Prop. 1.145]), and the stabilizer ofH+

SOa,2

in G(R) = SOa,2(R) is the identity component G(R)+ = SOa,2(R)+; namely, theconnected component containing the identity element (in the real analytic topol-ogy). The group SOa,2(R) can be viewed as the group of R-points of the algebraicgroup SOa,2 defined over R, which admits a simply-connected two-fold coveringgroup Spina,2. This is only a covering of algebraic groups, which means we canonly expect a surjection on C-points. In fact, the induced morphism

(3.4.3.1) Spina,2(R)→ SOa,2(R)

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 possible, with-out introducing the spin groups in detail—see, for example, [GW09, Ch. 6], fora review of their construction based on Clifford algebras. But the connectednessof Spina,2(R), nevertheless, is a general feature of the real points of any isotropicsimply-connected connected simple algebraic group over Q—see [PR94, Sec. 7.3,Thm. 7.6].

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

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

which induces by taking Galois cohomology the long exact sequence

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

where the homomorphism

(3.4.3.4) SOa,2(R)→ {±1}

(induced by the connecting homomorphism) is the spinor norm. The kernel of thespinor norm is SOa,2(R)+, which is at the same time the image of Spina,2(R).

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

,

with stabilizers isomorphic to

(3.4.3.5) SOa(R)×SO2(R).

Accordingly, SOa,2(R) acts transitively on HSOa,2, with stabilizers also isomorphic

to SOa(R)×SO2(R). We shall also omit the description of some bounded realiza-tions D+

SOa,2and DSOa,2

of H+SOa,2

and HSOa,2, respectively.

In the classification in [Hel01, Ch. X, Sec. 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 notationthere). In Cartan’s classification, it is a bounded symmetric domain of type IVa.

26 KAI-WEN LAN


(3.4.4.1) h0 : U1 → SOa,2(R) : eiθ 7→

1acos 2θ − sin 2θsin 2θ cos 2θ

.

Note that the image of h0 is connected because U1 is, and hence lies in SOa,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

(3.4.4.3) H+SOa,2

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

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

(3.4.4.4) h0 : U1 → Spina,2(R),

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

(3.4.4.5) Spina,2(R) ↪→ Sp2n(R),

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 [Bai70], [Kim93], [GL97], and [Eie09,Ch. 12].

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

(3.5.1.1) O = R⊕Re1⊕ · · ·⊕Re7

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

(3.5.1.2) e2i = −1

and

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

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

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

and

(3.5.1.5) N(x) := xx = x20 + x2

1 + ·+ x27 ≥ 0

for each

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

(3.5.2.1) A := Herm3(O),

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

(3.5.2.2) X · Y = 12 (XY + Y X),

where XY and Y X are the usual matrix multiplications. (See, for example, [FK95,Ch. II–V and VIII] for a general introduction to Jordan algebras and the classifi-cation of formally real ones. See also [AMRT75, Ch. II, Rem. at the end of Sec.1] or [AMRT10, Ch. II, Rem. 1.11], and the sections following there, for why theymatter.)

For

(3.5.2.3) X =

a x yx b zy z c

∈ A

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

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

and the determinant

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

Then we also have an inner product

(3.5.2.6) (X,Y ) := tr(X · Y )

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

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

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

(3.5.2.8) (X × Y,Z) = 3(X,Y, Z),

for X,Y, Z ∈ A, which specializes to

(3.5.2.9) X ×X =

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

when X = Y .

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

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

(3.5.3.1) M := {(g, r) ∈ GL(A)×R× : det(gX) = r det(X)},so that we have a homomorphism

(3.5.3.2) ν : M → R× : (g, r) 7→ r.

Then we have

(3.5.3.3) ker(ν) = {g ∈ GL(A) : det(gX) = det(X)}.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 notation of e6(−26)

28 KAI-WEN LAN

means it is the real form of the complex simple Lie algebra of type E6 such that theKilling form has signature (a, b) with a−b = −26; or, equivalent, that every Cartaninvolution has trace 26 = −(−26). For complex simple Lie algebras of exceptionaltype, such numbers already classify their real forms up to isomorphism. (The sameexplanation applies to other real Lie algebras of exceptional type below.) Also, thecenter of M is

(3.5.3.4) Z(M) ∼= R×

with x ∈ R× acting as multiplication by the scalar x on A, so that ν(x) = x3. Inthis case, the Lie algebra of M is the direct sum of those of Z(M) and ker(ν), andker(ν) is the derived group of M . The ker(ν)-orbit of 1 in A is isomorphic to thequotient of ker(ν) by a maximal compact subgroup with Lie algebra isomorphicto f4. In the classification in [Hel01, Ch. X, Sec. 6], this orbit is a noncompactRiemannian symmetric space of type E IV, which is not Hermitian.

Moreover, g ∈M induces g∗ ∈M by

(3.5.3.5) (gX, g∗Y ) = (X,Y ).

Then M acts on

(3.5.3.6) W := A⊕R⊕A⊕Rby

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

There is also an additive action

(3.5.3.8) ρ : A ↪→ GL(W) : B 7→ ρB

with ρB defined by

(3.5.3.9) ρB

XεX ′

ε′

:=

X + 2B ×X ′ + εB ×B

εX ′ + εB

ε′ + (B,X) + (B ×B,X ′) + εdet(B)

,

with elements of W represented by column vectors (cf. (3.5.3.6)).Consider the Lie group

(3.5.3.10) U := image(ρ).

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

(3.5.3.11) ι(X, ε,X ′, ε′) := (−X ′,−ε′, X, ε).Consider the Lie groups

(3.5.3.12) P := M n U

and

(3.5.3.13) G := 〈ι, P 〉(i.e., G is generated by ι and P ) as subgroups of GL(W). It is known that Gis a connected semisimple Lie group of real rank 3 with Lie algebra isomorphic toe7(−25), and that P is the maximal parabolic subgroup of G stabilizing the subspace{(0, 0, 0, ε′) : ε′ ∈ R} of W.

These definitions might seem rather complicated and unmotivated, but it isworth 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) consistingof elements preserving both the quartic form

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

4 ((X,X ′)− εε′)2(3.5.3.14)

and the skew-symmetric bilinear form

(3.5.3.15) 〈(X1, ε1, X′1, ε′1), (X2, ε2, X

′2, ε′2)〉 := (X1, X

′2)− (X2, X

′1) + ε1ε

′2 − ε2ε

′1

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

(3.5.4.1) HE7 := {Z ∈ A⊗RC : Im(Z) > 0},

where Im(Z) > 0 means the left multiplication of Im(Z) on A induces a positivedefinite linear transformation, or equivalently that Im(Z) lies in the interior of theset {X2 : X ∈ A} with its natural real analytic topology (see [FK95, Thm. III.2.1]).(For an analogy, consider the case of symmetric matrices over R, where positivedefinite matrices are exactly those matrices in the interior of the set of squares ofsymmetric matrices with its natural real analytic topology.) This is the most exoticgeneralization of the various classical upper half-spaces in Sections 3.1.1, 3.2.3, and3.3.2 defined by Hermitian matrices over R, C, and H, respectively. Together withthe “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

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

where J is a formally real Jordan algebra (see [FK95, Ch. VIII–X]; see also[AMRT75, Ch. III, Sec. 1] or [AMRT10, Ch. III, Sec. 1]), where the symboli denotes the action of 1⊗ i on J⊗

RC. In the definition of HE7

, we have

J = A = Herm3(O).Note that

(3.5.4.3) dimCHE7 = dimC(A⊗RC) = dimR(A) = 27.

The group U acts on HE7 by the translation

(3.5.4.4) Z 7→ Z +B,

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 HE7by

(3.5.4.5) ι(Z) := −Z−1,

30 KAI-WEN LAN

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 define an actionof G on HE7

.Given any Z = X + iY ∈ HE7 , where X,Y ∈ A and where the symbol i denotes

the action of 1⊗ i on A⊗RC, we have

(3.5.4.6) ρ−X(Z) = iY

for some Y > 0. By Freudenthal’s theorem (which intriguingly asserts that matricesin Herm3(O) can also be diagonalized ; see [FK95, Thm. V.2.5]), there exists someg ∈M such that

(3.5.4.7) gY = 13,

and so that

(3.5.4.8) g(iY ) = i13.

This shows that

(3.5.4.9) HE7= G · i13

and so that the action of G on HE7 is transitive.Let K denote the stabilizer of i13 in G. Then

(3.5.4.10) K = Centι(G)

as subgroups of G, which is a maximal compact subgroup of G and is a connectedcompact Lie group with Lie algebra e6⊕R. Consider the homomorphism

(3.5.4.11) h0 : U1 → G

defined in block matrix form by setting

(3.5.4.12) h0(cos θ + i sin θ) =

cos θ − sin θ

cos 3θ sin 3θsin θ cos θ

− sin 3θ cos 3θ

∈ GL(W)

(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

(3.5.4.13) HE7= G · h0

∼= G/K.

There is no homomorphism of the form G → Sp2n(R), for any n ≥ 1, whosepre-composition with h0 : U1 → G lies in the conjugacy class of the analogoushomomorphism for Sp2n(R) in (3.1.4.1) (see [Del79, 1.3.10(i)] or [Mil13, Sec. 10, p.531]; cf. the end of Section 5.2.2 below).

3.5.5. Bounded realization. There is also a bounded realization DE7of HE7

, alsowith coordinates in A⊗

RC, with a change of coordinates

(3.5.5.1) HE7

∼→ DE7: Z 7→ (Z − i13)(Z + i13)−1,

where (Z+ i13)−1 is (as before) defined by inverting the Jordan algebra multiplica-tion in A⊗

RC, which is yet another generalization of the classical Cayley transform

(cf. Sections 3.1.3 and 3.2.4).


In the classification in [Hel01, Ch. X, Sec. 6], HE7∼= DE7

is a Hermitian sym-metric domain of type E VII. In Cartan’s classification, it is a bounded symmetricdomain of type VI.

3.6. The case of E6.

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

(3.6.1.1) DE6:= DE7

∩

0 x yx 0 0y 0 0

: x, y ∈ O⊗RC

in A⊗

RC (see Section 3.5.5), where x and y are the conjugations induced by the

ones in O, which can be embedded as an open submanifold of O2⊗RC. Let us also

let HE6 denote the preimage of DE6 under the isomorphism (3.5.5.1), which is thenan unbounded realization.

Essentially by construction, we have

(3.6.1.2) dimCHE6= dimCDE6

= 2 dimC(O⊗RC) = 2 dimR O = 16.

By [Hir66, Sec. 3], DE6and hence HE6

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 connected Liegroups with Lie algebras isomorphic to so10⊕R.

In the classification in [Hel01, Ch. X, Sec. 6], HE6∼= DE6

is a Hermitian sym-metric domain of type E III. In Cartan’s classification, it is a bounded symmetricdomain of type V.

3.6.2. A real Lie group of Lie algebra e6(−14). It is not easy to describe DE6and

HE6, together with the actions of certain G as above, in explicit coordinates. Nev-

ertheless, let us define a real Lie group G with Lie algebra e6(−14), which can begiven the structure of the R-points of an algebraic group over Q, and which acts onDE6

and HE6by the abstract theory of Hermitian symmetric domains.

Consider the pairing

(3.6.2.1) 〈X,Y 〉 := (Xc, Y ),

for X,Y ∈ A⊗RC, where c denotes the complex conjugation with respect to the

tensor factor C; consider the involution σ on A = Herm3(O) (which extends linearlyto A⊗

RC) defined by

(3.6.2.2)

a x yx b zy z c

σ

=

a −x −y−x b z−y z c

;

and consider

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

where

(3.6.2.4) 〈X,Y 〉σ := 〈Xσ, Y 〉.

32 KAI-WEN LAN

We note that

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

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

(3.6.2.6) 〈gX, gY 〉σ = 〈X,Y 〉σdefines a real form of rank 2 of this complex Lie group with Lie algebra isomorphicto e6(−14), which is not isomorphic to either its compact form (see (3.5.4.10) andthe remark following it) or the noncompact real form

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

(see (3.5.3.3) and the remark following it) whose Lie algebra is isomorphic to 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.)


(3.6.3.1) h0 : U1 → G

defined by

(3.6.3.2) h0(α)

a x yx b zy z c

=

α4a αx αyαx α−2b α−2zαy α−2z α−2c

,

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

(3.6.3.3) K := Centh0(G),

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

(3.6.3.4) HE6= G · h0

∼= G/K.

There is no homomorphism of the form G → Sp2n(R), for some n ≥ 1, whosepre-composition with h0 : U1 → G lies in the conjugacy class of the analogoushomomorphism for Sp2n(R) in (3.1.4.1) (see [Del79, 1.3.10(i)] or [Mil13, Sec. 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 wehave considered as follows. Let us also include the information of the correspondingspecial node on each Dynkin diagram, marked with a unique double circle on thediagram. Such special nodes have not been explained, but should be useful for thereaders when they consult more advanced texts.

(1) G(R) = Sp2n(R), K ∼= Un, D = Hn:

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

(3.7.2) d d · · · da · · · d dHere 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:

(3.7.3)d d · · · d · · · d��

HH

dda

Again, the special node is only up to automorphisms of the Dynkin diagram.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:

(3.7.4) da d · · · d d > d(b) a = 2n− 2 is even, with n ≥ 3:

(3.7.5)da d · · · d · · · d��

HH

dd

(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:

(3.7.6) da d dd

d dYet 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:

(3.7.7) da d d dd

d dThese exhaust all the possibilities of irreducible Hermitian symmetric domains.

In the cases (1), (2), and (3), there exist injective homomorphisms of the formG(R)→ Sp2n′(R), for some n′ ≥ 1, whose pre-composition with h0 : U1 → G(R) liesin the conjugacy class of the analogous homomorphism for Sp2n′(R) as in (3.1.4.1).In the case (4), the analogous assertion is true after replacing SOa,2(R)+ with itsdouble cover Spina,2(R). In the cases (5) and (6), the analogous assertion is simplyfalse. (In Section 5.2.2 below, we will encounter this issue again in the classificationof abelian-type Shimura data, although the question there is more subtle becausewe need to also consider 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 varieties incases where D are Hermitian symmetric domains or their finite unions.

34 KAI-WEN LAN

4.1. Examples of rational boundary components. Let us start by explainingthe notion of rational boundary components in many examples, based on the Her-mitian symmetric domains we have seen in Section 3. We will begin with the caseswith D = H in Section 4.1.1, followed by the cases with D = H×H in Section4.1.2; the cases with D = Hn in Section 4.1.3; the cases with D = Ha,b, witha ≥ b, in Section 4.1.4; the cases with 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 in Examples2.2.7 and 2.2.23, and we have to consider quotients of the form Γ\D, where Γ is anarithmetic subgroup of SL2(Q). Note that the action of SL2(Q) preserves

(4.1.1.1) H∗ := H∪P1(Q)

as a subset of P1(C).In the natural topology of P1(C), the boundary of H is P1(R) = SL2(R) · ∞,

where∞ means just

(10

)in projective coordinates. The reason to consider instead

the rational subset P1(Q) = SL2(Q) · ∞ is the reduction theory : The fundamentaldomain of H under the action of Γ can be compactified by adding only points inP1(Q) = SL2(Q) ·∞. The irrational points are completely irrelevant. Therefore, bytopologizing H∗ at the points of P1(Q) using open discs bounded by the so-calledhorocycles, namely, circles tangent to P1(R) at the points of P1(Q) (see [Shi71, Sec.1.3]), we obtain a compactification

(4.1.1.2) Γ\H ↪→ Γ\H∗

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

(4.1.1.3) Γ\H∗ − Γ\H ∼= Γ\P1(Q)

called cusps. (Here the finiteness of Γ\P1(Q) can be deduced from the elementaryfact that P1(Q) = SL2(Q) ·∞ = SL2(Z) ·∞.) The points of P1(Q) = SL2(Q) ·∞ arethe so-called rational boundary components of H, which are prototypical examplesof their analogues for more general Hermitian symmetric domains.

Note that the notions of cusps and rational boundary components depend on thestructure of G = SL2 as an algebraic group over Q in an essential way. There existother algebraic groups G′ over Q such that G′(R) ∼= G(R) = SL2(R) and such thatthe fundamental domains for the actions of the corresponding arithmetic subgroupsΓ′ of G′(Q) on D are already compact, in which case there is no need to add anycusps (cf. the example of Shimura curves in Section 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 real quadraticextension field F of Q. It is the group scheme G over Q whose R-points are

(4.1.2.2) G(R) = SL2(R⊗QF ),

for each Q-algebra R. In particular, we have G(Q) = SL2(F ). This is the caseof Hilbert modular surfaces, which are generalized to higher-dimensional Hilbertmodular varieties by replacing F with totally real extension fields of Q of higherdegrees. Then we should take

(4.1.2.3) D∗ := (H×H)∪(G(Q) · (∞×∞)

),

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

In both cases, we can topologize D∗ such that Γ\D∗ is a compactification of Γ\Dfor each arithmetic subgroup Γ of G(Q). When G is simply SL2× SL2, this is justdoubling the case where G = SL2. When G is ResF/Q SL2, this is essentially provedin [Shi63a], in a different language. This generalizes the case where G = SL2 butwill be superseded by the introduction of Satake topology in general.

Here are some general principles, which are valid for not only these two examples,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 whole G′

or a maximal proper parabolic subgroup P′. When the group G is Q-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 subgroupsof G are always stabilizers of increasing sequences of totally isotropicsubspaces.

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

When n = 2, there are three G(Q)-orbits of proper parabolic subgroups of G(Q),with the following representatives, respectively: (We warn the readers that, al-though we will write down only the groups of Q-points, we will still be workingwith algebraic groups over Q. This is a somewhat misleading abuse of language,which we adopt only for simplicity of exposition.)

(1) A Borel subgroup

(4.1.3.1) B(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗∗ ∗

(which is a minimal parabolic subgroup) stabilizing the increasing sequence

(4.1.3.2)

∗000

⊂

∗∗00

36 KAI-WEN LAN

of totally isotropic subspaces of Q4. Note that we have

(4.1.3.3)

∗000

⊥

=

∗∗0∗

and

(4.1.3.4)

∗∗00

⊥

=

∗∗00

with respect to the alternating pairing

(4.1.3.5) 〈 · , · 〉 : Q4×Q4 → Q

defined by J2 =

(12

−12

), and so we can complete the above sequence

(4.1.3.2) into a flag

(4.1.3.6)

0000

⊂

∗000

⊂

∗∗00

⊂

∗∗0∗

⊂

∗∗∗∗

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 indeedanti-diagonal, not just block anti-diagonal, as remarked in Example 2.1.2.

(2) The Klingen parabolic subgroup

(4.1.3.7) P(1)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

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

(4.1.3.8)

∗000

of Q4.(3) The Siegel parabolic subgroup

(4.1.3.9) P(2)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗∗ ∗


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

(4.1.3.10)

∗∗00

of Q4.

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

(4.1.3.11) B(Q) = P(1)(Q)∩P(2)(Q)

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 Hn are given byelements of the G(Q)-orbits of

(4.1.3.12)

(∞r

Hn−r

)∼= Hn−r,

which means

(4.1.3.13)

1rZ

0r1n−r

: Z ∈ Hn−r

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

There is a topology, called the Satake topology (see [Sat60], [BB66, Thm. 4.9],and [BJ06, Sec. III.3] for systematic treatments not just for this case), on the set

(4.1.3.14) H∗n := Hn ∪(∪

1≤r≤nG(Q) · Hn−r

),

where Hn−r stands for

(∞r

Hn−r

)as in (4.1.3.12), such that Hn−r lies in the

closure of Hn−s exactly when r ≥ s, and such that Γ\H∗n is a compactificationof Γ\Hn as a topological space, for every arithmetic subgroup Γ of G(Q) (by thereduction theory, as in [BHC62] and [Bor62]). According to [BB66], Γ\H∗n is anormal projective variety over C, which is called the Satake–Baily–Borel or minimalcompactification of Γ\Hn. (The references given above are not the historical onesmainly for H∗n and Γ\H∗n, such as [Sat56], [Bai58], [Car58], and [SC58]—see [Sat01,Sec. 2] for a nice review. Rather, they are more recent ones that also work for ourlater examples.)

Each

(∞r

Hn−r

), or rather

1rZ

0r1n−r

: Z ∈ Hn−r

in generalized pro-

jective coordinates (see (4.1.3.13)), is stabilized by a maximal parabolic subgroup of

38 KAI-WEN LAN

G(Q) of the form

(4.1.3.15) P(r)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

because

(4.1.3.16)

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

1∗

01

=

∗ ∗∗

01

∼

1∗

01

,

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

(4.1.3.17)

x000

: x ∈ Qr

of

(4.1.3.18) Q2n ∼= Qr ⊕Qn−r ⊕Qr ⊕Qn−r.

We have

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

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

(4.1.3.21) U(r)(Q) =

1 ∗ ∗ ∗1 ∗

1∗ 1

is the unipotent radical . Note that L(r)(Q) acts on the rational boundary componentH2n−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

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

where

(4.1.3.23) W(r)(Q) =

1 Y1

11

: Y ∈ Symr(Q)

∼= Symr(Q)


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

(4.1.3.24) V(r)(Q) ∼= Q2n−2r

(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 imaginary quadraticextension field of Q. Suppose G is a group scheme over Q defined by

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

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

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

(4.1.4.2)

(∞r

Ha−r,b−r

)∼= Ha−r,b−r,

which means

(4.1.4.3)

1r

ZW

0r1b−r

:

(ZW

)∈ Ha−r,b−r

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

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

(4.1.4.4) H∗a,b := Ha,b ∪(∪

1≤r≤bG(Q) · Ha−r,b−r

),

where Ha−r,b−r stands for

(∞r

Ha−r,b−r

)as in (4.1.4.2), Ha−r,b−r lies in the

closure of Ha−s,b−s exactly when r ≥ s, and the quotient Γ\H∗a,b is a compactifi-

cation of Γ\Ha,b as a topological space, for every arithmetic subgroup Γ of G(Q).Again, according to [BB66], 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

1r

ZW

0r1b−r

:

(ZW

)∈ Ha−r,b−r

in gen-

eralized projective coordinates (see (4.1.4.3)), is stabilized by a maximal parabolic

40 KAI-WEN LAN

subgroup of G(Q) of the form

(4.1.4.5) P(r)(Q) =

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗∗ ∗ ∗ ∗

,

where the sizes of the block matrices are given by r, b− r, a− b, r, and b− r, bothhorizontally and vertically. This P(r)(Q) is the stabilizer of the totally isotropicE-subspace

(4.1.4.6)

x0000

: x ∈ Er

of

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

We have

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

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 smaller matrix

(4.1.4.10) Ja−r,b−r =

1b−rS

−1b−r

with the same S; and where

(4.1.4.11) U(r)(Q) =

1 ∗ ∗ ∗ ∗

1 ∗1 ∗

1∗ 1

is the unipotent radical. Note that L(r)(Q) acts on the rational boundary compo-nent Ha−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

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


where

(4.1.4.13) W(r)(Q) =

1 Y

11

11

: Y ∈ Hermr(E)

∼= Hermr(E)


(4.1.4.14) V(r)(Q) ∼= E(a−r)+(b−r)

(by viewing Hermr(E) and E(a−r)+(b−r) as commutative algebraic groups over Q).We note that the overall picture is very similar to the case of Hn in Section 4.1.3,although we need to work with coordinates over E and hence with GLr(E) as analgebraic group over Q (which means we are working with the restriction of scalarsResE/Q GLr), and with the matrix S when a > b.

Let us mention two important variants which frequently appear in the literature.The first variant replaces the imaginary quadratic extension field E of Q in the

above definition of G with a CM extension (namely, a totally imaginary quadraticextension of a totally real extension) F of Q, and replaces Ja,b with a nondegenerateskew-Hermitian matrix J over OF such that −iJ defines a Hermitian pairing ofsignatures (a1, b1), (a2, b2), . . . , (ad, bd) at the d real places of the maximal totallyreal subextension F+ of Q in F , where d := [F+ : Q], for some integers aj ≥ bj ≥ 0and n ≥ 0 such that aj + bj = n, for all j = 1, . . . , d. Then

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

acts on

(4.1.4.16) D ∼= Ha1,b1 ×Ha2,b2 × · · ·×Had,bd ,and the rational boundary components of D are given by elements of theG(Q)-orbits of some

(4.1.4.17) D(r) ∼= Ha1−r,b1−r ×Ha2−r,b2−r × · · ·×Had−r,bd−r,for r = 1, . . . ,min(b1, b2, . . . , bd). Note that there is no r at all if at least one ofb1, b2, . . . , bd is zero.

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

(4.1.4.18) D ∼= Ha,b.If d = 1, then this is just as before. If d ≥ 2, then D∗ = D and the quotient Γ\D isautomatically compact, for every arithmetic subgroup Γ of G(Q). Whether d ≥ 2or not, when (a, b) = (2, 1) (and when (4.1.4.18) holds), the quotient Γ\D is calleda Picard modular surface.

The second variant is a further generalization of the above first variant, byreplacing F with a central simple algebra B over a CM extension over Q thatis equipped with a positive involution ? (namely, an anti-automorphism of order2) such that trB/Q(xx?) > 0 for every x 6= 0 in B, and by defining G using anondegenerate skew-Hermitian pairing over a finitely generated B-module. Then westill have an isomorphism as in (4.1.4.16), and the rational boundary components ofD are still of the same form as in (4.1.4.17), but the values of r allowed in (4.1.4.17)are more restrictive, depending on the possible Q-dimensions of totally isotropic

42 KAI-WEN LAN

B-submodules. If B is a division algebra and the pairing is defined over a simpleB-module, then no r > 0 is allowed, in which case Γ\D is again automaticallycompact, for every arithmetic 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 that G(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

(4.1.5.1) H∗SO∗2n:= HSO∗2n

∪(

∪1≤r≤bn2 c

G(Q) · HSO∗2n−4r

),

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

exactly when r ≥ s, and the quotient

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

, for every arithmetic sub-

group Γ of G(Q).When n = 2k, an example of G is given by

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

for each Q-algebra R, where B is a quaternion algebra over Q such that B⊗QR ∼= H,

and where Jk =

(1k

−1k

)is as before. Then we can describe the rational

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

as

(4.1.5.3)

{(∞r

Z

): Z ∈ Hermk−r(H)⊗

RC, Im(Z) > 0

},

which means

(4.1.5.4)

1rZ

0r1k−r

: Z ∈ Hermk−r(H)⊗RC, Im(Z) > 0

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

as above is stabilized by a maximal parabolic subgroup of G(Q) of the

form

(4.1.5.5) P(r)(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

because

(4.1.5.6)

∗ ∗ ∗ ∗∗ ∗ ∗∗

∗ ∗ ∗

1∗

01

=

∗ ∗∗

01

∼

1∗

01

,

where the sizes of the block matrices are given by r, n − r, r, and n − r, bothhorizontally and vertically. This P(r)(Q) is the stabilizer of the totally isotropic


B-submodule

(4.1.5.7)

x000

: x ∈ Br

of

(4.1.5.8) B2n ∼= Br ⊕Bn−r ⊕Br ⊕Bn−r,with left B-module structures given by having each b ∈ B act by right multiplicationof its conjugate b, so that they do not intervene with left multiplications of matrixentries. We have

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

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)

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

(4.1.5.11) Jk−r =

(1k−r

−1k−r

);

and where

(4.1.5.12) U(r)(Q) =

1 ∗ ∗ ∗1 ∗

1∗ 1

is the unipotent radical. Note that L(r)(Q) acts on the rational boundary compo-nent 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 extension

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

where

(4.1.5.14) W(r)(Q) =

1 Y1

11

: Y ∈ Hermr(B)

∼= Hermr(B)


(4.1.5.15) V(r)(Q) ∼= B2k−2r

(by viewing Hermr(B) and B2k−2r as commutative algebraic groups over Q). Wenote that, again, the overall picture is very similar to the one in Section 4.1.3,although we need to work with coordinates over B and hence with GLr(B) as analgebraic group over Q.

44 KAI-WEN LAN

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

, with a ≥ 2. Suppose G is a group scheme over Qdefined by some Q-valued symmetric bilinear form Q on Qa+2 which has signature(a, 2) over R, so that G(R) ∼= SOa,2(R) and G(R)+ ∼= SOa,2(R)+. (The precisechoice of Q will not be important for our exposition.)

The maximal parabolic subgroups of G(Q) = SO(Qa+2, Q), where SO(Qa+2, Q)denotes the subgroup of SLa+2(Q) consisting of elements preserving the pairing Q(and we shall adopt similar notation in what follows), are in bijection with thenonzero subspaces I of Qa+2 that are totally isotropic with respect to the bilinearform Q. Then we must have 1 ≤ dimQ(I) ≤ 2 because the signature of Q is (a, 2)over R. (But it might happen that no such I exists.) For each such I, we have aparabolic subgroup of G(Q) of the form

(4.1.6.1) PI(Q) ∼= LI(Q) n UI(Q),

where

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

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

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

When dimQ(I) = 1, we have

(4.1.6.4) LI(R) ∼= SOa−1,1(R)×GL1(R),

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

(4.1.6.5) H+I∼= H0.

When dimQ(I) = 2, we have

(4.1.6.6) LI(R) ∼= SOa−2(R)×GL2(R),

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

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

on some one-dimensional rational boundary component

(4.1.6.8) H+I∼= H1.

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

(4.1.6.9)

1a−2

0 11 0

0 11 0

,

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

(4.1.6.10) I =

000∗0

,


which is one-dimensional, then

(4.1.6.11) PI(Q) =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗

∼= LI(Q) n UI(Q),

where(4.1.6.12)

LI(Q) =

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

XX−1

:

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

∈ SOa−1,1(Q), X ∈ Q×

and(4.1.6.13)

UI(Q) =

1 x

1 y1 z

− tx −z −y 1 − 12txx− yz

1

: x ∈ Qa−2, y, z ∈ Q

;

and PI(Q) stabilizes

(4.1.6.14) H+I =

00010

because

(4.1.6.15)

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗

00010

=

000∗0

∼

00010

in projective coordinates. Alternatively, if

(4.1.6.16) I =

0∗0∗0

,

which is two-dimensional, then

(4.1.6.17) PI(Q) =

∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗ ∗∗ ∗ ∗ ∗ ∗

∗ ∗

∼= LI(Q) n UI(Q),

46 KAI-WEN LAN

where(4.1.6.18)

LI(Q) =

X

a ba′ b′

c dc′ d′

:

X ∈ SOa−2(Q),

t

(a′ b′

c′ d′

)−1

=

(a bc d

)∈ GL2(Q)

and(4.1.6.19)

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

;

and PI(Q)∩G(R)+ stabilizes

(4.1.6.20) H+I =

0z010

: z ∈ C, Im z > 0

because

(4.1.6.21)

∗ ∗ ∗∗ a b ∗

∗ ∗∗ c ∗ d ∗

∗ ∗

0z010

=

0

az + b0

cz + d0

∼

0(az + b)(cz + d)−1

010

in projective coordinates.

There is a systematic way to combine these into a union

(4.1.6.22) (H+SOa,2

)∗ = H+SOa,2

∪

∪06=I totally

isotropic

H+I

,

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

I′ (for nonzeroI and I ′) exactly when I ⊂ I ′ (which is qualitatively quite different from the pre-vious cases in Sections 4.1.3, 4.1.4, and 4.1.5, where rational boundary componentsassociated with larger totally isotropic subspaces are smaller) and such that thequotient Γ\(H+

SOa,2)∗ gives the minimal compactification 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 two cases.For HE6

, there exists a group scheme G over Q such that G(R) is as in Section3.6.2 and such that the rational boundary components of HE6

are isomorphic toeither H5,1 or H0. For HE7

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

areisomorphic to either H+

SO10,2, H1, or H0. In each of these two cases, by [PR06,

Sec. 3, Prop. 3], there exists some G such that G(R) is as above and such thatrankQ GQ = rankR GR; or, in other words, GQ is as split as GR allows. In both


cases, we have the minimal compactifications, as usual, for arithmetic subgroups Γof G(Q).

4.2. Compactifications and algebraicity.

4.2.1. Overview. In all cases considered so far, we have the following diagram ofuseful compactifications of Γ\D, with canonical morphisms between them denotedby solid arrows:

(4.2.1.1) (Γ\D)torΣ

&&

not directly related(Γ\D)BS

��

(Γ\D)RBS

uu(Γ\D)

min

Let us explain the objects in this diagram.

4.2.2. (Γ\D)min

= Γ\D∗. This is the Satake–Baily–Borel or minimal compactifica-tion of Γ\D, which is a normal projective variety over C (see [BB66]). Here D∗ isthe union of D with its rational boundary components, equipped with the Sataketopology, as explained in Section 4.1. The transition from the analytic quotientΓ\D∗ to an algebraic variety is achieved by first endowing Γ\D∗ with the structureof a complex analytic space (see [Ser56, §1]) in which the closed complement of theopen subset Γ\D of Γ\D∗ is a closed complex analytic subspace; and then by con-structing a closed immersion of Γ\D∗ into a projective space PN (C) for some N , andby applying Chow’s theorem (or “GAGA” for closed complex analytic subspaces ofPNC , showing that they are all complex analytifications of algebraic subvarieties—

see [Ser56, §3, 19, Prop. 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 the closedcomplement. (This is the only known general method to show that Γ\D is an al-gebraic variety.) Moreover, it is proved in [Bor72] that any holomorphic map froma quasi-projective variety to Γ\D is the complex analytification of a morphism ofalgebraic varieties. In particular, the structure of Γ\D as an algebraic variety isunique up to isomorphism. (These are the key results used in the proof of Theorem2.4.1.)

The compactification (Γ\D)min

is canonical and does not depend on any choices,but is highly singular in general. As we have seen in the examples in Section 4.1,

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

minin general.

When the connected components of D are irreducible in the sense explained in thebeginning of Section 3 but are not one-dimensional, and when Γ\D is not alreadycompact, this codimension is always at least two. The intuition is that we arecollapsing a lot into a small neighborhood of each boundary point.

4.2.3. (Γ\D)torΣ . This is the toroidal compactification of Γ\D (see [AMRT75] or

[AMRT10], and see also [Mum75]), which is an algebraic space depending on thechoice of some combinatorial data Σ called a cone decomposition (or fan). (Moreprecisely, Σ is a compatible collection of cone decompositions, but we prefer not

48 KAI-WEN LAN

to go into the details here.) This means (Γ\D)torΣ is not a canonical compactifi-

cation 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 by replac-ing any given Σ with a refinement . In particular, we obtain meaningful smoothprojective compactifications by paying the price of working with the noncanonicalΣ’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 theoryin [Hir64a, Hir64b].)

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 subgroups P of thealgebraic group G over Q whose pullback to every Q-simple (almost) factor G′ ofG is either the whole G′ or a maximal parabolic subgroup P′. This stratification isfurther refined by a stratification in terms of the cones in the cone decompositionΣ. The incidence relations of the strata in this finer stratification is dual to theincidence relations of the cones in the cone 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 be compat-ible with all Hecke correspondences. For arithmetic applications, it is necessary tosystematically work with refinements of Σ.

4.2.4. (Γ\D)BS. This is the Borel–Serre compactification of Γ\D (see [BS73] and[BJ06, Sec. III.5]; see also [Gor05, Sec. 4]), which is a (real analytic) manifold withcorners. It can be defined more generally when D is a Riemannian symmetricspace that is not Hermitian, but is almost never an algebraic variety. Alreadywhen G = SL2 and D = H (and when Γ is neat), the compactification (Γ\D)BS isachieved by adding circles to D (instead of adding points to D as in the case of D∗).In general, its boundary is stratified by locally closed subsets of G correspondingto the Γ-orbits of all proper rational parabolic subgroups P of the algebraic groupG over Q, and the construction of 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 embedding

Γ\D ↪→ (Γ\D)BS is a homotopy equivalence, which makes (Γ\D)BS useful for study-ing the cohomology of Γ\D, and also that of the arithmetic subgroup Γ of G(Q),which are closely related to the theory of automorphic representations and Eisen-stein series (see, for example, [BW00], [Sch90b], [Fra98], [FS98], and [LS04]). Butlet us remark here, as an addendum to Section 4.2.2 above, that cuspidal automor-phic representations are more closely related to the L2-cohomology of Γ\D, and this

L2-cohomology is canonical isomorphic to the intersection cohomology of (Γ\D)min

.This was conjectured by Zucker (see [Zuc82, (6.20)]) and proved by Looijenga (see[Loo88]), Saper–Stern (see [SS90]), and Looijenga–Rapoport (see [LR91]).

4.2.5. (Γ\D)RBS. This is the reductive Borel–Serre compactification of Γ\D (see[Zuc82, Sec. 4], [GHM94, Part I], and [BJ06, Sec. III.6]; see also [Gor05, Sec. 5]),which is similar to (Γ\D)BS in the following aspects: It can also be defined moregenerally when D is a Riemannian symmetric space that is not Hermitian, andits boundary is also stratified by locally closed subsets of G corresponding to theΓ-orbits of all proper rational parabolic subgroups P of the algebraic group G over


Q. But there is a key difference—the subset associated with P is constructed usingonly the (reductive) Levi quotient L of P. The reductive Borel–Serre compactifi-cation (Γ\D)RBS is useful for studying, for example, the weighted cohomology of Γ(see [GHM94, Parts II–IV]; see also [Gor05, Sec. 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 to applications requiringalgebraic varieties (or at least algebraic spaces or algebraic 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 [Pin89]),and they extend to good models over the integers in many important special cases(see Sections 5.1.4 and 5.2.4 below).

All assertions so far in this Section 4.2 have their counterparts for Shimuravarieties; i.e., the double coset spaces XU defined by Shimura data. (As we explainedin Section 4.2.2, the results there are used in the proof of Theorem 2.4.1.) Moreover,the minimal and toroidal compactifications also have canonical models over thesame reflex field F0 determined by (G,D) as in Theorem 2.4.3 (see [Pin89] again).

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

of the minimal compactification (Γ\D)min

= Γ\D∗ are given by (holomorphic)modular forms, namely, holomorphic functions f : D → C satisfying the followingtwo 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 of f ,satisfying the cocycle condition j(γ′γ, Z) = j(γ′, Z)j(γ, Z) for all γ, γ′ ∈ Γand Z ∈ D. (This means f represents a section of an automorphic linebundle over Γ\D defined by j. The line bundles used in [BB66] are powersof the canonical bundle, namely, the top exterior power of the cotangentbundle, 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 condition thatf(Z)→ 0 as Z → D∗ −D, then f is called a cusp form.

Ratios of such modular forms f define modular functions which are in generalhighly 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 of integers, wecannot just naively use the coordinates of D.

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

(G(C)),so that it makes sense to consider modular forms with weights 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 automorphic vector bundle W over Γ\Ddoes not extend to a vector bundle over (Γ\D)

min. (This is consistent with the

fact that, in general, the boundary (Γ\D)min − (Γ\D) has a high codimension in

(Γ\D)min

.) Nevertheless, the automorphic vector bundle W has canonical exten-sions (see [Mum77] and [Har89]) to vector bundles W can over the toroidal com-

pactifications (Γ\D)torΣ , whose sections are independent of Σ. (When W is the

50 KAI-WEN LAN

restriction of a representation of G(C), the vector bundle W is equipped with anintegrable connection, and its canonical extension W can is equipped with an inte-grable connection with log poles along the boundary and coincides with the onegiven in [Del70].) The global sections or more generally the cohomology classes of

such canonical extensions 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 approach that will allowus to define modular forms in mixed or positive characteristics and study congru-ences among them using integral models. Even in characteristic zero, there havebeen some results which have only been proved with such an algebro-geometricdefinition. (See [Har90], [Mil90], [Lan12b], and [Lan16a] for some surveys on thistopic.)

4.2.8. Mixed Shimura varieties. Just as Shimura varieties parameterize certain vari-ations of Hodge structures with additional structures, there is also a notion of mixedShimura varieties parameterizing certain variations of mixed Hodge structures withadditional structures. Then there is also a theory of canonical models for suchmixed Shimura varieties, and even for their toroidal compactifications—see [Mil90]and [Pin89]. Many mixed Shimura varieties naturally appear along the bound-ary of the toroidal compactifications of pure Shimura varieties (i.e., the usual onesassociated with reductive groups as in Sections 2.3 and 2.4).

The theory of mixed Shimura varieties is even heavier in notation than thetheory of (pure) Shimura varieties, and is beyond the scope of this introductoryarticle. But let us at least mention the following prototypical example. Consider theShimura varieties XU associated to (G,D) = (GL2,H±), where H± := H±1 , whichare modular curves. For each integer m ≥ 1, consider the (non-reductive) semidirect

product G = GL2 n (G2a)m, with G = GL2 acting diagonally on (G2

a)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 we obtain by considering double coset spaces as in Section

2.2 the mixed Shimura varieties XU over XU for neat open compact subgroups Uwith image in U under the canonical homomorphism G(A∞)→ G(A∞), which aretorsors under abelian schemes over XU that are isogenous to the m-fold self-fiber

product of the universal elliptic curve over XU . Specifically, there exists some Usuch that XU is exactly isomorphic to the m-fold self-fiber product of the universalelliptic curve over XU . The projective smooth toroidal compactifications of suchmixed Shimura varieties then provide generalizations of the classical Kuga–Satovarieties, whose cohomology with trivial coefficients are useful for studying thecohomology with nontrivial coefficients over the usual modular curves XU , or evenfor the construction of related motives—see, for example, [Del71a, Sec. 5] and[Sch90a]. (We can also reduce questions about the cohomology of Shimura varietieswith nontrivial coefficients to questions about the cohomology of mixed Shimuravarieties with trivial coefficients in many more general cases—see the end of Section5.1.4 below.)


5. Integral models

5.1. PEL-type cases. All currently known constructions of integral models ofShimura varieties (which we can describe in some reasonable detail) rely on theimportant coincidence that, as explained in Section 3.1.5, when G = Sp2n and Γis a congruence subgroup of G(Z), the Siegel modular variety Γ\Hn parameterizespolarized abelian varieties with level structures; and on the fact that we can definemoduli problems over the integers that naturally extend such complex varieties.More generally, we can define the so-called PEL moduli problems, which param-eterize abelian schemes with additional PEL structures. As we shall see below,the abbreviation PEL stands for polarizations, endomorphism structures, and levelstructures. This is the main topic of this subsection.

5.1.1. PEL datum. The definition of PEL moduli problems require some linearalgebraic data. Suppose we are given an integral PEL datum

(5.1.1.1) (O, ?, L, 〈 · , · 〉, h0),

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 order 2

such that trO⊗ZR/R(xx?) > 0 for every x 6= 0 in O⊗

ZR.

(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 condition that

(5.1.1.2) 〈bx, y〉 = 〈x, b?y〉

for all x, y ∈ L and b ∈ O. Here

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

is the formal Tate twist of Z.(5) h0 is an R-algebra homomorphism

(5.1.1.4) h0 : C→ EndO⊗ZR(L⊗

ZR),

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⊗

ZR is

positive definite.

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

(5.1.1.5) A0 := (L⊗ZR)/L

with a complex structure given by h0 : C→ EndO⊗ZR(L⊗

ZR), which can be shown

to be an abelian variety (not just a complex torus) with a polarization

(5.1.1.6) λ0 : A0 → A∨0∼= (L⊗

ZR)/L#,

52 KAI-WEN LAN

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

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

the dual lattice of L with respect to 〈 · , · 〉, and where the λ0 is induced by thenatural inclusion L ⊂ L# (see [Mum70, Ch. I, Sec. 2, and Ch. II, Sec. 9] and[Lan12a, Sec. 2.2]; cf. [Lan13, Sec. 1.3.2]). The assumptions that L is an O-latticeand that the compatibility (5.1.1.2) holds imply that we have an endomorphismstructure

(5.1.1.8) i0 : O → EndC(A0)

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

(5.1.1.9) A0λ0 //

i(b?)

��

A∨0

(i(b))∨

��

A0λ0

// A∨0

is commutative (cf. [Lan13, Sec. 1.3.3]). Moreover, for each integer n ≥ 1, we havea principal level-n structure

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

∼→ µn),

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

nL)/L because of (5.1.1.5), where α0,n is induced by this canonical isomorphism,and where ν0,n is induced by the exponential function exp : C → C× (with kernelZ(1) by definition), so that the diagram

(5.1.1.11) (L/nL)×(L/nL)

α0,n×α0,n o��

〈 · , · 〉// (Z/nZ)(1)

ν0,no��

A0[n]×A0[n]λ0-Weil pairing

// µn

is commutative (see [Lan12a, Sec. 2.2 and 2.3]; cf. [Lan13, Sec. 1.3.6]), and so thatsuch a level structure is liftable to all higher levels n′ divisible by n. (The secondentry ν0,n in (α0,n, ν0,n) is often omitted in the notation, although ν0,n is notdetermined by α0,n when 〈 · , · 〉 is not a perfect pairing modulo n.) Furthermore,the homomorphism h0 : C→ EndO⊗

ZR(L⊗

ZR) defines a Hodge decomposition

(5.1.1.12) H1(A0,C) ∼= L⊗ZC = V0⊕V 0,

where

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

and

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

Then we also have

(5.1.1.15) LieA0∼= V0


as O⊗ZC-modules. The reflex field F0 for h0 (which is defined in this context

without referring to any Shimura datum) is the field of definition of the isomorphismclass of V0 as an O⊗

ZC-module, or more precisely the subfield

(5.1.1.16) F0 := Q(tr(b|V0) : b ∈ O)

of C (see [Lan13, Sec. 1.2.5]). This is the smallest subfield of C over which wecan formulate a trace or determinantal condition on the Lie algebra of an abelianvariety, to ensure that the pullback to C of its Lie algebra is isomorphic to V0 asan O⊗

ZC-module (see [Lan13, Sec. 1.2.5 and 1.3.4]).

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

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

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

5.1.2. Smooth PEL moduli problems. Given any integral PEL datum as in (5.1.1.1),we can define the associated PEL moduli problems (at varying levels) over (thecategory of schemes over) F0, or over suitable localizations of the ring of integersOF0

. Roughly speaking, they are the smallest moduli problems (over the respectivebase rings) parameterizing tuples of abelian schemes with additional PEL structuresas above, including the one we wrote down in (5.1.1.17). See [Lan13, Sec. 1.4.1] formore details on defining smooth PEL moduli problems over localizations of OF0 ofgood residue characteristics. (A prime number p is good for (O, ?, L, 〈 · , · 〉, h0) ifit satisfies all of the following conditions: p is unramified in O, as in [Lan13, Def.1.1.1.18]; p - [L# : L]; p 6= 2 whenever (O⊗

ZR, ?) has a simple factor isomorphic

to a matrix algebra over the Hamiltonian numbers H with its canonical positiveinvolution; and there is no nontrivial level structure at p. See [Lan13, Def. 1.4.1.1].)Note that people often talk about good reductions instead of smoothness whendescribing integral models, even though it is not always clear what reductions shouldmean without some properness assumption.

The moduli problems above are defined using the language of isomorphismclasses. If we use the language of isogeny classes instead, then it is more natu-ral to work with rational PEL datum

(5.1.2.1) (O⊗ZQ, ?, L⊗

ZQ, 〈 · , · 〉, h0),

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

(5.1.2.2) (O⊗ZZ(p), ?, L⊗

ZZ(p), 〈 · , · 〉, h0),

which is good for defining PEL moduli problems over OF0,(p) only up to isogeniesof degrees prime to p. The latter was done in [Kot92, Sec. 5] and generalized in[Lan13, Sec. 1.4.2], and then compared in [Lan13, Sec. 1.4.3] with the definition byisomorphism classes in [Lan13, Sec. 1.4.3]. (While the comparison was not difficult,the author still found it helpful to have the details recorded in the literature.)

54 KAI-WEN LAN

5.1.3. PEL-type Shimura varieties. The PEL moduli problems as above are re-lated to the double coset spaces in Section 2 as follows. Firstly, the subtuple(O, ?, L, 〈 · , · 〉) of (5.1.1.1) defines a group scheme G over Z by(5.1.3.1)

G(R) := {(g, r) ∈ EndO(L⊗ZR)×R× : 〈gx, gy〉 = r〈x, y〉, for all x, y ∈ L⊗

ZR},

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 hyperspecial opencompact subgroup of G(Qp) when GQp is connected as an algebraic group over Qp.Secondly, the remaining entry h0 : C→ EndO⊗

ZR(L⊗

ZR) defines a homomorphism

(5.1.3.2) h0 : C× → G(R),

whose G(R)-conjugacy class defines a manifold

(5.1.3.3) D := G(R) · h0.

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, how-ever, not always the case), we say it is a PEL-type Shimura datum, and say XU isa PEL-type Shimura variety . In this case, the reflex field F0 for h0 coincides withthe reflex field of (G,D) (see Theorem 2.4.3).

Example 5.1.3.4. Suppose O = Z; ? is trivial; L = Z2n; 〈 · , · 〉 is defined by the

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 quadratic extensionfield E of Q, with a fixed isomorphism E⊗

QR ∼= C; ? is the complex conjuga-

tion of E over Q; L = Oa+b for some integers a ≥ b ≥ 0; 〈 · , · 〉 is defined by2πi 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 θe−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. Suppose O is an order in a quaternion algebra B over Q such thatB⊗

QR ∼= H and O is stabilized by the canonical positive involution ?; L = O2n;

〈 · , · 〉 is 2πi times the composition of the skew-Hermitian pairing defined 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, and D = G(R) · h0

contains HSO∗2nas 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 the associatedPEL moduli problems. By the classification of positive involutions (see the review


in [Lan13, Prop. 1.2.1.13 and 1.2.1.14] based on Albert’s classification, or just theclassification over R in [Kot92, Sec. 2]), up to cocenter (i.e., up to replacing thegroup with a subgroup containing the derived group), G(R) factorizes as a productof groups of the form Sp2n(R), Ua,b, or SO∗2n (with varying values of n ≥ 0 anda ≥ b ≥ 0). Thus, the connected components of D are products of the Hermitiansymmetric domains we have seen in Sections 3.1, 3.2, and 3.3. However, we never seethe Hermitian symmetric domains in Sections 3.4, 3.6, and 3.5 as factors, exceptwhen it is because 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 [Hel01, Ch. X, Sec. 6]).For a PEL moduli problem defined by an integral PEL datum with O⊗

ZQ a

simple algebra over Q, the complex Lie group G(C) factorizes up to cocenter (i.e.,again, up to replacing the group with a subgroup containing the derived group) asa product of symplectic, special linear group, or special orthogonal groups, but onlyone type of them can occur. Accordingly, we say the PEL moduli problem is of typeC, type A, or type D. People often also say that a PEL moduli problem of type A isa unitary Shimura variety, even when G violates Condition (3) in Section 2.3 and so(G,D) is not a Shimura datum. However, by orthogonal Shimura varieties, peoplemight not mean PEL moduli problems of type D. On one hand, this is becausethere are useful abelian-type Shimura varieties associated with special orthogonalgroups, as we shall see in Example 5.2.2.11 below. On the other hand, since thegroups G associated with PEL moduli problems of type D are not even connected,the corresponding pairs (G,D) violate the conditions for being Shimura data ina more fundamental way, and hence there is some ambiguity in what PEL-typeShimura varieties of type D should mean.

In all cases, whether (G,D) is a Shimura datum or not, when the corresponding(smooth) PEL moduli problem MU is defined at level U , there is an open and closedimmersion

(5.1.3.7) XU ↪→ MU (C)

(see [Lan12a, Sec. 2.4 and 2.5]), which is generally not an isomorphism, due to theso-called failure of Hasse’s principle (see [Lan13, Rem. 1.4.3.12] and [Kot92, Sec.7–8]). More concretely, this is because the definition of MU uses only the adelicgroup G(A)—the level structures use G(A∞), while the Lie algebra condition usesG(R). But there can be more than one mutually nonisomorphic algebraic groupsG(1), . . . ,G(m) (including G) over Q such that 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 thedense subset of the so-called CM points, 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 [Kot92,Sec. 8]). However, in cases of type D, such a failure can be quite serious. It can even

happen that some of the X(j)U ’s are noncompact, while others are not (see [Lan15,

Ex. A.7.2]).With the tacit running assumption that U is neat (see [Pin89, 0.6] again, or see

[Lan13, Def. 1.4.1.8]), MU is a (smooth) quasi-projective scheme because it is finite

56 KAI-WEN LAN

over the base change of some Siegel moduli scheme (see Section 3.1.5), and hencean open and closed subscheme of MU serves as an integral model of XU . (When U isnot neat, we still obtain a Deligne–Mumford stack whose associated coarse modulispace 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 with PELstructures has a long history—see, for example, Shimura’s articles [Shi63b] and[Shi66]. The theory over the complex numbers is already quite complicated andheavy in notation, which seems inevitable for anything involving semisimplealgebras and positive involutions, but there is an additional subtlety whenformulating the PEL structures over the integers. That is, we can no longer talkabout the singular homology or cohomology of an abelian variety over a generalbase field, let alone their relative versions for an abelian scheme over a generalbase scheme. Instead, we have resorted to the (relative) de Rham and etalehomology or cohomology. This is the reason that PEL moduli problems sufferfrom the failure of Hasse’s principle, and that these moduli problems can only beeasily defined in mixed characteristics (0, p) when there is no ramification or levelat p. (In Section 5.1.5 below, we will summarize the known constructions whenthere are indeed some ramification and levels at p.)

5.1.4. Toroidal and minimal compactifications of PEL moduli. The above PELmoduli MU carry universal or tautological objects, which are abelian schemes withadditional PEL structures, essentially by definition.

In the important special case of Siegel moduli problems considered in Exam-ple 5.1.3.4, the tautological objects are just principally polarized abelian schemeswith level structures (but with no nontrivial endomorphism structures or polar-ization degrees specified in the moduli problem). By vastly generalizing resultsdue to Mumford (see [Mum72]), Faltings and Chai constructed toroidal compact-ifications of the Siegel moduli problems by studying the degeneration of polarizedabelian schemes into semi-abelian schemes (see [FC90, Ch. III and IV]). Over suchtoroidal compactifications, they can algebro-geometrically define modular forms,and construct minimal compactifications of Siegel moduli problems as the projec-tive spectra of some graded algebras of modular forms (with inputs of positivityresults from [MB85]; see [FC90, Ch. V]). The main difficulty in Faltings and Chai’sconstruction was that they needed to glue together boundary strata parameterizingdegenerations of different ranks. This was much more complicated than the earlierboundary construction in [Rap78], based on the original results of [Mum72].

The generalization of the results of [FC90, Ch. III–V] to all smooth PEL moduliproblems was carried out in [Lan13]. Later in [Lan12a], it was shown that thetoroidal and minimal compactifications in [Lan13] (and also in [FC90, Ch. III–V],both based on the theory of degeneration) are indeed compatible with the toroidaland minimal compactifications in [Har89] and [Pin89] (based on the constructionsin [BB66] and [AMRT75] using analytic coordinate charts). In other words, toroidaland minimal compactifications of PEL moduli problems provide integral models oftoroidal and minimal compactifications of PEL-type Shimura varieties.

Over the integral models of PEL-type Shimura varieties and their toroidalcompactifications, by considering the relative cohomology of certain Kuga families(which are abelian scheme torsors which are integral models of certain mixedShimura varieties as in Section 4.2.8) and their toroidal compactifications, we


can also define automorphic vector bundles and their canonical and subcanonicalextensions over these integral models (see [FC90, Ch. VI], [Lan12c, especially Sec.6], [LS12, Sec. 1–3], and [LS13, Sec. 4–5]), whose global sections and cohomologyclasses then provide an algebro-geometric definition of modular forms in mixedcharacteristics, generalizing the theory over C in Section 4.2.7. See [Lan12b] and[Lan16a] again for some surveys on this topic. See also [Hid02] and [Hid04], forexample, for some application to the theory of p-adic modular forms, with theassumptions on the integral models of toroidal and minimal compactificationsjustified by the constructions in [Lan13].

5.1.5. Integral models not defined by smooth moduli. Let p be any prime number,which is not necessarily good for the integral PEL datum. (The conditions forbeing good for the integral PEL datum are the same as the conditions for beinggood for defining the PEL moduli, except that we ignore the condition of havingno 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, whereUp ⊂ G(A∞,p) and Up ⊂ G(Qp) such that Up is a parahoric open compact sub-group of G(Qp)—that is, when Up is the identity component of the stabilizer of somemultichain of Zp-lattices in L⊗

ZQp—there is also a natural way to define a PEL

moduli problem parameterizing not just individual abelian schemes with additionalPEL structures, but rather multichains of isogenies of them. (See [RZ96, Ch. 3 and6].) This generalizes the idea that, in the case of modular curves, the moduli prob-lem at level Γ0(p) parameterizes degree-p cyclic isogenies of elliptic curves. (See,for example, [KM85, Sec. 3.4 and 6.6], which gave a moduli interpretation of thecorresponding construction in [DR73] that was by merely taking normalizations.)

PEL moduli problems at parahoric levels at p are generally far from smooth inmixed characteristics (0, p), in which case we say that the integral models have badreductions. Nevertheless, people have developed a theory of local models which de-scribes the singularities of such moduli problems—see, for example, [RZ96], [PR03],[PR05], [PR09], and [PZ13]. (See also [Rap05], [Hai05], and [PRS13] for some sur-veys.) Beginners should be warned that the forgetful morphisms from a higherparahoric level to a lower one, or more precisely from one such moduli problemdefined by some multichain of Zp-lattices to another one defined by a simpler mul-tichain, is usually just projective but not finite. This is very different from what wehave seen in characteristic zero (or in the classical modular curve case in [DR73]and [KM85]).

For higher levels at p, the older ideas in [DR73] by taking normalizations, and in[KM85] by generalizing the so-called Drinfeld level structures, are both useful. Butthey are useful in rather different contexts—exactly which strategy is more usefuldepends on the intended applications.

The idea of generalizing Drinfeld level structures played important roles in[HT01], [Man04], [Man05], and [Man11], and in subsequent works based on themsuch as [Shi11], where people needed to compute the cohomology of certain nearbycycles and count points, and hence would prefer to work with moduli problemseven when they are not flat . (It was known much earlier—see [CN90, AppendixA]—that moduli problems defined by Drinfeld level structures are generallynot even flat over the base rings.) But such moduli problems tend to be very

58 KAI-WEN LAN

complicated, and might be too cumbersome when it comes to the constructionof compactifications, because most known techniques of gluing require at leastsome noetherian normal base schemes. (The results in [HT01], [Man04], [Man05],[Man11], and [Shi11] were all for compact PEL-type Shimura varieties.)

On the contrary, the idea of taking normalizations produces, essentially by defini-tion, noetherian normal schemes flat over the base rings. It had been overlooked fora long time because of its lack of moduli interpretations, and because no theory oflocal models was available for them. While these two problems still remain, recentdevelopments beyond the PEL-type cases (see the discussion on the Hodge-typecases below in Section 5.2.3), and also developments in the construction of p-adicmodular forms (as in [HLTT16], which was based on [Lan18a] and did not requirethe integral models to be constructed as moduli problems) motivated people to sys-tematically reconsider the constructions by taking normalizations. In the PEL-typecases, we can construct good compactifications for all such integral models (andtheir variants)—see [Lan16b], [Lan17], and [Lan18b]. Such compactifications aregood in the sense that the complications in describing its local structures and itsboundary stratifications are completely independent of each other. This is useful in,for example, [LS18b, Sec. 6.3], which generalized the results of [Man05] and [Man11]to all noncompact cases, without having to compactify any moduli problems definedby Drinfeld level structures.

In [LS18a], it is shown that many familiar subsets of the reductions of the integralmodels of Shimura varieties in characteristic p > 0, such as p-rank strata, Newtonstrata, Oort central leaves, Ekedahl–Oort strata, and Kottwitz–Rapoport strata, andtheir pullbacks to the integral models defined by taking normalizations at arbitrarilyhigher levels at p, admit partial toroidal and minimal compactifications with manyproperties similar to the whole toroidal and minimal compactifications.

5.1.6. Langlands–Rapoport conjecture. In [Lan76] and [Lan77], where the terminol-ogy Shimura varieties first appeared in the literature, Langlands proposed to studyHasse–Weil zeta functions of Shimura varieties and relate them to the automor-phic L-functions, generalizing earlier works for GL2 and its inner forms by Eichler,Shimura, Kuga, Sato, and Ihara. This is very ambitious—see the table of contentsof [LR92] for the types of difficulties that already showed up in the case of Picardmodular surfaces. This motivated the Langlands–Rapoport conjecture, which givesa detailed description of the points of the reduction modulo p of a general Shimuravariety at a hyperspecial level at p—see [LR87], [Pfa96], [Rei97], and [Rap05]. Seealso [Kot90] for a variant of the conjecture due to Kottwitz, which was proved byhimself for all PEL-type Shimura varieties of types A and C in [Kot92].

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

(5.2.1.2) (G,D) ↪→ (GSp2n,H±n );

namely, an injective homomorphism G ↪→ GSp2n of algebraic groups over Q induc-ing an (automatically closed) embedding D ↪→ H±n , for some n ≥ 0 (cf. Remarks2.4.11 and 2.4.13). A Hodge-type Shimura variety is a Shimura variety asso-ciated with a Hodge-type Shimura datum.


If D = G(R) · h0 is the conjugacy class of some homomorphism h0 : C× → G(R)as before, then the above means the composition of h0 with G(R) ↪→ GSp2n(R) liesin the conjugacy class of an analogous homomorphism for GSp2n(R) as in (3.1.4.7).For each neat open compact subgroup U of G(A∞), which we may assume to be thepullback of a neat open compact subgroup U ′ of GSp2n(A∞), the Shimura varietyXU associated with (G,D) at level U is a special subvariety of the Siegel modularvariety at level U ′. Then the points of XU are equipped with polarized abelian vari-eties with level structures, which are pulled back from the Siegel modular varieties(and hence depend on the choice of the Siegel embedding above).

Example 5.2.1.3. All the PEL-type Shimura data are, essentially by definition, ofHodge type, because of the definition of G in (5.1.3.1), and because of the conditionon h0 in Section 5.1.1. But there are many Hodge-type Shimura data which arenot of PEL type. Important examples of this kind are given by general spin groups(with associated Hermitian symmetric domains as in Section 3.4.3), which admitSiegel embeddings thanks to the so-called Kuga–Satake construction (see [Sat66],[Sat68], and [KS67]; see also the reviews in [Del72, Sec. 3–4] and [MP16, Sec. 1 and3]).

The injectivity of the homomorphism G ↪→ GSp2n means we have a faithful2n-dimensional symplectic representation of G. By [DMOS82, Ch. I, Prop. 3.1], thefaithfulness 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 a Hodge structure on R2n, by [DMOS82,Ch. I, Prop. 3.4], these tensors (si) are Hodge tensors, namely tensors of weight 0and Hodge type (0, 0) with respect to the induced Hodge structure on the tensoralgebra of R2n.

Then the Shimura varieties XU associated with (G,D) can be interpreted as pa-rameter spaces of polarized abelian varieties (over C) equipped with some Hodgecycles and level structures. According to Deligne (see [DMOS82, Ch. I, MainThm. 2.11]), Hodge cycles on complex abelian varieties are absolute Hodge cycles.Roughly speaking, this means that the Hodge cycles can be conjugated, as if theywere known to be defined by algebraic cycles (as if the Hodge conjecture were truefor them), when the underlying abelian varieties are conjugated by an automor-phism of C. This defines a canonical action of Aut(C/F0) on the points of XU byconjugating the Hodge cycles of the corresponding abelian varieties, which coincideswith the action given by the structure of XU as the C-points of its canonical modelover F0. Because of this, people can formulate some moduli interpretation for thepoints of the canonical model of XU over F0.

Such a pointwise moduli interpretation has not really helped people write downany useful moduli problem, but they are still important for studying the less directconstruction by taking normalizations, to be introduced in Section 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 Shimuradatum such that there exist a Hodge-type Shimura datum (G1,D1) and a centralisogeny

(5.2.2.2) Gder1 → Gder

60 KAI-WEN LAN

between the derived groups which induces an isomorphism

(5.2.2.3) (Gad1 ,Dad

1 )∼→ (Gad,Dad)

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) isof 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 often provedby 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 Shimura curves,which are associated with Shimura data (G,D) with G(Q) = B×, where B is aquaternion algebra over a totally real extension F of Q of degree d > 1 such that

(5.2.2.5) B⊗QR ∼= M2(R)×Hd−1,

in which case

(5.2.2.6) D ∼= H±

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

(5.2.2.7) B⊗QR ∼= M2(R)a×Hb

for some a, b ≥ 1, in which case

(5.2.2.8) D ∼= (H±)a

(cf. [Shi71, Sec. 9.2]). Even more generally, as explained in [Del71b, Sec. 6], with thesame B and F (and a, b ≥ 1), for each integer n ≥ 1, there are abelian-type Shimuradata (G,D) such that G(Q) is the group of similitudes over F of a symmetric bilinearpairing over Bn, and such that

(5.2.2.9) G(R) ∼= GSp2n(R)a×Hb,

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

(5.2.2.10) D ∼= (H±n )a.

In all these cases, the related Hodge-type Shimura data (as in Definition 5.2.2.1)and their Siegel embeddings (as in Definition 5.2.1.1) can be constructed with someauxiliary choice of imaginary quadratic field extensions of Q (see [Del71b, Sec. 6],and see also [Car86, Sec. 2] and [KS16, Sec. 12]). However, none of these areof Hodge type (let alone of PEL type) because the Hodge structures defined byrational representations of G are not rational Hodge structures (see [Mil05, Sec. 9,part (c) of the last example]).

Example 5.2.2.11. There are also the abelian-type Shimura varieties associated withShimura data (G,D) such that the derived group of G as an algebraic group over Qis the special orthogonal group defined by some symmetric bilinear form over Q ofsignature (a, 2) over R, for some a ≥ 1, so that G(R)+ is up to cocenter the groupSOa,2(R)+ studied in Section 3.4.3. (See also the last paragraph of Section 3.4.4.)


In this case, the related Hodge-type Shimura data (as in Definition 5.2.2.1) aregeneral spin groups, with their Siegel embeddings (as in Definition 5.2.1.1) givenby the Kuga–Satake construction, as in Example 5.2.1.3, mapping G into GSp2n

with n = 2a (see the same references in Example 5.2.1.3). Such Shimura varietiesassociated with special orthogonal groups are important because they are relatedto the moduli of interesting algebraic varieties. For example, when a = 19 in theabove, the corresponding Shimura varieties contain as open subspaces the modulispaces of polarized K3 surfaces with level structures (see, for example, [Riz06, Sec.6], [MP15, Sec. 2 and 4], and [She17, Sec. 4.2]).

In [Del79, 2.3], based on earlier works due to Satake (see [Sat65]; and see also[Sat67] and [Sat80, Ch. IV]), Deligne analyzed all abelian-type Shimura data (G,D)that are adjoint and simple in the sense that G is adjoint and simple as an algebraicgroup over Q. (See also [Mil13, Sec. 10].) When G is simple over Q, there is upto isomorphism a unique irreducible Dynkin diagram associated with all the simplefactors of GC, or rather Lie G(C). (In fact, in this case, Gad ∼= Resk/Q H for somenumber field k and some connected semisimple algebraic group algebraic group Hover k such that HQ remains simple over the algebraic closure Q of Q in C; see,

for example, [BT65, 6.21(ii)].) We say that the group is of type An, Bn, Cn, Dn,E6, or E7 accordingly to the type of such a Dynkin diagram (cf. the summary inSection 3.7). In cases of type Dn, for n ≥ 5, we further single out the special case oftype 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 ofabelian type exactly when it is of types A, B, C, DR, or DH. In cases of types A,B, C, and DR, the related Hodge-type Shimura datum (G1,D1) (as in Definition5.2.1.1) can be chosen such that the derived group Gder

1 is simply-connected as analgebraic group over Q. However, in the remaining cases of DH

n , where n ≥ 4, therelated Hodge-type Shimura datum can only be chosen such that the noncompactsimple factors of Gder

1 (R) are all isomorphic to SO∗2n. (Although this could havebeen a more precise statement in terms of algebraic groups, we are content withthe simpler statement here in terms of real Lie groups.)

More generally, suppose that (G,D) is a Shimura datum that is simple in thesense that the adjoint quotient Gad is simple. (But we no longer assume that Gitself is adjoint.) Then we can still classify the type of (G,D) by classifying thetype of (Gad,Dad) as above, and we have the following possibilities:

(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 quotient of

Gder1 for the Hodge-type datum (G1,D1) mentioned in the last paragraph.

In particular, Gder1 (R) cannot have factors isomorphic to the spin group

cover Spin∗2n of SO∗2n.(3) Cases of mixed type D, namely those that are neither of type DR nor of

type DH, are not of abelian type. Such cases exist by a general result dueto Borel and Harder (see [BH78, Thm. B]).

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

Thus, roughly speaking, all Shimura varieties associated with symplectic and uni-tary groups are of abelian type, and none of the Shimura varieties associated with

62 KAI-WEN LAN

exceptional groups can be of abelian type, while there are some complications forShimura varieties associated with spin groups or special orthogonal groups.

5.2.3. Integral models of these Shimura varieties. The strategy for constructingintegral models of Hodge-type Shimura varieties is as follows. Suppose (G,D) is aHodge-type Shimura datum with Siegel embedding

(5.2.3.1) (G,D) ↪→ (GSp2n,H±n ),

and suppose U is an open compact subgroup of G(A∞) that is the pullback of someopen compact subgroup U ′ of GSp2n(A∞) such that U ′ = U ′,pU ′p, where U ′,p issome neat open compact subgroup of GSp2n(A∞,p), and where U ′p = GSp2n(Zp).(In the literature, people have considered the more general setting where U ′p is thestabilizer in GSp2n(Qp) of some Zp-lattice, which can nevertheless be reduced toour setting here by modifying the Siegel embedding (5.2.3.1), at the expense ofreplacing n with 8n, using “Zarhin’s trick” as in [Lan16b, Lem. 4.9] or [Lan18a,Lem. 2.1.1.9].) Let v|p be a place of F0, and let OF0,(v) demote the localization ofF0 at v. Then we have a canonical open immersion

(5.2.3.2) XU,F0 ↪→ XU ′,OF0,(v),

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

(5.2.3.3) XU,OF0,(v)

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 sense when U is

just a subgroup of the pullback of U ′. This is what we did in the PEL-type casesin [Lan18a] and [Lan16b], as reviewed in Section 5.1.5, and similar ideas can betraced back to [DR73].

But when U is of the form UpUp such that Up is a neat open compact subgroupof G(A∞,p) and such that Up is a hyperspecial maximal open compact subgroup ofG(Qp), Vasiu, Kisin, and others (see [Mil92, Rem. 2.15], [Vas99], [Moo98], [Kis10],and [KMP16]) proved that XU,OF0,(v)

is smooth and satisfies certain extension prop-

erties making it an integral canonical model . (The formulations of the extensionproperties vary from author to author. See, in particular, [Mil92, Sec. 2], [Moo98,Sec. 3], and [Kis10, (2.3.7)]. An integral canonical model then just means an inte-gral model satisfying some formulation of the extension property. This notion is notexpected to be useful when the levels at p are not hyperspecial.) They also showedthat, roughly speaking, by working with connected components, by taking suitablequotients of integral models of Hodge-type ones by finite groups, and by descent,we also obtain integral models for all abelian-type Shimura varieties at hyperspeciallevels at p, which are smooth and satisfy the same extension properties.

When p > 2, and when G splits over a tamely ramified extension of Qp, Kisinand Pappas proved in [KP18] that all abelian-type Shimura varieties at parahoriclevels at p have integral models which have local models as predicted by the group-theoretic construction in [PZ13], and which satisfy a weaker extension property.


In [Kis17], Kisin proved a variant of the Langlands–Rapoport conjecture (partlyfollowing Kottwitz’s variant; cf. Section 5.1.6) for all abelian-type Shimura varietiesat hyperspecial levels at p, when p > 2.

5.2.4. Integral models of toroidal and minimal compactifications. In [MP18], it isproved that, by taking normalizations over the toroidal and minimal compactifica-tions of Siegel moduli schemes constructed in [FC90] and [Lan13], we also have goodintegral models of toroidal and minimal compactifications of the integral models ofHodge-type Shimura varieties constructed in the previous subsection.

The cone decompositions for the toroidal compactifications thus obtained arepullbacks of certain smooth ones for the Siegel moduli, which are rather inexplicit.But it is believed that, by considering normalizations of blowups of the minimalcompactifications as in [Lan17], we can obtain toroidal compactifications associatedwith more general projective and smooth cone decompositions as well.

As in the PEL-type cases in Section 5.1.5, such compactifications are good inthe sense that the complications in describing its local structures and its bound-ary stratifications are completely independent of each other. Using the toroidalcompactifications thus constructed, we showed in [LS18b] and [LS18c] that, foreach integral model of Hodge-type or abelian-type Shimura varieties constructedin [Kis10], [KMP16], and [KP18], and for any prime ` different from the residuecharacteristics, the `-adic etale nearby cycle cohomology over the special fiber iscanonically isomorphic to the `-adic etale cohomology over the (geometric) genericfiber, without resorting to the proper base change theorem (cf. [AGV73, XII, 5.1]and [DK73a, XIII, (2.1.7.1) and (2.1.7.3)]). (In [LS18c], the proof in the case ofabelian-type Shimura varieties is achieved by reduction to the case of Hodge-typeShimura varieties.) Intuitively speaking, the special fibers of these integral mod-els have as many points as there should be, at least for studying the `-adic etalecohomology of Shimura varieties.

5.3. Beyond abelian-type cases. Although abelian-type Shimura varieties arealready very useful and general, as we have seen in Section 5.2.2, they do not coverall possibilities of Shimura varieties. Among those associated with simple Shimuradata, we are not just missing those of types E6 and E7—we are also missing themajority (depending on one’s viewpoint) of type D cases. But these missing casescan still be quite interesting, both for geometric and arithmetic reasons.

For example, in [MS10], Milne and Suh showed that there are many examples ofgeometrically connected quasi-projective varieties X over number fields F such thatthe complex analytifications of X ⊗

F,σC and X ⊗

F,τC have nonisomorphic fundamental

groups for different embeddings σ and τ of F into C, by considering connectedShimura varieties of mixed type D (which exist by [BH78, Thm. B]).

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

Theorem 5.3.1 (see [LZ17, Thm. 1.2]). Suppose XU is a Shimura variety asso-ciated with a general Shimura datum (G,D) (which can certainly be simple oftypes E6, E7, or mixed D) at some level U . Suppose V is an etale Qp-local systemassociated with some finite-dimensional representation of G, whose restriction tothe largest anisotropic subtorus in the center of G that is R-split is trivial. Sup-pose x : Spec(F ) → XU is a closed point of XU , together with a geometric pointx : Spec(F ) → XU lifting x. Then the stalk Vx, when regarded as a representation

64 KAI-WEN LAN

of Gal(F/F ), is geometric in the sense of Fontaine–Mazur (see [FM97, Part I,Sec. 1]). That is, it is unramified at all but finitely many places of F , and is deRham at the places of F above p.

Recall that Fontaine and Mazur conjectured that every irreducible geometricGalois representation comes from geometry in the sense that it appears as a sub-quotient of the etale cohomology of an algebraic variety (see [FM97, Part I, Sec.1] again). Theorem 5.3.1 would have been less surprising if it only consideredabelian-type Shimura varieties with certain assumptions such that the stalks as inthe theorem are just the p-adic realizations of some abelian motives. But the theo-rem does consider non-abelian-type Shimura varieties as well, and we do not knowany families of motives parameterized by these general Shimura varieties. We hopethat there are such families, but this is a difficult problem that is still wide open.

Both results mentioned above are about Shimura varieties or connected Shimuravarieties in characteristic zero. But what can we say about their integral models?Certainly, any quasi-projective variety over the reflex field F0 has some integralmodel over the rings of integers OF0

in F0, but we are not interested in sucha general construction. It would be desirable to have a construction such thatthe pullback to OF0,(p) of the integral model at level U is smooth and satisfiessome reasonable extension property, when U is of the form UpUp such that Upis a neat open compact subgroup of G(A∞,p) and such that Up is a hyperspecialmaximal open compact subgroup of G(Qp). Moreover, it would be desirable thatthe reduction modulo p of such integral models can be described by some variantof the Langlands–Rapoport conjecture (see Section 5.1.6 and the end of Section5.2.3). These are also difficult and wide open problems, but their nature mightbe very different from that of the problem (in the previous paragraph) of findingmotives parameterized by such Shimura varieties.

5.4. Beyond Shimura varieties? There are some exciting recent developmentswhich are not about integral models of Shimura varieties, or not even about Shimuravarieties, but it would have been a shame not to mention them at all.

In [Sch15], Scholze constructed perfectoid Shimura varieties in all Hodge-typecases, which are limits at the infinite level at p (in a subtle sense) of Shimuravarieties at finite levels. Base on these, in [She17], Shen also constructed perfectoidShimura varieties in all abelian-type cases. Although the constructions in [Sch15]used Siegel moduli schemes over the integers, these perfectoid Shimura varieties areobjects (called adic spaces) defined over the characteristic zero base field Cp (the

completion of an algebraic closure Qp of Qp). But the property of being perfectoid(see [Sch12]) makes them much more powerful than integral models at finite levelsfor studying many questions about torsion in the cohomology of Shimura varietiesand related geometric objects—see, for example, [Sch15], [NT16], and [CS17].

Even the concept of Shimura varieties itself has been challenged and generalized.In his Berkeley–MSRI lectures in the Fall of 2014 (see [SW17]), Scholze defined cer-tain moduli spaces of mixed-characteristic shtukas, which can be viewed as a localanalogue of Shimura varieties, but is much more general than the previously stud-ied local geometric objects such as the so-called Rapoport–Zink spaces (see [RZ96])and can be defined for all reductive linear algebraic groups over Qp. He proposedto use them to construct local Langlands parameters for all reductive linear alge-braic groups over Qp, along the lines of Vincent Lafforgues’s construction of global


Langlands parameters for global fields of positive characteristics (see [Laf18]), butin the much fancier category of diamonds (see [SW17] and [Sch17]).

Acknowledgements

Some materials in this article are based on introductory lectures given at theNational Center for Theoretical Sciences (NCTS) in Taipei in 2014; at the Mathe-matical Sciences Research Institute (MSRI) in Berkeley in 2014; at the Universityof Minnesota, Twin Cities, in 2015; and at the summer school sponsored by ETHZurich in Ascona, Switzerland, in 2016. We heartily thank the organizers for theirinvitations, and thank the audiences for their valuable feedbacks. We also thankBrian Conrad and Sug Woo Shin, and the anonymous referees, for their helpfulcomments and suggestions.

References

[AEK05] J. Arthur, D. Ellwood, and R. Kottwitz (eds.), Harmonic analysis, the trace formula,

and Shimura varieties, Clay Mathematics Proceedings, vol. 4, Proceedings of the Clay

Mathematics Institute 2003 Summer School, The Fields Institute, Toronto, Canada,June 2–27, 2003, American Mathematical Society, Providence, Rhode Island, Clay

Mathematics Institute, Cambridge, Massachusetts, 2005.

[AGV73] M. Artin, A. Grothendieck, and J.-L. Verdier (eds.), Theorie des topos et cohomologieetale des schemas (SGA 4), Tome 3, Lecture Notes in Mathematics, vol. 305, Springer-

Verlag, Berlin, Heidelberg, New York, 1973.

[AMRT75] A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locallysymmetric varieties, Lie Groups: History Frontiers and Applications, vol. 4, Math Sci

Press, Brookline, Massachusetts, 1975.

[AMRT10] , Smooth compactification of locally symmetric varieties, 2nd ed., Cambridge

Mathematical Library, Cambridge University Press, Cambridge, New York, 2010.

[Art69] M. Artin, Algebraization of formal moduli: I, in Spencer and Iyanaga [SI69], pp. 21–71.[AT83] M. Artin and J. Tate (eds.), Arithmetic and geometry, Vol. I. arithmetic. Papers

dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, Progress in

Mathematics, vol. 35, Birkhauser, Boston, 1983.[Bai58] W. L. Baily, Jr., Satake’s compactification of Vn, Amer. J. Math. 80 (1958), no. 2,

348–364.

[Bai70] , An exceptional arithmetic group and its eisenstein series, Ann. Math. (2) 91(1970), no. 3, 512–549.

[BB66] W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded

symmetric domains, Ann. Math. (2) 84 (1966), no. 3, 442–528.[BC79] A. Borel and W. Casselman (eds.), Automorphic forms, representations and L-

functions, Proceedings of Symposia in Pure Mathematics, vol. 33, Part 2, held atOregon State University, Corvallis, Oregon, July 11–August 5, 1977, American Math-

ematical Society, Providence, Rhode Island, 1979.[BG84] M.-J. Bertin and C. Goldstein (eds.), Seminaire de theorie des nombres, Paris 1982–

1983, Progress in Mathematics, vol. 51, Birkhauser, Boston, Basel, Stuttgart, 1984.[BH78] A. Borel and G. Harder, Existence of discrete cocompact subgroups of reductive groups

over local fields, J. Reine Angew. Math. 298 (1978), 53–64.[BHC62] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math.

(2) 75 (1962), no. 3, 483–535.[BJ06] A. Borel and L. Ji, Compactifications of symmetric and locally symmetric spaces,

Mathematics: Theory & Applications, Birkhauser, Boston, 2006.[Bor62] A. Borel, Ensembles fondamentaux pour les groupes arithmetiques, Colloq. Theorie des

Groupes Algebriques (Bruxelles, 1962), Librairie Universitaire, Louvain; GauthierVil-lars, Paris, 1962, pp. 23–40.

[Bor63] , Some finiteness properties of adele groups over number fields, Publ. Math.

Inst. Hautes Etud. Sci. 16 (1963), 5–30.

66 KAI-WEN LAN

[Bor69] , Introduction aux groupes arithmetiques, Publications de l’Institut de

Mathematique de l’Universite de Strasbourg XV, Actualites scientifiques et indus-

trielles, vol. 1341, Hermann, Paris, 1969.[Bor72] , Some metric properties of arithmetic quotients of symmetric spaces and an

extension theorem, J. Differential Geometry 6 (1972), 543–560.

[Bor87] M. V. Borovoı, The group of points of a semisimple group over a totally real-closedfield, Problems in group theory and homological algebra (Russian), Mathematika,

Yaroslav. Gos. Univ., Yaroslavl′, 1987, pp. 142–149.

[Bor91] A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126,Springer-Verlag, Berlin, Heidelberg, New York, 1991.

[Bor84] M. V. Borovoı, Langlands’ conjecture concerning conjugation of connected Shimura

varieties, Selecta Math. Soviet. 3 (1983/84), no. 1, 3–39.[BS73] A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48

(1973), 436–491, with an appendix by A. Douady et L. Herault.

[BT65] A. Borel and J. Tits, Groupes reductifs, Publ. Math. Inst. Hautes Etud. Sci. 27 (1965),

55–151.[BW00] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and represen-

tations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67,

American Mathematical Society, Providence, Rhode Island, 2000.[Car58] H. Cartan, Prolongement des espaces analytiques normaux, Math. Ann. (1958), 97–

110.

[Car86] H. Carayol, Sur la mauvaise reduction des courbes de Shimura, Compositio Math. 59(1986), no. 2, 151–230.

[CCM91] J. A. Carlson, C. H. Clemens, and D. R. Morrison (eds.), Complex geometry and

Lie theory, Proceedings of Symposia in Pure Mathematics, vol. 53, Proceedings ofa Symposium on Complex Geometry and Lie Theory held at Sundance, Utah, May

26–30, 1989, Springer-Verlag, Berlin, Heidelberg, New York, 1991.[CHKS18] J. Cogdell, G. Harder, S. Kudla, and F. Shahidi (eds.), Cohomology of arithmetic

groups: On the occasion of Joachim Schwermer’s 66th birthday, Bonn, Germany,

June 2016, Springer Proceedings in Mathematics & Statistics, vol. 245, Springer In-ternational Publishing AG, 2018.

[CM90a] L. Clozel and J. S. Milne (eds.), Automorphic forms, Shimura varieties, and L-

functions. Volume II, Perspectives in Mathematics, vol. 11, Proceedings of a Con-ference held at the University of Michigan, Ann Arbor, July 6–16, 1988, Academic

Press Inc., Boston, 1990.

[CM90b] L. Clozel and J. S. Milne (eds.), Automorphic forms, Shimura varieties, and L-functions. Volume I, Perspectives in Mathematics, vol. 10, Proceedings of a Conference

held at the University of Michigan, Ann Arbor, July 6–16, 1988, Academic Press Inc.,

Boston, 1990.[CN90] C.-L. Chai and P. Norman, Bad reduction of the Siegel moduli scheme of genus two

with Γo(p)-level structure, Amer. J. Math. 112 (1990), no. 6, 1003–1071.[Con12] B. Conrad, Weil and Grothendieck approaches to adelic points, Enseign. Math. (2) 58

(2012), no. 1–2, 61–97.[CS17] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary

Shimura varieties, Ann. Math. (2) 186 (2017), no. 3, 649–766.

[CY97] J. Coates and S. T. Yau (eds.), Elliptic curves, modular forms & Fermat’s last theo-

rem, Proceedings of a conference held in the Institute of Mathematics of the ChineseUniverisity of Hong Kong, International Press, Cambridge, Massachusetts, 1997.

[Del70] P. Deligne, Equations differentielles a points singuliers reguliers, Lecture Notes inMathematics, vol. 163, Springer-Verlag, Berlin, Heidelberg, New York, 1970.

[Del71a] , Formes modulaires et representations `-adiques, Seminaire Bourbaki, expose

n◦ 355 (fevrier 1969), Lecture Notes in Mathematics, vol. 179, Springer-Verlag, Berlin,Heidelberg, New York, 1971, pp. 139–172.

[Del71b] , Travaux de Shimura, Seminaire Bourbaki, expose n◦ 389 (fevrier 1971), Lec-ture Notes in Mathematics, vol. 244, Springer-Verlag, Berlin, Heidelberg, New York,1971, pp. 123–165.

[Del72] , La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226.


[Del77] P. Deligne (ed.), Cohomologie etale (SGA 4 12

), Lecture Notes in Mathematics, vol.

569, Springer-Verlag, Berlin, Heidelberg, New York, 1977.

[Del79] , Varietes de Shimura: Interpretation modulaire, et techniques de constructionde modeles canoniques, in Borel and Casselman [BC79], pp. 247–290.

[DK73a] P. Deligne and N. Katz (eds.), Groupes de monodromie en geometrie algebrique (SGA

7 II), Lecture Notes in Mathematics, vol. 340, Springer-Verlag, Berlin, Heidelberg,New York, 1973.

[DK73b] P. Deligne and W. Kuyk (eds.), Modular functions of one variable II, Lecture Notes

in Mathematics, vol. 349, Springer-Verlag, Berlin, Heidelberg, New York, 1973.[DMOS82] P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Hodge cycles, motives, and Shimura

varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, Heidelberg,

New York, 1982.[DR73] P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, in Deligne

and Kuyk [DK73b], pp. 143–316.[EHLS16] E. Eischen, M. Harris, J.-S. Li, and C. Skinner, p-adic L-functions for unitary groups,

preprint, 2016.

[Eie09] M. Eie, Topics in number theory, Monographs in Number Theory, vol. 2, World Sci-entific, Singapore, 2009.

[Fal84] G. Faltings, Arithmetic varieties and rigidity, in Bertin and Goldstein [BG84], pp. 64–

77.[FC90] G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergebnisse der Math-

ematik und ihrer Grenzgebiete, 3. Folge, vol. 22, Springer-Verlag, Berlin, Heidelberg,

New York, 1990.[FK95] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford Mathematical Mono-

graphs, Clarendon Press, Oxford University Press, Oxford, 1995.

[FM97] J.-M. Fontaine and B. Mazur, Geometric Galois representations, in Coates and Yau[CY97], pp. 190–227.

[FM04] L. Fargues and E. Mantovan, Varietes de Shimura, espaces de Rapoport–Zink, etcorrespondances de Langlands locales, Asterisque, no. 291, Societe Mathematique de

France, Paris, 2004.

[FM13a] Gavril Farkas and Ian Morrison (eds.), Handbook of moduli: Volume II, Advanced Lec-tures in Mathematics, vol. 25, International Press, Somerville, Massachusetts, Higher

Education Press, Beijing, 2013.

[FM13b] Gavril Farkas and Ian Morrison (eds.), Handbook of moduli: Volume III, AdvancedLectures in Mathematics, vol. 26, International Press, Somerville, Massachusetts,

Higher Education Press, Beijing, 2013.

[Fra98] J. Franke, Harmonic analysis in weighted L2-spaces, Ann. Sci. Ecole Norm. Sup. (4)31 (1998), 181–279.

[FS98] J. Franke and J. Schwermer, A decomposition of spaces of automorphic forms, and

the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), 765–790.[GHM94] M. Goresky, G. Harder, and R. MacPherson, Weighted cohomology, Invent. Math. 116

(1994), 139–213.

[GL97] W. T. Gan and H. Y. Loke, Modular forms of level p on the exceptional tube domain,J. Ramanujan Math. Soc. 12 (1997), no. 2, 162–202.

[Gor05] M. Goresky, Compactifications and cohomology of modular varieties, in Arthur et al.[AEK05], pp. 551–582.

[GW09] R. Goodman and N. R. Wallach, Symmetry, representations, and invariants, GraduateTexts in Mathematics, vol. 255, Springer-Verlag, Berlin, Heidelberg, New York, 2009.

[Hai05] T. J. Haines, Introduction to Shimura varieties with bad reductions of parahoric type,

in Arthur et al. [AEK05], pp. 583–658.

[Har89] M. Harris, Functorial properties of toroidal compactifications of locally symmetricvarieties, Proc. London Math. Soc. (3) 59 (1989), 1–22.

[Har90] , Automorphic forms and the cohomology of vector bundles on Shimura vari-eties, in Clozel and Milne [CM90a], pp. 41–91.

[Hel01] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Stud-

ies in Mathematics, vol. 34, American Mathematical Society, Providence, Rhode Is-land, 2001.

68 KAI-WEN LAN

[Hid02] H. Hida, Control theorems of coherent sheaves on Shimura varieties of PEL type, J.

Inst. Math. Jussieu 1 (2002), no. 1, 1–76.

[Hid04] , p-adic automorphic forms on Shimura varieties, Springer Monographs inMathematics, Springer-Verlag, Berlin, Heidelberg, New York, 2004.

[Hir64a] H. Hironaka, Resolution of singularities of an algebraic variety over a field of charac-

teristic zero: I, Ann. Math. (2) 79 (1964), no. 1, 109–203.[Hir64b] , Resolution of singularities of an algebraic variety over a field of characteristic

zero: II, Ann. Math. (2) 79 (1964), no. 2, 205–326.

[Hir66] U. Hirzebruch, Uber Jordan-Algebren und beschrankte symmetrische Gebiete, Math.

Z. 94 (1966), 387–390.

[HLS06] M. Harris, J.-S. Li, and C. M. Skinner, p-adic L-functions for unitary Shimura vari-eties. I. Construction of the Eisenstein measure, Doc. Math. (2006), no. Extra Vol.,

393–464.

[HLTT16] M. Harris, K.-W. Lan, R. Taylor, and J. Thorne, On the rigid cohomology of certainShimura varieties, Res. Math. Sci. 3 (2016), article no. 37, 308 pp.

[HT01] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura vari-

eties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton,2001.

[Int75] International Colloquium on Discrete Subgroups of Lie Groups and Applications to

Moduli, Bombay, 8–15 January 1973, Discrete subgroups of Lie groups and applica-tions to moduli: Papers presented at the Bombay Colloquium, 1973, Tata Institute

of Fundamental Research Studies in Mathematics, vol. 7, Oxford University Press,Oxford, 1975.

[JPYY12] L. Ji, Y. S. Poon, L. Yang, and S.-T. Yau (eds.), Fifth International Congress of

Chinese Mathematicians, AMS/IP Studies in Advanced Mathematics, vol. 51, part1, Proceedings of the Fifth International Congress of Chinese Mathematicians held

at Tsinghua University, Beijing, in December 2010, American Mathematical Society,

Providence, Rhode Island, and International Press, Cambridge, MA, 2012.[Kim93] H. H. Kim, Exceptional modular form of weight 4 on an exceptional domain contained

in C27, Rev. Mat. Iberoamericana 9 (1993), no. 1, 139–200.

[Kis10] M. Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc.23 (2010), no. 4, 967–1012.

[Kis17] , Mod p points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30(2017), no. 3, 819–914.

[KM85] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics

Studies, vol. 108, Princeton University Press, Princeton, 1985.[KMP16] W. Kim and K. Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma

4 (2016), e28, 34 pp.

[Kna01] A. W. Knapp, Representation theory of semisimple groups: An overview based on ex-amples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton,

2001.

[Kna02] , Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol.140, Birkhauser, Boston, 2002.

[Kot90] R. E. Kottwitz, Shimura varieties and λ-adic representations, in Clozel and Milne[CM90b], pp. 161–209.

[Kot92] , Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5

(1992), no. 2, 373–444.[KP18] M. Kisin and G. Pappas, Integral models of Shimura varieties with parahoric level

structure, Publ. Math. Inst. Hautes Etud. Sci. 128 (2018), no. 1, 121–218.[KS67] M. Kuga and I. Satake, Abelian varieties attached to polarized K3-surfaces, Math.

Ann. 169 (1967), 239–242.[KS16] A. Kret and S. W. Shin, Galois representations for general symplectic groups, preprint,

2016.

[Laf18] V. Lafforgue, Chtoucas pour les groupes reductifs et parametrisation de Langlands

globale, J. Amer. Math. Soc. 31 (2018), no. 3, 719–891.[Lan76] R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Math-

ematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol.


XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), American Mathematical Society,

Providence, Rhode Island, 1976, pp. 401–418.

[Lan77] , Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1977),no. 6, 1292–1299.

[Lan12a] K.-W. Lan, Comparison between analytic and algebraic constructions of toroidal com-

pactifications of PEL-type Shimura varieties, J. Reine Angew. Math. 664 (2012), 163–228.

[Lan12b] , Geometric modular forms and the cohomology of torsion automorphic

sheaves, in Ji et al. [JPYY12], pp. 183–208.[Lan12c] , Toroidal compactifications of PEL-type Kuga families, Algebra Number The-

ory 6 (2012), no. 5, 885–966.

[Lan13] , Arithmetic compactification of PEL-type Shimura varieties, London Math-

ematical Society Monographs, vol. 36, Princeton University Press, Princeton, 2013,

errata available on the author’s website.[Lan15] , Boundary strata of connected components in positive characteristics, Algebra

Number Theory 9 (2015), no. 9, 1955–2054, an appendix to the article “Families of

nearly ordinary Eisenstein series on unitary groups” by Xin Wan.[Lan16a] , Cohomology of automorphic bundles, preprint, 2016, to appear in the pro-

ceedings of the Seventh International Congress of Chinese Mathematicians.

[Lan16b] , Compactifications of PEL-type Shimura varieties in ramified characteristics,Forum Math. Sigma 4 (2016), e1, 98 pp.

[Lan17] , Integral models of toroidal compactifications with projective cone decomposi-

tions, Int. Math. Res. Not. IMRN 2017 (2017), no. 11, 3237–3280.[Lan18a] , Compactifications of PEL-type Shimura varieties and Kuga families with

ordinary loci, World Scientific, Singapore, 2018.[Lan18b] , Compactifications of splitting models of PEL-type Shimura varieties, Trans.

Amer. Math. Soc. 370 (2018), no. 4, 2463–2515.

[Loo88] E. Looijenga, L2-cohomology of locally symmetric varieties, Compositio Math. 67(1988), 3–20.

[LR87] R. P. Langlands and M. Rapoport, Shimuravarietaten und Gerben, J. Reine Angew.

Math. 378 (1987), 113–220.[LR91] E. Looijenga and M. Rapoport, Weights in the local cohomology of a Baily–Borel

compactification, in Carlson et al. [CCM91], pp. 223–260.

[LR92] R. P. Langlands and D. Ramakrishnan (eds.), The zeta functions of Picard modularsurfaces, based on lectures delivered at a CRM Workshop in the spring of 1988, Les

Publications CRM, Montreal, 1992.

[LS90] J.-P. Labesse and J. Schwermer (eds.), Cohomology of arithmetic groups and automor-phic forms, Lecture Notes in Mathematics, vol. 1447, Proceedings of a Conference held

in Luminy/Marseille, France, May 22–27, 1989, Springer-Verlag, Berlin, Heidelberg,New York, 1990.

[LS04] J.-S. Li and J. Schwermer, On the Eisenstein cohomology of arithmetic groups, Duke

Math. J. 123 (2004), no. 1, 141–169.[LS12] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on com-

pact PEL-type Shimura varieties, Duke Math. J. 161 (2012), no. 6, 1113–1170.[LS13] , Vanishing theorems for torsion automorphic sheaves on general PEL-type

Shimura varieties, Adv. Math. 242 (2013), 228–286.

[LS18a] K.-W. Lan and B. Stroh, Compactifications of subschemes of integral models of

Shimura varieties, Forum Math. Sigma 6 (2018), e18, 105.[LS18b] , Nearby cycles of automorphic etale sheaves, Compos. Math. 154 (2018), no. 1,

80–119.[LS18c] , Nearby cycles of automorphic etale sheaves, II, in Cogdell et al. [CHKS18],

pp. 83–106.

[LZ17] R. Liu and X. Zhu, Rigidity and a Riemann–Hilbert correspondence for p-adic localsystems, Invent. Math. 207 (2017), 291–343.

[Man04] E. Mantovan, On certain unitary group Shimura varieties, in Varietes de Shimura,

espaces de Rapoport–Zink, et correspondances de Langlands locales [FM04], pp. 201–331.

70 KAI-WEN LAN

[Man05] , On the cohomology of certain PEL-type Shimura varieties, Duke Math. J.

129 (2005), no. 3, 573–610.

[Man11] , `-adic etale cohomology of PEL type Shimura varieties with non-trivial coeffi-cients, WIN–Woman in Numbers, Fields Institute Communications, vol. 60, American

Mathematical Society, Providence, Rhode Island, 2011, pp. 61–83.

[MB85] L. Moret-Bailly, Pinceaux de varietes abeliennes, Asterisque, vol. 129, SocieteMathematique de France, Paris, 1985.

[MFK94] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergeb-

nisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer-Verlag, Berlin, Heidel-berg, New York, 1994.

[Mil83] J. S. Milne, The action of an automorphism of C on a Shimura variety and its special

points, in Artin and Tate [AT83], pp. 239–265.[Mil90] , Canonical models of (mixed) Shimura varieties and automorphic vector bun-

dles, in Clozel and Milne [CM90b], pp. 283–414.[Mil92] , The points on a Shimura variety modulo a prime of good reduction, in Lang-

lands and Ramakrishnan [LR92], pp. 151–253.

[Mil05] , Introduction to Shimura varieties, in Arthur et al. [AEK05], pp. 265–378.[Mil13] , Shimura varieties and moduli, in Farkas and Morrison [FM13a], pp. 467–548.

[Moo98] B. Moonen, Models of Shimura varieties in mixed characteristic, in Galois Represen-

tations in Arithmetic Algebraic Geometry [ST98], pp. 267–350.[MP15] K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent.

Math. 201 (2015), no. 2, 625–668.

[MP16] , Integral canonical models for spin Shimura varieties, Compos. Math. 152(2016), no. 4, 769–824.

[MP18] , Toroidal compactifications of integral models of Shimura varieties of Hodgetype, preprint, 2018, to appear in Ann. Sci. Ecole Norm. Sup. (4).

[MS10] J. S. Milne and J. Suh, Nonhomeomorphic conjugates of connected Shimura varieties,

Amer. J. Math. 132 (2010), no. 3, 731–750.[Mum70] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in

Mathematics, vol. 5, Oxford University Press, Oxford, 1970, with appendices by C. P.

Ramanujam and Yuri Manin.[Mum72] , An analytic construction of degenerating abelian varieties over complete

rings, Compositio Math. 24 (1972), no. 3, 239–272.

[Mum75] , A new approach to compactifying locally symmetric varieties, in DiscreteSubgroups of Lie Groups and Applications to Moduli: Papers presented at the Bombay

Colloquium, 1973 [Int75], pp. 211–224.

[Mum77] , Hirzebruch’s proportionality theorem in the non-compact case, Invent. Math.

42 (1977), 239–272.

[Mum83] , Tata lectures on theta I, Progress in Mathematics, vol. 28, Birkhauser, Boston,1983.

[NT16] J. Newton and J. A. Thorne, Torsion Galois representations over CM fields and Hecke

algebras in the derived category, Forum Math. Sigma 4 (2016), e21, 88 pp.[Pfa96] M. Pfau, The conjecture of Langlands and Rapoport for certain Shimura varieties of

non-rational weight, J. Reine Angew. Math. 471 (1996), 165–199.

[Pin89] R. Pink, Arithmetic compactification of mixed Shimura varieties, Ph.D. thesis,Rheinischen Friedrich-Wilhelms-Universitat, Bonn, 1989.

[PR94] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Applied

Mathematics, vol. 139, Academic Press Inc., New York and London, 1994, translatedby Rachel Rowen.

[PR03] G. Pappas and M. Rapoport, Local models in the ramified case, I. The EL-case, J.

Algebraic Geom. 12 (2003), no. 1, 107–145.[PR05] , Local models in the ramified case, II. Splitting models, Duke Math. J. 127

(2005), no. 2, 193–250.[PR06] 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

(2006), no. 2, 646–660.[PR09] G. Pappas and M. Rapoport, Local models in the ramified case, III. Unitary groups,

J. Inst. Math. Jussieu 8 (2009), no. 3, 507–564.


[PRS13] G. Pappas, M. Rapoport, and B. Smithling, Local models of Shimura varieties, I.

Geometry and combinatorics, in Farkas and Morrison [FM13b], pp. 135–217.

[PZ13] G. Pappas and X. Zhu, Local models of Shimura varieties and a conjecture of Kottwitz,Invent. Math. 194 (2013), 147–254.

[Rap78] M. Rapoport, Compactifications de l’espace de modules de Hilbert–Blumenthal, Com-

positio Math. 36 (1978), no. 3, 255–335.[Rap05] , A guide to the reduction modulo p of Shimura varieties, in Tilouine et al.

[TCHV05], pp. 271–318.

[Rei97] H. Reimann, The semi-simple zeta function of quaternionic Shimura varieties, LectureNotes in Mathematics, vol. 1657, Springer-Verlag, Berlin, Heidelberg, New York, 1997.

[Riz06] J. Rizov, Moduli stacks of polarized K3 surfaces in mixed characteristic, Serdica Math.

J. 32 (2006), no. 2–3, 131–178.[RZ96] M. Rapoport and T. Zink, Period spaces for p-divisible groups, Annals of Mathematics

Studies, vol. 141, Princeton University Press, Princeton, 1996.[Sat56] I. Satake, On the compactification of the Siegel space, J. Indian Math. Soc. (N.S.) 20

(1956), 259–281.

[Sat60] , On compactifications of the quotient spaces for arithmetically defined discon-tinuous groups, Ann. Math. (2) 72 (1960), no. 3, 555–580.

[Sat65] , Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J.

Math. 87 (1965), no. 2, 425–461.[Sat66] , Clifford algebras and families of abelian varieties, Nagoya Math. J. 27 (1966),

435–446.

[Sat67] , Symplectic representations of algebraic groups satisfying a certain analyticitycondition, Acta Math. 117 (1967), 215–279.

[Sat68] , Corrections to “Clifford algebras and families of abelian varieties”, NagoyaMath. J. 31 (1968), 295–296.

[Sat80] , Algebraic structures of symmetric domains, Publications of the Mathematical

Society of Japan, vol. 14, Princeton University Press, Princeton, 1980.[Sat01] , Compactifications, old and new, Sugaku Expositions 14 (2001), no. 2, 175–

189.

[SC58] I. Satake and H. Cartan, Demonstration du theoreme fondamental, Seminaire HenriCartan 10e annee: 1957/1958, expose 15 (14 avril 1958), Secretariat mathematique,

11 rue Pierre Curie, Paris 5e, 1958.

[Sch90a] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430.[Sch90b] J. Schwermer, Cohomology of arithmetic groups, automorphic forms and L-functions,

in Labesse and Schwermer [LS90], pp. 1–29.

[Sch12] P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Etud. Sci. 116 (2012), 245–313.

[Sch15] , On torsion in the cohomology of locally symmetric varieties, Ann. Math. (2)182 (2015), no. 3, 946–1066.

[Sch17] , Etale cohomology of diamonds, preprint, 2017.[Ser56] J.-P. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier. Grenoble

6 (1955–1956), 1–42.

[She17] X. Shen, Perfectoid Shimura varieties of abelian type, Int. Math. Res. Not. IMRN2017 (2017), no. 21, 6599–6653.

[Shi63a] H. Shimizu, On discontinuous groups operating on the product of the upper half planes,

Ann. Math. (2) 77 (1963), no. 1, 33–71.[Shi63b] G. Shimura, On analytic families of polarized abelian varities and automorphic func-

tions, Ann. Math. (2) 78 (1963), no. 1, 149–192.[Shi66] , Moduli and fibre systems of abelian varieties, Ann. Math. (2) 83 (1966), no. 2,

294–338.

[Shi70] , On canonical models of arithmetic quotients of bounded symmetric domains,Ann. Math. (2) 91 (1970), no. 1, 144–222.

[Shi71] , Introduction to the arithmetic theory of automorphic functions, Publicationsof the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton,1971.

[Shi78] , The arithmetic of automorphic forms with respect to a unitary group, Ann.

Math. (2) 107 (1978), no. 3, 569–605.

[Shi98] , Abelian varieties with complex multiplication and modular functions, Prince-

ton University Press, Princeton, 1998.

[Shi11] S. W. Shin, Galois representations arising from some compact Shimura varieties, Ann.Math. (2) 173 (2011), no. 3, 1645–1741.

[SI69] D. C. Spencer and S. Iyanaga (eds.), Global analysis. Papers in honor of K. Kodaira,

Princeton University Press, Princeton, 1969.[Spr98] T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9,

Birkhauser, Boston, 1998.

[SS90] L. Saper and M. Stern, L2-cohomology of arithmetic varieties, Ann. Math. (2) 132(1990), no. 1, 1–69.

[ST98] A. J. Scholl and R. L. Taylor, Galois representations in arithmetic algebraic geometry,

London Mathematical Society Lecture Note Series, vol. 254, Cambridge UniversityPress, Cambridge, New York, 1998.

[SU14] C. Skinner and E. Urban, The main conjecture for GL(2), Invent. Math. 195 (2014),no. 1, 1–277.

[SW17] P. Scholze and J. Weinstein, Berkeley lectures on p-adic geometry, preprint, 2017, to

appear in Annals of Mathematics Studies.[TCHV05] J. Tilouine, H. Carayol, M. Harris, and M.-F. Vigneras (eds.), Formes automorphes

(I): Actes du semestre du Centre emile Borel, printemps 2000, Asterisque, no. 298,

Societe Mathematique de France, Paris, 2005.[Vas99] A. Vasiu, Integral canonical models of Shimura varieties of preabelian type, Asian J.

Mathematics 3 (1999), 401–518.

[Zuc82] S. Zucker, L2 cohomology of warpped products and arithmetic groups, Invent. Math.70 (1982), 169–218.

Index

〈 · , · 〉, 5, 17, 31, 36, 51, 54〈 · , · 〉σ, 310n, 51a,b, 6, 18, 231n, 5, 6

(1) (Tate twist), 51

A, 27A, 4A0, 51, 53A∨0 , 51, 52A0[n], 52A∞, 4A∞,p, 57abelian motive, 64abelian scheme, 17, 50, 51, 53

univeral, 56abelian variety, 16, 17, 51, 56, 59

dual, 52abelian-type

Shimura datum, 33, 59–62Shimura variety, 60, 62–64

absolute Hodge cycle, 59adelic topology, 4adic space, 64adjoint quotient, 9, 18, 60, 61adjoint representation, 9

adjoint Shimura datum, 60, 61

affine group scheme, 4

Albert algebra, 27

Albert’s classification, 55

algebraic curve, 9, 34

algebraic cycle, 59

algebraic group, 4, 9, 10

simply-connected, 25, 61

algebraic space, 17, 47

algebraic stack, 17, 56

algebraicity, 9–11, 17, 33, 34, 47, 49, 64

α0,n, 52, 53

αN , 11

analytification, 10, 11, 17, 47, 63

approximation, 6

arithmetic subgroup, 8, 10, 12, 37, 39,42, 46–48

neat, 8

Artin’s criterion, 17

Aut(C/F0), 55, 59

automatic compactness, 34, 41, 42

automorphic L-function, 58

automorphic line bundle, 49

automorphic representation, 3, 7, 48

automorphic vector bundle, 49, 57

automorphy condition, 49

automorphy factor, 7, 49

72

INDEX 73

B, 41–43, 54, 60bad reduction, 57blowup, 63Borel subgroup, 35Borel–Serre compactification, 47–49boundary, 34, 48, 50

codimension, 47, 49stratification, 48, 58, 63

boundary component, 4, 34, 35, 37, 39,41, 42, 44, 46, 47

bounded realization, 15, 18, 22, 25, 30,31

Cp, 64canonical bundle, 49canonical extension, 49, 57canonical model, 3, 4, 11, 12, 49, 50,

54, 55, 62integral, 62

Cartan involution, 9Cartan’s classification of Hermitian

symmetric domaintype Iba, 21type IIn, 22type IIIn, 15type IVa, 25type V, 31type VI, 31

Cayley numbers, 20, 26Cayley transform, 15, 20, 30Cent (centralizer), 16, 21, 26, 30, 32, 49center of unipotent radical, 39, 41, 43central leaves, 58Chow’s theorem, 47classical group, 35Clifford algebra, 25closed complex analytic subspace, 47CM elliptic curve, 12CM field, 41CM point, 12, 55coarse moduli space, 56cocenter, 55, 60coherent cohomology, 50, 57cohomology, 7, 48, 50

coherent, 50, 57etale, 11, 63, 64intersection, 48L2, 48nearby cycle, 57, 63relative, 56weighted, 49

commensurable, 8compact factor, 9

compact Riemann surface, 9, 34compactification

Borel–Serre, 47–49reductive Borel–Serre, 47–49Satake–Baily–Borel or minimal, 4, 11,

37, 39, 42, 46, 47, 47, 47, 49integral model, 17, 56, 58, 63

toroidal, 47–50integral model, 56, 58, 63

complete algebraic curve, 9, 34complex analytic space, 47complex analytic subspace, 47complex analytification, 10, 11, 17, 47,

63complex manifold, 7, 9complex structure, 7, 17, 51complex torus, 16, 51condition on Lie algebra, 53, 55cone decomposition, 47, 63

refinement, 48congruence subgroup, 8, 9, 12, 17

principal, 8, 9, 11conjugacy class, 9, 10, 12, 16, 21–22,

26, 30, 32–33, 54, 59connected, 6, 9, 10, 54, 55connected component, 4, 6, 8–10, 12,

23–25, 54, 55, 60, 62connected Shimura datum, 9, 10–12, 16connected Shimura variety, 12, 56, 63connection, 50cotangent bundle, 49counting point, 57cusp, 9, 34cusp form, 49, 50cyclic isogeny, 57

D, 6, 9, 54Da,b, 18, 21D±a,b, 21

Dad, 60DE6 , 31DE7 , 30, 31Dn, 15D+, 8–10D(r), 41, 42DSO∗2n

, 22DSOa,2 , 25

D+SOa,2

, 25

D∗, 34, 35, 47de Rham, 64degeneration, 56Deligne torus, 9Deligne–Mumford stack, 17, 56

74 INDEX

derived group, 28, 54, 55, 60, 61determinantal condition, 53diamond, 65division algebra, 42

normed, 20octonion, 26

double coset space, 4, 7, 8, 33, 54Drinfeld level structure, 57, 58dual abelian variety, 52dual lattice, 52Dynkin diagram, 32, 33, 61

E, 39, 54e6, 30, 33e6(−14), 31–33e6(−26), 27, 32e6(C), 32e7(−25), 28, 33Ez, 11Eisenstein series, 48Ekedahl–Oort stratification, 58elliptic curve, 11, 16, 57

CM, 12universal, 50

endomorphism structure, 51–53eq-SO-2n-C, 22eq-SO-2n-star, 22eq-so-2n-star, 22etale cohomology, 11, 63, 64etale local system, 63exceptional group, 27, 28, 30–33, 46,

61–63extension property, 62, 64

F , 41, 60F0, 10, 49, 53, 54, 64f4, 28F+, 41failure of Hasse’s principle, 55, 56faithful representation, 4, 8, 59fan, 47flag, 36flat, 57, 58Fontaine–Mazur conjecture, 64formally real Jordan algebra, 27, 29free action, 8Freudenthal’s theorem, 30fundamental domain, 34fundamental group, 63

G, 28, 30, 31, 33G, 4, 6, 9, 32, 33, 39, 41, 42, 44, 46, 54g, 9Gad, 9, 60

Gad(Q)+, 12

Gad(R)+, 10(G,D), 7, 9–11, 54, 58(G,D+), 9–12

Gder, 59, 61G(j), 55Gm, 4G(Q)+, 8

G(r), 40, 43G(R)+, 8, 25, 33, 60g∗, 28

G, 50GAGA, 47Gal(Q/F0), 11Galois cohomology, 25Galois representation, 3, 11

geometric, 64Galois symmetry, 3, 11Γ, 10, 11, 37, 39, 42, 46–48Γ0(p), 57Γ\D, 47, 49(Γ\D)BS, 47–49

(Γ\D)min, 47–49Γ\D+, 10, 11(Γ\D)RBS, 47–49

(Γ\D)torΣ , 47, 49

Γ\D∗, 47, 49Γi, 8Γ(N), 9, 11general linear group, 3, 4, 50, 58general spin group, 59, 61generalized projective coordinates, 13,

15, 18, 19, 23, 34, 37, 39, 42geometric Galois representation, 64geometric invariant theory, 17GL1, 4GL2, 3, 50, 58GL2k(H), 23GLn, 3, 4GLr(B), 43global Langlands correspondence, 3global Langlands parameter, 65good prime, 53, 54, 56, 57good reduction, 53Gram–Schmidt process, 23Grassmannian, 14group scheme, 4growth condition, 49GSp2n, 5, 54, 58, 60, 61GUa,b, 21, 54GU′a,b, 21, 54

H, 7, 9, 11, 13, 15, 25, 34, 48, 55

INDEX 75

H, 20, 23, 53, 60h, 9, 10, 17H0, 44, 46h0, 16, 17, 21, 22, 26, 30, 32, 51, 53, 54,

59h′0, 21

h0, 26H1, 15, 25, 44, 46H1,1, 55H±1 , 25, 50H2, 55H2,2, 55H5,1, 46Ha,b, 19, 21, 32, 39H±a,b, 21, 54

H∗a,b, 39Ha−r,b−r, 39

h, 10HE6 , 31, 33, 46HE7 , 29, 31, 33, 46H+I , 44Hn, 13, 15, 32, 35H±n , 16, 54, 60H∗n, 37Hn−r, 37H±, 50, 60HSO1,2 , 25

H+SO1,2

, 25, 55

H+SO10,2

, 46

H+SO2,2

, 33, 55

HSO∗2n, 22, 33, 42, 54

H∗SO∗2n, 42

HSO∗2n−4r, 42

H+SO3,2

, 55

H+SO4,2

, 55

HSO∗4k

, 23

H+SO6,2

, 55

HSO∗8, 55

HSOa,2 , 24

H−SOa,2, 24

H+SOa,2

, 24, 25, 33, 59

(H+SOa,2

)∗, 46

HSOa,2 , 24

HSOa,b , 23H∗, 34H∗×H∗, 34H×H, 34, 55Hamiltonian numbers, 20, 23, 53, 60Hasse–Weil zeta function, 58Hecke action, 7

Hecke symmetry, 3, 7, 11Herm3(O), 27Hermb(C), 19Hermk(H), 23Hermitian

imaginary part, 19real part, 19

Hermitian part of Levi subgroup, 38,40, 43

Hermitian symmetric domain, 4, 9, 12,13, 23, 24, 32–33

irreducible, 13, 31, 33of type A III, 21of type BD I, 25of type C I, 15of type D III, 22of type E VII, 31

Hermitian upper half-space, 20Hilbert modular surface, 35Hilbert modular variety, 35Hodge conjecture, 59Hodge cycle, 59

absolute, 59Hodge decomposition, 52Hodge filtration, 14Hodge structure

rational, 60variation, 3, 9, 12, 16, 17, 50

Hodge tensor, 59Hodge-type

Shimura datum, 58–62Shimura variety, 58, 60, 62–64

homothety of lattices, 17homotopy equivalence, 48horocycle, 34hyperspecial, 54, 62–64

i0, 52, 53identity component, 6, 8, 10, 25imaginary quadratic field, 12, 39, 54, 60infinite level, 64∞, 34∞r, 37, 39, 42integrable connection, 50integral canonical model, 62integral model, 4, 17, 18, 49–51, 56, 62,

64minimal compactification, 56, 58, 63toroidal compactification, 56, 58, 63

integral PEL datum, 51, 53intersection cohomology, 48ι, 28, 29isogeny class, 53

76 INDEX

isomorphism class, 53Iwasawa main conjecture, 3

J, 29J2, 36Ja,b, 19, 39Jn, 5Jordan algebra, 27, 29, 30

K, 14, 18, 22, 26, 30, 32, 33k, 9K3 surface, 61Klingen parabolic subgroup, 36Kottwitz conjecture, 58, 63Kottwitz–Rapoport stratification, 58Kuga family, 56Kuga–Satake construction, 59, 61Kuga–Sato variety, 50

L, 51, 54L2-cohomology, 48LI(Q), 44

L#, 51, 52L(r)(Q), 38, 40, 43λ0, 51, 53Langlands parameter, 64, 65Langlands program, 3Langlands–Rapoport conjecture, 58, 63,

64lattice, 16, 17, 51

dual, 52homothety, 17

level, 11level structure, 17, 51–53, 55, 59, 61

Drinfeld, 57, 58Levi quotient, 44, 48, 49Levi subgroup, 38, 40, 43, 44

Hermitian part, 38, 40, 43LieA0, 52, 53Lie algebra condition, 53, 55light cone, 24line bundle, 49linear algebraic group, 4local analogue of Shimura variety, 64local Langlands conjecture, 3local Langlands parameter, 64local model, 57, 58, 62local system, 9

etale, 63

M , 27, 29Ma,b, 18Mn, 6, 13MU , 55

manifold, 4, 6–9complex, 7, 9with corners, 48

maximal compact subgroup, 15, 26, 28,30–32

maximal parabolic subgroup, 28, 35–37,40, 42, 44

minimal compactification, 4, 11, 37, 39,42, 46, 47, 47, 47, 49

integral model, 17, 56, 58, 63minimal parabolic subgroup, 35mixed Hodge structure, 50mixed Shimura variety, 50

integral model, 56mixed type D, 61, 63mixed-characteristic shtuka, 64Mobius transformation, 7, 44modular curve, 9, 11, 12, 50, 57modular form, 7, 49, 50, 56, 57p-adic, 57, 58

modular function, 49moduli interpretation, 58, 59moduli problem, 17, 51, 61, 64

PEL, 51, 53, 55morphism of Shimura data, 12morphism of Shimura varieties, 12motive, 64multichain of isogenies, 57multichain of lattices, 57

N(X), 26nearby cycle, 57, 63neat, 8, 12, 55Newton stratification, 58normal, 37, 39, 47, 58normal crossings divisor, 48normalization, 57–59, 62, 63ν, 5, 27ν0,n, 52, 53number field, 9–11, 17, 49, 63

O, 51, 54O, 20, 26

norm, 26Oa,b, 6OE , 54OF , 41OF0 , 53, 64OF0,(p), 64OF0,(v), 62O-lattice, 51On, 6octonion division algebra, 26

INDEX 77

octonion upper half-space, 29Oort central leaves, 58open compact subgroup, 7, 8, 10, 12,

50, 62hyperspecial, 54, 62–64neat, 8, 12, 55parahoric, 57, 62

orthogonal group, 6orthogonal Shimura variety, 54, 55, 62

P , 28PI(Q), 44p−, 9p+, 9P(r)(Q), 38, 40, 42p-adic L-function, 3p-adic modular form, 57, 58p-adic realization, 64p-rank stratification, 58parabolic subgroup, 35, 48

Klingen, 36maximal, 35–37, 40, 42, 44minimal, 35Siegel, 36

parahoric, 57, 62parameter space, 11, 17, 51, 56, 59partial minimal compactification, 58partial toroidal compactification, 58PEL datum

integral, 51, 53rational, 53

PEL moduli problem, 51, 53, 55, 57of type A, 55of type C, 55of type D, 55

PEL structure, 51, 53, 56PEL-type

Shimura datum, 54, 59Shimura variety, 54, 56

perfectoid Shimura variety, 64Picard modular surface, 41Poincare upper half-plane, 7, 12, 13, 15,

25polarization, 17, 51, 53polarized abelian scheme, 17polarized abelian variety, 16, 17, 59polarized K3 surface, 61positive involution, 41, 51, 54, 56principal congruence subgroup, 8, 9, 11principal level structure, 52principal polarization, 17projective coordinates, 7, 13, 15, 18, 19,

23, 34, 37, 39, 42, 49

projective spectrum, 56projective variety, 11, 37, 39, 47, 48proper base change theorem, 63

Q, 44Q, 11Qp, 64quasi-projective scheme, 55quasi-projective variety, 4, 10, 47, 63, 64quaternion algebra, 42, 60quaternion upper half-space, 23

Rapoport–Zink space, 64rational boundary component, 4, 34, 35,

37, 39, 41, 42, 44, 46, 47rational Hodge structure, 60rational PEL datum, 53real analytic topology, 6, 8, 10, 25real quadratic extension, 35real torus, 16, 51realization

bounded, 15, 18, 22, 25, 30, 31unbounded, 13, 18, 22, 24, 25, 29, 31

reduction, 58bad, 57good, 53

reduction theory, 34, 37reductive algebraic group, 4, 9, 64reductive Borel–Serre compactification,

47–49reductive Lie group, 18refinement of cone decompositions, 48reflex field, 10, 49, 53, 54, 64relative cohomology, 56representation

automorphic, 3, 7, 48Galois, 3, 11

geometric, 64resolution of singularity, 48restricted product, 4restriction of scalars, 9, 35, 41ρ, 28, 30ρB , 28, 29Riemann surface, 9, 34Riemannian symmetric space, 48

noncompact of type A I, 6noncompact of type E IV, 28

Rosati condition, 52

S, 19, 39, 40S, 9S2, 7S3, 7Sn, 6, 7

78 INDEX

Satake topology, 35, 37, 39, 42, 46, 47Satake–Baily–Borel compactification, 4,

11, 37, 39, 42, 46, 47, 47, 47, 49integral model, 17, 56, 58, 63

semi-abelian scheme, 56semisimple algebra, 35, 51, 56semisimple algebraic group, 4, 10semisimple Lie group, 18Shimura curve, 34, 60Shimura datum, 3, 4, 9, 10, 11, 16, 22,

49, 54, 55adjoint, 60, 61connected, 9, 10–12, 16morphism, 12of abelian type, 33, 59–62of Hodge type, 58–62of PEL type, 54, 59simple, 61

Shimura reciprocity law, 12Shimura variety, 3, 4, 9, 11, 49, 50, 58

CM point, 12, 55connected, 12, 63integral canonical model, 62integral model, 4, 49–51, 56, 62, 64level, 11local analogue, 64morphism, 12of abelian type, 60, 62–64of Hodge type, 58, 62–64of PEL type, 54, 56perfectoid, 64special point, 12special subvariety, 12zero-dimensional, 12

shtuka, 64Siegel embedding, 22, 26, 30, 32, 33, 58,

59–62Siegel modular variety, 17, 18, 51, 59Siegel moduli scheme, 17, 51, 54, 56, 64Siegel parabolic subgroup, 36Siegel upper half-space, 13Σ, 47σ, 31signature, 6, 18, 24, 41, 44similitude character, 5simple algebra, 55simple normal crossings divisor, 48simple Shimura datum, 61simply-connected algebraic group, 25,

61SL2, 7, 9, 11, 15, 34, 48SL2×SL2, 34SLn, 4, 6

smooth, 10, 48, 53, 62, 64so10, 31–33SO2, 7, 15, 25SO∗2n, 22, 33, 42, 54, 55, 61so∗2n, 22, 61so2n−2,2, 61SO∗4k, 23so6,2, 61so∗8, 61SOa,2(R)+, 25, 33, 44, 60SOa,b, 6, 23, 25, 44SOn, 6, 22, 25SO(Qa+2, Q), 44Sp2, 15Sp2k(H), 23Sp2n, 5, 13, 32, 35, 51, 55special linear group, 4, 6, 7, 9, 11, 15,

34, 48, 55special node, 32special orthogonal group, 6, 6, 7, 15, 22,

23, 25, 33, 42, 44, 54, 55, 60–62special point, 12special subvariety, 12special unitary group, 18Spin2,2, 33Spin∗2n, 61Spina,2, 25, 26, 33spin group, 25, 33, 59, 61, 62spinor norm, 25?, 51, 54stratification

boundary, 48, 58, 63Ekedahl–Oort, 58Kottwitz–Rapoport, 58Newton, 58p-rank, 58

SUa,b, 18subcanonical extension, 50, 57sufficiently small, 8Symn, 6, 13symmetric space

Hermitian, 4, 9, 12, 13, 23, 24, 32–33noncompact of type A III, 21noncompact of type BD I, 25noncompact of type C I, 15noncompact of type D III, 22noncompact of type E III, 31noncompact of type E VII, 31

irreducible, 13, 29, 31, 33Riemannian, 48

noncompact of type A I, 6noncompact of type E IV, 28

symmetry

INDEX 79

Galois, 3, 11Hecke, 3, 7, 11

symplectic group, 5, 13, 15, 32, 35, 51,54, 55, 58, 60, 61

symplectic Shimura variety, 54, 60, 61

tamely ramified, 62Tate twist, 51topology

adelic, 4of H∗, 34real analytic, 6, 8, 10, 25Satake, 35, 37, 39, 42, 46Zariski, 6

toroidal compactification, 47–50integral model, 56, 58, 63

torsion cohomology class, 3, 64torsion subgroup, 11, 52torus, 12totally imaginary extension, 41totally isotropic subspace, 35–38, 40,

43, 44totally real field, 35, 41, 60trace condition, 53transitivity, 6–8, 14, 21, 22, 25, 30tube domain, 13, 20, 23–25, 29type A, 18–22, 32, 39–42, 55, 58, 61type B, 23–26, 33, 44–46, 61type C, 13–18, 32, 35–39, 55, 58, 61type D, 22–26, 33, 42–46, 55, 61

mixed, 61, 63type DH, 61type DR, 61

type E6, 31–33, 46, 61, 63type E7, 26–31, 33, 46, 61, 63

U , 28, 29U , 7, 10, 11, 54, 59, 62U1, 10, 15Ua,b, 18, 32, 41, 55U′a,b, 19, 22, 39UI(Q), 44Un, 14, 32, 33Up, 57, 62, 64Up, 57, 62, 64U ′, 59, 62U ′p, 62U ′,p, 62U(r)(Q), 38, 40, 43

U , 50unbounded realization, 13, 18, 22, 24,

25, 29, 31

unipotent radical, 38–41, 43, 44unit disc, 15unitary group, 3, 10, 14, 15, 18, 19, 22,

32, 33, 39, 41–42, 54, 55, 61unitary Shimura variety, 54, 55, 61universal abelian scheme, 56universal elliptic curve, 50unramified, 53, 64upper half-plane

Poincare, 7, 12, 13, 15, 25upper half-space

Hermitian, 20octonion, 29quaternion, 23Siegel, 13

V , 23, 24V0, 52V 0, 52VC, 23, 24V(r)(Q), 38, 40, 43variation of Hodge structures, 3, 9, 12,

16, 17, 50variation of mixed Hodge structures, 50vector bundle, 9, 49, 57

W , 49W , 49W, 28W can, 49W(r)(Q), 38, 40, 43W sub, 50weak approximation, 6weight, 49weighted cohomology, 49Weil pairing, 11, 52

XU , 50XU , 7, 8–11, 33, 49, 50, 54, 59, 62XU,F0 , 62XU,OF0,(v)

, 62

X(j)U , 55

XU′,OF0,(v), 62

Z, 4

Zp, 57Z(M), 28Zarhin’s trick, 62Zariski topology, 6zero-dimensional Shimura varieties, 12ζN , 11Zucker’s conjecture, 48

80 INDEX

University of Minnesota, Minneapolis, MN 55455, USA

Email address: [email protected]

AN EXAMPLE-BASED INTRODUCTION TO SHIMURA ...kwlan/articles/intro-sh-ex.pdfAN EXAMPLE-BASED...

Documents

Transcript of AN EXAMPLE-BASED INTRODUCTION TO SHIMURA ...kwlan/articles/intro-sh-ex.pdfAN EXAMPLE-BASED...