INTEGRAL MODELS OF CERTAIN PEL SHIMURA
VARIETIES WITH Γ1(p)-TYPE LEVEL STRUCTURE
By
Richard Shadrach

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics – Doctor of Philosophy
2014

ABSTRACT
INTEGRAL MODELS OF CERTAIN PEL SHIMURA
VARIETIES WITH Γ1(p)-TYPE LEVEL STRUCTURE
By
Richard Shadrach

We study p-adic integral models of certain PEL-Shimura varieties with level
subgroup at p given by the pro-unipotent radical of an Iwahori. We will consider
two cases: the case of Shimura varieties associated to unitary groups that split
over an unramified extension of Qp and the case of Siegel modular varieties. We
construct local models, i.e. simpler schemes which are ´etale locally isomorphic to
the integral models. Our integral models are defined by a moduli scheme using the
notion of an Oort-Tate generator of a group scheme. We use these local models
to find a resolution of the integral model in the case of the Siegel modular variety
of genus 2. The resolution is regular with special fiber a nonreduced divisor with
normal crossings.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1 Shimura Varieties
1.1 Shimura datum . . . . . .
1.2 PEL Shimura varieties . .
1.2.1 Unitary case . . . .
1.2.2 Symplectic case . .
Chapter 2 Integral and
2.1 PEL case . . . . .
2.2 Unitary case . . . .
2.3 Symplectic case . .
2.4 Representability . .
Chapter 3

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Stratification of A0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 4 Integral and local models of
4.1 The group schemes Gi . . . . . . . .
4.2 Integral and local model of A1 . . . .
4.3 Modification of U1 . . . . . . . . . .

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Chapter 5 Resolution of A1 associated with GSp4 . . .
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . .
´
5.1.1 Etale
cover . . . . . . . . . . . . . . . . . . . .
5.1.2 Notation . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Connected components . . . . . . . . . . . . . .
5.1.4 KR Strata . . . . . . . . . . . . . . . . . . . . .
5.2 Step 0: A0 , U0 , and A1 . . . . . . . . . . . . . . . . . .
5.2.1 Description of the local model U0 . . . . . . . .
5.2.2 Description of the integral model A0 . . . . . .
5.2.3 Description of A1 . . . . . . . . . . . . . . . . .
5.3 Step I: Semi-stable resolution of A0 . . . . . . . . . . .
5.3.1 Description of the local model U0 . . . . . . . .
5.3.2 Description of the integral model A0 . . . . . .
5.4 Step II: Fiber with A1 . . . . . . . . . . . . . . . . . .
5.4.1 Description of the local model U1 . . . . . . . .

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iii

85
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108
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115
115

5.5

5.6

5.4.2 Description of the integral model A1
Step III: Blowup of Z3 . . . . . . . . . . . . .
5.5.1 Description of the local model U1 . .
5.5.2 Description of the integral model A1
Step IV: p − 2 blowups of Z11 . . . . . . . . .
[i]
5.6.1 Description of the local model U1 . .
5.6.2 Description of the integral model . .

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116
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APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

iv

LIST OF TABLES

Table 4.1

Dimension of invariant differentials of group schemes of order p. . . . .

Table 5.1

Irreducible components of A0 . . . . . . . . . . . . . . . . . . . . . . . 104

Table 5.2

Subschemes of A0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table 5.3

Number of connected and irreducible components of subschemes of A0

107

Table 5.4

Number of connected and irreducible components of subschemes of A0

113

Table 5.5

Number of connected and irreducible components of subschemes of A1

118

Table 5.6

Number of connected and irreducible components of subschemes of A1

124

Table 5.7

Ideal sheaf of Z11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Table 5.8

Images of irreducible components of A1

[p+1]

v

77

. . . . . . . . . . . . . . . . 135

LIST OF FIGURES

Figure 5.1

µ-permissible set for GSp4 . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 5.2

Dual complex of U1

Figure 5.3

Batons of Γp,K p where p = 5, ns2 τ = 2 . . . . . . . . . . . . . . . . . . 132

Figure 5.4

Base of Γp,K p where p = 5, ns2 τ = 2, ns0 s2 τ = 3 . . . . . . . . . . . . . 133

Figure 5.5

Γp,K p where p = 5, ns2 τ = 2, ns0 s2 τ = 3 . . . . . . . . . . . . . . . . . 133

[p+2]

for p = 5 . . . . . . . . . . . . . . . . . . . . . 130

vi

Introduction

In the arithmetic study of Shimura varieties, one seeks to have a model of the Shimura variety
over the ring of integers OE , where E is the completion of the reflex field E at some finite
place p. Denote by ShK (G, X) the Shimura variety given by the Shimura datum (G, X)
and choice of an open compact subgroup K =

K ⊂ G(Af ), where Af is the ring of

finite rational ad`eles. For Shimura varieties of PEL-type, which are moduli spaces of abelian
varieties with certain (polarization, endomorphism, and level) structures, one can define
such an integral model by proposing a moduli problem over OE . The study of such models
began with modular curves by Shimura and Deligne-Rapoport. More generally, Langlands,
Kottwitz, Rapoport-Zink, Chai, and others studied these models for various types of PEL
Shimura varieties. The reduction modulo p of these integral models is nonsingular if the
factor Kp ⊂ G(Qp ) is chosen to be “hyperspecial” for the rational prime p lying under
p. However if the level subgroup Kp is not hyperspecial, usually (although not always)
singularities occur. It is important to determine what kinds of singularities can occur, and
this is expected to be influenced by the level subgroup Kp .
In order to study the singularities of these integral models, significant progress has been made
by finding “local models”. These are schemes defined in simpler terms which control the
singularities of the integral model. They first appeared in [DP] for Hilbert modular varieties

1

and in [dJ2] for Siegel modular varieties with Iwahori level subgroup. More generally in [RZ],
Rapoport and Zink constructed local models for PEL Shimura varieties with parahoric level
subgroup.
In [G¨or1] G¨ortz showed that in the case of a Shimura variety of PEL-type associated with
a unitary group which splits over an unramified extension of Qp , the Rapoport-Zink local
models are flat with reduced special fiber. In [G¨or2], the same is shown for the local models
of Siegel modular varieties. On the other hand, Pappas has shown that these local models
can fail to be flat in the case of a ramified extension [Pap2]. In [PR1], [PR2], and [PR3],
Pappas and Rapoport give alternative definitions of the local models which are flat. More
recently in [PZ], Pappas and Zhu have given a general group-theoretic definition of the local
models which, for PEL cases, agree with Rapoport-Zink local models in the unramified case
and the alternative definitions in the ramified case.
Throughout this article, Kp is assumed to be either an Iwahori subgroup of G(Qp ) or the
pro-unipotent radical of an Iwahori subgroup. There is some ambiguity in calling these Γ0 (p)level structure and Γ1 (p)-level structure respectively; indeed one may consider more generally
a parahoric subgroup. As such, we will call the former Iw0 (p)-level structure and the latter
Iw1 (p)-level structure. In all the situations we consider, G = GQp extends to a reductive
group over Zp and one can take an Iwahori subgroup as being the inverse image of a Borel
subgroup of G(Fp ) under the reduction G(Zp ) → G(Fp ). We will also take K p =

=p

K

to be a sufficiently small open compact subgroup of G(Apf ) so that the moduli problems we
consider below are represented by schemes.
In [HR] Haines and Rapoport, interested in determining the local factor of the zeta function associated with the Shimura variety, constructed affine schemes which are ´etale locally
isomorphic to integral models of certain Shimura varieties with Iw1 (p)-level structure. This

2

follows the older works of Pappas [Pap1] and Harris-Taylor [HT]. Haines and Rapoport
consider the case of a Shimura variety associated with a unitary group which splits locally
at p given by a division algebra B defined over an imaginary quadratic extension of Q. The
cocharacter associated with the Shimura datum is assumed to be of “Drinfeld type”.
In this article, we will consider Iw1 (p)-level structure for two particular types of Shimura
varieties. First the unitary case, where the division algebra B has center F , an imaginary
quadratic extension of a totally real finite extension F + of Q which is unramified at p. We
will make assumptions on p so that the unitary group G in the Shimura datum splits over
an unramified extension of Qp as GLn × Gm . The second case is that of the Siegel modular
varieties where the group in the Shimura datum is G = GSp2n . We will refer to this as the
symplectic case.
The moduli problem defining the integral model with Iw0 (p)-level structure is given in terms
of chains of isogenies of abelian schemes with certain additional structures. We write AGL
0
and AGSp
for the scheme representing this moduli problem in the unitary and symplectic
0
cases respectively. Then in these two cases, the moduli problem defining the integral model
with Iw1 (p)-level structure is given by also including choices of “Oort-Tate generators” for
certain group schemes associated with the kernels of the isogenies (see Section A.4.2 for the
GSp
notion of an Oort-Tate generator). Let AGL
denote the schemes representing these
1 and A1

moduli problems in each case respectively.
To study the singularities of AGL
and AGSp
we will construct ´etale local models.
1
1

Definition 0.1. Let X and M be schemes. We say that M is an ´etale local model of X if
there exists an ´etale cover V → X and an ´etale morphism V → M .

3

In order to describe our results in the unitary case, we begin by recalling the local model
of AGL
as constructed in [RZ, Chapter 3]. In this introduction, we assume for simplicity
0
that F + = Q. The local model in the general case will be a product of such local models
after an unramified base extension. As mentioned above, we also make assumptions so that
GQp = GLn,Qp × Gm,Qp . We can choose an isomorphism BQp ∼
= Mn (Qp ) × Mn (Qp ) so that
the minuscule cocharacter µ : Gm,Qp → GQp is identified with
µ(z) = diag(1n−r , (z −1 )r ) × diag((z −1 )n−r , 1r ),

1 ≤ r ≤ n − 1,

which we will write concisely as µ = (0n−r , (−1)r ). Then for a Zp -scheme S, an S-valued
loc
is determined by giving a diagram
of AGL
point of the local model MGL
0

ϕ0

OSn

F0

OSn

F1

ϕ1

ϕn−2

···

···

OSn

ϕn−1

Fn−1

OSn

Fn

where ϕi is given by the matrix diag((p−1 )i+1 , 1n−i−1 ) with respect to the standard basis, Fi
is an OS -submodule of OSn , and Zariski locally on S, Fi is a direct summand of OSn of rank
loc
, the determinants
r. With S = MGL
top

top

Fi →

top

Fi+1

top

OSn /Fi

and

→

OSn /Fi+1

determine global sections qi and qi∗ of the universal line bundles
−1

top

Qi =

Fi

top

⊗

−1

top

Fi+1

and

Q∗i

respectively.

4

=

OSn /Fi

top

⊗

OSn /Fi+1

As shown in [G¨or1], the special fiber of the local model can be embedded into the affine
loc
flag variety for SLn and identified with a disjoint union of Schubert cells. Let U ⊂ MGL

be an affine open neighborhood of the “worst point”, i.e. the unique cell which consists of
a single closed point, with U sufficiently small so that each Q∗i is trivial. Choosing such a
trivialization, we can then identify the sections qi∗ with regular functions on U .

Theorem 0.2. The scheme

∗
U1 = SpecU O[u0 , . . . , un−1 ]/(up−1
− q0∗ , . . . , up−1
0
n−1 − qn−1 )

is an ´etale local model of AGL
1 .

By loc. cit. we can take U = Spec(BGL ) where
BGL = Zp [aijk , i = 0, . . . , n − 1, j = 1, . . . , n − r, k = 1, . . . , r]/I

and I is the ideal generated by the entries of certain matrices. In this chosen presentation,
we will show that, up to a unit, qi∗ = ai+1
n−r,r for 0 ≤ i ≤ n − 1 where the upper index is taken
modulo n.
For the symplectic case, the integral model AGSp
is again given in terms of chains of isogenies
0
of abelian schemes with certain additional structures. Our construction of the local models
for AGSp
is similar to that of the unitary case. In particular, they are explicitly defined as
1
well.
It is also of interest to have certain resolutions of the integral model of the Shimura variety
with “nice” singularities, for example one which is semi-stable or locally toroidal. This

5

problem was considered in the case of Iw0 (p)-level structure by Genestier [Gen], Faltings
[Fal], de Jong [dJ1], and G¨ortz [G¨or3] among others. Using the explicitly defined local
model, and in particular the rather simple expression for qi∗ , we will construct a resolution
of AGSp
in the case n = 2. By a “nonreduced divisor with normal crossings” we mean a
1
divisor D such that in the completion of the local ring at every closed point, D is given by
Z(f1e1 · · · ftet ) where {f1 , . . . , ft } are part of a regular system of parameters and the integers
ei are greater than zero.

Theorem 0.3. Let A1 denote the moduli scheme for the Siegel modular variety of genus
2 with Iw1 (p)-level structure. There is a regular scheme A1 with special fiber a nonreduced
divisor with normal crossings that supports a birational morphism A1 → A1 .

Moreover, we will describe the irreducible components of A1 ⊗ Fp and how they intersect
using a “dual complex”, see Theorem 5.6.6 for details.
Let us outline the construction of A1 . We begin with the known semi-stable resolution
A0 → A0 [dJ1]. This gives a modification (i.e. proper birational morphism) A1 ×A0 A0 →
A1 . The scheme A1 ×A0 A0 is not normal. Let Z be the reduced closed subscheme of A0
whose support is the locus of closed points where all of the corresponding group schemes
are infinitesimal. Take the strict transform of Z with respect to the morphism A0 → A0
followed by the reduced inverse image of this with respect to the projection A1 ×A0 A0 → A0
and denote the resulting scheme by Z . Consider the modification given by the blowup of
A1 ×A0 A0 along Z :
BlZ (A1 ×A0 A0 ) → A1 ×A0 A0 .
We will see that BlZ (A1 ×A0 A0 ) is normal. In A0 , denote by W the unique irreducible component of the special fiber where each corresponding group scheme is generically isomorphic

6

to µp . Transform W via the morphisms

BlZ (A1 ×A0 A0 ) → A1 ×A0 A0 → A0 → A0

by taking the strict transform with respect to the first and third morphisms, and the reduced
inverse image with respect to the second morphism. Denote the resulting subscheme of
BlZ (A1 ×A0 A0 ) by W . We arrive at A1 by first blowing up BlZ (A1 ×A0 A0 ) along W and
then blowing up each resulting modification along the strict transform of W , stopping after
a total of p − 2 blowups. Carrying out the corresponding process on the local model, by
explicit computation we will show that the resulting resolution of the local model is regular
with special fiber a nonreduced divisor with normal crossings. It will then follow that A1 has
these properties as well. By keeping track of how certain subschemes transform in each step
of the above process, with much of this information coming from the explicit computation
of the modifications of the local model, we will be able to describe certain aspects of the
irreducible components of A1 ⊗ Fp as mentioned above.
In closing we mention that as this article was prepared, T. Haines and B. Stroh announced
a similar construction of local models in order to prove the analogue of the Kottwitz nearby
cycles conjecture. They relate their local models to “enhanced” affine flag varieties.
Finally, I would like to thank G. Pappas for introducing me to this area of mathematics and
for his invaluable support. I would also like to thank M. Rapoport for a useful conversation,
T. Haines and B. Stroh for communicating their results, and U. G¨ortz for providing the
source for Figure 5.1.4 to which some modifications were made.

7

Chapter 1
Shimura Varieties

In this section we review the definition of a Shimura datum, the group theoretic data which
is used to construct a Shimura variety. We specialize this data in the unitary and symplectic
cases.

1.1

Shimura datum

Let S denote ResC/R (Gm ), where Res(·) is the Weil restriction of scalars. Note that S(R) =
C× and S(C) = C× × C× and we have the homomorphism

S(R) → S(C) sending z → (z, z).

8

For a connected algebraic group H defined over R, a Cartan involution of H is an involution
θ of H as an algebraic group over R such that
H (θ) (R) := {g ∈ H(C) : g = θ(¯
g )}

is compact, where g¯ denotes complex conjugation. The following definition uses the language
introduced by Deligne in [Del].

Definition 1.1.1. A Shimura datum is a collection (G, {h} , K) where
• G is a reductive group defined over Q;
• {h} a G(R)-conjugacy class of homomorphisms of real algebraic groups S → GR ; and
• K is a sufficiently small compact open subgroup of G(Af )
such that the following conditions hold.
(SV1) For any h : S → GR , only the characters 1, z/¯
z , and z¯/z occur in the induced
representation of S on Lie(Gad )C .
(SV2) The adjoint action of h(i) induces a Cartan involution on the adjoint group of GR .
(SV3) The adjoint group Gad
R does not admit a factor H defined over Q such that the
projection of h on H is trivial.

Remark 1.1.2.
• For h ∈ {h}, h : S → GR , the action of g ∈ G(R) is given as follows. For an R-algebra
A, we define the homomorphism

(g · h)(A) : S(A) → GR (A)

sending

9

α → g · h(A)(α) · g −1

where we are identifying g ∈ G(R) with its image under G(R) → G(A).
• Condition (SV1) means the following. Given a homomorphism h : S → GR , we can
compose this with the adjoint representation GR → GL(Lie(Gad
R )) and then complexify
so that we have S(R) → S(C) → GL(Lie(Gad )C ) where the first homomorphism is as
described above. Thus for z ∈ S(R) = C× we have a natural action on Lie(Gad )C . The
condition is that
Lie(Gad )C = V 0 ⊕ V 1 ⊕ V −1
with

V 0 := v ∈ VC : z · v = v for all z ∈ C×
V 1 := v ∈ VC : z · v = zz −1 v for all z ∈ C×
V −1 := v ∈ VC : z · v = zz −1 v for all z ∈ C×

where on the right hand side the product is given by the natural action of C on the
complex vector space Lie(Gad )C .
• The condition that K is sufficiently small will be explained in Section 2.4.

1.2

PEL Shimura varieties

We now specialize to the case of PEL Shimura varieties. Fix once and for all a choice
√
i = −1.

Definition 1.2.1. A PEL Shimura datum is given by a tuple (B, ι, V, (·, ·), h0 , K) satisfying
the following conditions.

10

• B is a finite-dimensional semi-simple Q-algebra with positive involution ι.
• V = 0 is a finitely-generated left B-module.
• (·, ·) is a non-degenerate alternating form V × V → Q such that (bv, w) = (v, bι w) for
all b ∈ B and v, w ∈ V .
• h0 is given as follows. The form (·, ·) determines an involution ∗ on EndB (V ) where
for f ∈ EndB (V ), f ∗ ∈ EndB (V ) is the unique element such that
(f (x), y) = (x, f ∗ (y)) for all x, y ∈ V.

In particular, for b ∈ B we have b∗ = bι . Here were are making the identification
b ∈ EndB (V ) by left multiplication. We require that h0 : C → EndB⊗R (V ⊗ R) be an
R-algebra homomorphism satisfying h0 (z) = h0 (z)∗ for all z ∈ C and is such that the
symmetric bilinear form
(·, h0 (i)·) : VR × VR → R
is positive definite.
• We define the Q-group G on a Q-algebra R by
G(R) = g ∈ GLB⊗R (V ⊗ R) : g ∗ g ∈ R× .
We require K be a sufficiently small compact open subgroup of G(Apf ).

Remark 1.2.2. In the above definition, given a PEL Shimura datum (B, ι, V, (·, ·), h0 , K)
we defined the involution ∗ on EndB (V ) and the algebraic group G defined over Q. We will
also define the objects h, µh , E, and E below. Henceforth we will implicitly associate these
objects with a PEL Shimura datum.

11

We define h to be the homomorphism of real algebraic groups
z→z −1

h

0
h : C× −−−−→ C× −→
G(R)

and the cocharacter µh : Gm,C → GC as follows. By base change hC : SC → GC where for
any C-algebra R we have
∼

SC (R) = ResC/R (Gm )(R) = Gm,C (C ⊗R R) −
→ Gm,C (R) × Gm,C (R).
∼

The isomorphism above is induced by C ⊗R R −
→ R × R sending z ⊗ r → (zr, zr). We thus
have SC ∼
= Gm,C × Gm,C and we define µh by restricting the map hC : SC → GC to the factor
of SC corresponding to the identity (as opposed to complex conjugation).
Define the field E to be the field of definition of the G(R)-conjugacy class {µh }.

We will

postpone the definition of E until after the proof of Proposition 1.2.4.

Remark 1.2.3. Let k be an algebraically closed field. For B a semisimple k-algebra with
involution ι, (B, ι) is isomorphic to a product of the following three types [Kot2, Section 1].

(A)
(C)

Mn (k) × Mn (k), (a, b)∗ = (bt , at )
Mn (k),

b∗ = bt


(BD)

Mn (k),

b∗ = Jbt J −1 ,



0 −I 
J =

I 0

Now suppose B is a semisimple Q-algebra with involution ι and center a field F . Set
F0 = {x ∈ F : x∗ = x}. Then for all Q-homomorphisms ρ : F0 → Q, the Q-algebra with
involution (B ⊗F0 ,ρ Q, ι) is a product of types (A), (C), or (BD). We will refer to this multiset

12

of types as the type of (B ⊗F0 ,ρ Q, ι). Since Gal(F0 /Q) acts transitively on the collection
(B ⊗F0 ,ρ Q, ι)ρ by isomorphisms, we define the type of (B, ι) to be the type of any extension
(B ⊗F0 ,ρ Q, ι).
From here on we will assume that (B, ι) is a product of types (A) or (C).

Proposition 1.2.4. Given a PEL Shimura datum (B, ι, V, (·, ·), h0 , K), the induced (G, h, K)
(see Remark 1.2.2) is a Shimura datum.

Proof. To see that G is reductive, we consider GQ¯ . Then (B, ι) decomposes into a product
where each factor is of type (A) or (C). Hence GQ¯ decomposes into a product of reductive
groups, each being GLn × Gm or GSpn depending on whether the corresponding factor of
(B, ι) is of type (A) or (C).
(SV1) Set J = h(i) ∈ G(R) ⊂ GLB⊗R (V ⊗R). The action of J makes V ⊗Q R into a complex
vector space with complex structure h : C → EndR (V ⊗Q R). Now consider the the
action of C× on the Lie algebra of GLC (VC ) through h and the adjoint action. Note
that VC = V + ⊕ V − where
V + = {v ∈ VC : Jv = iv}
V − = {v ∈ VC : Jv = −iv}

which induces the decomposition

Hom(VC , VC ) = Hom(V + , V + ) ⊕ Hom(V + , V − ) ⊕ Hom(V − , V + ) ⊕ Hom(V − , V − ).

The adjoint action of h(z) on Hom(VC , VC ) is by conjugation, i.e. for g : VC → VC and

13

z = a + bi ∈ C× , we have that h(a + bi) · g = (a + bJ) ◦ g ◦ (a + bJ)−1 . Therefore in the
decomposition above, h(z) acts as 1 on Hom(V + , V + ), zz −1 on Hom(V + , V − ), z −1 z on
Hom(V − , V + ), and 1 on Hom(V − , V − ) as required.
(SV2) We must show that the group

(Gad )(h(i)) (R) := g ∈ Gad (C) : gg ∗ = 1, h(i)−1 gh(i) = g
is compact, where g denotes complex conjugation. For g ∈ Gad (C), denote by g → g
the involution given by the tensor product of ∗ on EndB (V ) and complex conjugation
on C, i.e. for α ⊗ λ ∈ EndB⊗C (V ⊗ C) we have
(α ⊗ λ) = α∗ ⊗ λ.

As ∗ and complex conjugation are both positive involutions, it follows that g → g
is also positive [Kot2, Lemma 2.3]. Note that (Gad )(h(i)) (R) is a closed subgroup of
g ∈ Gad (C) : gg = 1 , namely (Gad )(h(i)) (R) is given by enforcing the condition gg ∗ =
1. Since g → g is positive, by [Kot2, Lemma 2.2] there is a faithful positive definite
Hermitian Gad (C)-module W . Denote its Hermitian form by (·, ·)W . Now the transpose
defined by (·, ·)W is precisely g → g and thus (Gad )(h(i)) (R) can be viewed as a closed
subgroup of the orthogonal group with respect to (·, ·)W . As this orthogonal group is
compact, it therefore follows that (Gad )(h(i)) (R) is compact.
(SV3) If Gad
R has a Q-factor on which h is trivial, then the form (·, h0 (i)·) on VR × VR could
not be positive definite since (·, ·) is alternating.

Remark 1.2.5. From (SV1), we get that µh induces a decomposition VC = V + ⊕ V − where

14

µh acts as z −1 on V + and 1 on V − . We define E to be the finite extension of E over which
this decomposition is defined. Fix an isomorphism BC ∼
= Mn (C) × Mn (C) so that µh (z) is
identified with
diag(1n − r, (z −1 )r ) × diag(z −1 )n−r , 1r ).
We will write this as µh = (0n−r , (−1)r ).

1.2.1

Unitary case

We now give a specialized set of data for which the group G in the induced Shimura datum
is a unitary group.

Definition 1.2.6. A unitary PEL Shimura datum is a tuple (D, ∗, h0 ) where
• D is a finite dimensional division algebra with center a field F , where F is an imaginary
quadratic extension of some totally real field F + /Q;
• ∗ is an involution of D which induces on F the nontrivial element of Gal(F/F + ); and
• h0 : C → D ⊗Q R is an R-algebra homomorphism such that h0 (z)∗ = h0 (z) and the
involution x → h0 (i)−1 x∗ h0 (i) is positive.

A datum (D, ∗, h0 ) induces a PEL datum (B, ι, V, (·, ·), h0 , K), up to a choice of K, as follows.
Set B = Dopp and V = D where we view V as a left B-module using right multiplications.
That is, for v ∈ D and b ∈ Dopp we define b · v = vb, where on the right hand side the
multiplication is given by the multiplication in D. Then EndB (V ) can be identified with D
using left multiplications. It remains to define the involution ι on B = Dopp and the pairing
(·, ·) on V = D. We will use the following lemma.

15

Lemma 1.2.7. There exists ξ ∈ D× such that ξ ∗ = −ξ and the involution x → ξx∗ ξ −1 is
positive. With such a ξ, the alternating pairing (·, ·) : D × D → Q defined by
(x, y) = TrD/Q (xξy ∗ )

is nondegenerate. We may also choose ξ, still subject to the above conditions, such that the
pairing (·, h0 (i)·) is positive definite.

Proof. Let σ be any involution of D of the second kind, meaning σ restricts to the nontrivial
element of Gal(F/F + ). Then ∗ ◦ σ fixes F . Thus we may apply Skolem-Noether to the
F -algebra homomorphisms ∗ ◦ σ : D → D and IdD : D → D. Hence there is a unit u
such that (∗ ◦ σ)(d) = u−1 du for all d ∈ D. Applying the involution ∗ to both sides gives
σ(d) = u∗ d∗ (u−1 )∗ for all d ∈ D. As u∗ is also a unit of D, we will replace u with u∗ and
write this as σ(d) = ud∗ u−1 . Then for all d ∈ D,

σ(σ(d)) = u(ud∗ u−1 )∗ u−1 = u(u−1 )∗ du∗ u−1 .

The condition that σ is an involution implies that for all d ∈ D, u(u−1 )∗ du∗ u−1 = d. This
condition is satisfied if and only if u(u∗ )−1 lies in the center F . Conversely, for any unit u
such that u(u∗ )−1 ∈ F , we have that d → ud∗ u−1 is an involution of the second kind.
By [Mum, pg. 201-2], positive involutions of the second kind exist. So there is a u ∈ D such
that d → ud∗ u−1 is a positive involution. Then

u(u∗ )−1 · (u(u∗ )−1 )∗ = u(u∗ )−1 · (u−1 u∗ ) = uu−1 u∗ (u∗ )−1 = 1

where the second to last equality is using that u(u∗ )−1 is in the center of D. Since ∗ restricted

16

to F is the nontrivial element in Gal(F/F + ), this says precisely that NF/F + (u(u∗ )−1 ) = 1
where NF/F + is the norm of F over F + . By Hilbert’s Theorem 90 [Hil, Theorem 90], there
exists f ∈ F × such that u(u∗ )−1 = f ∗ f −1 . Using this equation we have (uf )∗ = f ∗ u∗ = uf .
Finally, there exists ε ∈ F × such that ε∗ = −ε (take any nonzero element of F \ F + and
complete the square). We (temporarily, see below) set ξ = εf u, and denote the involution
d → ξd∗ ξ −1 by ι. Note that since εf is in the center of D, this is the same involution as
d → ud∗ u−1 and hence is positive and of the second kind.
Since TrD/Q is invariant with respect to ∗ [Kot2, Lemma 2.7], we have that for all x, y ∈ D
(x, y) = TrD/Q (xξy ∗ ) = TrD/Q ((xξy ∗ )∗ ) = − TrD/Q (yξx∗ ) = −(y, x)

so the claimed pairing is indeed alternating. It is also non-degenerate because the pairing
(x, y) → TrD/Q (xy) is non-degenerate, ∗ is bijective, and ξ is a unit.
We claim that (·, h0 (i)·) is either positive or negative definite. To see this, fix an isomorphism
∼

D ⊗F0 R −
→ Mn (C)
t

such that the involution ι goes over to the standard involution X → X on Mn (C). Denote
by H the image of ξh0 (i)−1 under this isomorphism. Since ξh0 (i)−1 is invertible in D, H is
invertible in Mn (C). Furthermore
ι(ξh0 (i)−1 ) = ξ(ξh0 (i)−1 )∗ ξ −1 = ξh0 (i)(−ξ)ξ −1 = −ξh0 (i) = ξh0 (i)−1
t

and thus H = H, i.e. H is Hermitian. Unwinding the definition of the pairing (·, h0 (i)·) we

17

have for all x, y ∈ DR
(x, h0 (i)y) = TrDR /R (xξy ∗ h0 (i)−1 ) = TrDR /R (x(ξy ∗ ξ −1 )ξh0 (i)−1 ) = TrDR /R (xι(y)ξh0 (i)−1 )

and hence under the fixed isomorphism this pairing becomes
t

X, Y = TrMn (C)/R (XY H)

for X, Y ∈ Mn (C).

Let U ∈ Mn (C) be a unitary matrix such that U −1 HU = D where D = diag(λ1 , · · · , λn )
is some diagonal matrix with λi ∈ C. In fact, since H is Hermitian and hence has real
eigenvalues, we have λi ∈ R for all i. Then since the involution x → h0 (i)−1 x∗ h0 (i) is
positive by hypothesis, we have

TrDR /R (xh0 (i)−1 x∗ h0 (i)) > 0 for all 0 = x ∈ DR .

Now

xh0 (i)−1 x∗ h0 (i) = xh0 (i)−1 ξ −1 (ξx∗ ξ −1 )ξh0 (i) = x(ξh0 (i)−1 )−1 ι(x)(ξh0 (i)−1 )

where the last equality is using that −h0 (i) = h0 (i)−1 . Thus under the fixed isomorphism
t

TrMn (C)/R (XH −1 X H) > 0 for all 0 = X ∈ Mn (C).

18

As U is unitary,
t

t

TrMn (C)/R (XH −1 X H) = TrMn (C)/R ((U XU −1 )H −1 (U X U −1 )H)
t

= TrMn (C)/R (X(U −1 H −1 U )X (U −1 HU ))
t

= TrMn (C)/R (XD−1 X D)
> 0.

Finally, we calculate
n
t

TrMn (C)/R (XD−1 X D) = 2

|xij |2
i,j

λj
λi

where X = (xij ).

Since this last quantity must be positive for any X ∈ Mn (C), it must be that every λi ∈ R
has the same sign.
We now show that ·, · is either positive or negative definite.

X, X = U X, U X
t

= TrMn (C)/R (U XX U −1 H)
t

= TrMn (C)/R (XX U −1 HU )
t

= TrMn (C)/R (XX D)

Letting X = (xij ), one can calculate that
n
t

TrMn (C)/R (XX D) = 2

n

|xij |2 λr ,

xij xij λr = 2
i,j=1

i,j=1

and with all λi ’s possessing the same sign, the claim follows. Therefore, by possibly replacing
ξ with −ξ, we have that ·, · and hence (·, h0 (i)·) is positive definite.

19

We take the involution ι and the pairing (·, ·) defined by ξ as in the lemma.

Proposition 1.2.8. Let (D, ∗, h0 ) induce (B, ι, V, (·, ·), h0 ) as described above. This datum
satisfies all the conditions of being a PEL datum, up to a choice of K.

Proof. All claims have already been shown except that (·, ·) is a Hermitian form. It remains
to see that (bx, y) = (x, bι y) for all x, y ∈ D and b ∈ Dopp . Regarding b as an element of D,
we need to show (xb, y) = (x, ybι ). Recall that ξ is chosen so that ξ ∗ = −ξ.

(x, ybι ) = (x, yξb∗ ξ −1 )
= TrD/Q (xξ(yξb∗ ξ −1 )∗ )
= TrD/Q (xξ(−ξ −1 )b(−ξ)y ∗ )
= TrD/Q (xbξy ∗ )
= (xb, y)

1.2.2

Symplectic case

In this section we will describe the PEL datum (B, ι, V, (·, ·), h0 , K) for the Siegel modular
varieties. The data here is given by
• B = Q;
• ι is the trivial involution on Q;
• V = Q2n ;

20

• (·, ·) is the alternating pairing on V given by the 2n × 2n matrix






J =

Jn 


−Jn

where





Jn = 




1
1
..
1

.











• h0 : C → EndR (VR ) is the unique R-algebra homomorphism with h0 (i) = J; and
• K is a sufficiently small compact open subgroup of G.

Proposition 1.2.9. The datum (B, ι, V, (·, ·), h0 , K) described above is a PEL datum.

Proof. B is a finite semi-simple Q-algebra and ι, being trivial, is positive since α2 > 0 for
α ∈ R. The pairing (·, ·) induced by J is certainly non-degenerate and alternating, and the
equality (bv, w) = (v, bι w) for v, w ∈ V and b ∈ B is an immediate consequence of (·, ·) being
bilinear.
It remains to show that h0 (z) = h0 (z)∗ , where ∗ is the involution on EndB (V ) induced by
(·, ·), and (·, J·) is positive definite. Note that h0 (a + bi) = a + bJ and h0 (a + bi) = a − bJ,
where on the right hand side we are writing a and b as the linear map given as scalar
multiplication by a and b respectively. The involution ∗ is given by






t

t



A B 
 D − B

→

C D
− tC tA
where t A denotes the transpose of A along the anti-diagonal. Thus it follows that (a+bJ)∗ =
(a−bJ). One can compute the pairing (·, J·) in standard coordinates and see that it is positive
definite.

21

Remark 1.2.10. From the definition of h0 , we have µ = (0n , (−1)n ).

22

Chapter 2
Integral and local models of A0
In this chapter we describe the integral and local models for PEL Shimura varieties where
the level subgroup at an odd rational prime p is given by a parahoric subgroup. We then
specialize the description of the integral and local models to the unitary and symplectic
cases where the level subgroup at p is given by an Iwahori subgroup. Finally we prove the
representability of the moduli problems defining the integral models in these two special
cases.

2.1

PEL case

In order to define the integral model, we first need to specify additional integral data.

Definition 2.1.1. Fix an odd rational prime p. An integral PEL Shimura datum is a tuple

(B, ι, V, (·, ·), h0 , OB , L, K p )

23

where
• (B, ι, V, (·, ·), h0 , K) is a PEL Shimura datum with K = Kp K p (see below);
• OB is a Z(p) -order in B whose p-adic completion OB ⊗ Zp is a maximal order in BQp
that is stable under ι;
• L is a self-dual multichain of OB ⊗ Zp -lattices in VQp , where duality is with respect to
the pairing induced by (·, ·);
• Kp = Aut(L) ⊂ G(Qp ); and
• K p ⊂ G(Apf ) is an open compact subgroup.
Furthermore, if Kp ⊂ G(Qp ) is an Iwahori subgroup, we say that the integral datum is of
Iwahori-type.

Remark 2.1.2. We will not recall the definition of a self-dual multichain of OB ⊗Zp -lattices,
see [RZ, Defintions 3.1, 3.4, 3.13]. However in the two cases we consider, we will make the
definition of L explicit.

For the remainder of this section we fix an odd rational prime p and an integral PEL Shimura
datum. Recall that associated with a PEL Shimura datum is the reflex field E described in
Remark 1.2.2. Fix once and for all embeddings Q → C and Q → Qp , and let p denote the
corresponding prime of OE lying over p. Set E = Ep , E = Ep , and G = GQp . Using these
fixed embeddings, we have that the conjugacy class µ : Gm,C → GC induces, by abuse of
notation, the conjugacy class µ : Gm,Qp → GQp .
Definition 2.1.3. The moduli functor A0 is defined over Spec(OE ) as follows. For an OE ¯ η¯) up to isomorphism
scheme S, the set A0 (S) is given by the collection of tuples ({AΛ } , i, λ,
where

24

• {AΛ }Λ∈L is an L-set of abelian schemes with an action of OB
i : OB ⊗ Z(p) → End(A) ⊗ Z(p) ;

• λ is a Q-homogeneous principal polarization of the L-set A; and
∼

• η : V ⊗ Apf −
→ H1 (A, Apf ) mod K p is a K p -level structure that respects the bilinear
forms on both sides up to a constant in (Apf )× (see below)
such that the determinant condition of Kottwitz holds: for b ∈ OB and Λ ∈ L we have

detOS (b|Lie(AΛ )) = detE (b|V + )

(see below).

¯ η¯) and ({A } , i , λ
¯ , η¯ ) is an isoAn isomorphism between two S-valued points ({AΛ } , i, λ,
Λ
∼
¯ i, and η¯ to λ
¯ , i , and η¯ respectively.
morphism of the L-sets {AΛ } −
→ {AΛ } which carries λ,

Remark 2.1.4. We will not recall the definition of an L-set of abelian schemes or a Qhomogeneous principal polarization, see [RZ, Definitions 6.5,6.7]. We will make these notions
explicit in the two cases we consider (Definitions 2.2.1 and 2.3.1).

K p -level structures

Let (A, λ, i) be a polarized abelian scheme over S with OB -action by i as in the definition
above. Fix a geometric point s of S and consider an isomorphism
∼

η : V ⊗ Apf −
→ H1 (As , Apf ).

25

The pairing (·, ·) on V induces a pairing on V ⊗ Apf , we again denote this by (·, ·). The
polarization λ induces the pairing

·, · : H1 (As , Apf ) × H1 (As , Apf ) → Apf (1).
p
Noncanonically, Apf (1) ∼
= Af , where the isomorphism is well-defined up to some scalar mul-

ˆ (p) )× . We say that η respects the pairings if there exists cη ∈ (Ap )× such that for
tiple in (Z
f
all x, y ∈ V ⊗ Apf we have
(x, y) = cη η(x), η(y) .
Now consider the OB -action on A given by i. This induces an OB -action on H1 (A, Apf )
which we again denote by i. Then we say that η respects the OB -action if for all b ∈ OB
and x ∈ V ⊗ Apf we have
i(b) · η(x) = η(b · x).

Proposition 2.1.5. If η : V ⊗ Apf → H1 (A, Apf ) respects the pairings and OB -action, then
so does η ◦ g for g ∈ G(Apf ).

Proof. This follows immediately from the definition of G(Apf ) consisting of elements g ∈
GLB⊗Apf (V ⊗ Apf ) such that there exists c ∈ (Apf )× with (gx, gy) = c · (x, y) for all x, y ∈
V ⊗Q Apf .

Definition 2.1.6. Let S be an OE -scheme and (A, λ, i) be a principally polarized abelian
scheme over S with OB -action by i. Then a K p -level structure on (A, λ, i) is a choice of
geometric point s in S for each connected component of S and a K p -orbit η of isomorphisms
∼

η : V ⊗ Apf −
→ H1 (As , Apf ) respecting the pairings and OB -action such that the orbit is fixed
under the action of π1 (S, s).

26

Here is what is meant by the π1 -action in the definition above. For this, we suppose S is
connected and choose a geometric point s of S. View A[ n ] as a locally constant constructible
´
etale

Z/ n Z-sheaf on the ´etale site Se´t . That is, A[ n ](U −−→ S) = HomS (U, A[ n ]). Denote by
Fs the functor from finite ´etale covers of S to sets given by
f

Fs (T →
− S) = {geometric points t of T : f (t) = s}

with the monodromy action of π1 (S, s) on Fs (T → S). Then we have the canonical identification A[ n ]s = Fs (A[ n ], S) = As [ n ].
Given ϕ ∈ π1 (S, s), we have ϕ(A[ n ] → S) : Fs (A[ n ] → S) → Fs (A[ n ] → S) and this gives
the isomorphism
ϕ

[ϕ] : As [ n ] = Fs (A[ n ] → S) −
→ Fs (A[ n ] → S) = As [ n ].

Now let s be another geometric point of the (connected) scheme S. Then there exists an
∼

isomorphism Φ : Fs −
→ Fs of the fiber functions and hence as above, we get an isomorphism
∼

[Φ] : As [ n ] −
→ As [ n ].

Taking the inverse limit over n ∈ Z>0 of these isomorphisms we have [ϕ] : T (As ) → T (As )
and [Φ] : T (As ) → T (As ). Taking the product over all

= p and tensoring with Q gives

[ϕ] : H1 (As , Apf ) → H1 (As , Apf ) and [Φ] : H1 (As , Apf ) → H1 (As , Apf ).

Proposition 2.1.7. Let (A, λ, i) a principally polarized abelian scheme over an OE -scheme
S with OB -action. The collection of geometric points of S where there exists a K p -level

27

structure is a union of connected components of S.

Proof. It suffices to show that in a connected component of S, if a K p -level structure exists
at a single geometric point, then it exists at all geometric points. So let S be connected,
suppose there is a K p -level structure at some geometric point s, and let s be any geometric
∼

point of S. Choose an isomorphism Φ : Fs −
→ Fs of the fiber functions. This induces
∼

the isomorphism [Φ] : H1 (As , Apf ) −
→ H1 (As , Apf ) described above. We define η to be the
collection of symplectic similitudes
[Φ]

η

V ⊗ Apf →
− H1 (As , Apf ) −→ H1 (As , Apf )
for all η ∈ η. That this is a K p -orbit follows immediately from the fact that η is. To see
that η is also fixed under the action of π1 (S, s ), first note that the map
π1 (S, s) → π1 (S, s ) sending ϕ → Φ ◦ ϕ ◦ Φ−1 ∈ π1 (S, s )

is an isomorphism. Letting Φ ◦ ϕ ◦ Φ−1 be an arbitrary element of π1 (S, s ), its action on an
element Φ ◦ η ∈ η is given by
η

Φ

Φ−1

ϕ

Φ

V ⊗ Apf →
− H1 (As , Apf ) −
→ H1 (As , Apf ) −−→ H1 (As , Apf ) −
→ H1 (As , Apf ) −
→ H1 (As , Apf ).
This is also an element of η because ϕ ◦ η ∈ η.

28

The determinant condition of Kottwitz

Recalling the decomposition VC = V + ⊕V − , we have that V + is a BC -module. Thus it makes
sense to consider
detV + : VBC → A1C
where VBC is the functor on C-algebras sending S to S ⊗C BC (see Section A.1 with R = C
and A = BC ).

Proposition 2.1.8. detV + is defined over OE .

Proof. The B-module structure of V + gives a Q-algebra homomorphism ϕ : B → EndC (V + ).
Choosing a C-basis of V + gives an embedding of EndC (V + ) → Mm (C) where m is the
dimension of V + over C. With B finite-dimensional, we in fact have ϕ0 : B → Mn (E) for
some finite extension E of Q. By enlarging E if necessary, assume E ⊂ E. Set W = Em
with ϕ0 giving W the structure of a BE -module. Note that W ⊗E C is isomorphic to V + as
BC -modules.
Choose a basis of W over E and let M be the OB ⊗Z OE -module generated by this basis.
Then we can recover the BE -module W by M ⊗OE E. Note that M is finite and locally free
over OE , so we may consider detM . Since M ⊗OE E ⊗E C = V as BC -modules, we have
detM ⊗O

E

C

= detV + .

Therefore detV + is defined over OE .
It remains to show that detV + is defined over E, as then it is defined over OE ∩ E = OE .
It is therefore sufficient to show that detC (x|V + ) ∈ E for all x ∈ BE . Since we have

29

TrC (x|V + ) ∈ E, the result will follow if we are able to show that we can express detC (x|V + )
as a polynomial in
TrC (x|V + ), TrC (x2 |V + ), . . . , TrC (xm |V + ) .

Let pi ∈ Z[X1 , . . . , Xm ] denote the ith power sum, so p1 = X1 + X2 + · · · + Xm , p2 =
2
, etc. Let ei denote the ith complete symmetric polynomial.
X12 + X1 X2 + · · · + Xm−1 Xt + Xm

With λ1 , . . . , λm denoting the eigenvalues of x, we have
TrC (xi |V + ) = pi (λ1 , . . . , λm )
detC (x|V + ) = e1 (λ1 , . . . , λm ).
By [Mac, I.2.12], Z[p1 , . . . , pm ] = Z[e1 , . . . , em ] and therefore e1 can be expressed as a polynomial in p1 , . . . , pm , giving the result.

Let S be an OE -scheme, b1 , . . . , bt be a set of generators of OB as a Z(p) -module, and
(A, λ, i, η) be a principally polarized abelian scheme over S equipped with an OB -action and
K p -level structure. The action i : OB ⊗ Z(p) → End(A) ⊗ Z(p) induces an action of OB ⊗ Z(p)
on Lie(A), a locally free OS -module. Thus on each affine open U ⊂ S, we have detLie(A) (U ) ∈
Γ(U, OU )[X1 , . . . , Xt ]. Since the determinant respects localization (Proposition A.1.2), these
sections glue to define
detLieA ∈ Γ(S, OS )[X1 , . . . , Xt ].
Likewise, from the above proposition the OB -action on V + gives

detV + ∈ OE [X1 , . . . , Xt ].

By applying the ring homomorphism OE → Γ(S, OS ) to the coefficients of detV + , we can

30

compare these two determinants. Then the determinant condition is that detV + = detLie(A) .

Local model

Definition 2.1.9. A local model of a scheme X is a scheme M such that there exists an
´etale cover V → X and an ´etale morphism V → M .

For the integral models described above, [RZ, Chapter 3] constructs a local model diagram.

A0
Ψ

Φ

M loc

A0

1
This gives, in particular, a local model of the integral model. To construct A0 , let HdR
(Ai )∨

denote the OS -dual of the de Rham cohomology sheaf (see Section A.5). It is a locally free
1
OS -module of rank 2n2 [BBM, Section 2.5]. Then (HdR
(Ai )∨ )i gives a polarized multichain

of OB ⊗Zp OS -modules of type (L) in the sense of [RZ, Definition 3.14]. Define A0 to be the
OE -scheme that represents the functor defined as follows. We associate with an OE -scheme
¯ η¯, {γi }) up to isomorphism where ({Ai } , λ,
¯ η¯) ∈ A0 (S) and
S the set of tuples ({Ai } , λ,
∼

1
→ Λi ⊗Zp OS
γi : HdR
(Ai )∨ −

is an isomorphism of polarized multichains of OB ⊗Zp OS -modules. Then we define the
morphism
Φ : A0 → A0

¯ η¯, {γi }) → ({Ai } , λ,
¯ η¯).
by ({Ai } , λ,

31

Proposition 2.1.10. [RZ, Theorems 3.11, 3.16] Let L = {Λ} be a self-dual multichain of
OB ⊗ Zp lattices in V in the sense of loc. cit. Let S be any Zp -scheme where p is locally
nilpotent. Then any polarized multichain {MΛ } of OB ⊗Zp OS -modules of type (L) is locally
(for the ´etale topology on S) isomorphic to the polarized multichain Λ ⊗Zp OS .
Moreover, the functor sending

T → Isom({MΛ ⊗ OT } , {Λ ⊗ OT }),

is represented by a smooth affine scheme over S.

The proposition holds for any Zp -scheme S [Pap2, Theorem 2.2]. Let G be the smooth affine
S-group scheme given by

G(T ) = Aut({Λ ⊗ OT }) for an S-scheme T.

It follows from the proposition above that Φ : A0 → A0 is a smooth surjective G-torsor.

Definition 2.1.11. [RZ, Definition 3.27] With S an OE -scheme, an S-valued point of M loc
is given by the following data.
(i). A functor from the category L to the category of OB ⊗Zp OS -modules on S

Λ → ωΛ ,

Λ ∈ L.

(ii). A morphism of functors ψΛ : ωΛ → Λ ⊗Zp OS .
We require the following conditions are satisfied:

32

(i). the morphisms ψΛ are inclusions;
(ii). the quotient tΛ := Λ ⊗Zp OS /ψ(ωΛ ) is a locally free OS -module of finite rank. For the
action of OB on tΛ , we have the Kottwitz condition

detOS (b|tΛ ) = detE (b|V + ),

b ∈ OB ; and

(iii). for each Λ ∈ L, ωΛperp = ωΛ⊥ where
ωΛperp = y ∈ Λ⊥ ⊗ OS : (x, y) = 0 for all x ∈ ωΛ .

Remark 2.1.12. The above definition of the local model is the subobject variant. Denoting
ΛS = Λ ⊗ OS , the moduli problem remains the same if one replaces ωΛ and the injective
morphisms ψΛ : ωΛ → ΛS with tΛ and surjective morphisms ϕΛ : ΛS → tΛ . In such a case,
condition (iii) can be restated as follows. For each Λ ∈ L the composition
ψ

(·,·)
ϕ
ˆΛ
Λ⊥
tˆΛ −→
ΛS −−→ Λ⊥
S −−→ tΛ⊥

is zero. Here ˆ· = HomS (·, OS ).

Note that G acts on the local model by acting on ψΛ through its natural action on Λ ⊗ OS .
For an abelian scheme A/S we have the Hodge filtration (Proposition A.5.1)

1
0 → ωAˆ → HdR
(A)∨ → Lie(A) → 0.

¯ η¯, {γi }) ∈ A0 (S) a collection of injective
We can now associate with each point ({Ai } , λ,
morphisms
∼

1
0 → ωAˆi → HdR
(Ai )∨ → Λi ⊗Zp OS
γi

33

and this defines the map Ψ in the following diagram.

A0
Ψ

Φ

M loc

A0

Proposition 2.1.13. [RZ, Chapter 3] The diagram above is a local model diagram. Specifically the morphisms Φ and Ψ are smooth, Φ is surjective, and ´etale locally A0 ∼
= M loc : there
exists an ´etale cover V → A0 and a section s : V → A0 of Φ such that Ψ ◦ s is ´etale.
Also the morphism Φ is a torsor for the smooth affine group scheme G and Ψ is G-equivariant.

2.2

Unitary case

We now specialize the moduli problem of Definition 2.1.3 defining the integral model to
the unitary case. Let (D, ∗, h0 ) be a unitary datum as in Definition 1.2.6. As described in
Section 1.2.1, this induces the PEL Shimura datum (B, ι, V, (·, ·), h0 , K) up to the choice of
K. As we are considering the split unitary case, we make the following two assumptions on
the odd rational prime p.
(i). (p) is unramifed in F + and each factor of (p) in F + splits in F .
(ii). DQp splits.
In view of the first assumption, write (p) =

Fpj × Fp∗j

FQp =

j

pj in F + and pj = pj p∗j in F . Then

Dpj × Dp∗j

making DQp =

j

j

34

where Dpj and Dp∗j are respectively a central simple Fpj and Fp∗j algebra for each j. Recalling
∼

that ∗ induces on F the nontrivial element of Gal(F/F + ), we have ∗ : Dpj −
→ Dpopp
∗ . The
j
second assumption means Dpj ∼
= Mn (Fpj ) for every j.
The splitting of DQp makes
G=

Gj
j

where each factor is given on a Qp -algebra R as
Gj (R) = (x1 , x2 ) ∈ (Dpj × R)× × (Dp∗j × R)× : x1 = c(x∗2 )−1 for some c ∈ R× .
Thus Gj ∼
= GLn,Qp × Gm,Qp . Finally we define µj : Gm,Qp → Gj,Qp by
= Dp×j × Gm,Qp ∼
composing µ : Gm,Qp → GQp with the jth projection.
With these decompositions, by taking the product over all j it suffices to describe the order
OB and lattice chain L on each factor. Set Dj = Dpj × Dp∗j and fix an isomorphism
∼

→ Mn (Fpj ) × Mn (Fpj )
Dpj × Dp∗j −
such that the involution ι becomes (X, Y ) → (Y t , X t ). With ξ as in Lemma 1.2.7, set
(χt , −χ) to be the image of ξ under this isomorphism where χ ∈ GLn (Fpj ). Then the pairing
(·, ·) becomes
(X1 , X2 ), (Y1 , Y2 ) = TrDj /Qp (X1 Y2t χt , −X2 Y1t χ).
Letting πj be a uniformizer of OFpj , the OFpj -lattice chain Lj in Dpj × Dp∗j is given by
Λj,i = diag((πj−1 )i , 1n−i )Mn (OFpj ) and Λ∗j,i = χ−1 diag(1n−i , (πj−1 )i )Mn (OFpj )

35

where again we are using the fixed isomorphism above. Of course the description of Λj,i and
Λ∗j,i is independent of the choice of uniformizer πj . By definition we have
(Λj,0 ⊕ Λ∗j,0 ) ⊂ (Λj,1 ⊕ Λ∗j,1 ) ⊂ · · · ⊂ (Λj,n ⊕ Λ∗j,n ) = πj−1 (Λj,0 ⊕ Λ∗j,0 )

and one can compute
(Λj,i ⊕ Λ∗j,i )⊥ = Λj,−i ⊕ Λ∗j,−i .
Recalling that Bj = Djopp , take OBj ⊂ Bj to be the unique maximal Z(p) -order such that
under the fixed isomorphism we have
∼

OBj ⊗ Zp −
→ Mnopp (OFpj ) × Mnopp (OFpj ).
Then it is immediate that Λj,i ⊕ Λ∗j,i is an OBj ⊗ Zp -lattice and OBj ⊗ Zp is invariant under
ι.
Set L = (Λi )i with Λi =

j

Λj,i ⊕ Λ∗j,i and OB =

j

OBj . Finally take K = K p Kp where

Kp = Aut(L) and K p is a sufficiently small open compact subgroup of G(Apf ). With this data,
the moduli problem for the integral model given in Definition 2.1.3 becomes the following.

GL
Definition 2.2.1. For an OE -scheme S, AGL
0 (S) = A0,K p (S) is the collection of tuples

¯ i, η¯) up to isomorphism where
(A• , λ,
• A• is a chain of abelian schemes
α

α

αn−1

0
1
· · · → A0 −→
A1 −→
· · · −−−→ An → · · ·

of relative dimension n2 over S, indexed by L by setting Ai = AΛi for i ∈ Z;

36

• Each Ai is equipped with an OB -action i : OB ⊗ Z(p) → End(Ai ) ⊗ Z(p) ;
¯ is a Q-homogeneous class of principal polarizations. That is, λ
¯ is a collection of
• λ
isogenies λi : Ai → Aˆ−i
···

α−2

A−1

making the diagram
α−1

λ−1
···

α1∨

Aˆ1

α0

A0

A1

λ0
α0∨

Aˆ0

α1

···

αn−1

λ1
∨
α−1

Aˆ−1

∨
α−2

···

∨
α−n

An

αn

λn
∨
α−n−1

Aˆ−n

···

···

commute and satisfying the following two conditions: up to some Q-multiple every λi
λi ˆ
is an isomorphism and for each i we have Ai −
→ A−i → Aˆi is a rational multiple of a

polarization of Ai ; and
• η¯ is a K p -level structure
∼

η¯ : H1 (A0 , Apf ) −
→ V ⊗ Apf

mod K p .

We require that the following conditions hold:
(i). Each αi is an isogeny of degree p2n ;
∼

→ Ai such that for each i the map
(ii). There are “periodicity isomorphisms” θp : Ai+n −
θp

Ai → Ai+1 → · · · → Ai+n −
→ Ai

is multiplication by p;
(iii). αi commutes with the OB ⊗ Z(p) actions;
(iv). For all i and b ∈ OB
detOS (b|Lie(Ai )) = detE (b|V + ).

37

∼

¯ i, η¯) −
¯ , i, η¯ ) is a collection of
An isomorphism of S-valued points f : ({Ai } , λ,
→ ({Ai } , λ
Z(p) -isogenies fi : Ai → Ai each making the diagram
αi

Ai

Ai+1

fi

fi+1
αi

Ai

Ai+1

commute such that
• for each i there exists a locally constant function ri with values in Z×
(p) such that
λi = ri · (fi∨ ◦ λi ◦ fi );
• for each i the morphism End(Ai ) ⊗ Z(p) → End(Ai ) ⊗ Z(p) induced by fi , which is again
denoted by fi , is such that fi ◦ i(b) = i (b) for all b ∈ OB ; and
• H1 (f0 ) ◦ η = η .

We now turn our attention to the local model. Let F Gal denote the Galois closure of F
inside F0sep . Under our fixed embedding Q → Qp (see Section 2.1), we will identify F0 ⊂
F ⊂ F Gal ⊂ Qp . As we have that GF Gal splits, E ⊂ F Gal by [Kot1, 1.2]. We can thus consider
M loc ⊗OE OF Gal . An S-valued point of this scheme is given by a functor Λ → ωΛ from the
category L to the category of OB ⊗Zp OS -modules, satisfying the additional conditions as in
Definition 2.1.11. Now we have the decompositions

F ⊗Qp F Gal =

F Gal ,
ϕ:F →F Gal

V ⊗Qp F Gal =

Vϕ ,
ϕ

V+ =

Vϕ+
ϕ

38

where dimF Gal Vϕ = dimF V = n and the number of summands is [F : Qp ]. Since F/Qp is
unramified, there is a ring isomorphism

OF ⊗Zp OF Gal =

OF Gal .
ϕ

We therefore have that an OF ⊗Zp OF Gal -module M is a family (Mϕ )ϕ of OF Gal -modules
and likewise, homomorphisms M → N are families (Mϕ → Nϕ )ϕ of homomorphisms of
OF Gal -modules. The following proposition immediately follows.

Proposition 2.2.2. M loc ⊗OE OF Gal is isomorphic to the base change of a product of local
models in the case F + = Q.

For the remainder of this section, we will work on a single factor, taking the product over
all such factors. We thus omit any subscript ϕ or j and we will assume that F + = Q. Then
F is an imaginary quadratic extension of Q and the prime p splits in F . Thus F = F Gal and
GQp splits so E = Qp . Furthermore Fp = Fp∗ = Qp .
Recall from earlier in this section that the assumptions on p give the splitting
OB ⊗ Zp ∼
= Mnopp (Zp ) × Mnopp (Zp ).

For a scheme S over Spec(OE ), the sheaves on S induced by a S-valued point of the local
model carry an OB ⊗ Zp -action, and as such we get a corresponding splitting. Using Morita
equivalence, we will be able to reduce the “size” of the local model data. We will now
describe this in more detail.

39

loc
An S-valued point of the local model MGL
is determined by the following commutative

diagram.

∗
L0,S ⊕ L0,S

ϕ0

∗
L1,S ⊕ L1,S

ω0

ϕ1

···

ϕn−1

∗
Ln,S ⊕ Ln,S

···

ω1

ωn

∗
Here we are writing Li,S ⊕ Li,S
for what was Λi,S in Definition 2.1.11, where the splitting is
∗
given by OB ⊗ Zp = Mnopp (Zp ) × Mnopp (Zp ). In particular, Li,S and Li,S
are locally free OS ∗
, ωi is Zariski-locally
sheaves of rank n2 . Now ωi is a locally free OS -submodule of Li,S ⊕ Li,S
∗
a direct summand of Li,S ⊕ Li,S
of rank n2 , and the vertical arrows are inclusions,.

The action of Mnopp (Zp ) × Mnopp (Zp ) gives
e11 Li,S

Li,S =

∗
f11 Li,S

∗
and Li,S
=
n

n

where e11 and f11 are respectively idempotents of the first and second factors of Mnopp (Zp ) ×
Mnopp (Zp ).
Let W and W ∗ be Znp , viewed as left OB ⊗ Zp -modules via right multiplications by elements
of the first and second factor of Mnopp (Zp ) × Mnopp (Zp ) respectively. Recall the decomposition
VE = V + ⊕ V − induced by µh = (0n−r , (−1)r ).
V + = WEn−r ⊕ (WE∗ )r .
With tLi = Li /ωi , the determinant condition
detOS (a; tLi ) = detE (a; V + ),

40

a ∈ OB

is equivalent to the splitting of tLi into two factors of rank n2 − nr and nr, each respectively
a quotient of Li and Li∗ . Thus, we have the splitting of ωi into two summands of ranks
nr and n2 − nr contained in Li and Li∗ respectively. The action of each copy of Mnopp (Zp )
splits these into a direct sum of n copies of Fi of ranks r and Fi∗ of rank n − r respectively.
As such, we write Λi,S = e11 Li,S , Λ∗i,S = f11 Li,S , Fi = e11 ωi , and Fi∗ = f11 ωi . Therefore, an
loc
S-valued point of the local model MGL
is determined by the commutative diagram

Λ0,S ⊕ Λ∗0,S

ϕ0

Λ1,S ⊕ Λ∗1,S

F0 ⊕ F0∗

ϕ1

F1 ⊕ F1∗

···

ϕn−1

Λn,S ⊕ Λ∗n,S

Fn ⊕ Fn∗

···

where Fi ⊕ Fi∗ is an OS -submodule of Λi,S ⊕ Λ∗i,S , the vertical arrows are inclusions, and
Zariski locally over S, Fi is a direct summand of Λi,S of rank r and Fi∗ is a direct summand
of Λ∗i,S of rank n − r.
Condition (iii) of the local model
ωΛperp = ωΛ⊥
∗
is simply (Fi ⊕ Fi∗ )perp = F−i ⊕ F−i
. From the explicit definition of ·, · , we have (Fi ⊕

Fi∗ )perp = (Fi∗ )perp ⊕ Fiperp . Therefore {Fi } (or alternatively, {Fi∗ }) determines {Fi ⊕ Fi∗ }.
The following proposition summarizes this discussion.

Proposition 2.2.3. In the unitary case, an S-valued point of the local model is determined
by a diagram

Λ0,S

F0

ϕ0

Λ1,S

ϕ1

F1

···

···

41

ϕn−1

Λn,S

Fn

where
• Λi,S = e11 (Li ⊕ Li∗ );
• ϕi is the morphism induced from the inclusions of the lattice chain;
• Fi is a locally free OS -module which is Zariski locally a direct summand of Λi,S of rank
r; and
• the vertical arrows are inclusions.

Note that in the above diagram we are, by abuse of notation, labeling the restriction ϕi |Λi,S
as ϕi .

2.3

Symplectic case

We now consider the symplectic case. With the datum (B, ι, V, (·, ·), h0 , K) as in Section 1.2.2, we take OB = Z(p) and L to be the standard Zp -lattice chain as follows. Let
{e1 , . . . , e2n } be the standard basis of V = Q2n
p and define the Zp -lattice chain

Λ0 ⊂ Λ1 ⊂ · · · ⊂ Λn−1

where for 0 ≤ i ≤ 2n − 1,

Λi = p−1 e1 , . . . , p−1 ei , ei+1 , . . . , e2n ⊂ Q2n
p as a Zp -module
extended periodically by Λi+n = p−1 Λi for all integers i. Note that Λ⊥
i = Λ−i . Let Kp =
Aut(L) and K p be a sufficiently small open compact subgroup of G(Apf ). With G split over
Qp , we have E = Qp . The moduli problem in Definition 2.1.3 for the integral model AGSp
0

42

may be described as follows.

Definition 2.3.1. For any Zp -scheme S, AGSp
(S) = AGSp
0
0,K p (S) is the collection of tuples
(A• , λ0 , λn , η¯) up to isomorphism (defined below) where
(i). A• is a chain of abelian schemes
α

αn−1

α

0
1
· · · → A0 −→
A1 −→
· · · −−−→ An → · · ·

over S of relative dimension n, indexed by L by setting Ai = AΛi for i ∈ Z, where each
morphism αi : Ai → Ai+1 is an isogeny of degree p;
(ii). the maps λ0 : A0 → Aˆ0 and λn : An → Aˆn are principal polarizations making the loop
starting at any Ai or Aˆi in the diagram
α0

A0

A1

α1

···

αn−1

λ−1
0

An
λn

α0∨

Aˆ0

Aˆ1

α1∨

···

∨
αn−1

Aˆn

multiplication by p; and
(iii). η¯ is a K p -level structure on A0 .
∼

An isomorphism of S-valued points f : (A• , λ0 , λn , η¯) −
→ (A• , λ0 , λn , η¯ ) is a collection of
Z(p) -isogenies fi : Ai → Ai making the diagrams

Ai

αi

fi
Ai

Ai+1
fi+1

αi

commute and are such that

43

Ai+1

• for i = 0 and i = n there exists a locally constant function si with values in Z×
(p) such
that λi = si · (fi∨ ◦ λi ◦ f ); and
• H1 (f0 ) ◦ η = η .

There is an alternative description of this moduli problem in terms of chains of finite flat
group subschemes instead of chains of isogenies [dJ2, Section 1].
loc
Turning our attention to the local model, we have that an S-valued point of MGSp
is given

by a commutative diagram

Λ0,S

Λ1,S

···

Λ2n−1,S

p−1 Λ0,S

F0

F1

···

F2n−1

p−1 Λ0

satisfying the three conditions as in Definition 2.1.11. Condition (iii) is equivalent to the
following condition. Set
ˆ i,S = HomO (Λi,S , OS )
Λ
S
Fˆi = HomOS (Fi , OS ).

(iii’) For any i the composition
∼

ˆ 2n−i,S → Fˆ2n−i
Fi → Λi,S −
→Λ

is zero, where the middle isomorphism is induced by the isomorphism
∼ ˆ
Λi −
→Λ
2n−i

sending x → (px, ·).

44

To see this, first assume (iii) making Fiperp = pF2n−i . Then for a morphism S → M loc and
an open subscheme U ⊂ S, the composition above sends a section x ∈ Fi (U ) to the map

(px, ·)|F2n−i (U ) : F2n−i (U ) → OS (U )

which is clearly zero. Conversely, assume condition (iii’). Then in the same way we see that
pF2n−i ⊂ Fiperp . But since they have the same rank (namely, n), we get equality as required.
The following proposition summarizes this discussion.

loc
is given by a commutative diagram
Proposition 2.3.2. An S-valued point of MGSp

Λ0,S

F0

ϕ0

Λ1,S

F1

ϕ1

···

ϕ2n−1

···

Λ2n−1,S

F2n−1

ϕ2n

p−1 Λ0,S

p−1 Λ0

where Λi,S = Λi ⊗Zp OS , the ϕi are induced by the inclusions of the lattice chain, Fi are
locally free OS -submodules of rank n which are Zariski-locally direct summands of Λi,S , the
vertical arrows are inclusions, and the Fi satisfy the following duality condition. We require
the map
∼ ˆ
ˆ
→Λ
Fi → Λi,S −
2n−i,S → F2n−i
∼ ˆ
∼ ˆ
is zero for all i. Here Λi,S −
→Λ
→Λ
2n−i,S is induced from the duality Λi −
2n−i .

2.4

Representability

Let An,1,N denote the moduli space of principally polarized abelian schemes of relative dimension n equipped with a full symplectic level N structure. Then for N ≥ 3 with p N ,

45

An,1,N is represented by a scheme (Theorem A.6.7). To prove the representability of AGL
0
and AGSp
, we will show that they are relatively representable over An,1,N . We first give an
0
equivalent moduli problem for An,1,N involving a K p -level structure.
Fix a integral PEL Shimura datum (B, ι, V, (·, ·), h0 , OB , L, K p ) induced from the data in
either the unitary or symplectic case and an integer N ≥ 3 with p N . We assume that K p
is the principle level N structure with respect to the integral PEL Shimura datum. That is,

ˆ (p) ) ⊂ N · Λ0 ⊗ Z
ˆ (p)
K p = K(N ) = g ∈ G(Apf ) : (g − 1)(Λ0 ⊗ Z

Note that K(N ) =

=p

.

K (N ) with K (N ) ⊂ G(Q ).

Remark 2.4.1. Let s be a geometric point of a Zp -scheme S. For any Z(p) -isogeny f :
∼

A → A of abelian schemes over S, there is an induced isomorphism H1 (f ) : H1 (As , Apf ) −
→
∼

H1 (As , Apf ), and hence isomorphisms V (f ) : V (As ) −
→ V (As ) for all primes = p. Therefore
given such a Z(p) -isogeny f we will implicitly identify H1 (As , Apf ) and H1 (As , Apf ) using this
isomorphism, and similarly with V (As ) and V (As ).

Proposition 2.4.2. Let S be a connected Zp -scheme, s a geometric point of S, and A/S an
ˆ (p) -lattice that is fixed by the action of
abelian scheme. Let Λ ⊂ H1 (As , Apf ) be a self-dual Z
π1 (S, s). Then there exists a unique abelian scheme B, up to isomorphism, equipped with a
ˆ (p) ) = Λ.
Z(p) -isogeny A → B such that H1 (Bs , Z

Proof. We first show uniqueness. Suppose that there are two such abelian schemes B and B
ˆ (p) ) = Λ =
over S. Then we have the Z(p) -isogenies f : A → B and f : A → B with H1 (Bs , Z
ˆ (p) ). Therefore we have that the Z(p) -isogeny f −1 ◦ f : B → B carries H1 (Bs , Z
ˆ (p) )
H1 (Bs , Z
ˆ (p) ), and hence it must be that f −1 ◦ f is an isomorphism.
isomorphically onto H1 (Bs , Z

46

To show existence, we must produce an abelian scheme B and a Z(p) -isogeny A → B such
ˆ (p) ) = Λ. Note that Λ ∩ H1 (As , Z
ˆ (p) ) ⊂ H1 (As , Ap ) is a Z
ˆ (p) -lattice. Thus there
that H1 (Bs , Z
f
ˆ (p) such that
exists α ∈ Z
ˆ (p) ) ⊂ Λ ∩ H1 (As , Z
ˆ (p) ) ⊂ H1 (As , Z
ˆ (p) )
αH1 (As , Z

and writing α = (α ) ∈

=p

Z , we have α is a unit in Z for almost every prime . In the

following, we will work on a single factor where α is not a unit in Z , taking the product
over all such factors. Denote by Λ ⊂ V (As ) the factor of Λ corresponding to .
Now for such an , there exists an integer k > 0 such that

k

T (As ) ⊂ Λ ∩ T (As ) ⊂ T (As ).

Thus we can consider the quotients

Λ ∩ T (As ) / k T (As ) ⊂ T (As )/ k T (As ) = As [ k ].

Set C = (Λ ∩ T (As ))/ k T (As ) ⊂ As [ k ]. Denote the order of C by
isogeny A/C → A with kernel A[

m

m

and consider the

]/C . Then from Proposition A.6.12, we have the exact

sequence
0 → T (A/C ) → T (A) → A[
Since A[

m

m

]/C → 0.

]/C = T (A)/Λ ∩ T (As ), we have T (A/C ) = Λ ∩ T (As ).

As mentioned above, set C =

=p

C . Then A → A/C is a Z(p) -isogeny with

ˆ (p) ) = Λ ∩ H1 (As , Z
ˆ (p) ).
H1 ((A/C)s , Z

47

ˆ (p) ) ⊂ Λ.
Thus we are reduced to the case where H1 (As , Z
Similar to the above, for each prime
k

= p there exists an integer k ≤ 0 such that Λ ⊂

T (As ). Note that k = 0 for almost every prime . By using the multiplication by

k

ˆ (p) ).
map, we may assume that Λ ⊂ H1 (As , Z
ˆ (p) such that αH1 (As , Z
ˆ (p) ) ⊂ Λ ⊂ H1 (As , Z
ˆ (p) ). For each prime
Now there exists an α ∈ Z
set C = Λ /α T (As ). Then as above, T (As /C ) = Λ and taking C =

=p

C gives the

ˆ (p) ) = Λ as required.
equality H1 (As /C, Z

Proposition 2.4.3. Let S be an OE -scheme and A/S a principally polarized abelian scheme.
Fix a geometric point s of S. In the following statements, we take

= p to be a rational

prime.
(i). Let

∼

N . Then giving a K (N )-orbit of symplectic similitudes η : V ⊗ Q −
→ V (As ),

with similitude in Z× , is equivalent to giving a self-dual Z -lattice Λ ⊂ V (As ).
(ii). Let

| N and set N = a

k

with

a. Then giving a K (N )-orbit of symplectic

∼

→ V (As ), with similitude in Z× , is equivalent to giving a
similitudes η : V ⊗ Q −
self-dual Z -lattice Λ ⊂ V (As ) and an isomorphism

Λ0 ⊗ Z /

k

∼

→ Λ/ k Λ.
· (Λ0 ⊗ Z ) −

Proof.
(i). Suppose first that η is such a K (N )-orbit. Define Λ by choosing η ∈ η and setting
Λ = η(Λ0 ⊗ Z ). Since η preserves the pairing up to some Z× -multiple and Λ0 ⊗ Z is
self-dual, it follows that Λ is self-dual. As g(Λ0 ⊗ Z ) = Λ0 ⊗ Z for all g ∈ K (N ), we
have Λ is well-defined.

48

Conversely, let Λ ⊂ V (As ) be a self-dual lattice. Then we may choose a symplectic
∼

basis of Λ. The choice of basis of Λ gives a symplectic isomorphism η : V ⊗Q −
→ V (As )
sending Λ0 ⊗ Z onto Λ. While the isomorphism η depends on the choice of basis, the
K (N )-orbit {η ◦ g : g ∈ K (N )} does not.
(ii). Suppose now that η is a K (N )-orbit with N = a

k

a. Choose η ∈ η and set

and

Λ = η(Λ0 ⊗ Z ) as before. Since (g − 1)(Λ0 ⊗ Z ) ⊂ N · (Λ0 ⊗ Z ), for all possible choices
of η ∈ η the induced isomorphism Λ0 ⊗ Z /(

k

∼

· Λ0 ⊗ Z ) −
→ Λ/ k Λ is fixed.

Conversely, fix a self-dual Λ ⊂ V (As ) and an isomorphism Λ0 ⊗ Z /

k

∼

· (Λ0 ⊗ Z ) −
→

Λ/ k Λ. As in (i), the choice of a symplectic basis of Λ determines an isomorphism
∼

∼

→
→ V (As ). We choose a symplectic basis so that the induced map V ⊗ Q −
V ⊗Q −
V (As ) extends the fixed isomorphism Λ0 ⊗ Z /(

k

∼

· Λ0 ⊗ Z ) −
→ Λ/ k Λ. This is well-

defined up to an element of K (N ).

Proposition 2.4.4. The functor An,1,N on the category of Zp -schemes is isomorphic to the
following functor. For a Zp -scheme S, let An,1,N (S) be the set of all tuples (A, λ, η) up to
isomorphism where
• A is an abelian scheme of dimension n over S;
• λ : A → Aˆ is a polarization which is also a Z(p) -isogeny; and
∼

• η is a K(N )-orbit of symplectic similitudes η : V ⊗ Apf −
→ H1 (As , Apf ).
We furthermore require that there be some representative (A, λ, η) with λ a principal polarization.
Here an isomorphism f : (A, λ, η) → (A , λ , η ) between two objects in An,1,N (S) consists of

49

a Z(p) -isogeny f : A → A such that
∨
• there exists a locally constant function r with values in Z×
(p) such that λ = r(f ◦ λ ◦ f );

and
• H1 (f ) ◦ η = η .

Proof. We start by defining a natural transformation Φ : An,1,N → An,1,N . Let (A, λ, α)
∼

be a representative of an element of An,1,N (S). Then α : (Z/N Z)2n −
→ A[N ] induces an
∼

isomorphism αs : (Z/N Z)2n −
→ As [N ] that is invariant under the action of π1 (S, s). To
define η, we need to define it at every place = p, i.e. a K (N )-orbit of symplectic similitudes
∼

V ⊗Q −
→ V (As ) that is invariant under the action of π1 (S, s). By Proposition 2.4.3, for
N such a K (N )-orbit is given by the self-dual lattice H1 (As , Z ). Note this lattice is
also fixed under the action of π1 (S, s). For

| N with N = a

k

and

a, a K (N )-orbit of
∼

→ As [ k ] which is
symplectic similitudes is given by T (As ) and the isomorphism (Z/ k Z)2n −
induced from α.
Suppose that the S-valued points (A, λ, α) and (A , λ , α ) of An,1,N are isomorphic. Then
there is an isomorphism f : A → A such that λ = f ∨ ◦ λ ◦ f and α = f ◦ α. Hence f also
serves as an isomorphism between (A, λ, η) and (A , λ , η ) constructed as above.
Suppose now that two objects (A, λ, α) and (A , λ , α ) of An,1,N (S) are sent to isomorphic
objects (A, λ, η) and (A , λ , η ) of An,1,N (S). We may assume that λ and λ are principal
polarizations. Let f be an isomorphism between these two objects, i.e. a Z(p) -isogeny respecting the polarization and K p -level structures. Then since the objects (A, λ, η) and (A , λ , η )
ˆ (p) ) isomorphically onto H1 (A , Z
ˆ (p) ).
arise from objects of An,1,N (S), H1 (f ) maps H1 (As , Z
s
Therefore f must be an isomorphism. Now λ = r · f ∨ ◦ λ ◦ f for some locally constant
function r taking values in Z×
(p) . But since λ, λ , and f are isomorphisms, it must be that

50

r = 1. That α = f ◦ α follows from Proposition 2.4.3. Therefore f induces an isomorphism
between the objects (A, λ, α) and (A , λ , α ) of An,1,N (S).
Now let (A, λ, η) be a representative of some element in An,1,N (S). Then we must find an
object in the same isomorphism class of (A, λ, η) that arises from some object of An,1,N (S)
ˆ (p) -lattice
via the functor constructed above. The K p -level structure provides a self-dual Z
Λ ⊂ H1 (As , Apf ) invariant under the action of π1 (S, s). From Proposition 2.4.2 we can find
an abelian scheme B/S, unique up isomorphism (of abelian schemes), and a Z(p) -isogeny
ˆ (p) ). We claim that B suffices, and to show this
f : A → B such that Λ = H1 (Bs , Z
it remains to equip B with a principal polarization λ and K p -level structure η so that
f : A → B induces an isomorphism of objects in An,1,N (S) and (B, λ , η ) arises from an
object of An,1,N (S). To equip B with a principal polarization we first define the Z(p) -isogeny
ˆ (p) ) = Λ and hence H1 (B
ˆ (p) ) = Λ since
ˆs , Z
λ = (f ∨ )−1 ◦ λ ◦ f −1 . By construction, H1 (Bs , Z
ˆ (p) ) onto H1 (B
ˆ (p) ) and hence λ
ˆs , Z
the lattice Λ is self-dual. Therefore λ sends H1 (Bs , Z
ˆ (p) ) we have the
must be an isomorphism. We also define η = H1 (f ) ◦ η. With Λ = H1 (Bs , Z
canonical identification Λ/N Λ = Bs [N ]. Thus (B, λ , η ) induces (B, λ , α ) as required.

Remark 2.4.5. We will henceforth use either description of the functor An,1,N .

GSp
As an intermediate step in proving the representability of AGL
, we will show that
0 and A0

the moduli problem defined below is representable by a quasi-projective scheme.

Definition 2.4.6. [Kot2, Section 5] Let Ahyp denote the the following functor on schemes
over Spec(Zp ). For a Zp -scheme S, let Ahyp (S) be the set of all tuples (A, λ, i, η) up to
isomorphism, where
• A is an abelian scheme of dimension n over S;

51

• λ : A → Aˆ is a Z(p) -isogeny, which at every geometric point s of S is a polarization of
A;
• i : OB ⊗ Z(p) → End(A) ⊗ Z(p) a homomorphism;
• detS (b|LieA) = detE (b|V + ) for all b ∈ OB ; and
∼

→ H1 (As , Apf ).
• η is a K(N )-orbit of symplectic similitudes V ⊗ Apf −
We furthermore require that there be some representative (A, λ, i, η) with λ a principal
polarization.
An isomorphism f : (A, λ, i, η) → (A , λ , i , η ) between two objects in Ahyp (S) consists of a
Z(p) -isogeny f : A → A such that
∨
• there exists a locally constant function r with values in Z×
(p) such that λ = r(f ◦λ ◦f );

• the morphism End(A) ⊗ Z(p) → End(A ) ⊗ Z(p) induced by f , which is again denoted
by f , is such that f ◦ i(b) = i (b) for all b ∈ OB ; and
• H1 (f ) ◦ η = η .

Proposition 2.4.7. The functor Ahyp is representable by a quasi-projective scheme.

Proof. We have the forgetful functor Ahyp → An,1,N by forgetting the OB -action. We will
show that Ahyp is relatively representable over An,1,N by a projective scheme.
Fix a Zp -scheme S and a morphism S → An,1,N inducing (A, λ, η). Consider the functor
Ahyp ×An,1,N S. By [Hid, Section 6.1] the functor on S-schemes sending T → End(AT ) ⊗ Z(p)
is representable by a union of projective schemes over S, denote the scheme representing this
functor by E. Now the polarization λ induces the Rosati involution on End(A)⊗Z(p) , and this
in turn gives an involution r : E → E. Choose a set of generators {a1 , . . . , a2m } of OB ⊗ Z(p)
as a Z(p) -algebra such that am+i = ι(ai ) for 1 ≤ i ≤ m and define the closed subscheme Z

52

of E 2m as follows. For an S-scheme T , a point (x1 , . . . , x2m ) ∈ E 2m (T ) is in Z if and only if
any relationship satisfied by (a1 , . . . , a2m ) is also satisfied by (x1 , . . . , x2m ) and r(xi ) = xm+i .
Note that any morphism T → E 2m induced by a T -valued point of Ahyp ×An,1,N S factors
through Z.
With this, AZ = A ×S Z is an abelian scheme over Z and the morphism AZ → Z induces
{x1 , . . . , x2m } ⊂ End(AZ ) ⊗ Z(p) giving the algebra homomorphism OB ⊗ Z(p) → End(AZ ) ⊗
Z(p) by sending ai to xi . By the conditions defining Z, this homomorphism is compatible
with the Rosati involution.
By Proposition 2.1.7, the locus where there is a K p -level structure is a union of connected
components of Z. Now consider the determinant condition. With Lie(AZ ) locally free, we
have detLie(AZ ) ∈ Γ(Z, OZ )[X1 , . . . , X2m ]. The condition detLie(AZ ) = detE (V + ) is an equality
of global sections of Z. Thus enforcing both of these conditions gives a closed subscheme
X ⊂ Z. Again, any morphism T → E 2m induced by a T -valued point of Ahyp ×An,1,N S
factors through X.
Therefore X, with the universal abelian scheme AX , represents the functor Ahyp ×An,1,N S.
Since X is projective over S, we have that the scheme representing Ahyp is quasi-projective.

Lemma 2.4.8. Let S be a Zp -scheme. Any object (A, λ, i, η) of Ahyp (S) only has the trivial
automorphism.

Proof. An automorphism of (A, λ, i, η) is given by f : (A, λ, i, η) → (A , λ , i , η ) where
(A , λ , i , η ) is in the same isomorphism class as (A, λ, i, η). With the forgetful morphism
of functors Ahyp → An,1,N , the automorphism f induces an automorphism of the induced

53

objects of An,1,N (S). By [Ser2], this induced automorphism must be the identity. It follows
that f : A → A is the identity.

Proposition 2.4.9. AGL
is representable by a quasi-projective scheme.
0

GL
Proof. One has the forgetful functor AGL
is
0 → Ahyp and hence it suffices to show that A0

relatively representable over Ahyp .
Fix S → Ahyp inducing (A0 , λ0 , η) and consider the functor AGL
0 ×Ahyp S. Note that we may
choose the representative (A0 , λ0 , η) such that λ0 is a principal polarization.
Let F H = F HAn0 denote the Flag Hilbert scheme with respect to the projective scheme A0
and the Hilbert polynomials

2

p2n , p4n , . . . , p2n

[Ser1, Section 4.5]. This is a projective

scheme representing the following functor: to give a T valued point of F H is to give a chain
H1 ⊂ H2 ⊂ · · · ⊂ Hn of T -flat closed subschemes of AT such that Hi has Hilbert polynomial
p2ni . Let Z• = (Zi )i denote the universal object over F H, so in particular for each i we have
Zi → A0 ×S F H.
Given an object of (AGL
0 ×Ahyp S)(T ), we claim that there is an induced morphism T → F H.
Choosing a representative (A• , λ, i, η), we have the chain of finite flat subgroup schemes
Hi = ker(αi : A0 → Ai ). We must show that this chain is well-defined, i.e. that it does not
depend on the choice of representative. To see this, recall that an isomorphism of objects
f : (A• , λ, i, η) → (A• , λ , i , η ) of (AGL
0 ×Ahyp S)(T ) is given by a collection fi : Ai → Ai of
Z(p) -isogenies such that, in particular, the diagram

54

A0

αi

Ai
fi

f0
i

A0

(α )

Ai

commutes. By the lemma above, f0 : A0 → A0 must be the identity. Furthermore fi : Ai →
Ai , if it exists, is determined uniquely by f0 . We may identify Ai = A0 /Hi , and likewise
Ai = A0 /Hi . Then we see that fi : A0 /Hi → A0 /Hi must descend from f0 : A0 → A0 .
Thus f0 (Hi ) = Hi and so the chain is well-defined. Therefore an object of (AGL
0 ×Ahyp S)(T )
induces a morphism T → F H.
We define the closed subscheme F H1 ⊂ F H as follows. A geometric point x of F H is a
µ

ε

point of F H1 if and only if for all i, the morphisms x →
− A0,x , Zi,x ×F H Zi,x →
− A0,x , and
ι

Zi,x →
− A0,x factor through Zi,x → A0,x . Here Zi,x and A0,x denote the geometric fiber with
respect to x and ε, µ, and ι are the restrictions of the identity, multiplication, and inverse of
A0,x respectively. Any morphism T → F H induced by an object of (AGL
0 ×Ahyp S)(T ) factors
through F H1 .
Now define the closed scheme F H2 ⊂ F H1 as follows. A geometric point x of F H1 is a point
of F H2 if and only if Zn,x = A0,x [p] as closed subschemes of A0,x . Any morphism T → F H1
induced by an object of (AGL
0 ×Ahyp S)(T ) factors through F H2 .
Define Zi = Zi ×F H F H2 for 1 ≤ i ≤ n − 1. Then each Zi is a flat subgroup scheme of
A0 ×S F H2 and moreover each Zi is finite over F H2 . So any morphism T → F H2 induces a
chain
α

α

αn−2

αn−1

0
1
A0,T −→
A0,T /H1 −→
. . . −−−→ A0,T /Hn−1 −−−→ A0,T /A0 [p]

where A0,T = A0 ×S T and each morphism is the canonical quotient map and hence an isogeny
of degree p2n . By abuse of notation write A0 for A0,T and set Ai = A0 /Hi for 0 ≤ i ≤ n − 1.

55

αn−1

Since Hn−1 ⊂ A0 [p], we can define An = A0 with the map An−1 −−−→ An being induced by
[p] : A0 → A0 . We then take the periodicity isomorphism θp : An → A0 to be the identity.
Now extend this chain periodically to be an infinite chain (Ai )i for i ∈ Z.
It remains to enforce the condition that there exists a Q-homogeneous class of principal
polarizations. Consider the diagram
αi

A−i

A0

αi

Ai

λ0
∨ i

(α ) ˆ
Aˆ−i
A0
and write Ai = A−i /Hi where we now denote by Hi a finite flat subgroup scheme of A−i .
∼
We claim that an isomorphism λi : Ai −
→ Aˆ−i making the diagram

A0

αi

λ0

Ai
λi

∨ i

(α ) ˆ
Aˆ0
A−i

i

∨ i

(α )
λ0
α
commute, if it exists, must be descended from the map A−i −
→ A0 −→
Aˆ0 −−−→ Aˆ−i . To
∼

see this, denoting θp : An −
→ A0 the periodicity isomorphism and [p] multiplication by p, we
have

(α∨ )i ◦ λ0 ◦ θp ◦ αn−i ◦ α2i = (α∨ )i ◦ λ0 ◦ [p] ◦ αi
= λi ◦ αi ◦ [p] ◦ αi

Since [p] is an isogeny, this implies that (α∨ )i ◦ λ0 ◦ αi = λi ◦ α2i which is precisely to say

56

that λi descends from (α∨ )i ◦ λ0 ◦ αi .
∼
The map (α∨ )i ◦ λ0 ◦ αi will descend to an isomorphism Ai −
→ Aˆ−i if and only if its kernel is

precisely Hi . Enforcing this condition for each i, we see that there exists a closed subscheme
F H3 of F H2 such that a morphism T → F H2 will factor through F H3 if and only if the
corresponding point of T → F H2 may be equipped with a Q-homogeneous class of principal
GL
polarizations. Therefore F H3 represents AGL
0 . With F H3 a projective over S, A0 is quasi-

projective.

Proposition 2.4.10. AGSp
is representably by a scheme.
0

Proof. As in the previous theorem, one has the forgetful functor AGSp
→ Ahyp and hence
0
it suffices to show that AGSp
is relatively representable over Ahyp . Fix S → Ahyp inducing
0
×Ahyp S. Note that we may choose the representative
(A0 , λ0 , η) and consider the functor AGSp
0
(A0 , λ0 , η) so that λ0 is a principal polarization. As in the above theorem, an object of
(AGSp
×Ahyp S)(T ) induces a morphism T → F HAn0 where we now take {p, p2 , . . . , pn } to
0
be the collection of Hilbert polynomials. There is a closed subscheme Z of F HAn0 such that
T → F HAn0 factors through Z if and only if the corresponding chain of flat subschemes forms
a chain of finite flat subgroup schemes of AT contained in AT [p].
By abuse of notation denote A0,T by A0 and set Ai = A0 /Hi for 1 ≤ i ≤ n − 1. Then we
have the diagram
A0

α

A1

α

...

α

λ0
Aˆ0

An−1
λn

α

∨

Aˆ1

α

∨

57

...

α

∨

Aˆn−1

It remains to enforce the condition that there exists principal polarizations λn : An → Aˆn
making the loop in the diagram multiplication by p. Such a polarization exists if and only
if the kernel Hn−1 = ker(αn−1 ) is totally isotropic with respect to the Weil pairing induced
by λ0 . Thus there exists a closed subscheme of X ⊂ Z such that T → Z factors through X
if and only if there exists such a λn as above.
Therefore Z represents AGSp
.
0

58

Chapter 3
Stratification of A0
In this chapter we will describe the following result.

Theorem 3.1 ([G¨or1],[G¨or2]). Let A0 and M loc respectively denote the integral model and
local model in either the unitary or symplectic case. Then M loc ⊗ Fp can be embedded into
an affine flag variety associated with SLn and Sp2n respectively. Each affine flag variety is
stratified by Schubert cells which induces a stratification of M loc ⊗ Fp . There is a unique
stratum of M loc ⊗ Fp which consists of a single closed point, called the “worst point”. Any
open subscheme of M loc containing the worst point is an ´etale local model for A0 .

Following loc. cit., we will also make a specific choice of open subscheme U0 ⊂ M loc (denoted
Uτ in loc. cit.) containing the worst point and give an explicit presentation of U0 .
Consider the local model M loc associated with an integral Shimura datum of Iwahori-type
in either the unitary or symplectic case. As before, let G = GQp where G is the reductive

59

group defined over Q in the Shimura datum. In the unitary case, recall that the assumptions
on p gave the splitting of G =

j

Gj which in turn makes the local model (after some finite

base extension) isomorphic to a product of local models in the case F + = Q (see Proposition
2.2.2). We thus assume that in the unitary case, F + = Q.
Throughout this chapter, we will fix notation in such a way that we may describe the
stratifications in both the unitary and the symplectic cases simultaneously. In the unitary
case we have G = GLn,Qp × Gm,Qp with µ = (0n−r , (−1)r ) and in the symplectic case we
have G = GSpn,Qp where n = 2r for r ∈ N with µ = (0r , (−1)r ). Thus in either case,
µ = (0n−r , (−1)r ). Note that this notation differs from slightly from Section 2.3 where we
had GSp2n,Qp .
Let k be an algebraically closure of Fp and let F denote the affine flag variety over k associated
with SLn in the unitary case and Spn in the symplectic case. We identify FSL with the space
of complete (n−r)-special lattice chains and likewise FSp with the space of complete self-dual
(n − r)-special lattice chains (see Propositions A.3.5 and A.3.7). Let R be a k-algebra and
set
λi = (t−1 R[[t]])i ⊕ R[[t]]n−i ,

0≤i≤n−1

noting that λ⊥
i = λ−i .

Proposition 3.2. [G¨or1, Proposition 3.5],[G¨or2, Proposition 3.2]
loc
(i). Let (Fi )i be an R-valued point of MGL
⊗k so that Fi is a subspace of Λi,R = Rn ∼
= λi /tλi .

Let Li be the inverse image of Fi under the canonical projection λi → λi /tλi . Then
(Li )i is a complete (n − r)-special lattice chain and the map
loc
MGL
⊗ k → FSL

sending

60

(Fi )i → (Li )i

is a closed immersion. The image is precisely the lattice chains (Li )i ∈ FSL such that
tλi ⊂ Li ⊂ λi .
loc
(ii). Let (Fi )i be an R-valued point of MGSp
⊗ k inducing (Li )i as in (i). Then (Li )i is
loc
a complete self-dual r-special lattice chain and the map MGSp
⊗ k → FSp is a closed

immersion. The image is precisely the lattice chains (Li )i ∈ FSp such that tλi ⊂ Li ⊂
λi .

Proof.
(i). That (Li )i forms a complete lattice chain is immediate from the definition of (Fi )i . We
now show that L0 is (n − r)-special. By localization, we can assume that F0 ⊂ Rn is
a direct summand. Without loss of generality, suppose F0 = Rr → Rr ⊕ Rn−r = Rn .
Then under the map π0 : λ0 → λ0 /tλ0 ∼
= Rn we have that π0−1 (F0 ) is the R[[t]]-lattice
generated by the elements of λ0 where the last n − r coordinates are contained in
tR[[t]]. As L0 → ∧n L0 is given by the determinant, it follows that ∧n Li = tn−r R[[t]]
as required.
loc
⊗ k the induced lattice chain (Li ) is such that tλi ⊂
By definition, for (Fi ) ∈ MGL

Li ⊂ λi for all i. Now suppose (Li )i ∈ FSL such that tλi ⊂ Li ⊂ λi for all i. Then
∼

the image of Li under the map λi → λi /tλi −
→ Rn is a locally free R-module of rank
loc
r, denote it by Fi . Thus (Fi )i is a point of MGL
⊗ k that maps onto (Li )i as required.

(ii). Noting that n − r = r here, all of the claims will follow from (i) by showing that
enforcing the duality condition on (Fi )i is equivalent to enforcing the condition that
(Li )i is self-dual. Since (Fi )i and hence (Li )i are periodic, it suffices to show the claim
on the indices i = 0, . . . , n−1. We claim that in fact the duality condition is equivalent
to L⊥
i = Ln−i for 0 ≤ i ≤ n − 1.

61

⊥
−1 ⊥
⊥
Since tλi ⊂ Li ⊂ λi , certainly we have λ⊥
i ⊂ Li ⊂ t λi . Using λi = tλn−i , we have
⊥
⊥
tλn−i ⊂ L⊥
i ⊂ λn−i . This implies that Ln−i ⊂ Li if and only if Ln−i = Li .

We have the following commutative diagram
Li × L−i

R[[t]]

Fi,R × F−i,R

R

where the horizontal map on the bottom is the restriction of Λi,R × Λ−i,R → R induced
by the duality Λ⊥
i = Λ−i .
The duality condition is equivalent to the pairing Li × L−i → R[[t]] having image in
⊥
tR[[t]], and this in turn is equivalent to t−1 L−i ⊂ L⊥
i . Since (Li )i is periodic, Ln−i = Li

and the result follows.

As a result of the proposition, we will frequently identify M loc ⊗k with its image in F. Recall
the description of the stratification of F (see Section A.3 for details). Set

ω = (ω0 , . . . , ωn−1 ) where ωi = (1i , 0n−i ).
τ = (1r , 0n−r ), (1r+1 , 0n−r−1 ), . . . , (2r−2 , 1n−r ), (2r−1 , 1n−r+1 )

With τ as a base alcove, the extended affine Weyl group W acts simply transitively on the
set of alcoves and hence we may identify W with the set of alcoves. Thus given (Li )i ∈ F,
there is a uniquely determined element w ∈ W and an element b in the Iwahori subgroup
I, such that (Li )i = bwτ . We say that wτ is the alcove associated to the point (Fi )i . This

62

gives a stratification
F=

Swτ ,

where Swτ = IwI/I.

w∈W aff

From the description of M loc ⊗ k in F in Proposition 3.2, we immediately have that M loc ⊗ k
is invariant under the action of the Iwahori subgroup I, and hence that M loc ⊗ k is settheoretically a disjoint union of Schubert cells Swτ .
Now let (Fi )i ∈ M loc (k) and (Li )i ∈ F the corresponding point of the affine flag variety. Let
x = (x0 , . . . , xn−1 ) be an alcove and suppose (Li )i ∈ Sx . Then





Li = b · 






−xi (1)+1

t

t−xi (2)+1
..

.
t−xi (n)+1










for some b ∈ I. Here we are writing the matrix on the right with respect to the standard
basis {e1 , . . . , en } of k((t))n and identifying this matrix with the k[[t]]-submodule of k((t))n
given by its column space.

Remark 3.3. Note that for an alcove x such that 0 ≤ xi (j) − ωi (j) ≤ 1, the quotient λi /Li
is generated by the eij such that xi (j) = ωi (j). Indeed, we certainly have that the claim holds
for b = 1. Now let b ∈ I be arbitrary. Since I stabilizes the standard lattice chain, we have
λi /Li is generated by beij : xi (j) = ωi (j) . Of course in the quotient λi /Li , the subspace
generated by this set is the same as that generated by eij : xi (j) = ωi (j) .

Definition 3.4. Let x = (x0 , . . . , xn−1 ) be an alcove. Then
(i). The number

j

xi (j) −

j

ωi (j) is independent of i and is called the size of x.

63

(ii). We say that x is minuscule if 0 ≤ xi (j) − ωi (j) ≤ 1 for all 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ n.
(iii). We say that x is µ-permissible if x is minuscule of size r.
We will denote the collection of µ-permissible alcoves by Perm(µ).

Remark 3.5. τ is µ-permissible.

Proposition 3.6. Sx ∩ (M loc ⊗ k) = ∅ if and only if x is a µ-permissible alcove and hence,
set-theoretically,
M loc ⊗ k =

Sx .
x∈Perm(µ)

Proof. For all 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ n, the condition tλi ⊂ Li ⊂ λi is equivalent to
the condition 0 ≤ xi (j) − ωi (j) ≤ 1. Thus we may assume that this is indeed the case. Now
(Li )i is (n − r)-special is equivalent to λi /Li has rank n − r. By Remark 3.3, this is in turn
equivalent to # {j : xi (j) = ωi (j)} = n − r for every 0 ≤ i ≤ n − 1, and hence equivalent to
n

xi (j) − ωi (j) = r.
j=1

Definition 3.7. Let x = (x0 , . . . , xn−1 ) be a µ-permissible alcove. Define Ux to be the subset
of M loc ⊗ k which consists of all the points (Fi )i such that for all i, the quotient Λi /Fi is
generated by those eij with ωi (j) = xi (j).

Proposition 3.8. Let x = (x0 , . . . , xn−1 ) be a µ-permissible alcove and Ux be as in the
definition above. Then we have the following.
(i). Ux is an open subscheme of M loc ⊗ k.

64

(ii). The stratum Sx is contained in Ux .
(iii). The irreducible components of M loc ⊗k are the closures of the Sx where x is an extreme
alcove, i.e. x = tw(µ) for some w ∈ W .
(iv). The stratum Sτ consists of only one point.
(v). For any open subscheme U ⊂ M loc ⊗ k containing Sτ , U intersects every stratum.

Proof.
(i). Let x ∈ Perm(µ), (Fi )i ∈ M loc (k), and (Li )i the associated point of F. For any fixed i,
as shown in the proof of Proposition 3.6, the collection of eij with ωi (j) = xi (j) consists
of n − r elements, and this set is a subset of a basis of λi . Since λi /Li is free of rank
n − r, in general for an arbitrary set {s1 , . . . , sn−r } ⊂ λi , the set {s1 , . . . , sn−r } forms
a basis of λi /Li if and only if {s1 , . . . , sn−r } ∈
/ Li and {s1 , . . . , sn−r } is part of a basis
of λi .
Let {jk }n−r
k=1 be a collection of distinct integers with 1 ≤ jk ≤ n. Consider the collection
T = eijk : 0 ≤ i ≤ n − 1, 1 ≤ k ≤ n − r .
/ Li for
By the above it suffices to show that the collection (Li )i ∈ M loc such that eijk ∈
all eijk ∈ T is open. This is the intersection of finitely many sets, each defined by the
condition that for some fixed i and k, eijk ∈
/ Li . As each such set is open, it follows
that Ux is open.
(ii). This is Remark 3.3.
(iii). From [KR], the extreme alcoves are precisely the µ-permissible alcoves x such that
x ≤ y implies x = y for y a µ-permissible alcove. Recalling that the stratification has
the property Sx ⊂ Sy if and only if x ≤ y, the result follows.

65

(iv). With I the Iwahori subgroup of LG(k), consider the action of I on M loc ⊗ k given by
I ×M loc ⊗k → M loc ⊗k. As Sτ is an I-orbit, for any p ∈ Sτ we have that the restriction
I × {p} → Sτ is surjective. Furthermore, since the action of I on M loc ⊗ k is continuous
and I is connected, we get that Sτ is connected. Finally, Sτ is zero dimensional since
(τ ) = 0 (Proposition A.3.9). Therefore Sτ consists of a single point.
(v). Let U ⊂ M loc ⊗ k be an open subscheme with Sτ ∈ U . From Proposition A.3.9, as
τ ≤ x for all µ-permissible x, we have Sτ ∈ Sx . Hence U intersects Sx and thus U
meets Sx .

Remark 3.9. Let U be an open subscheme of M loc ⊗ k with Sτ ∈ U and let x be µpermissible. Then since U meets Sx and Sx is an I-orbit, we have that the image of I × U →
M loc contains all of Sx . As M loc ⊗ k is the disjoint union of Sx for x ∈ Perm(µ), we have
I × U → M loc is surjective. It follows immediately that any open subscheme U ⊂ M loc
containing the point Sτ serves as an ´etale local model of A0 . In particular, Uτ is a local
model of A0 .

As in [G¨or1] and [G¨or2], we now write down a presentation of Uτ , which we will henceforth
denote by U0 . With M loc being a closed subscheme of a product of Grassmannians, we
represent a point of M loc by giving (Fi )n−1
i=0 where each Fi is an r-dimensional subspace, and
we represent Fi as the column space of the n × r matrix (aijk ) with respect to the basis eij .
It is then easy to check (Fi )i ∈ U0 implies that the r × r minor given by rows i + 1 to r + i
(taken cyclically, so row n + 1 is row 1) of Fi is invertible for 0 ≤ i < n. As such, we require

66

this submatrix to be the identity matrix. For example, F0 and F1 are represented by





1




1


..

.




1


 a0
a012 . . . a01r
 11
 .
..
..
 .
.
.
 .

a0n−r,1 a0n−r,2 . . . a0n−r,r











































and

a1n−r,2

a1n−r,1

...

a1n−r,r

1
1
..

.
1

a111
..
.

a112
..
.

...

a11r
..
.

a1n−r−1,1 a1n−r−1,2 . . . a1n−r−1,r













.











Note that by requiring the matrix to have a specific r × r submatrix which is the identity, all
the entries of the matrix are uniquely determined by its column space. By abuse of notation,
we will use Fi to denote both the subspace and the matrix representing it. To express the
condition Fi is mapped into Fi+1 , we must have

ϕi (Fi ) = Fi+1 Ai

for some r × r matrix Ai . However Fi+1 has an r × r submatrix which is given by the identity
matrix, and so Ai is determined:





Ai = 




0

1




.
.

.

0
1 

i
i
a12 . . . a1r
..

ai11



Proposition 3.10.

67

..

(i). U0GL ∼
= Spec(BGL ) with
BGL = Zp [aijk ; i = 0, . . . , n − 1, j = 1, . . . , n, k = 1, . . . , r]/I

where I is the ideal generated by following two collections of relations. The first collection is given for 0 ≤ i ≤ n − 1 by the entries of the matrices





ai21
..
.

ai22
..
.

...

ai2r
..
.





i+1
i+1
. . . a1r



 a11



 .


.
 ..
. .  Ai − 
.



 i
i
i


a
a
.
.
.
a
 n−r,1 n−r,2
n−r,r 
i+1
i+1


an−r,1 . . . an−r,r
p
0
...
0
The second collection is given by the entries of the matrices

An−1 An−2 · · · A0 −p·Id,

An−2 · · · A0 An−1 −p·Id,

...,

A0 An−1 · · · A1 −p·Id.

(ii). [G¨or2, Section 5] U0GSp ∼
= Spec(BGSp ) with BGSp = BGL /J where J is the ideal generated by

a2n−i
− εjk ain−k+1,n−j+1
jk

with

εjk



 1 j, k ≤ i or j, k ≥ i + 1
=

 −1
otherwise

for each 0 ≤ i ≤ n − 1.

Proof.
(i). The first collection of equations is equivalent to the requirement that ϕ(Fi ) ⊂ Fi+1

68

and the second collection is equivalent to the compositions

ϕn−1 ϕn−2 . . . ϕ0 ,

ϕn−2 ϕn−3 . . . ϕ0 ϕn−1 ,

...,

ϕ0 ϕn−1 . . . ϕ1

are all multiplication by p.
loc
loc
(ii). MGSp
is a closed subscheme of MGL
given by enforcing the duality condition. That

this condition is given by the equations above is computed in [G¨or2, Section 5.1].

Remark 3.11. We mention here without proof the vastly reduced presentation from [G¨or3,
Section 3]: UτGL is isomorphic to the spectrum of
Zp [ai1k ; i = 0, . . . , n − 1, k = 1, . . . , r]/I

where I is the ideal generated by the entries of the matrices

An−1 An−2 · · · A0 − p · Id,

An−2 · · · A0 An−1 − p · Id,

...,

A0 An−1 · · · A1 − p · Id.

The following lemma will be used in Chapter 4.

Lemma 3.12. With the presentation of U0 as in the above proposition, let x be a closed
point of U0 associated with {Fi ⊂ Λi }.
(i). The map Fi → Fi+1 is an isomorphism if and only if ai11 = 0.
(ii). The map Λi /Fi → Λi+1 /Fi+1 is an isomorphism if and only if ai+1
n−r,r = 0.

Proof.

69

(i). The map Fi → Fi+1 will be an isomorphism if and only if det(Ai ) = 0, which is if and
only if ai11 = 0.
(ii). Before proceeding, let us make the following remark on the indices for eij and aijk . The
upper indices refer to the flag and so are taken modulo n with the standard set of
representatives {0, . . . , n − 1}. The lower indices refer to the position in a vector or
matrix, and as such are also taken modulo n however with the set of representatives
{1, . . . , n}.
The relations defining the quotient Λi /Fi are
n−r

eii+j

aikj eii+r+k

=−

for 1 ≤ j ≤ r.

k=1

Thus we may take eii+r+1 , . . . , eii+n as a basis of Λi /Fi . By abuse of notation, let ϕi
be the induced map Λi /Fi → Λi+1 /Fi+1 and note we have following equations.
n−r

ϕi (eii+r+1 )

=

ei+1
i+r+1

i+1
ai+1
kr ei+r+k+1

=−
k=1

and
ϕi (eii+r+j ) = ei+1
i+r+j

for 2 ≤ j ≤ n − r.

Therefore the matrix representing the map Λi /Fi → Λi+1 /Fi+1 with respect to these
bases is





−ai+1
1r

1




 −ai+1

1


2r


.


..
..

.
.






1




−ai+1
n−r,r

70

From this, we see that the map Λi /Fi → Λi+1 /Fi+1 is an isomorphism if and only if
ai+1
n−r,r = 0.

71

Chapter 4
Integral and local models of A1
In this section we use the theory of Oort-Tate to define the integral model of A1 and construct
affine local models. Throughout this chapter, let S be an OE -scheme and k an algebraically
closed field. In the unitary case we let G = GLn,Qp × Gm,Qp with minuscule cocharacter
µ = (1r , 0n−r ) and in the unitary case we let G = GSp2n,Qp with minuscule cocharacter
µ = (1n , 0n ).

4.1

The group schemes Gi

An S-valued point x : S → A0 is given by the data ({Ai } , λ, i, η). In this section we will
associate to x a collection {Gi }n−1
i=0 of finite flat group schemes of rank p corresponding to
the kernel of Ai → Ai+1 . Furthermore, in the case where S = Spec(k), there exists an ´etale
neighborhood ϕ : V → A0 with a closed point p of V and a section σ : V → A0

72

σ

A0
Ψ

Φ

M loc

A0

V

such that ϕ(p) = x and Ψ ◦ σ is ´etale at p. Let y = Ψ ◦ σ(p). In such a case, we will be able
to tell the isomorphism type of Gi from the data ({Fi → Λi,S }) given by y.
We start by first restricting our attention to the unitary case. Recall that we have the
splittings
∗
Li,S ⊕ Li,S
= (Λi,S )n ⊕ (Λ∗i,S )n

and ωi = (Fi )n ⊕ (Fi∗ )n

induced by the splitting OB ⊗Zp OS ∼
= Mnopp (OFp ) × Mnopp (OFp ). Consider the p-divisible
group defined by
Ai (p∞ ) = lim Ai [pn ]
−→
n

and note Hi = ker(Ai → Ai+1 ) is contained in Ai (p∞ ). Now the action of OB ⊗ Zp gives
n
∞

n

f11 Ai (p∞ )

∞

e11 Ai (p ) ⊕

Ai (p ) =
j=1

j=1

where e11 and f11 are idempotents of the first, respectively second, factor of Mnopp (Zp ) ×
Mnopp (Zp ). By functoriality this gives a chain
e11 A0 (p∞ ) → e11 A1 (p∞ ) → · · · → e11 An−1 (p∞ ) → e11 A0 (p∞ )

of isogenies of degree p with the composition e11 A0 (p∞ ) → e11 A0 (p∞ ) being multiplication
by p. We then set
Gi = ker (e11 Ai (p∞ ) → e11 Ai+1 (p∞ ))
noting that Gi is a finite flat group scheme of order p.

73

Now consider the symplectic case. Here the maps Ai → Ai+1 are isogenies of degree p, and
we set Gi = ker(Ai → Ai+1 ).

Definition 4.1.1. For a group scheme G/S, define ωG = ωG/S to be the sheaf on S given
by ε∗ (Ω1G/S ) where ε : S → G is the identity section.

In the following proposition in the unitary case, recall that we are using ϕi for both the
∗
∗
morphism Li,S ⊕ Li,S
→ Li+1,S ⊕ Li+1,S
and its restriction to Λi,S → Λi+1,S .

Proposition 4.1.2. Let S = Spec(k), x : S → A0 , and y : S → M loc corresponding to x as
described above inducing the data {Fi → Λi,S }. Then
(i) In the unitary case, dimk ωHi∗ = dimk (ωi+1 /ϕi (ωi )), where Hi∗ the Cartier dual of Hi .
∗
∗
(ii) In the unitary case, dimk ωHi = dimk (Li+1,S ⊕ Li+1,S
)/ ϕi (Li,S ⊕ Li,S
) + ωi+1 .

(iii) In both the unitary and symplectic cases, dimk ωG∗i = dimk (Fi+1 /ϕi (Fi )).
(iv) In both the unitary and symplectic cases, dimk ωGi = dimk Λi+1,S / (ϕi (Λi,S ) + Fi+1 ).

Proof. (i) Hi∗ is given by the exact sequence
α
ˆ
0 → Hi∗ → Aˆi+1 −
→ Aˆi .

α
ˆ
The sequence of morphisms Aˆi+1 −
→ Aˆi → Spec(k) gives the standard exact sequence of

K¨ahler differentials
α
ˆ ∗ (Ω1Aˆi ) → Ω1Aˆi+1 → Ω1Aˆi+1 /Aˆi → 0.

74

Now pull back this sequence by the identity section εAˆi+1
ωAˆi → ωAˆi+1 → ε∗Aˆi+1 (Ω1Aˆi+1 /Aˆi ) → 0.
As Hi∗ can be described by the fibered product
Hi∗

i

Aˆi+1
α

Spec(k)

εAˆi

Aˆi

and i ◦ εHi∗ = εAˆi+1 , we have the canonical isomorphisms
Ω1Hi∗ = i∗ (Ω1Aˆi+1 /Aˆi ) and ε∗Aˆi (Ω1Aˆi+1 /Aˆi ) = (i ◦ εHi∗ )∗ (Ω1Aˆi+1 /Aˆi ) = ωHi∗ .
Therefore the exact sequence of invariant differentials becomes

ωAˆi → ωAˆi+1 → ωHi∗ → 0.
Now (i) follows since by definition ωi = ωAˆi .
(ii) Starting with the exact sequence
α

0 → Hi → Ai −
→ Ai+1

as above we have
ωAi+1 → ωAi → ωHi → 0.

75

From the Hodge filtration, ωAi ∼
= M (Ai )/ωAˆi

∨

. As the map ϕi sends ωi into ωi+1 we have

∗
)/ (ϕi (Li ⊕ Li∗ ) + ωi+1 ) .
dimk ωHi = dimk Li+1 ⊕ Li+1

In the unitary case, parts (iii) and (iv) then follow from (i) and (ii) by the functoriality of
our decompositions. In the symplectic case, (iii) and (iv) follow as in the proof of (i) and
(ii) by replacing Hi with Gi , ωAi with Fi , and Li ⊕ Li∗ with Λi .

For a general S-valued point y : S → M loc , the maps ϕi : Fi → Fi+1 and ϕ∗i : Λi /Fi →
Λi+1 /Fi+1 induce global sections qi and qi∗ of the line bundles
−1

top

Qi =

Fi

top

⊗

−1

top

Fi+1

and

Q∗i

=

Λi /Fi

top

⊗

Λi+1 /Fi+1

respectively. In the case S = Spec(k), qi and qi∗ are the determinants of the corresponding
linear maps and it is immediate that qi ⊗ qi∗ = π.
Proposition 4.1.3. Let x : Spec(k) → A0 and y : Spec(k) → M loc a corresponding geometric point inducing the sections qi and qi∗ as described above.
(i) qi = 0 if and only if dimk ωG∗i = 1.
(ii) qi∗ = 0 if and only if dimk ωGi = 1.

Proof. This follows immediately as qi = 0 if and only if Fi is carried isomorphically onto
Fi+1 , and similarly with qi∗ .

We now recall the classification of finite flat group schemes of order p over an algebraically
closed field (see Section A.4.1 for details). If char(k) = p, then any finite flat group scheme

76

of order p over Spec(k) is isomorphic to the constant group scheme Z/pZ. If char(k) = p,
then there are, up to isomorphism, three finite flat group schemes G of order p over Spec(k):
Z/pZ, µp , and αp . The Cartier dual of Z/pZ is µp , and the Cartier dual of αp is αp itself.
The following table is computed in Section A.4.1.
G
(dimk ωG , dimk ωG∗ )

µp
(1,0)

Z/pZ
(0,1)

αp
(1,1)

Table 4.1: Dimension of invariant differentials of group schemes of order p.

Thus knowing qi and qi∗ , one can determine the isomorphism type of Gi .

Corollary 4.1.4. Given a morphism S → A0 , consider the divisor on S defined by the
vanishing of qi∗ . Then the support of this divisor is precisely the locus given by the collection
of closed points
{x ∈ S ⊗ Fp : x : Spec(k(x)) → A0 induces Gi ∼
= µp or Gi ∼
= αp } .

4.2

Integral and local model of A1

Oort-Tate theory, as described in Section A.4.3, can be summarized as follows.

Theorem 4.2.1. [HR, Theorem 3.3.1] Let OT be the Zp -stack representing finite flat group
schemes of order p.
(i). OT is an Artin stack isomorphic to

[(Spec Zp [X, Y ]/(XY − wp ))/Gm ]

77

where Gm acts via λ · (X, Y ) = (λp−1 X, λ1−p Y ). Here wp denotes an explicit element
of pZ×
p given in loc. cit.
(ii). The universal group scheme GOT over OT is
GOT = [(SpecOT O[Z]/(Z p − XZ))/Gm ],

(where Gm acts via Z → λZ), with zero section Z = 0.
(iii). Cartier duality acts on OT by interchanging X and Y .

×
As in [HR], we denote GOT
to be the closed subscheme of GOT defined by the ideal (Z p−1 −X).
×
The morphism GOT
→ OT is relatively representable, finite, and flat of degree p − 1. The

notation is justified by the following proposition.

Proposition 4.2.2. [HR, 3.3.2] (cf. [Pap1, 5.1]) Let S and G be as in Definition A.4.11,
so G corresponds to a morphism ϕ : S → OT determined by the condition G = ϕ∗ (GOT ).
×
Set G× = ϕ∗ (GOT
). For c ∈ G(S), c ∈ G× (S) if and only if c is an Oort-Tate generator.

Remark 4.2.3. Let ϕ : S → OT correspond to G/S with G finite flat of order p. Then
ϕ∗ (X) cuts out the locus of closed point of S where G is infinitesimal.

Given an S-valued point of A0 , we have associated with it the group schemes {Gi }n−1
i=0 over
S in the previous section. This defines a morphism
n times

ϕ : A0 → OT ×Zp · · · ×Zp OT .

Definition 4.2.4. A1 is the fibered product

78

×
×
GOT
× · · · × GOT

A1
π

n times

A0

ϕ

OT × · · · × OT .

×
Remark 4.2.5. With GOT
→ OT relatively representable, we have A1 is represented by a

quasi-projective scheme. It is also immediate from the definition that A1 is finite and flat
over A0 . Since A0 is flat over Spec(Zp ) [G¨or1] [G¨or2], we have A1 is flat over Spec(Zp ) as
well.

Given a geometric point x : Spec(k) → A0 , as noted in the previous section there exists an
´etale neighborhood V → A0 and an ´etale morphism ψ : V → M loc . Assume that ψ factors
through an open subscheme U ⊂ M loc where each Q∗i is trivial. Choosing a trivialization, by
abuse of notation we will write qi∗ ∈ Γ(U, OU ), where qi∗ is the global section of Q∗i defined in
the previous section. Note that if Q∗i is trivial then so is Qi . Thus we also have qi ∈ Γ(U, OU ).
Consider the following diagram
V

ϕi

A0

OT

ψ
U
where ϕi : A0 → OT is ϕ followed by the ith projection.

Proposition 4.2.6. The morphism ρi : V → OT in the diagram above is given by
ρ∗i (Xi ) = εi ψ ∗ (qi∗ )

∗
ρ∗i (Yi ) = wp ε−1
i ψ (qi )

where εi is a unit in V and wp is from the description of the Oort-Tate stack in Theorem

79

4.2.1.

Proof. The special fiber of M loc , and therefore of V , is reduced [G¨or1], [G¨or2]. From the
equalities qi qi∗ = π and Xi Yi = ωp π we have that the divisors defined by the vanishing
of the global sections Z(ψ ∗ (qi∗ )) and Z(ρ∗i (Xi )) are reduced. By Corollary 4.1.4, Example
A.4.14, and Remark 4.2.3, the locus where ψ ∗ (qi∗ ) vanishes agrees with the locus where ρ∗i (Xi )
vanishes. Therefore Z(ψ ∗ (qi∗ )) = Z(ρ∗i (Xi )).
M loc is flat over OE with reduced special fiber [G¨or1],[G¨or2]. Furthermore, the generic
fiber is normal (smooth, even). It follows that M loc is normal [PZ, Proposition 8.2]. Since
V → M loc is finite ´etale, we have that V is normal as well. Thus the equality of divisors
above implies ψ ∗ (qi∗ ) and ρ∗i (Xi ) are equal up to a unit, say εi . The same proof applies for
the statement regarding ρ∗i (Yi ), and so ψ ∗ (qi ) and ρ∗i (Yi ) are equal up to a unit. This unit
must be wp ε−1
because Xi Yi = wp π and qi qi∗ = π.
i

Proposition 4.2.7. Let x : Spec(k) → A1 with π(x) = x : Spec(k) → A0 and V → A0 be
an ´etale neighborhood of x which carries an ´etale morphism V → M loc . Suppose V → M loc
factors through an open affine subscheme U ⊂ M loc on which Q∗i is trivial for each i. Set
∗
U1 = SpecU OU [u0 , . . . , un−1 ]/ up−1
− q0∗ , . . . , up−1
0
n−1 − qn−1

.

Then there exists an ´etale neighborhood V of x and an ´etale morphism ψ : V → U1 .

Proof. Define V = V ×A0 A1 . Consider the diagram

80

U1

V

A1

×
×
GOT
× · · · × GOT

U

V

A0

OT × · · · × OT

×
×
and denote by η the morphism V → GOT
× · · · × GOT
in the diagram. The two right squares

of the diagram are Cartesian and the morphism on the top left is defined by sending ui to
×
η ∗ (Zi ). The diagram commutes by the proposition above. The morphism GOT
→ OT is

relatively representable and thus
p−1
p−1
V ∼
= SpecV OV [u0 , . . . , un−1 ]/(u0 − η ∗ (Z0 ), . . . , un−1 − η ∗ (Zn−1 )) .

As the top left morphism of the diagram above is given by sending ui to η ∗ (Zi ), it follows
that V ∼
= U1 ×U V . Therefore by the above diagram, the morphism V → U1 is ´etale.

Remark 4.2.8. Given a covering of affine open subschemes {Uj } of A0 such that Q∗i is trivial
on every Uj for each i, it is tempting to hope that one may glue together the corresponding
affine schemes defined in the proposition above to get a scheme “M1loc ” with a morphism
M1loc → M loc which is a local model for A1 . However this is not possible. Indeed, let {Uj } be
any open cover of M loc so that for each j, Q∗i |Uj is trivial. Suppose that M1loc is a connected
scheme which is a local model for A1 with a morphism M1loc → M loc such that for each j,
p−1
p−1
∗
) .
Uj ×M loc M1loc ∼
= SpecUj OUj [u0 , . . . , un−1 ]/(u0 − q0∗ , . . . , un−1 − qn−1

∗
Recall that the sections q0∗ , . . . , qn−1
vanish only on the special fiber. Therefore, the restriction

of M1loc → M loc to the generic fiber is finite ´etale. However the generic fiber of M loc is the
Grassmannian Gr(n, r), and is therefore simply connected. It follows that M1loc → M loc is
an isomorphism on the generic fiber, which is a contradiction.

81

Let U ⊂ M loc be any open subset containing Sτ and consider the special fiber U ⊗ Fp . From
the above proposition, U ⊗ Fp intersects every stratum. From Proposition 3.8, the action of
I on M loc ⊗ Fp gives a surjective map I × (U ⊗ Fp ) → M loc ⊗ Fp . Thus every closed point of
M loc has a Zariski neighborhood isomorphic to a neighborhood of some closed point of U .
Combining this with Proposition 4.2.7, assume in addition that U is small enough so that
each Q∗i is trivial.

Theorem 4.2.9. The scheme

p−1
∗
U1 = SpecU OU [u0 , . . . , un−1 ]/ up−1
− q0∗ , . . . , un−1
− qn−1
0

is ´etale locally isomorphic to A1 . More precisely, for every closed point x of A0 , there exists
an ´etale neighborhood V of x and an ´etale morphism V → U1 .

By choosing U = U0 from Chapter 3, Lemma 3.12 states that up to a unit, qi∗ = ai+1
n−r,r for
0 ≤ i ≤ n − 1. With this chosen presentation, the above theorem becomes the following.

Theorem 4.2.10. The scheme

p−1
n−1
0
U1 = Spec B[u0 , . . . , un−1 ]/ up−1
− a1n−r,r , . . . , un−2
− an−r,r
, up−1
0
n−1 − an−r,r

is ´etale locally isomorphic to A1 . More precisely, for every closed point x of A0 , there exists
an ´etale neighborhood V of x and an ´etale morphism V → U1 .

82

4.3

Modification of U1

As in the previous section, fix an affine open subscheme U ⊂ M loc where U = Spec(B) such
that U contains the “worst point” and for each i the line bundle Qi |U is trivial. We therefore
identify qi∗ ∈ Γ(U, Qi |U ) with a regular function on U . Assume we have a modification (i.e.
proper birational morphism) U → U which is an isomorphism on the generic fiber and such
that U is Zariski locally of the form

Spec(Zp [x1 , . . . , xt ]/(x1 · · · xs − p)).

Such modifications are known to exist for the local models associated with GSp4 and GL4 ,
given by blowing up irreducible components of the special fiber [dJ1, G¨or3]. We then get a
modification of U1 given by
U1 = U1 ×U U → U1 .

Proposition 4.3.1. Let x ∈ U1 be a closed point. Then there exists a Zariski open neighborhood of x of the form

∗
Spec(Zp [x1 , . . . , xt , u0 , . . . , un−1 ]/(x1 · · · xs − p, up−1
− q0∗ , . . . , up−1
0
n−1 − qn−1 ))

where each qi∗ is, up to a unit, a monomial in x1 , . . . , xs with each xj occurring with multiplicity at most 1.

Proof. Let x ∈ U1 map to y ∈ U and let V ⊂ U be an affine open neighborhood of y of the
form
V = Spec(C),

C = Zp [x1 , . . . , xt ]/(x1 · · · xs − p).

83

By the preceding section, U1 is given by
∗
Spec(B[u0 , . . . , un−1 ]/(up−1
− q0∗ , . . . , up−1
n−1 − qn−1 )).
0

For each j, by abuse of notation write qj and qj∗ for their images under the ring homomorphism
B → C. We then have an affine open chart of U1 given by
∗
Spec(C[u0 , . . . , un−1 ]/(u0p−1 − q0∗ , . . . , up−1
n−1 − qn−1 )).

It remains to show that the qj∗ are, up to a unit, monomials in the variables x1 , . . . , xs with
multiplicity at most 1. With qj qj∗ = p we have that, as a divisor on V = Spec(C), Z(qj∗ )
has support in the special fiber and hence must be a sum of the irreducible components of
V ⊗ Fp . These are given by Z(xi ) for 1 ≤ i ≤ s. Suppose
Z(qj∗ ) =

ni Z(xi ).
i

Then Z(qj∗ ) = Z(xn1 1 · · · xns s ) and since V is normal we get qj∗ = εj xn1 1 · · · xns s where εj is a
unit in C. As the special fiber of V is reduced and qi qi∗ = 0, each ni is either 0 or 1 for all
i.

Remark 4.3.2. Such a modification U1 → U1 is not in general normal. An example of this
will be seen when we consider such a modification for the group GSp4 in the next chapter.

84

Chapter 5
Resolution of A1 associated with GSp4
All schemes in this section are of finite type over Spec(Zp ) and all subschemes are locally
closed, where p is an odd rational prime. We denote by A1 the integral model of the Shimura
variety associated with GSp4 equipped K(N ) level structure, where N ≥ 3 and p N . Will
construct and describe the resolution of singularities of A1 mentioned in the introduction.
To reach a resolution of A1 we will start with the known semi-stable resolution A0 of A0 . This
is obtained by blowing up an irreducible component in the special fiber of A0 (see below).
By fibering A0 → A0 with A1 → A0 , the resulting modification (i.e. proper birational
morphism) for A1 will take the form mentioned in Proposition 4.3.1; however it will fail to
be even normal. A sequence of p − 1 further blowups will produce the regular resolution
described in the introduction.
At each stage of the above process, we will first work on a local model and then carry this
over to the integral model. When performing a blowup of a local model, we require that the
subscheme being blown up corresponds to a subscheme of the integral model. In order to

85

understand more of the global structure of the resolution, such as the number of irreducible
components and how they intersect, it is also necessary to track how certain subschemes
transform with each modification. We begin by stating definitions and lemmas that will be
used throughout the process.

5.1

Preliminaries

Notation 5.1.1. Let f : X → Y be a morphism of schemes and Z ⊂ Y a subscheme. Then
• Z red denotes the subscheme of Y given as a set by Z with reduced scheme structure;
and
• f −1 (Z) denotes the scheme-theoretic inverse image under f .
In particular, f −1 (Z)red denotes the reduced inverse image of Z in X.

Definition 5.1.2 ([EH]). Let X be a scheme, Z ⊂ X a subscheme. We say that Z is Cartier
at a closed point p in X if in an affine open neighborhood of p it is the zero locus of a single
regular function which is not a zero divisor. We say that Z is a Cartier subscheme of X if
Z is Cartier at all closed points of X.

Definition 5.1.3. Let ρ : X → X be a modification.
• If ρ is given by the blowup of a closed subscheme Z of X, then Z is called a center of
ρ.
• The true center of ρ, denoted by Cρ , is defined to be the closed reduced subscheme of
X given set-theoretically by the complement of the maximal open subscheme where ρ

86

is an isomorphism.
• The fundamental center of ρ, denoted by Cρfund , is defined to be the reduced subscheme
of Cρ whose support is given by the closed points with fiber of dimension at least one.
• The residual locus of ρ, denoted by Cρres , is defined to be the reduced subscheme of Cρ
whose support is given by the closed points with fiber of dimension zero.
• The exceptional locus of ρ is ρ−1 (Cρ )red .

Remark 5.1.4. For a modification ρ with a center Z, we will often say “the center” when
there is a canonical choice of Z. The subscript ρ will be dropped from Cρ , Cρfund , and Cρres
when the morphism ρ is understood. By upper semi-continuity of the fiber, the fundamental
center C fund is a closed subscheme of C. By definition C res = C \ C fund , an open subscheme
of C. Since all schemes are assumed to be of finite type over Spec(Zp ), the fiber over a closed
point of C res is a finite collection of closed points.

Definition 5.1.5. Let ρ : X → X be a modification and W ⊂ X be a closed subscheme.
Set U = X \ Cρ . Then the strict transform of W with respect to ρ, denoted by STρ (W ) or
by ST(W ) if ρ is understood, is defined to be either
(i). the Zariski closure of ρ−1 (W \ Cρ ) inside of X if W ⊂ Cρ ; or
(ii). ρ−1 (W ) if W ⊂ Cρ .

Definition 5.1.6. Let M be an ´etale local model of X. We say that a subscheme Z of M
´etale locally corresponds to a subscheme Z of X if there exists an ´etale cover V → X with
an ´etale morphism V → M such that the scheme-theoretic pullback of Z and Z to V are
equal as subschemes of V .

Remark 5.1.7. In the situation of the definition, we have in particular that Z is an ´etale

87

local model of Z.

Lemma 5.1.8. Let M be an ´etale local model of X. Fix an ´etale cover V → X with an
´etale morphism V → M . Suppose that with these fixed morphisms, for i = 1, 2 the reduced
subscheme Zi ⊂ X ´etale locally corresponds to the reduced subscheme Zi ⊂ M . Then the
following pairs ´etale locally correspond, where each is given the reduced scheme structure.

Z1 ∪ Z 2

and Z1 ∪ Z2

Z1 ∩ Z 2

and Z1 ∩ Z2

Z1 \ Z 2

and

Z1 \ Z2

Z1

and

Z1

Here, Z1 denotes the Zariski closure inside of X, and similarly with Z1 inside of M .

Proof. Let VZi denote the pullback of Zi along the morphism V → X, VZi the pullback of
Zi along V → M , and similarly with VZ1 ∪Z2 and VZ1 ∪Z2 . Then VZ1 ∪Z2 = VZ1 ∪ VZ2 and
VZ1 ∪Z2 = VZ1 ∪ VZ2 as sets. With VZi = VZi for i = 1, 2 we have VZ1 ∪Z2 = VZ1 ∪Z2 as sets.
Now since Z1 ∪ Z2 is given the reduced scheme structure and V → X is ´etale, VZ1 ∪Z2 is
reduced. Likewise VZ1 ∪Z2 is reduced. Therefore they are equal as subschemes of V .
The statement for the next two pairs of subschemes follows in a similar manner. The statement for Z1 and Z1 follows from the fact that ´etale morphisms are flat and hence open,
giving that the pullback of Z1 to V is the Zariski closure of the pullback of Z1 to V .

Lemma 5.1.9. Let X be a scheme with ´etale local model M , Z ⊂ X a closed subscheme
´etale locally corresponding to the closed subscheme Z ⊂ M . Then BlM (Z) is an ´etale local
model of BlX (Z).

88

Proof. Let ϕ : V → X be an ´etale cover of X with an ´etale morphism ψ : V → M such that
ZV = ϕ−1 (Z) is equal to ZV = ψ −1 (Z) as closed subschemes of V . Then since blowing up
commutes with flat (and hence, ´etale) base extension, we have that the diagrams

BlV (ZV )

BlX (Z)

BlV (ZV )

BlM (Z)

V

X

V

M

are cartesian. Of course, BlV (ZV ) = BlV (ZV ) and so BlV (ZV ) → BlX (Z) is an ´etale cover
with ´etale morphism BlV (ZV ) = BlV (ZV ) → BlM (Z). Therefore BlM (Z) is an ´etale local
model of BlX (Z).

Remark 5.1.10. In the proof of the above lemma, given an ´etale cover V → X and ´etale
morphism V → M there is constructed a canonical ´etale cover BlV (ZV ) → BlX (Z) and ´etale
morphism BlV (ZV ) → BlM (Z). This will be vital for applying Lemma 5.1.8 throughout the
construction.

Lemma 5.1.11. Let Z ⊂ X be a closed subscheme and ϕ : V → X be an ´etale morphism.
Then Z is Cartier at a closed point x ∈ X if and only if ϕ−1 (Z) is Cartier at some closed
point of ϕ−1 (x).

Proof. Let Z be Cartier at a closed point x, so in fact Z is Cartier on some open affine
neighborhood U of x. Since ϕ is ´etale, we have in particular that ϕ|ϕ−1 (U ) : ϕ−1 (U ) → U is
flat. It follows that ϕ−1 (Z) is Cartier on ϕ−1 (U ), and in particular, at every closed point of
ϕ−1 (x).
Conversely, suppose ϕ−1 (Z) is Cartier at some closed point y of ϕ−1 (x). Let Spec(A) ⊂ X be
an affine open neighborhood of x, and let Spec(B) ⊂ ϕ−1 (S) be an affine open neighborhood

89

of y where ϕ−1 (Z) is Cartier. Let η : A → B be the ring homomorphism corresponding to
ϕ|Spec(B) , I ⊂ A the ideal corresponding to Z, mx ⊂ A the maximal ideal corresponding to x,
ˆ denoting the completion
and my ⊂ B the maximal ideal corresponding to y. With Aˆ and B
ˆ is an isomorphism after possibly
with respect to mx and my respectively, we have ηˆ : Aˆ → B
some base extension. As η(I)B is principal and generated by a nonzero divisor, the same
ˆ By Nakayama’s lemma, it suffices to show that IAm ⊂ Am is principal and
is true for I A.
generated by a nonzero divisor. This is the content of the following lemma.

Lemma 5.1.12. Let (R, m) be a Noetherian local ring and I ⊂ R an ideal. Denote by ˆ
ˆ is principal and generated by a
the completion with respect to m. Suppose that Iˆ = I ⊗R R
nonzero divisor. Then I is principal and generated by a nonzero divisor as well.

Proof. Once it is shown that I = (t) for some t ∈ R, it follows that t is a nonzero divisor
ˆ
from the faithfully flat map R → R.
By Nakayama’s lemma, it suffices to show that I/mI is generated by a single element as an
R/m-module. In the following, all isomorphisms are as R/m-modules.
I/mI ∼
= I ⊗R R/m
∼
ˆ m
ˆ
= I ⊗R R/
∼
ˆ ⊗ ˆ R/
ˆ m
ˆ
= (I ⊗R R)
R
∼
ˆm
ˆ Iˆ
= I/
ˆ
ˆm
ˆ mNow Iˆ is generated by a single element as an R-module,
and hence so is I/
ˆ Iˆ as an R/
ˆ
ˆ m
ˆm
ˆ and hence I/mI, is generated by a single
module. But as R/
ˆ = R/m, we have that I/
ˆ I,
element as an R/m-module.

90

Lemma 5.1.13. Let Z ⊂ X be a closed subscheme, ϕ : V → X ´etale, ZV = ϕ−1 (Z), and
ρX : BlX (Z) → X (respectively ρV : BlV (ZV ) → V ) be the blowing up of Z in X (respectively
ZV in V ). Then
CρV = ϕ−1 (CρX ),

Cρfund
= ϕ−1 (Cρfund
),
V
X

and

Cρres
= ϕ−1 (Cρres
).
V
X

Proof. Note that since ϕ is ´etale, the pullback of a reduced subscheme is reduced. As such,
all subschemes of V in the above equalities have reduced scheme structure, and it suffices to
verify the equalities as sets.
=
That CρV = ϕ−1 (CρX ) follows immediately from Lemma 5.1.11. The second equality Cρfund
V
) is clear from the cartesian diagram
ϕ−1 (Cρfund
X
BlV (ZV )

BlX (Z)

V

X

) follows immediately from the
= ϕ−1 (Cρres
as mentioned in Lemma 5.1.9. Finally, Cρres
X
V
previous two statements.

Corollary 5.1.14. Let X be a scheme with ´etale local model M , Z ⊂ X a closed subscheme
´etale locally corresponding to the closed subscheme Z ⊂ M . Then the true center, fundamental center, and residual locus of BlX (Z) → X ´etale locally corresponds respectively to the
true center, fundamental center, and residual locus of BlM (Z) → M .

The next proposition says that strict transforms may be calculated ´etale locally.

91

Proposition 5.1.15. Let Z ⊂ X be a closed subscheme, ϕ : V → X be ´etale, and ZV =
ϕ−1 (Z) giving the following cartesian diagram.

BlV (ZV )

ϕ

ρV

BlX (Z)
ρX

V

ϕ

X

Then
STρV (ϕ−1 (Z)) = (ϕ )−1 (STρX (Z)).

Proof. In the case that Z is contained in the true center of ρX , the strict transform of Z
under ρX is defined to be ρ−1
X (Z). Then the claim follows immediately from the definitions
and Lemma 5.1.13.
So assume that Z is not contained in the true center CρX . Then
−1
STρV (ϕ−1 (Z)) = ρ−1
V (ϕ (Z) \ CρV )
−1
= ρ−1
V (ϕ (Z \ CρX ))

by Lemma 5.1.13

= (ϕ )−1 (ρ−1
X (Z \ CρX ))
= (ϕ )−1 ρ−1
X (Z \ CρX )

since ϕ is flat and hence open

= (ϕ )−1 (STρX (Z)).

Lemma 5.1.16. Let M and M be ´etale local models of X and X respectively. Suppose there
is an ´etale cover ϕ : V → X with an ´etale morphism ψ : V → M along with morphisms ρX
and ρM giving the diagram

92

X

ϕ

ρX
X

ψ

V

M

ρV
ϕ

V

ρM
ψ

M

where the left and right squares are cartesian.
(i). Let Z ⊂ X be a closed subscheme ´etale locally corresponding to a closed subscheme
Z ⊂ M with respect to ϕ and ψ. Then ρ−1
etale locally corresponds to ρ−1
X (Z) ´
M (Z).
(ii). Let v ∈ V be a closed point, and set

x = ϕ (v ),

y = ψ (v ),

v = ρV (v ),

x = ϕ(v),

y = ψ(v).

Suppose further that k(x) = k(v). Then k(x ) = k(v ) and
∼ −1
ρ−1
X (x) = ρM (y) ×Spec(k(y)) Spec(k(x)).

−1
−1
Proof. (i) Pulling back ρ−1
X (Z) (respectively ρM (Z)) to V is the same as pulling back ϕ (Z)

(respectively ψ −1 (Z)) to V . With ϕ−1 (Z) = ψ −1 (Z) as subschemes of V , the claim follows.
(ii) With v ∈ V a closed point mapping to x ∈ X and v ∈ V , k(v ) is a quotient of
k(x ) ⊗k(x) k(v) = k(x ). Thus k(x ) = k(v ).
For the second statement, note that there is an inclusion k(y) ⊂ k(v) = k(x) giving the

93

morphism Spec(k(x)) → Spec(k(y)). Thus

ρ−1
X (x) = X ×X Spec(k(x))
= (X ×X V ) ×V Spec(k(x))
= (M ×M V ) ×V Spec(k(x))
= M ×M Spec(k(x))
= (M ×M Spec(k(y))) ×Spec(k(y)) Spec(k(x))
= ρ−1
M (y) ×Spec(k(y)) Spec(k(x)).

Lemma 5.1.17. Let f : X → Y be a morphism of schemes. Suppose there exists a closed
point y ∈ Y such that f −1 (y) is connected and the set-theoretic image of every connected
component of X under f contains y. Then X is connected.

Proof. Let X = X1

X2 where Xi is open and closed in X for i = 1, 2. Then
f −1 (y) = (f −1 (y) ∩ X1 ) ∪ (f −1 (y) ∩ X2 )

with f −1 (y) ∩ Xi being open and closed in f −1 (y) for i = 1, 2. Since f −1 (y) is connected,
without loss of generality suppose f −1 (y) = f −1 (y) ∩ X1 . Then f −1 (y) ∩ X2 = ∅. As the
image of each connected of X contains y, it must be that X2 is not a union of connected
components; i.e. X2 = ∅. Therefore X is connected.

94

5.1.1

´
Etale
cover

In view of Proposition 5.1.8, it will be important that we work with a fixed ´etale cover of
the integral model and a fixed ´etale morphism to the local model in each step. It is also
important that we chose such a cover so that we are able to apply Proposition 5.1.16 part
(ii) as well as Proposition 4.2.6. We now describe the covers and morphisms to be used
throughout the construction.

Proposition 5.1.18. There exists an ´etale cover ϕ : V → A0 and an ´etale morphism
ψ : V → U0 such that for each closed point x ∈ A0 , there exists a closed point v ∈ V with
x = ϕ(v) and k(x) = k(v).

Proof. This follows from modifying the argument given in [DP, Section 3]. Let us briefly
sketch how. Let x : Spec(k) → A0 be a closed point with k = k(x) and
Auniv
→ Auniv
→ Auniv
0
1
2

be the chain of universal abelian schemes over A0 . By [RZ, Proposition A.56], there exists
1
a Zariski open neighborhood U ⊂ A0 of x where HdR
(Auniv
) is free for all i. This gives a
i

section U → A0 of A0 → A0 and hence, restricting if necessary, a morphism

ϕ : U → A0 → U0 .

By taking V to be the Zariski open subset of U where ϕ is ´etale, the proposition will follow
from showing ϕ is ´etale at x.
∧

∧

Let y = ϕ(x), A∧0 x the completion of A0 at x, U0 y the completion of U0 at y, and U0 y ⊗ k

95

∧

be the extension of residue field of U0 y to k. It suffices to show that the induced map
∧

A∧0 x → U0 y ⊗ k is an isomorphism.
Let B = k[ε]/(ε2 ) and
(A0 → A1 → A2 , λ0 , λn , η)
correspond to x : Spec(k) → A0 . As in [dJ2, Section 2] the principal polarizations induce
the nondegenerate alternating pairings

eλj : D(Aj ) × D(Aj ) → Okcrys

for j = 0, n.

Let A0 → · · · → An be a chain of abelian schemes over Spec(B) with Ai a deformation
of Ai for every i. Then as shown in [dJ2, Proposition 4.5], the polarization induced by x
lifts giving a deformation Spec(B) → A∧0,x of x if and only if for j = 0, n the corresponding
filtration Fil1j ⊂ D(Aj )Spec(B) is totally isotropic with respect to the pairing eλj ,Spec(B) . It
∧

follows that the Spec(B)-valued points of A∧0 x and U0 y ⊗ k are the same and hence the map
∧

A∧0 x → U0 y ⊗ k induces an isomorphism on Zariski tangent spaces.
Now the proof given in [DP, Theorem 3.3] below Lemma 3.5 shows that the map A∧0 x →
∧

U0 y ⊗ k is indeed an isomorphism.

Remark 5.1.19. In the course of the proof of the above proposition, ϕ and ψ were chosen
so that we have the diagram

A0
Φ
V

ϕ

A0

Ψ
U0

96

where ψ is given by the composition V → A0 → U0 . Thus Proposition 4.2.6 may be applied.
In fact, since Ψ is surjective [Gen, Proposition 1.3.2], we also choose V → U0 such that ψ is
surjective.

We fix such an ´etale cover ϕ : V → A0 and ´etale morphism ψ : V → U0 as in the proposition.
In Step I we construct
A0 = BlA0 (Z1 ) and U0 = BlU0 (Z1 )
taking the induced ´etale cover ϕ : V → A0 with ´etale morphism ψ : V → U0 as described
in the proof of Proposition 5.6.1. In Step II, the schemes constructed are

A1 = A1 ×A0 A0

U1 = U1 ×U0 U0 .

The ´etale cover and morphism in Step II are given by the top horizontal arrows of the
diagram

A1

ϕ

ρV

ρA
A0

ψ

V

ϕ

V

U1
ρU

ψ

U0

where both squares are cartesian. With each step after II being given by a blowup, we again
take the ´etale cover and ´etale morphism described in the proof of Proposition 5.6.1.
Repeated applications of Proposition 5.1.16 part (ii) give the following.
[i−1]

Proposition 5.1.20. Let 1 ≤ i ≤ p + 1 and x[i−1] ∈ Aj

be a closed point, where j = 0

if i = 1, 2 and j = 1 if i > 2. Then there exists a closed point v [i−1] ∈ V [i−1] with x[i−1] =

97

ϕ[i−1] (v [i−1] ) such that k(x[i−1] ) = k(v [i−1] ), where k(·) denotes the residue field. Let v [i] ∈ V [i]
[i]

be any closed point such that ρV (v [i] ) = v [i−1] and set y [i−1] = ψ(v [i−1] ). Then
(ρA )−1 (x[i−1] ) ∼
= (ρU )−1 (y [i−1] ) ×Spec(k(y[i−1] )) Spec(k(x[i−1] )).
[i]

[i]

[i]

[i]

In particular if the fiber (ρU )−1 (y [i−1] ) is geometrically connected, then (ρA )−1 (x[i−1] ) is connected.

5.1.2

Notation

Before beginning the construction of the resolution, we explain some notation that will be
used throughout.
Roman letters such as Z, C, and E will be used to denote subschemes of the local models.
Calligraphic letters such as Z, C, and E denote subschemes of the integral models that ´etale
locally correspond to their Roman counterparts.
The construction will proceed in steps, where Step II is constructed by a fiber product and
every other step is constructed by a blowup. The schemes constructed in each additional
step will be decorated with an additional tick mark , and the superscript [i] denotes i tick
[0]

[1]

marks (e.g. A0 = A0 , A0 = A0 ). The integral models that will be constructed are
[p+1]

[p+1] ρA

A1

[p]

[p] ρ

ρ

[5]

[4] ρ

[4]

ρ

ρ

ρ

ρ

ρ

ρ

A
A
A
A
A
A
A0
−−−→ A1 −−
→ . . . −−
→ A1 −−
→ A1 −→
A1 −→
A0 −→

with their corresponding ´etale local models
[p+1]

[p+1] ρU
U1
−−−→

[p]

[p] ρU
U1 −−
→

[5]

ρU

. . . −−→

[4]

[4] ρU
U1 −−
→

98

U
U
U
U1 −→
U1 −→
U0 −→
U0 .

Moreover, any subscheme will also be decorated by tick marks, e.g. Z signifies that Z ⊂ A1 .
The subschemes that will be blown up in each step will arise from subschemes of A0 . As
such, it will be necessary to observe how these subschemes of A0 transform (either their
strict transform or scheme-theoretic inverse image) in each step. To keep track of this, we
will use a subscript to denote which step the subscheme will be used in. So for example, C4
is a subscheme of A0 . The C indicates it will transform to be the true center of some blowup,
and the 4 indicates that it will become the true center of Step 4. So in this example, we start
with the strict transform C4 = STρ (C4 ) and then C4 = (ρ )−1 (C4 )red . Finally C4 = STρ (C4 )
[4]

and, as we will show, this is the true center of the blowup A1 → A1 in Step 4.
In general, C and C will denote true centers, E and E will denote exceptional loci, and Zij and
Zij will denote irreducible components. For x ∈ Perm(µ), Ax will denote the KR-stratum
corresponding to x and nx the number of connected components of Ax (see Chapter 3 and
the next subsection).

5.1.3

Connected components

Let (A0 → A1 → A2 , λ0 , λ2 , η) correspond to a closed point Spec(k) → A0 . The functor
∼

A0 → An,1,N from Section 2.4 induces A0 [N ] −
→ (Z/N Z)4 . Combining this with the Weil
pairing induced by λ0 , we get the homomorphism
∼

(Z/N Z)4 × (Z/N Z)4 −
→ A[N ] × A[N ] → µN (k)
∼

where µN denotes the N th roots of unity. Taking the highest exterior power gives Z/N Z −
→
µN which is equivalent to choosing a primitive N th root of unity ζN . Therefore the structure

99

morphism of A0 over Spec(Zp ) factors as

A0 → Spec(Zp [ζN ]) → Spec(Zp ).

The fibers over the closed points of Spec(Zp [ζN ]) are connected [Hai, Lemma 13.2].
Furthermore, in each connected component every KR-stratum is nonempty [Hai, Lemma
13.2] and every KR-stratum has the same number of connected components [GY1]. In
view of the this, for our construction of the resolution it suffices to treat a single connected
component of A0 → Spec(Zp [ζN ]). By abuse of notation we will write A0 for such a connected
component over Spec(Zp [ζN ]). Similarly, we write A1 for the union of connected components
lying over A0 with respect to the map π : A1 → A0 .

5.1.4

KR Strata

The KR strata of A0 will play a fundamental role in what follows. Recall that the strata of
M loc ⊗ Fp correspond to the µ-permissible set of W aff .

100

s1 s0 s2 τ
s1 s2 τ

s0 s1 s0 τ

s1 s0 τ

s0 s1 τ

s1 τ
τ

s2 τ
s2 s1 s2 τ
=t

µ

s2 s1 τ

s0 τ

s0 s2 τ
s0 s2 s1 τ

Figure 5.1: µ-permissible set for GSp4

The alcoves that make up the µ-permissible set are shown above in various shades of gray.
The collection {si } are the standard generators of the affine Weyl group of GSp4 (see Section
A.2). The dimension of each stratum can be read off the corresponding w ∈ W aff by (w),
where (·) is the length with respect to the Bruhat order (see Section A.2 and Proposition
A.3.9). The stratum Sτ corresponding to the base alcove τ is the unique KR-stratum of
dimension zero, which corresponds to the worst point of the local model. The irreducible
components are the extreme alcoves which are shaded medium gray. By [GY2, Theorem 1.5]
the extreme alcoves are connected, and as they are smooth, it follows that there are precisely

101

four irreducible components of A0 ⊗ Fp . The supersingular locus is pictured in dark gray,
given by As0 s2 τ ∪ As1 τ . Note that the supersingular locus is precisely the locus where the
zero section is an Oort-Tate generator of both of the corresponding group schemes.

5.2

5.2.1

Step 0: A0, U0, and A1

Description of the local model U0

A presentation of U0 is given as the closed subscheme of Spec(Zp [aijk ; i = 0, . . . , 3, j, k = 1, 2])
cut out by the following two collections of equations. This first collection comes from the
equations of the local model associated with GL4 .

i
ai21 = ai+1
12 a11 ,

i+1 i
ai22 = ai+1
11 + a12 a12 ,

i+2 i+1 i
(ai+2
11 + a12 a12 )a11 − p,

i ∈ Z/4Z

i ∈ Z/4Z

i+1
i+2
i+2 i+1 i
ai+2
12 a11 + (a11 + a12 a12 )a12 ,

i ∈ Z/4Z

We also have to include those coming from the duality condition (see Proposition 3.10).

a022 = a011

a222 = a211

102

a311 = a122

a312 = −a112

a321 = −a121

a322 = a111 .

b = a212

c = −a112

By setting
x = a122

y = a011

a = a012

we arrive at the equations derived in [dJ2]:

U0 = Spec(B) where B = Zp [x, y, a, b, c]/(xy − p, ax + by + abc).

With this presentation we have that, up to a unit:

q0 = y,

q0∗ = x,

q1 = y + ac,

q1∗ = x + bc.

Following [dJ2, Section 5], we label the four irreducible components of U0 ⊗ Fp as

Z00 = Z(y, a)
Z01 = Z(y, x + bc)
Z10 = Z(x, y + ac)
Z11 = Z(x, b).

5.2.2

Description of the integral model A0

Still following [dJ2, Section 5], using the local model diagram

103

A0
Φ

Ψ

A0

U0

we define Zij = Φ(Ψ−1 (Zij )). Since there are precisely four irreducible components of A0 ⊗Fp
[Yu], the Zij make up all of the irreducible components of A0 ⊗ Fp .

Proposition 5.2.1. Zij ´etale locally corresponds to Zij . Moreover this holds for arbitrary
unions, intersections, and complements of the Zij , e.g. Z11 ∪ (Z01 ∩ Z10 ) ´etale locally corresponds to Z11 ∪ (Z01 ∩ Z10 ), where each is given the reduced scheme structure.

Proof. Let x ∈ A0 be a closed point of Zij . With A0 ← V → U0 our chosen ´etale cover
and ´etale morphism, by Remark 5.1.19 we may apply Proposition 4.2.6. Thus we have that
pullback of q0 and y to V induce the same divisor on V , and likewise for q0∗ , q1 , and q1∗ with
x, y + ac, x + bc respectively. Now each Zij can be described as the Zariski closure of the
locus given by certain conditions on the functions qi and qi∗ .
Irreducible component
Z00
Z01
Z10
Z11

q0
q0
q0
q0

= 0,
= 0,
= 0,
= 0,

Conditions
= 0, q1 = 0,
= 0, q1 = 0,
= 0, q1 = 0,
= 0, q1 = 0,

q0∗
q0∗
q0∗
q0∗

q1∗
q1∗
q1∗
q1∗

=0
=0
=0
=0

Table 5.1: Irreducible components of A0

Thus we have that the pullback of Zij and Zij to V define the same closed subscheme and
hence ´etale locally correspond. The claim for arbitrary unions, intersections, and complements follows by applying Lemma 5.1.8.

104

We now define subschemes that will be used in the steps throughout the construction of the
resolution.
Local
Z1 =
C1 =
Z3 =
C3 =
Z4 =
C4 =

model U0
Z(x, b)
Z(x, y, a, b)
Z(x, bc)
Z(x, bc)
Z(x, b)
Z(x, y, b, c)

Z1
C1
Z3
C3
Z4
C4

=
=
=
=
=
=

Integral model A0
Z11
Z00 ∩ Z01 ∩ Z10 ∩ Z11
Z11 ∪ (Z01 ∩ Z10 )
Z11 ∪ (Z01 ∩ Z10 )
Z11
(Z01 ∩ Z10 ∩ Z11 ) \ Z00

Table 5.2: Subschemes of A0

Proposition 5.2.1 shows that the subschemes on the left ´etale locally correspond to those on
the right.

Proposition 5.2.2. Writing each subscheme below as a disjoint union of KR-strata, we
have the following.
Z11
Z01 ∩ Z10
Z00 ∩ Z01 ∩ Z10 ∩ Z11
(Z01 ∩ Z10 ∩ Z11 ) \ Z00

As2 s1 s2 τ
=
= As0 s2 τ ∪ As1 τ
=
As1 τ
=
As2 τ

=
=
=
=

Aτ
Aτ
Aτ
Aτ

∪ A s1 τ ∪ A s2 τ ∪ A s2 s1 τ ∪ A s2 s1 s2 τ
∪ A s0 τ ∪ A s1 τ ∪ A s2 τ ∪ A s0 s2 τ
∪ A s1 τ
∪ A s2 τ

Proof. With our chosen presentation, the locus on U0 corresponding to the supersingular
locus of A1 is Z01 ∩ Z10 . By Proposition 5.2.1 it follows that Z01 ∩ Z10 is the supersingular
locus. The supersingular locus is given by As0 s2 τ ∪ As1 τ [GY1, Section 2] and the claim
follows.
To show Z11 = As2 s1 s2 τ we choose the point x = a = b = c = 0 and y = 1 of U0 which lies
solely on the irreducible component Z11 . Using the equations in Section 5.2.1 and Chapter

105

3, this point corresponds to the following flag.



1
0




 0 1 


F0 = 
,
 1 0 




0 1




0
0




 1 0 


F1 = 
,
 0 1 




1 0




0
0




 0 0 


F2 = 
,
 1 0 




0 1




0
1




 0 0 


F3 = 
.
 0 1 




1 0

In the notation of Proposition A.3.10 this gives the alcove



1





π
1 



,
 1

π




1



 1


1



1


1
π









,




 1


1



1





1





,






−1
 π


1


 π −1
1





1










where we omit any entry which is zero. Our chosen point lies in As2 s1 s2 τ if and only if there
is an element b in the Iwahori subgroup such that b · s2 s1 s2 τ gives the same alcove as above.
With s2 s1 s2 τ given by the alcove


 π


π



1





1





,






 1


π



1





1







,




 1


1



1



it is easy to check that



1
1





1
1 


b=



1




1

106



1





,






 1


1



π −1





1










suffices. Therefore Z11 = As2 s1 s2 τ . Since Z00 ∩ Z11 is the union of a two dimensional and
one dimensional scheme, we see from the µ-permissible set (Section 5.1.4) it must be that
Z00 = As0 s1 s0 τ . That Z00 ∩ Z01 ∩ Z10 ∩ Z11 = Aτ ∪ As1 τ again follows from the diagram in
Section 5.1.4. Finally
Z01 ∩ Z10 ∩ Z11 = Aτ ∪ As1 τ ∪ As2 τ
and thus we have
(Z01 ∩ Z10 ∩ Z11 ) \ Z00 = As2 τ .

Proposition 5.2.3. The number of connected and irreducible components of the subschemes
of A0 are as follows.
Subscheme of A0
Z1
C1
Z3
C3
Z4
C4

# of connected components
1
ns 1 τ
1
1
1
ns 2 τ

# irreducible components
1
ns1 τ
1 + ns0 s2 τ
1 + ns0 s2 τ
1
ns2 τ

Table 5.3: Number of connected and irreducible components of subschemes of A0

Proof. Z1 = Z4 = Z11 : This is an irreducible component.
Z3 = C3 = Z11 ∪ (Z01 ∩ Z10 ): To see this subscheme is connected, it suffices to show that
each connected component of Z01 ∩ Z10 meets Z11 . Let W be such a connected component.
With Z01 ∩ Z10 being a union of KR-strata, by possibly shrinking W we may assume W is
a connected component of some KR-stratum. By [GY2, Theorem 6.4] W ∩ Aτ = ∅, where
W is the Zariski closure of W inside of A0 . As Aτ ⊂ Z11 , the claim follows.

107

To find the number of irreducible components, note that Z11 ∪ (Z01 ∩ Z10 ) is a union of three
and two dimensional irreducible components: the unique three dimensional component is
Z11 and the two dimensional components are given by the irreducible components of

(Z01 ∩ Z10 ) \ Z11 = As0 s2 τ .

As (Z01 ∩ Z10 ) \ Z11 corresponds to the ideal (x, y, a, b) of B, we have that Z01 ∩ Z10 \ Z11
is smooth. Thus as Z01 ∩ Z10 \ Z11 has ns0 s2 τ connected components and each component is
irreducible, we have a total of 1 + ns0 s2 τ irreducible components of Z3 = C3 .
C1 = Z00 ∩ Z01 ∩ Z10 ∩ Z11 : From Proposition 5.2.2, this subscheme is given by As1 τ . Since
C1 and hence As1 τ is smooth, each connected component of As1 τ is irreducible and As1 τ has
the same number of connected components as As1 τ .
C4 : This follows by a similar argument to that given for the statement about C1 .

5.2.3

Description of A1

As remarked in section 5.1.4, A0 ⊗ Fp has four irreducible components. Using the notation
of [dJ2, Section 5], we denote these irreducible components by Z00 , Z01 , Z10 , and Z11 . To
determine the irreducible components of A1 ⊗ Fp , we will need the following.

Lemma 5.2.4. The irreducible components of U1 ⊗ Fp are normal.

Proof. The four irreducible components of U1 ⊗ Fp correspond to the following ideals in

108

Zp [x, y, a, b, c, u, v]/(xy − p, ax + by + abc, up−1 − x, v p−1 − x − bc).

(u, v, b),

(u, y + ac),

(y, v),

(y, a)

The first irreducible component is smooth, while the other three are the spectra of

Fp [y, a, b, c, v]/(v p−1 −bc),

Fp [a, b, c, u, v]/(up−1 −bc),

and Fp [b, c, u, v]/(v p−1 −up−1 −bc).

Note that each is a complete intersection and hence Cohen-Macaulay. The Jacobian Criterion
shows that the singular-locus has codimension greater than one. By Serre’s Criterion [Mat,
Theorem 23.8], each is normal.

Proposition 5.2.5. A1 ⊗ Fp is connected, equidimensional of dimension three, and the
irreducible components are normal. Furthermore A1 ⊗ Fp and has precisely four irreducible
components and these irreducible components are given by π −1 (Zij )red where π : A1 → A0 .

Proof. That A1 ⊗ Fp is equidimensional of dimension three is immediate from inspection of
the local model U1 ⊗ Fp . As Aτ is in the supersingular locus, the fiber above a closed point of
Aτ ⊂ A0 with respect to π consists of a single closed point. Since π is finite and surjective,
each irreducible component of A1 ⊗ Fp maps surjectively onto an irreducible component of
A0 ⊗ Fp . Finally as every irreducible component of A0 ⊗ Fp contains Aτ , we conclude that
A1 ⊗ Fp is connected by Lemma 5.1.17.
Let Zij denote a fixed irreducible component of A1 ⊗ Fp which maps onto Zij and suppose
there is a fifth irreducible component Z of A1 ⊗Fp . By [GY2, Theorem 6.4], each irreducible
component of A0 ⊗ Fp contains Aτ , and thus Z00 , Z01 , Z10 , Z11 , and Z simultaneously intersect at some closed point x of A1 ⊗ Fp . We now show that this is not possible.

109

ϕ

ψ

Let A1 ⊗ Fp ←
−V −
→ U1 ⊗ Fp be an ´etale cover of A1 ⊗ Fp with ´etale morphism to U1 ⊗ Fp .
Choose a closed point p ∈ V such that ϕ(p) = x. Set y = ψ(p). Since the irreducible
components of U1 ⊗ Fp are integral, normal, and excellent, the completion of any irreducible
component at y is also an integral domain and normal. Therefore, there are at most four
components of V passing through p. Thus the number of irreducible components passing
through x ∈ A1 ⊗ Fp is at most four, giving the contradiction that we are seeking.

5.3

Step I: Semi-stable resolution of A0

We define the integral and local models

A0 = BlA0 (Z1 )

5.3.1

and

U0 = BlU0 (Z1 ).

Description of the local model U0

With I the ideal sheaf on Spec(B) corresponding to the ideal (x, b) ⊂ B, the blowup is given
by
U0 = ProjU0 O ⊕ I ⊕ I 2 ⊕ . . . .
Note that we have the morphism

T = ProjU0 B[x, b]/(ax + by + abc, xb − xb) → U0

110

sending x and b respectively to x and b in grade one. The two standard affine open charts
of T are given by

x=0:

T1 = Spec Zp x, y, a, c,

b
b
b
/ xy − p, a + y + a c
x
x
x

and
b=0:

T2 = Spec Zp a, b, c,

x
b

/

x
b

b(−a)

x
b

+c −p

.

T1 is covered by the two open subschemes defined by the conditions b = 0 and 1 + bc = 0.
Noting that the condition b = 0 makes the first of these open subschemes a subscheme of
T2 , we merely write down the presentation of the second, given by 1 + bc = 0:
T1 = Spec Zp [x, y, c, b, (1 + bc)−1 ]/(xy − p) .

From the affine open cover T1 ∪ T2 , we see that T is integral. As the blowup U0 may be,
a priori, cut out by more equations, we at least have a closed immersion U0 → T . This
closed immersion is an isomorphism on the generic fiber and with T integral, it must be an
isomorphism.
The morphism U0 → U0 is given as homomorphisms of coordinate rings, sending the ordered
set of global sections to the ordered set of global sections, as

T1 :

T2 :

{x, y, a, b, c} → {xb, −a(x + c), a, b, c}

{x, y, a, b, c} → x, y, by(1 + bc)−1 , xb, c .

The true center of U0 → U0 is the closed subscheme where the ideal sheaf induced by the ideal

111

(x, b) is not Cartier. From the relation ax + by + abc = 0, we see that C1 must be contained
in the closed subscheme Z(x, y, a, b). From the presentation of the morphism above, we see
that the fiber over any closed point of Z(x, y, a, b) is the affine line given by the coordinate
x. Therefore, C1 = Z(x, y, a, b) and is of dimension one, the true center of ρU is equal to
the fundamental center, and the exceptional locus of ρU is two dimensional. The affine cover
T1 ∪ T2 also shows that U0 ⊗ Fp is equidimensional of dimension three. The strict transforms
of the closed subschemes given in Step 0 are as follows.
Z3 = C3 = Z(x, bc): The complement Z3 \ C1 a union of three subschemes defined by the
following conditions.
x = 0,

b = 0,

a=0

x = 0,

b = 0,

y=0

x = 0,

b = 0,

a = 0,

c=0

Now consider the inverse image of Z3 \ C1 under the morphisms ρU : U0 → U0 . The Zariski
closure of the inverse image of the first two subschemes give the same subscheme, namely the
subscheme corresponding to the ideals (x, b) and (x, b) in the coordinate rings of T1 and T2
respectively. Likewise the third corresponds to (x, y, c, x) and (1) in T1 and T2 respectively.
Thus Z3 = C3 is given by Z(x, b) ∪ Z(x, y, c, x).
Z4 = Z(x, b): This must necessarily be the irreducible component lying above Z(x, b), and
it is also given by Z(x, b).
C4 = Z(x, y, b, c): The complement Z3 \ C1 given by x = y = b = c = 0 and a = 0. The
Zariski closure of the inverse image is Z(x, y, b, c).
In summary, we have:

112

Z3 = Z(x, b) ∪ Z(x, y, c, x)
C3 = Z(x, b) ∪ Z(x, y, c, x)
Z4 = Z(x, b)
C4 = Z(x, y, b, c).

5.3.2

Description of the integral model A0

With U0 an ´etale local model of A0 , Lemma 5.1.13 and Proposition 5.2.1 give that the true
center of A0 → A0 is C1 . By the remarks in the previous section, A0 ⊗Fp is equidimensional of
dimension three. From Lemma 5.1.16, the exceptional locus of A0 → A0 is two dimensional.
Thus no irreducible component of A0 ⊗ Fp is contained in the exceptional locus. Therefore
A0 ⊗ Fp has four irreducible components, each being given by the strict transform of an
irreducible component of A0 ⊗ Fp . We denote these strict transforms by Z00 , Z01 , Z10 , and
Z11 .

Proposition 5.3.1. The number of connected and irreducible components of the subschemes
of A0 are as follows.
Closed subscheme of A0
Z3 = ST(Z3 )
C3 = ST(C3 )
Z4 = ST(Z4 )
C4 = ST(C4 )

# connected components
1
1
1
ns2 τ

# irreducible components
ns0 s2 τ + 1
ns0 s2 τ + 1
1
ns2 τ

Table 5.4: Number of connected and irreducible components of subschemes of A0

113

Proof. Z3 = C3 : We start by showing Z3 is connected. From Proposition 5.2.3, Z3 =
Z11 ∪ (Z01 ∩ Z10 ) is a union of three and two dimensional components intersecting in a one
dimensional closed subscheme. We claim that this one dimensional subscheme intersects
with the true center C1 = Z00 ∩ Z01 ∩ Z10 ∩ Z11 in a zero dimensional subscheme. This can
easily be seen by writing each as a union of KR-strata. Indeed, set W = Z01 ∩ Z10 \ Z11
which is equidimensional of dimension two. Then Z3 = Z11 ∪ W and from Proposition 5.2.2,
each is given as a union of KR-strata as follows.

Z11 = Aτ ∪ As1 τ ∪ As2 τ ∪ As2 s1 τ ∪ As2 s1 s2 τ
W = Aτ ∪ As0 τ ∪ As2 τ ∪ As0 s2 τ
C1 = Aτ ∪ As1 τ

Therefore the one dimensional subscheme Z11 ∩W intersects with C1 in Aτ , a zero dimensional
subscheme as claimed. With Z11 and W smooth it follows immediately that Z3 \ C1 is
connected, and thus so is the strict transform of Z3 . Therefore Z3 = C3 is connected.
Now we show that Z3 = C3 has ns0 s2 + 1 irreducible components. From the proof of Proposition 5.2.3 we have Z3 is a union ns0 s2 + 1 irreducible components, each of dimension two or
three. Since the true center of A0 → A0 is of dimension one, each irreducible component is
not contained in the true center. Hence the strict transform of each irreducible component
is irreducible as well and the claim follows.
Z4 : Note that Z4 is connected, smooth, and three dimensional giving that Z4 \C1 is connected
as well. It follows immediately that Z4 is connected and irreducible.
C4 : As C4 is smooth, each irreducible component is a connected component. Also, C4 intersects the true center of A0 → A0 in a zero-dimensional subscheme and therefore strict

114

transform of each irreducible component of C4 is irreducible.

5.4

Step II: Fiber with A1

We define the integral and local models

A1 = A1 ×A0 A0

5.4.1

and

U1 = U1 ×U0 U0 .

Description of the local model U1

As in Theorem 4.2.10, U1 is given in the chosen presentation by adjoining the variables u
and v along with the relations up−1 − x and v p−1 − (x + bc) to U0 . We thus have
U1 = ProjU0 B[u, v][x, b]/(ax + by + abc, xb − xb, up−1 − x, v p−1 − (x + bc)

where B[u, v] is of grade 0 and x and b are of grade 1. We define Zij to be the reduced inverse
image of Zij under the morphism ρU : U1 → U0 . These are the irreducible components of

115

U1 . The reduced inverse images under U1 → U0 are given by

Z00 = Z(y, a)
Z01 = Z(y, v)
Z10 = Z(u, y + ac)
Z11 = Z(u, v, b)
Z3 = Z(u, v, b) ∪ Z(u, v, y, c, x)
C3 = Z(u, v, b) ∪ Z(u, v, y, c, x)
Z4 = Z(u, v, b)
C4 = Z(u, v, y, b, c, x).
Note that Z(u, v, b) ∪ Z(u, v, y, c, x) = Z(u, v) as the relation v p−1 − up−1 − bc implies that
if u and v are zero, then bc = 0 giving the two components.

5.4.2

Description of the integral model A1

With A1 = A1 ×A0 A0 , the projection A1 → A1 is proper and birational and so it is a
modification. Also note that the projection ρ : A1 → A0 is finite and flat. As claimed in
the introduction, we have the following.

Proposition 5.4.1. A1 is not normal.

Proof. Since ψ : V → U0 and hence ψ : V

→ U1 is surjective, it suffices to show that

U1 is not normal. Consider the irreducible component Z11 of U1 ⊗ Fp . In the local ring of
the generic point of this component, the maximal ideal is given by (u, v). This ideal is not

116

principal since u ∈
/ (v) and v ∈
/ (u). Therefore U1 is not normal by Serre’s Criterion [Mat,
Theorem 23.8].

Proposition 5.4.2. Set Zij = (ρ )−1 (Zij )red . Each Zij is an irreducible component of
A1 ⊗ Fp , and these give all the irreducible components of A1 ⊗ Fp .

Proof. From Proposition 5.2.5 we have that Wij := π −1 (Zij )red is irreducible, where π : A1 →
A0 . Note that the morphism A1 → A1 is a modification with true center of dimension at
most one. As such, Wij is not contained in the true center and therefore its strict transform
Wij with respect to A1 → A1 is irreducible.
Set
U = A0 \ C1 ,

U = (ρA )−1 (U),

and U = (ρA )−1 (U ).

Then Zij ∩ U = Wij ∩ U because both can be described as the reduced inverse image of
Zij ∩ (A0 \ C1 ) under the two paths in the following cartesian diagram.
A1

A0

A1

A0

117

As sets we have

Zij = (ρA )−1 (Zij )
= (ρA )−1 (Zij ∩ U )

since Zij is irreducible

= (ρA )−1 (Zij ∩ U )

since ρA is flat

= (ρA )−1 (Zij ) ∩ (ρA )−1 (U )
= Zij ∩ U .

It thus suffices to show that Zij ∩ U is irreducible. But this is immediate since Zij ∩ U =
Wij ∩ U with Wij irreducible.
That the collection Zij gives all the irreducible components is immediate.

Note that by Lemma 5.1.16 part (i) we have that Zij ´etale locally corresponds to Zij .

Proposition 5.4.3. The number of connected and irreducible components of the subschemes
of A1 are as follows.
Closed subscheme of A1
Z3 = (ρA )−1 (Z3 )red
C3 = (ρA )−1 (C3 )red
Z4 = (ρA )−1 (Z4 )red
C4 = (ρA )−1 (C4 )red

# connected components
1
1
1
ns 2 τ

# irreducible components
ns0 s2 τ + 1
ns0 s2 τ + 1
1
ns2 τ

Table 5.5: Number of connected and irreducible components of subschemes of A1

Proof. Z3 = C3 : Let W ⊂ Z3 be an irreducible component. We claim that (ρA )−1 (W ) is
irreducible. As shown in the proof of Proposition 5.3.1, W arises as the strict transform of
an irreducible component of Z3 . First consider the case where W is the strict transform of
Z11 . Then W = Z11 and Proposition 5.4.2 says that (ρA )−1 (Z11 ) is irreducible.

118

So assume now that W is the strict transform of some two dimensional irreducible component
of Z3 . From the proof of Proposition 5.2.3 it must be that this two dimensional component
of Z3 is contained in Z01 ∩ Z10 and hence ρA (W ) ⊂ Z01 ∩ Z10 .
For convenience, we remind the reader of the following cartesian diagram for the next argument.

A1

ρA

A0
ρA

A1

π

A0

Let x ∈ W be a closed point and so x = ρA (x ) ∈ Z01 ∩Z10 . Then as x is in the supersingular
locus, π −1 (x) consists of a single closed point. Therefore the fiber (ρA )−1 (x ) also consists of
a single closed point. With ρ finite and flat, it must be that (ρA )−1 (W ) is irreducible.
Thus we conclude that each irreducible component of Z3 = C3 arises as the reduced inverse
image of an irreducible component of Z3 , and therefore the number of irreducible components
of Z3 is ns0 s2 τ + 1. That Z3 = C3 is connected follows from the fact that the fiber above any
closed point of a two dimension component of Z11 with respect to the morphism ρA consists
of a single closed point.
C4 : Let W be a connected component of C4 . From the proof of Proposition 5.2.3, W is
irreducible and arises as the strict transform of some irreducible component of C4 ⊂ A0 .
Let x ∈ W be a closed point. Then x = ρA (x ) ∈ C4 is contained in the supersingular
locus of A0 . As such, π −1 (x) consists of a single closed point. Thus it follows that the fiber
(ρA )−1 (W ) consists of a single closed point as well. Hence the reduced inverse image of
W under A1 → A0 is connected and irreducible. Therefore, C4 has the same number of
connected and irreducible components as C4 , namely ns2 τ .

119

Step III: Blowup of Z3 .

5.5

A1 = BlA1 (Z3 ) and U1 = BlU1 (Z3 )

5.5.1

Description of the local model U1

In each affine chart of U1 , the subscheme Z3 corresponds to the ideal (u, v). We start by
describing a scheme X which is given by, a priori, a subset of the equations defining U1 .
Once we show that X is integral, from an argument similar to that given in Step I it will
follow that X = U1 . A presentation of X is given by the closed subscheme of ProjU1 (O[u, v])
where u and v are of grade 1 cut out by the following equations.

uv − uv,

(x + bc)up−1 − xv p−1 ,

yup−1 − (y + ac)v p−1

Using these equation along with those in the presentation for U1 , we have that X is covered
by four standard affine charts.

X00

x = 1 u = 1 Zp [y, a, c, u, b, v]/(up−1 y − p, v p−1 − (1 + bc), a(1 + bc) + by)

X01

x=1 v=1

Zp [y, c, v, b, u]/(v p−1 up−1 y − p, up−1 (1 + bc) − 1)

X10

b=1 u=1

Zp [a, b, u, x, v]/(bx2 v p−1 (−a) − p, up−1 − bx)

X11

b=1 v=1

Zp [a, b, s, v, u]/(bs2 up−1 (−a) − p, v p−1 − bs).

Note: The last chart uses a change of coordinates s = x + c.
We can cover X00 with two open subschemes each respectively defined by the condition b
and 1 + bc is invertible. These open subschemes are

120

X00
X00

b=0

Zp [a, u, b±1 , v]/(up−1 (−a)v p−1 b−1 − p)

1 + bc = 0 Zp [y, c, u, b, v, (1 + bc)−1 ]/(up−1 y − p, v p−1 − (1 + bc)).

Since X00 ⊂ X10 as an open subscheme, X is covered by X00 , X01 , X10 , and X11 . Thus we
see that X is integral and so X ∼
= U1 .

Proposition 5.5.1. X is normal.

Proof. The statement is clear for the charts X00 and X01 . Focusing now on the chart X10 ,
note first that it is a complete intersection and hence Cohen-Macaulay. With the generic
fiber smooth, by Serre’s Criterion it suffices to check that the generic points of the irreducible
components of the special fiber are regular, i.e. their maximal ideals are generated by a single
element. Their maximal ideals, written in the corresponding local ring, are given by

(b, u) = (u),

(x, u) = (u),

(v),

(a).

The normality of the chart X11 follows from the argument just given by a change of variables.
Therefore X is normal.

As remarked in the previous section, Z(u, v) = Z(u, v, b) ∪ Z(u, v, y, c, x) inside of U1 . The
true center is a closed subscheme of Z(u, v), and we see from the above charts that the fiber
above a closed point of Z(u, v, y, c, x) consists of the projective line given by [u : v] which
lies inside Z10 ∪ Z11 . Also, the fiber over a closed point of Z(u, v, b) outside of Z(u, v, y, c, x)
consists of a p − 1 closed points. Therefore the true center is C3 , the fundamental center
is Z(u, v, x, y, c) = Z01 ∩ Z10 , and the residual locus is Z11 \ (Z01 ∩ Z10 ). Moreover the
fundamental center is two dimensional and smooth, the residual locus is three dimensional,
and the exceptional locus is equidimensional of dimension three. Taking the strict transform

121

under ρU we have

Z00 = Z(y, a)
Z01 = Z(y, v, x + bc, v)
Z10 = Z(u, y + ac, x, u)
Z11 = Z(u, v, b)
Z4 = Z(u, v, b)
C4 = Z(u, v, y, b, c, x).

We note that C4 is smooth of dimension two and C4 intersects with the fundamental center
of U1 → U1 in a smooth one dimensional subscheme.

5.5.2

Description of the integral model A1

A1 has U1 as an ´etale local model and it is immediate that A1 is normal. As the true
center of U1 → U1 is C3 , we have that the true center of A1 → A1 is C3 .

Proposition 5.5.2. A1 ⊗ Fp has precisely 4 + ns0 s2 τ irreducible components. Three are
given by the strict transforms of Z00 , Z01 , and Z10 . The other 1 + ns0 s2 τ are contained in the
exceptional locus: one lying above Z11 and one lying above each two dimensional irreducible
component of Z3 .

Proof. That the strict transforms of Z00 , Z01 , and Z10 are irreducible follows immediately
from the fact that they are not contained in the true center C3 .
From Section 5.5.1, the exceptional locus E

of A1 → A1 is equidimensional of dimension

122

three. As the exceptional locus maps surjectively onto the true center C3 and C3 has 1+ns0 s2 τ
irreducible components by Proposition 5.4.3, we conclude that E

must consist of at least

1 + ns0 s2 τ irreducible components. Denote these irreducible components by {Wi }. Without
loss of generality assume that ρA (W1 ) ⊂ Z11 and ρA (W2 ), . . . , ρA (W1+ns0 s2 τ ) are each
contained in a unique two dimensional irreducible component of Z3 .
We claim that if ρA (Wi ) ⊂ Z11 , then ρA (Wi ) = Z11 . Indeed, since ρA is proper it suffices
to show that ρA (Wi ) is three dimensional. As the fiber above any closed point of U1 with
respect to ρU is at most one dimensional, we conclude that the same is true for ρA and
thus ρA (Wi ) is at least two dimensional. So by way of contradiction, suppose ρA (Wi ) has
dimension exactly two. From Step II we have that Z11 = Z(u, v, b) and from the previous
section Cfund,3 = Z(u, v, x, y, c). Thus they intersect in a smooth one dimensional scheme.
It follows that the intersection of Z11 with any two dimensional component of Cfund,3 is
one dimensional. As such, there exists a closed point x ∈ ρA (Wi ) lying solely on the
component Z11 such that the fiber above x is one dimensional. Let y be a closed point of
U1 corresponding to x . Then it must be that y lies solely on the irreducible component
Z11 . Since no closed point of Cres,3 = Z11 \ (Z01 ∩ Z10 ) has fiber of dimension one with respect
to the morphism U1 → U1 , using Proposition 5.1.20 we arrive at a contradiction. Therefore,
for any i with Wi → Z11 it must be that the image is three dimensional as claimed.
Now consider a closed point x ∈ C3 \ Z11 . From Proposition 5.1.20 and U1 → U1 we see
that the fiber above x is connected and smooth. It follows from Lemma 5.1.17 that for each
irreducible component of C3 \ Z11 , there is a single irreducible component of the exceptional
locus of ρ

mapping surjectively onto it. Recall that we have labeled these components

W2 , . . . , W1+ns0 s2 τ .
We claim that (ρ )−1 (Z11 )red is irreducible. Indeed, since the reduced inverse image of

123

Z11 under the morphism U1 → U1 is smooth and equidimensional of dimension three, so is
(ρ )−1 (Z11 )red . This implies that (ρ )−1 (Z11 )red is a disjoint union of irreducible components
of A1 ⊗ Fp . As each of these irreducible components maps into Z11 , they must indeed map
surjectively onto Z11 . It follows that the image of each irreducible component contains a
closed point x ∈ Z11 ∩ C3 \ Z11 . From Proposition 5.1.20 we have the fiber above x is
connected, and hence (ρ )−1 (Z11 )red is connected. The claim follows immediately.
Now suppose there exists another irreducible component W2+ns0 s2 τ . By the above, it must
be that W2+ns0 s2 τ → Z11 . But then it follows that in fact
W2+ns0 s2 τ ⊂ (ρ )−1 (Z11 )red = W1
and therefore W2+ns0 s2 τ = W1 .
Proposition 5.5.3. The number of connected and irreducible components of the subschemes
of A1 are as follows.
Closed subscheme of A1
Z4 = ST(Z11 )
C4 = ST(C4 )

# connected components
1
ns2 τ

# irreducible components
1
ns 2 τ

Table 5.6: Number of connected and irreducible components of subschemes of A1

Proof. That Z4 is irreducible was shown in the proof of the previous proposition.
As C4 and C4 are both smooth, C4 and C4 are as well giving that each connected component
is irreducible. Let W ⊂ C4 be some connected component. Recalling that C4 has ns2 τ such
connected components, the proposition will follow by showing that the inverse image of
W with respect to A1 → A1 is connected. But this follows immediately since the fiber
above every closed point of C4 with respect to the morphism U1 → U1 is smooth and

124

connected.

5.6

Step IV: p − 2 blowups of Z11.
[i]

In this last step we define the integral models A1 for 4 ≤ i ≤ p + 1 by first blowing up Z4
in A1 and then blowing up the strict transform of Z4 in each successive step. Likewise, we
[i]

define the local models U1 for 4 ≤ i ≤ p + 1 by blowing up Z4 in U1 and then blowing up
the strict transform of Z4 in each successive step.

5.6.1

[i]

Description of the local model U1

Recall that Z11 is given by Z(u, v, b). Z11 may be described on each affine chart of X = U1
by giving its corresponding ideal.
Chart
Ideal

X00
(u)

X01
(v)

X10
(u, b)

X11
(v, b)

Table 5.7: Ideal sheaf of Z11

[3]

[3]

Write U1 = U1 and Z11 = Z11 . As Z11 is Cartier on X00 and X01 , the blowups of Z11 and
its strict transforms are isomorphisms over these open subschemes. Focusing now on X10
and X11 , each of these two charts are given by a scheme with the presentation

Y = Spec(A),

p−1
A = Zp [x1 , x2 , x3 , x4 , u]/(x1 x22 xp−1
− x1 x2 )
3 x4 − p, u

and Z3 is given by the subscheme W corresponding to the ideal (u, x1 ) in this presentation.
[0]

[1]

[2]

To describe the blowups, we write x1 = x1 , and x1 = x1 , x1 = x1 , etc. for projective

125

coordinates.

Proposition 5.6.1. Set Y [0] = Y , W0 = W , and for 1 ≤ i ≤ p − 2, define Y [i] inside
i−times

Y × P1 × · · · × P1 by
[i]

u[i] up−i−1 − x1 x2 ,
[j]

[j−1] [j]

uu[j−1] x1 − x1

u

[1]

ux1 − x1 u[1]
for

2 ≤ j ≤ i.

Let Wi be the strict transform of Wi−1 in Y [i] for each i ≥ 1. Then for 1 ≤ i ≤ p − 2 we
have the following.
(i). Y [i] ∼
= BlWi−1 (Y [i−1] ).
(ii). The true center of Y [i] → Y [i−1] is one dimensional and smooth.
(iii). The fundamental center of Y [i] → Y [i−1] is equal to the true center.
(iv). The exceptional locus of Y [i] → Y [i−1] is smooth and two dimensional.
Furthermore, Y [p−2] is regular with special fiber a divisor with normal crossings.

[i−1]

Proof. (i) We proceed by induction. So assume Wi−1 corresponds to the ideal (u, x1

),

which is certainly true for i = 1. By explicit computation, the claimed equations are part of
those defining BlWi−1 (Y [i−1] ). The standard affine charts of Y [i] , indexed by 1 ≤ k ≤ i + 1,
are described by the conditions

u[j] = 0 for 1 ≤ j < k

[j]

and x1 = 0 for k ≤ j ≤ i.

In order to explicitly write them, we must consider three cases.

126

k = 1: The equations of Y [i] become

x2 =

p−1

u[1]

xp−2
1

,

[1]

u = x1

x1

u[1]
[1]

u[j]

,

[j]

x1

=

x1

j

u[1]

xj−1
1

2 ≤ j ≤ i,

,

[1]

x1

and the coordinate ring is

Zp x1 ,

u

[1]

[1]
x1


u

, x3 , x4 / x12p−3



2p−2

[1]

x3p−1 x4 − p .

[1]
x1

1 < k ≤ i: The equations of Y [i] become
[k−1]

x1 =

x1
u[k−1]

k

u[k]

k−1

,

[k]

x2 =

x1
[j−1]

x1
u[k−1]

u[j]

x1
u[k−1]

[j]

k−j+1

[k−1]

x1
=
u[j−1]

=

u[k]

u[k]

p−k

[k]

[k−1]

,

u=

x1

x1
u[k]
u[k−1] x[k]
1

k−j

for 2 ≤ j < k,

,

[k]

x1
j−k

[k−1]

x1

p−k−1

[k−1]

x1
u[k−1]

j−k+1

u[k]

for k < j ≤ i,

[k]

x1

and the coordinate ring is

Zp

[k−1]
x1
,
u[k−1]

u



[k]

[k]
x1

, x3 , x4 / 

[k−1]
x1
u[k−1]

2p−k−2

u

[k]

2p−k−1

[k]
x1

k = i + 1: The equations of Y [i] become
[1]

x1 = u
[j]

x1
u[1]

[i]

up−i−1 −

x1
x2 = 0
u[i]

[i+1]

x1
i−j+1 x1
=
u
u[j]
u[i+1]

127

for 2 ≤ j ≤ i


.
xp−1
3 x4 − p

and the coordinate ring is
[i]

[i]

[i]

x1
x1
p−i−1
i x1 2 p−1
x
x
−
p,
u
−
,
x
,
,
x
,
x
,
u
/
u
x
x2 .
4
2
3
4
3
2
u[i]
u[i]
u[i]

Zp

Note that each chart is integral. Since the equations defining Y [i] are part of those defining
BlWi−1 (Y [i−1] ), there is a closed immersion ι : BlWi−1 (Y [i−1] ) → Y [i] which is an isomorphism
on the generic fiber. With Y [i] integral and of the same dimension as BlWi−1 (Y [i−1] ), this
implies ι is an isomorphism.
To complete the induction, we must show that the strict transform of the subscheme of
[i−1]

Y [i−1] given by Z(u, x1

[i]

) corresponds to the subscheme of Y [i] given by Z(u, x1 ). From
[i−1]

the charts above, the true center of the blowup in Y [i−1] is given by Z(u, x1

, x2 ). Thus

taking the inverse image away from the center we have that x2 is invertible, and so from the
[i]

[i]

relation u[i] up−i−1 − x1 x2 of Y [i] we get that x1 is in the ideal defining the strict transform.
[i−1]

As subschemes of Y [i] , Z(u, x1

[i]

[i]

, x1 ) = Z(u, x1 ). This subscheme is irreducible and of

dimension three and therefore we conclude it must be the strict transform of Wi−1 .
We now inspect these charts to deduce the remainder of the proposition. To calculate the
[i+1]

true center of U1

[i]

→ U1 , one need only to consider the chart indexed by k = i + 1

[i]

since Z(u, x1 ) is Cartier in all others. Here we see that the true center is contained in
[i]

[i]

Z(u, x1 , x2 ). Now consider the fiber over any closed point of Z(u, x1 , x2 ) with respect to
[i+1]

the morphism U1
[i+1]

uu[i] x1

[i]

[i+1]

→ U1 . We have that both of the relations u[i+1] up−i−2 − x1

[i]

x2 and

[i]

− x1 u[i+1] vanish, since we are assuming that u, x1 , and x2 are all zero. No other
[i+1]

relation involving u[i+1] and x1

exists and therefore the fiber over this closed point is

isomorphic to P1Fp . This gives (ii), (iii), and (iv).

128

Using the explicit equations above, we record the global structure of the irreducible components of the special fiber.
[p−2]

Lemma 5.6.2. The irreducible components of U1

⊗ Fp are described as follows.

• There are p + 3 components.
• Three components are given by Z(u), Z(v) and Z(a). We index the other components
by 1 ≤ i ≤ p. For 1 ≤ i ≤ p − 1, the ith irreducible component is given by the locus

Zi = Z(u, b, x, b[1] , b[2] , . . . , b[i−2] , u[i] , u[i+1] , . . . , u[p−2] )

and the pth irreducible component is given by the locus

Zp = Z(u, b, b[1] , b[2] , . . . , b[p−2] ).

• The components given by Z(u), Z(v), Z(a), and Zi have multiplicity p − 1, p − 1, 1,
and 2p − i − 1 respectively. In particular, Zp−1 is the only component with multiplicity
divisible by p.
• The components Z1 and Zp are isomorphic to A3Fp . The components Zi with 2 ≤ i ≤
p − 1 are isomorphic to P1Fp × A2Fp .
• The components intersect as indicated in the following “dual complex”, drawn for p = 5.
Each vertex represents an irreducible component where the label indicates the multiplicity of the irreducible component. Each edge indicates that the two irreducible components intersect.

129

[p+2]

Figure 5.2: Dual complex of U1

for p = 5

Moreover, consider a k-simplex appearing in the complex where every pair of vertices
within the simplex is directly connected by an edge. Such a simplex indicates a (k + 1)fold intersection of the irreducible components.
• A k-fold intersection of the components has dimension 3 − k over Spec(Fp ).

5.6.2

Description of the integral model
[i]

Proposition 5.6.3. For 3 ≤ i ≤ p + 1, the number of irreducible components of A1 is
4 + ns0 s2 τ + ns2 τ · (i − 3).

Proof. We recall the following facts:
[3]

(i). C4 has ns2 τ connected components and each is smooth of dimension two;
[i]

[i−1]

(ii). For 3 ≤ i ≤ p + 1, the fiber over a closed point of the true center of U1 → U1

130

is

one dimensional, smooth, and connected; and
[i]

(iii). For 3 ≤ i ≤ p + 1, U1 ⊗ Fp is equidimensional of dimension three.
[4]

[3]

We proceed by induction, starting with the modification A1 → A1 . Now (i) and (ii) imply
[4]

[3]

that the exceptional locus of A1 → A1 has the same number of connected components
[3]

as the true center C4 , and furthermore that each such connected component is three dimensional and smooth. By (iii) each of these components is an irreducible component of
[4]

[4]

A1 ⊗ Fp , with all of the other irreducible components of A1 ⊗ Fp being given by the strict
[3]

transform of the irreducible components of A1 ⊗ Fp . Therefore there are 4 + ns0 s2 τ + ns2 τ
[4]

irreducible components of A1 ⊗ Fp .
[i−1]

Now assume the result is true for i − 1 with 4 < i ≤ p + 1. We must show that, Ci

has ns2 τ

connected components and each is smooth of dimension two. Indeed, then the induction will
follow using the same argument as in the above paragraph. Note that from the local model
[i−1]

we have each connected component of Ci

is two dimensional and smooth, so it is left to
[i−1]

show that there are ns2 τ connected components of Ci
[i−1]

Now Zi

[i−1]

As Ci

[i−1]

, E [i−1] , and Ci
[i−1]

= Zi

.
[i−1]

´etale locally correspond to Zi
[i−1]

∩ E [i−1] we get that Ci

[i−1]

= Zi

[i−1]

, E [i−1] , and Ci

respectively.

∩ E [i−1] .

[i−2]

[i−2]

Consider a connected component of Ci−1 , which is irreducible because Ci−1 is smooth. The
[i−2]

fiber above this component is connected since A1

is normal by Zariski’s Main Theorem.
[i−2]

[i−1]

Thus E [i−1] has the same number of connected components as Ci−1 . Now Zi
[i−2]

[i−1]

maps surjectively onto Zi−1 via ρA
[i−2]

[i−1]

of Ci−1 . As such, Zi
[i−1]

Zi

[i−2]

= ST(Zi−1 )

and hence the image meets each connected component
[i−1]

meets each connected component of E [i−1] . Therefore Ci

=

[i−2]

∩ E [i−1] has the same number of connected components as Ci−1 ; namely ns2 τ .

We will use the following graph to describe how these irreducible components of the special

131

fiber intersect.

Definition 5.6.4. Let p be an odd rational prime and K p ⊂ G(Apf ) so that K p determines
the numbers ns2 τ and ns0 s2 τ of A0,K p , i.e. the number of connected components of the KR
strata s2 τ and s0 s2 τ . We then define the vertex-labeled graph Γp,K p as follows.
(i). Begin with ns2 τ batons, each having p−2 vertices. Label the vertices 2p−3, 2p−4, . . . , p
from head to tail.

Figure 5.3: Batons of Γp,K p where p = 5, ns2 τ = 2
(ii). Add one vertex labeled 2p − 2 (top left) and attach edges between this vertex and the
heads of the batons. Add two more vertices labeled p − 1 (bottom left and top right)
and connect these two vertices to every vertex in the batons, as well as the (unique)
vertex labeled 2p − 2. Add ns0 s2 τ vertices labeled p − 1 (bottom right) and attach
edges between these and the tails of the batons, as well as the two vertices labeled
p − 1 added in the previous sentence.

132

Figure 5.4: Base of Γp,K p where p = 5, ns2 τ = 2, ns0 s2 τ = 3
(iii). Add one vertex labeled 1 and attach edges from this to every vertex constructed in the
above two steps.

Figure 5.5: Γp,K p where p = 5, ns2 τ = 2, ns0 s2 τ = 3

Definition 5.6.5. We define the following subsets of the vertices of Γp,K p .
• The batons consist of the vertices given in step (i) above. They may be identified as
the vertices with label in [p, 2p − 3].
• The front consists of the vertices labeled p − 1 on the bottom right of the diagram

133

directly above. They may be identified as the vertices of label p − 1 and (edge) degree
3 + ns2 τ that share edges with precisely two vertices labeled 4.
• The sides consist of the vertices labeled p − 1 which are not in the front.

Recall that we are writing A0 for a single connected component of A0 → Spec(Zp [ζN ]) and
similarly with A1 (see Section 5.1.3 for details).
[p+1]

Theorem 5.6.6. A1

[p+1]

→ A1 is a resolution of singularities and the special fiber of A1
[p+1]

a nonreduced divisor with normal crossings. A1

is

⊗Fp has 4+ns0 s2 τ +ns2 τ (p−2) irreducible

components whose intersections are described by the vertex-labeled graph Γp,K p as follows.
(i). Each vertex represents an irreducible component. The label of the vertex is the multiplicity of the component.
(ii). A k-simplex of Γp,K p indicates a (k + 1)-fold intersection of irreducible components
corresponding to the vertices of the k-simplex. Such an intersection has dimension
3 − k over Spec(Fp ).
[p+1]

(iii). Let x[p+1] ∈ A1

be a closed point and {e1 , . . . , et } be the multiset of the multiplicities

of the irreducible components which x[p+1] lies on. Then there is an ´etale neighborhood
of x[p+1] of the form

Spec(Zp [x1 , x2 , x3 , x4 ]/(xe11 . . . xet t − p)).

(iv). The following table gives the image of each irreducible component under the map
[p+1]

A1

→ A0 .

134

Description
Front
Sides
Vertex labeled 2p − 2
Vertex labeled 1

Batons

Image
Each irreducible component surjects onto a connected
component of As0 s2 τ
These two irreducible components surject onto the
irreducible components Z01 and Z10 respectively.
Surjects onto Z11 .
Surjects onto Z00 .
Fix a baton B. The irreducible components
corresponding to a vertices in B all surject onto the
same connected component of As2 τ . This induces a
bijection between the set of batons and the connected
components of As2 τ .
[p+1]

Table 5.8: Images of irreducible components of A1

135

APPENDIX

136

A.1

Determinants

Let R be a ring and M a finite locally free R-module. For r ∈ R, denote by [r] : M → M
the homomorphism given by multiplication by r. We then define

det : EndR (M ) →

rkM

as follows. The R-homomorphism EndR (M ) →

∼

EndR (M ) −
→R
rkM

EndR (M ) is given by f →

rkM

f.

The second map in the above composition is the inverse of the isomorphism
rkM

R→

rkM

EndR (M ) sending r →

[r].

That this map is indeed an isomorphism can be verified locally, and hence we are reduced
to the case where M is free.
Let A be an R-algebra and let M be a left A-module which is finite and locally free as an Rmodule. Define VA to be the functor on the category of R-algebras given by VA (S) = A⊗R S.
We define the morphism detM,A : VA → A1R on S-valued points by

x → detS (x|M ⊗R S).

If A is finite and free as an R-algebra, let {a1 , . . . , at } be a R-basis of A. Then we have
∼

St −
→ A ⊗R S

by (x1 , . . . , xt ) → a1 ⊗ x1 + · · · + at ⊗ xt .

137

Therefore V is representable by AtR and detM corresponds to the polynomial

detR[X1 ,...,Xt ] a1 ⊗ X1 + · · · + at ⊗ Xt |M ⊗R R[X1 , . . . , Xt ] .

Proposition A.1.1. Suppose R = k is a field and A is a finite dimensional semisimple kalgebra. Let M and N be A-modules. Then M ∼
= N as A-modules if and only if detM = detN .

Proof. If M ∼
= N then certainly detM = detN . So suppose detM = detN . Write A =
A1 × · · · × Ar where each Ai as simple. Let {a1 , . . . ar } be a set of mutually orthogonal
idempotents with ai ∈ Ai , so a2i = ai and ai aj = 0 for i = j. Then we have the decompositions

M = M1 × · · · × Mr

and

N = N1 × · · · × Nr

where Mi = ai M and Ni = ai N . Set S = Ri [T ] and consider the element 1 ⊗ T ∈ VA (S) =
A ⊗R Ri [T ]. Then

rkRi Mi = detRi [T ] (T |M ⊗R Ri [T ]) = detRi [T ] (T |N ⊗R Ri [T ]) = rkRi Ni .

Since each Ai is simple there exists a unique irreducible Ai -module up to isomorphism, and
therefore Mi ∼
= Ni . The proposition immediately follows.

Proposition A.1.2. Let f ∈ R and Rf denote the localization of R with respect to the set
{1, f, f 2 , . . .}. Then detMf = detM ⊗R Rf , where detMf is with respect to the Rf -module Af .

Proof. Note that V ⊗ Rf = VA⊗R Rf is the functor on the category of Rf -algebras sending S
to Af ⊗Rf S. Then detM ⊗R Rf : V ⊗ Rf → A1Rf which on an Rf -algebra S sends x ∈ S to
detS (x|(M ⊗R Rf )S ). Since M ⊗R Rf = Mf by definition, the result is immediate.

138

A.2

Weyl groups

Let G be a split reductive linear algebraic group over a field k, B a Borel subgroup, and
T ⊂ B a maximal torus defined over k. Let Φ be the set of roots given by T , Φ+ denote the set
of positive roots distinguished by B, and Q denote the subgroup of the affine transformations
of V ∗ generated by Φ∨ . Write X∗ (T ) = Hom(Gm , T ), the cocharacter lattice of T .

Definition A.2.1. With respect to the above data, we define the following groups.
• The Weyl group W = NG (T )/T ;
• The affine Weyl group W aff = Q

W;

• The extended affine Weyl group W = W

X∗ (T ).

Example A.2.2.
• The Weyl group of G = SLn is isomorphic to Sn . The affine Weyl group Wa is the
semidirect product of Sn with the subgroup of Zn consisting of all tuples (a1 , . . . , an )
with

i

ai = 0, where Sn acts on Zn via permutation of the coordinates.

• The Weyl group of Sp2n can be realized as a subgroup of S2n consisting of the permutations that commute with the permutation (1, 2n)(2, 2n − 1) . . . (n, n + 1) ∈ S2n . The
affine Weyl group of Sp2n is the subgroup of Z2n

S2n generated by the permutations

(i, i + 1)(2n + 1 − i, 2n − i) for 1 ≤ i ≤ n − 1, the permutation (n, n + 1), and the
element (−1, 0, . . . , 0, 1)(1, 2n).

Definition A.2.3. Let (W, S) be a Coxeter system. Then for u, v ∈ W we write u ≤ v if
there is a reduced word v = s1 s2 . . . sd and a sequence 1 ≤ i1 < i2 < · · · < ir ≤ d such that

139

w = si1 si2 . . . sir is a reduced word for w. This is a partial order on W .

As W and W aff are both Coxeter groups, they can be equipped with the Bruhat order. The
Bruhat order, denoted by ≤, may be extended to W = W

X∗ as follows. With x, x ∈ W ,

they may be uniquely decomposed as x = wc and x = w c where w, w ∈ W and c, c ∈ X∗ .
Then x ≤ x means w ≤ w and c = c .

A.3

Affine flag variety

Let G be a reductive linear algebraic group over a field k, B a Borel subgroup of G.

Definition A.3.1. Define the following functors from k-algebra to sets.
• The loop group LG
R → G(R((T ))).
• The positive loop group L+ G
R → G(R[[T ]])).
• Let I denote the Iwahori subgroup of LG induced by B, given by identifying the inverse
image of B under L+ G → G with a subgroup of LG using L+ G → LG. Then the
affine flag variety FG = LG/I where the quotient is as fpqc-sheaves on k-schemes.

Proposition A.3.2. We have the following properties.
(i). L+ G is represented by an affine scheme over Spec(k).
(ii). LG is represented by an ind-scheme over Spec(k).

140

Proof.
(i). Let us first show this for G = GLn . Here we identify GLn (R[[t]]) with the set of
matrices
{(A, B) ∈ Mn (R[[t]]) × Mn (R[[t]]) : AB = 1} .
2

We can consider Mn (R[[t]]) as

i≥0

AnR , where each index i gives the coefficient of ti .

It then follows that GLn is a closed subscheme of the affine scheme

2

i≥0

2

An × An .

For an arbitrary linear group G, a closed embedding of G into GLn for some n gives a
closed embedding of L+ G into L+ GLn .
(ii). It is clear that
LG(R) = lim G(ti R[[t]])
−→
i≤0

with the directed system given by the inclusion homomorphisms, so that LG is an
ind-scheme.

The affine flag variety can be realized as a space of lattices subject to additional conditions.
We now explain this in detail in the case G = SLn or G = GSp2n .
Definition A.3.3. A lattice L ⊂ R((t))n is a locally free R[[t]]-submodule such that L ⊗R[[t]]
R((t)) = R((t))n .
We say that a lattice L is r-special if

n

L = t r ΛR .

Definition A.3.4. A sequence L0 ⊂ L1 ⊂ · · · ⊂ Ln−1 ⊂ t−1 L0 of lattices in R((t))n is called
a complete lattice chain if Li+1 /Li is a locally free R-module of rank one for all i.

141

Proposition A.3.5. Fix r ∈ Z. There is a functorial isomorphism
∼

F(R) −
→ {r-special complete lattice chains in R((t))n } .

Proof. The morphism is given by

g → g · (λi )i ,

where λi = R[[t]]n−r+i ⊕ (tR[[t]])r−i .

Since I is the stabilizer of the standard lattice chain (λi )i , this map is well-defined and
injective. It remains to show that if (Li )i is an r-special lattice chain, then Zariski-locally
on Spec(R) there exists an element g ∈ SLn (R[[t]]) such that (Li )i = g · (λi )i . Now each Li
is locally free and so there exists, Zariski-locally on R, g ∈ GLn (R[[t]]) such that (Li )i =
g · (λi )i . As L0 is r-special, we have
n

∧g

λ0

tr R[[t]]

∼

n

L0

tr R[[t]]

giving that det(g ) ∈ R[[t]]× . As such, we can find an element g ∈ SLn (R[[t]]) such that
(Li )i = g · (λi )i .

Definition A.3.6. Let ·, · denote the alternating pairing on R((t))n given by the matrix



J =


−Jn

Jn 
,





where Jn = 



1

142


1


1 

.
.

.
.



We say that a lattice chain L is self-dual if for all lattices Λ ∈ L, the dual

Λ⊥ = {x ∈ K n : x, y ∈ R[[t]] for all y ∈ Λ}

also occurs in the lattice chain.

Proposition A.3.7. There is a functorial isomorphism
∼

F(R) −
→ {0-special self-dual complete lattice chains in R((t))n } .

Proof. The morphism is given by

g → g · (λi )i ,

where λi = R[[t]]n−r+i ⊕ tR[[t]]r−i .

Since I is the stabilizer of the standard lattice chain (λi )i , this map is well-defined and
injective. It remains to show that if (Li )i is an r-special lattice chain, then Zariski-locally
on Spec(R) there exists an element g ∈ Spn (R[[t]]) such that (Li )i = g · (λi )i . This follows
from [RZ, Proposition A.21].

The affine flag varieties admit a stratification by Schubert cells. By a stratification of a space
X we mean that there exists a collection {Xi ⊂ X}i∈I where I has a partial order ≤ such
that
X=

Xi

and

Xj for every i ∈ I.

Xi =

i

j≤i

In the case of the affine flag variety associated with SLn or Sp2n , there is a canonical stratification where I = W and for w ∈ W , Xw is the associated Schubert cell.
In the following, we regard all algebraic groups as the group given by their k((t))-valued

143

points. Thus we will write G for G(k((t))), etc. We define an embedding ι : X∗ (T ) → G as
follows. For λ ∈ X∗ (T ), we have λ : Gm → T and thus set ι(λ) = λ(t) ∈ T ⊂ G.
When G = SLn we can identify W = NG T /T with the group of permutation matrices. When
G = Sp2n we can identify W with the subgroup of generalized permutation matrices. Thus
in either case we have an extension ι : W → G. From now on, we will identify W as a subset
of G via the embedding ι.

Definition A.3.8. Let w ∈ W . The Schubert cell associated to w is given by IwI/I ⊂ FG .
The Schubert variety associated with w, denoted by Xw , is the Zariski closure of IwI/I
inside of FG .

Proposition A.3.9. For w ∈ W , we have
(i). Xw = ∪v≤w IvI/I;
(ii). For v ∈ W aff , Xv ⊂ Xw if and only if v ≤ w;
(iii). dim Xw = (w), where (w) is the length of a reduced expression for w;
and FG admits a stratification
FG =

IwI/I.
w∈W

The standard apartment of (the Bruhat-Tits building associated with) F is defined as follows.
Let {e1 , . . . , en } be the standard basis of k((t))n . Then the vertices of the standard apartment
are given by lattices generated by

t−r1 e1 , . . . , t−rn en .

144

Identify such a lattice with the n-tuple (r1 , . . . , rn ) ∈ Zn . Two lattices (r1 , . . . , rn ) and
(s1 , . . . sn ) are considered equivalent if there exists an integer m such that

(s1 , . . . , sn ) = (r1 + m, . . . , rn + m).

Thus the set of vertices of the standard apartment can be identified with Zn modulo Z,
where Z acts diagonally by addition.
The alcoves of the standard apartment of F are by definition tuples (x0 , . . . , xn−1 ) where
each xi is a vertex (i.e. an element of Zn /Z) such that for some choice of lifts xi ∈ Zn we
have
x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn := x0 + (1, . . . , 1)
and
xi+1 (j) =
j

xi (j) + 1 for all i.
j

Here xi ≤ xj is defined coordinate-wise: xi (j) ≤ xi+1 (j) for all i, j. In the case G = Spn , we
also impose the additional condition that for 1 ≤ i ≤ 2n we have

xn−i = θ(xi ),

where θ(r1 , r2 , . . . , rn ) = (−rn , −rn−1 , . . . , −r1 ).

Note that such an alcove (x1 , . . . , xn ) naturally corresponds to a complete periodic (and in
the case GSp2n , self-dual) lattice chain, i.e. an element of F.
We now fix the alcoves
ω = (ω0 , . . . , ωn−1 ),

ωi = (1i , 0n−i )

τ = (1r , 0n−r ), (1r+1 , 0n−r−1 ), . . . , (2r−2 , 1n−r ), (2r−1 , 1n−r+1 )

145

noting that both are alcoves of FSp and hence also alcoves of FSL . We define the size of an
alcove x to be
n−1

xi (j) − ωi (j)
j=0

which is constant with respect to the choice of i. The affine Weyl group in each case naturally
acts on the set of alcoves of size r in the standard apartment, given by acting on each vertex.
This action is simply transitive. With our fixed base alcove τ we can thus identify the affine
Weyl group with the set of alcoves of size r in the standard apartment.

Proposition A.3.10. Let x = (x0 , . . . , xn−1 ) be an and (Li )i ∈ Sx . Then there exists b ∈ I
such that




−xi (1)+1

t





Li = b · 




t−xi (2)+1
..

.
t−xi (n)+1





.




Here we are identifying the matrix above with the lattice generated by its columns.

Proof. This is immediate as Sx is defined as the Iwahori-orbit of x in F.

A.4

Group schemes

Definition A.4.1. Let S be a scheme. A group scheme G over S is an S-scheme equipped
with S-morphisms
µ : G ×S G → G

ε:S→G

such that the following diagrams commute.

146

ι:G→G

(i). Associativity
G ×S G ×S G

µ × id

G ×S G
µ

id × µ
µ

G ×S G

G

(ii). Identity
S ×S G

ε × id

pr2
G

G ×S G
id × ε

µ

G ×S S

pr1

(iii). Inverse
G ×S G
(id, ι)

µ

G

S

ε

G
µ

(ι, id)
G ×S G

Definition A.4.2. Let G/S be a group scheme and σ : G ×S G → G ×S G be the morphism
which interchanges the factors. Then we say G is a commutative group scheme if the following
diagram commutes.

G ×S G
σ

µ

G

µ

G ×S G
Definition A.4.3. Let S be a Noetherian scheme. A finite flat group scheme over S is a
group scheme G/S such that the structure morphism G → S is finite and flat. Equivalently,
the structure morphism makes OG into a locally free OS -module of finite rank. The rank is
then a locally constant function on S, and when the rank function is constant, we will refer

147

to it as the order of G/S, or simply the order of G if S is understood.

Definition A.4.4. Let S = Spec(R) be a Noetherian scheme and G = Spec(A) a finite flat
group scheme over S. The Cartier dual of G is given by G∗ = Spec(HomR (A, R)) equipped
with the following maps.
• ε∗ : S → G∗ . This morphism is defined on the coordinate rings HomR (A, R) → R by
sending an R-module homomorphism ϕ : A → R to ϕ ◦ π # (1) where π # : R → A is
the structure morphism of G/S.
• µ∗ : G∗ × G∗ → G∗ . This morphism is defined on the coordinate rings HomR (A, R) →
HomR (A, R) ⊗R HomR (A, R) by first identifying

HomR (A, R) ⊗R HomR (A, R) = HomR (A ⊗R A, R)

and sending an R-module homomorphism ϕ : A → R to
m

ϕ

A ⊗R A −
→A−
→R

where m(a ⊗ b) = ab.
• ι∗ : G∗ → G∗ . This morphism is defined on the coordinate rings HomR (A, R) →
ι#

ϕ

HomR (A, R) by sending an R-module homomorphism ϕ : A → R to A −→ A −
→ R.
One can check that these maps give G∗ the structure of a finite flat group scheme over S,
and that (G∗ )∗ is canonically isomorphic to G.

Definition A.4.5. Let G/S be a group scheme, ε : S → G the identity section. The sheaf

148

of invariant differentials of G/S is defined to be

ωG/S = ε∗ (ΩG/S ).

A.4.1

Finite group schemes of order p

Throughout this section, k denotes an algebraically closed field and p a rational prime. Set
S = Spec(k). If the characteristic of k is not p, there is (up to isomorphism) precisely one
finite flat group scheme of order p over S. On the other hand, if the characteristic of k is p,
then there are (up to isomorphism) three finite flat group schemes of order p over S. We will
present these three group schemes, calculate their Cartier duals, and calculate the dimension
of their invariant differentials.

Example A.4.6. The constant group scheme G = (Z/pZ)S .
Description of G: As a scheme, G is given by the disjoint union of p copies of Spec(k) and
we fix an indexing by 0, 1, . . . , p − 1 which we will write as Si ⊂ G for 0 ≤ i ≤ p − 1. Then
this induces an indexing of G ×S G by pairs (i, j) with 0 ≤ i, j ≤ p − 1, which we write as
Sij ⊂ G ×S G. Define the multiplication map G ×S G → G by Sij → Si+j where the addition
is in Z/pZ and the morphism is the identity map. The identity and inverse are respectively
defined as
S → S0

and

Si → S−i

again using the identity maps. It is straightforward to see that this makes G/S into a group
scheme and that G is finite and flat over S.
The Cartier dual of G: The collection of (set) maps {ei : Z/pZ → k} where ei is defined by

149

ei (j) = δij gives a basis of the k-vector space Γ(G, OG ). The morphisms µ, ε, and ι are given
on the coordinate rings as

µ# (ei ⊗ ej ) = δij ei

ε# (ei ) = δ0i

ι# (ei ) = e−i .

To calculate the Cartier dual, let {e∗i } be the dual basis defined by e∗i (ej ) = δij . Then µ∗ ,
ε∗ , and ι∗ of the Cartier dual G∗ are given on the coordinate rings as

(µ∗ )# (e∗i ) = e∗i ⊗ e∗i

(ε∗ )# (e∗i ) = 1

(ι∗ )# (e∗i ) = e∗−i .

From this description, it is immediate that (Z/pZ)∗S ∼
= (µp )S (see the example below).
The invariant differentials of G: Since G → S is ´etale, it is immediate that ΩG/S = 0 and
hence ωG/S = 0.

Example A.4.7. The roots of unity G = (µp )S .
Description of G: Define G = Spec(k[T ]/(T p − 1)) over Spec(k) with the morphisms µ, ε,
and ι defined on the coordinate rings as

T →T ⊗T

T →1

T → T p−1 .

It is easy to check that this morphisms satisfy the required commutative diagrams, and
G is visibly flat and finite over Spec(k). Furthermore G is ´etale over S if and only if the
characteristic of k is not p.
The Cartier dual of G: As the double (Cartier) dual of a group scheme is canonically
isomorphic to the group scheme itself, we have that the Cartier dual of G is (Z/pZ)S .

150

The invariant differentials of G: If the characteristic of k is not p, then µS is ´etale over S
and thus the dimension of its invariant differentials is automatically zero. We thus assume
that the characteristic of k is p. Set I = (xp − 1) and A = k[x]/I. We have the standard
exact sequence
δ

I/I 2 →
− Ωk[x]/k ⊗k[x] A → ΩR/k → 0
where the first map is given by δ(α) = dα ⊗ 1. The image of the first map is thus generated
by pxp−1 which is zero since the characteristic of k is p. We therefore have that ΩG/S
is given by the module k[x]dx/(xp − 1). The pullback of this module to k is given by
k[x] dx /(xp − 1) ⊗k[x]/(xp −1) k. It follows that the invariant differentials are given by k dx
and therefore are of dimension one.

Example A.4.8. G = (αp )S , where p is a rational prime and k is of characteristic p.
Description of G: Define G = Spec(k[T ]/(T p )) with the morphisms µ, ε, and ι defined on
the coordinate rings as

T →T ⊗1+1⊗T

T →0

T → −T.

Note that the first map above is a ring homomorphism precisely because the characteristic
of k is p. Again, it is easy to check that this morphisms satisfy the required commutative
diagrams, and αp is visibly flat and finite over Spec(k), but certainly not ´etale since G is not
reduced.
The Cartier dual of G: Set R = k[T ]/(T p ). We have a k-basis of R given by {T i }. Let {ui }
denote the dual basis, i.e. ui (T j ) = δij . Then the k-linear map
ϕ : Homk (R, k) → R sending ui → T i /i!

151

is k-isomorphism. Furthermore, it preserves the morphisms giving the group structure of G.
To see this, first note that the multiplication map m is given on coordinate rings by
i

i
T j ⊗ T i−j .
j

m# (T i ) =
j=0

The multiplication map for the Cartier dual of G is given on coordinate rings by
i
∗ #

uj ⊗ ui−j .

(m ) (ui ) =
j=0

Thus we have that the diagram (given on generators)
ϕ

ui

T i /i!

(m∗ )#
i
j=0

m#

uj ⊗ ui−j

ϕ⊗ϕ

1
i!

i
i
j=0 j

T j ⊗ T i−j

commutes for all i. The inverse and identity maps may be checked in a similar fashion and
it therefore follows that G ∼
= G∗ .
The invariant differentials of G: Set I = (xp ) and A = k[x]/I. We have the standard exact
sequence
δ

− Ωk[x]/k ⊗k[x] A → ΩR/k → 0
I/I 2 →
where the first map is given by δ(α) = dα ⊗ 1. The image of the first map is thus generated
by pxp−1 = 0 since the characteristic of k is p. We therefore have that ΩG/S is given by the
module k[x] dx /(xp ). The pullback of this module to k is given by k[x] dx /(xp ) ⊗k[x]/(xp ) k.
It follows that the invariant differentials are given by k dx and therefore are of dimension
one.

152

Theorem A.4.9. [OT, Lemma 1] Let k be an algebraically closed field. If the characteristic
of k is not p, then (Z/pZ)k is the only finite flat group scheme of order p up to isomorphism.
If the characteristic of k is p, then there are three nonisomorphic finite flat group schemes
of order p:
(Z/pZ)k ,

A.4.2

(µp )k ,

and

(αp )k .

Oort-Tate generators

Using the notion of a “full-set of sections” [KM] for a finite flat group scheme G/S, we recall
the definition of an Oort-Tate generator from [HR].

Definition A.4.10. [KM, 1.8.2] Let G be finite flat group scheme over S of rank N ≥ 1.
Then we say that a set of sections P1 , . . . , PN in G(S) is a full set of sections of G/S if for
every affine S-scheme T = Spec(R), and for every function f ∈ B = Γ(G ×S T, OZ×S T ), we
have:
N

NormB/R (f ) =

f (Pi ).
i=1

Definition A.4.11. Let S be a Zp -scheme and π : G → S finite flat group scheme of order
p with finite presentation over S. Suppose σ : S → G is a section of π. Then we have a
collection of sections {ε, σ, [2]σ, . . . , [p − 1]σ}. We say that σ is an Oort-Tate generator if
this collection is a full set of sections of G/S.

Proposition A.4.12. [KM, Lemma 1.8.3] Let S be a connected scheme, G/S be a finite
´etale group scheme of order N , and σ1 , . . . , σN be a collection of sections S → G of G. Then
the following conditions are equivalent.

153

(i). The S-morphism
N

S→G
i=1

defined by the sections {σi } is an isomorphism of S schemes.
(ii). {σ1 , . . . , σN } form a full set of sections of G/S.

S → G is an isomorphism. Then for T = Spec(R) in

Proof. Suppose the morphism

the definition above, B = ⊕N
i=1 R. Choosing the standard R-basis {ei }, we have that for
f =

i ri ei

the matrix representing “multiplication by f ” in this basis is diag(r1 , . . . , rn )

and hence NormB/R (f ) =

i ri .

Since f (Pi ) = ri , the conclusion follows.

Conversely, suppose {σi } form a full set of sections of G/S. Then on each connected component of the source, the morphism
N

S→G
i=1

restricts to the identity map onto its image in G. Thus to show that this is an isomorphism,
we only need to show that no two connected components of the source are sent to the same
connected component of G.
It thus suffices to show that for every geometric point Spec(k) → S, the N points σi,k :
Spec(k) → Gk are all distinct. To see this, let Q1 , . . . , QN denote the reduced closed points
whose disjoint union forms Gk . Let f : Gk → A1k such that f (Qi ) are all distinct values.
Then the characteristic polynomial of f is
N

det(T − f ) =

(T − f (Qi ))
i=1

154

and since {σi } form a full set of sections, we also have
N

det(T − f ) =

(T − f (σi,k )).
i=1

It follows that f (σi,k ) are all distinct, meaning of course that σi,k are as well.

Proposition A.4.13. Let Z = Spec(k[x]/(x − a)n ) and S = Spec(k) where k is an algebraically closed field. Then the collection of sections {P1 , . . . , Pn } where each is defined on
coordinate rings as x → a gives a full set of sections. Furthermore, this collection is the only
full set of sections.

Proof. Let R be a k-algebra, B = R[x]/(x − a)n , and f ∈ B. We first calculate NormB/R (f ).
Note that {1, x − a, . . . , (x − a)n−1 } forms an R-basis of B and write f =

n
i=0

fi (x − a)i .

Then the matrix representing the map “multiplication by f ” with respect to this basis is












f0
f1
..
.

f0
..
.

..

.

fn−1 fn−2 . . . f0

Therefore NormB/R (f ) = f0n and this is precisely

n
i=1





.




f (a).

That this is the only collection forming a full set of sections is immediate, as a collection of
sections in this case is determined by the collections cardinality.

We now describe the Oort-Tate generators of the three group schemes we are primarily
interested in.

155

Example A.4.14.
(i). G = (Z/pZ)k , k = k a field. This group scheme is ´etale and hence the p − 1 nonzero
sections are all Oort-Tate generators.
(ii). G = (µp )k , k = k a field of characteristic p. This group scheme is the spectrum of a
nonreduced point, and hence the zero section is the only generator.
(iii). G = (αp )k , k = k a field of characteristic p. This group scheme is the spectrum of a
nonreduced point, and hence the zero section is the only generator.

A.4.3

Oort-Tate Theory

In [OT], Oort and Tate classify finite flat group schemes over

Spec(Λp ),

where Λp = Z ζ,

1
∩ Zp .
p(p − 1)

In particular, the classification applies to schemes S over Spec(Zp ), the case of interest. We
will state the classification and then recast it using stacks as in [HR].

Theorem A.4.15. [OT, Theorem 2] Let S be a scheme over Spec(Λp ). Then there is a
natural 1-1 correspondence finite flat group schemes of order p and the collection of (L, a, b)
where
• L is an invertible sheaf on S;
• a ∈ Γ(S, L⊗(p−1) ), b ∈ Γ(S, L⊗(1−p) ); and
• a ⊗ b = wp ∈ Γ(S, Os ) where wp = ε · p with ε ∈ Λ×
p.

Sketch. Let us merely indicate how to construct (L, a, b) from a group scheme G. We first

156

define
ei =

1
p−1

χ−i (j)[j] ∈ OS [Z/pZ× ].
j∈Z/pZ×

Here χ : Z/pZ → Zp is the Teichm¨
uller representative (whose image consists of the (p − 1)st
roots of unity) and [j] : G → G is multiplication by j. Let m0 = ker(ε# ) where ε : S → G
is the identity section. One can show that the ei are orthogonal idempotents, and thus we
have
p−1

ei m0 .

m0 =
i=1

Set I = e1 m0 . The above construction applies equally as well to the Cartier dual G∗ , and we
likewise define I D in the same manner but with respect to G∗ . Now it turns out the p-fold
multiplication of G sends I ⊗p → I which is to say that there is a homomorphism in

HomOS (I ⊗p , I) = HomOS (OS , I ⊗1−p ) = Γ(S, I ⊗1−p )
giving the global section a. Applying the same to G∗ we get b. Now one cay show a ⊗ b = wp
where wp is defined as follows. In the ring Λp [z]/(z p − 1) set
χ−j (m)(1 − z m )

yj =
m∈Z/pZ×

and define wp by the equation y1p = wp yp . Finally one shows that wp is indeed p · ε where ε
is a unit in Λp .

157

A.5

de Rham cohomology

Let F • be a complex of sheaves of abelian groups on a topological space X such that F i = 0
for i << 0. Denote by di : F i → F i+1 the differential of the complex, so that di+1 di = 0.
Then the ith cohomology of the complex F • is given by

hi (F • ) = ker di / Img di−1 .

We say that a map of complexes ϕ : F • → G• of sheaves of abelian groups on a topological
space X is called a quasi-isomorphism if it induces an isomorphism hi (ϕ) : hi (F • ) → hi (G• )
for all i.
An injective resolution is a quasi-isomorphism f • : F • → I • where I i is an injective sheaf
for all i. Then the hypercohomology of F • is

Hi (X, F • ) = hi (Γ(X, I • ))

where Γ is the global sections functor. The hypercohomology groups are independent of the
injective resolution chosen.
Now let S be a scheme and A/S be an abelian scheme of relative dimension n. The de Rham
complex of A/S is given by
d1

d2

dn−1

Ω1A/S −
→ Ω2A/S −
→ . . . −−−→ ΩnA/S
with differential di : ΩiA/S → Ωi+1
A/S . We define the de Rham cohomology of A/S to be the

158

hypercohomology of A/S with respect to the de Rham complex:

1
(A/S) = H1 (A, Ω•A/S ).
HdR

Proposition A.5.1. [BBM] Set ωA/S = e∗ (Ω1A/S ) where e : S → A is the identity section;
ωA/S is called the sheaf of invariant differentials of A/S. Let Lie(A∨ /S) denote the Lie
algebra of the dual abelian scheme A∨ /S. There is an exact sequence of locally free modules
over S
1
0 → ωA/S → HdR
(A/S) → Lie(A∨ /S) → 0

whose formation commutes with base change.

A.6

Abelian schemes

We collect here definitions and propositions relating to abelian schemes. We include only
the essentials needed.

Definition A.6.1. Let S be a Noetherian scheme, and π : A → S be an abelian scheme
of relative dimension n. That is, A/S is a group scheme where π is smooth, proper, and
the geometric fibers of π are connected. Denote the multiplication by µ : A ×S A → A, the
inverse by ι : A → A, and the identity by ε : S → A.

Definition A.6.2. The dual abelian scheme over S is

A∨ = Pic0 (A/S)

159

where Pic0 is the connected component of the identity of the scheme representing the Picard
functor.

Definition A.6.3. The Lie algebra of A/S is

Lie(A/S) = ε∗ TA/S

where TA/S = HomOA (ΩA/S , OA ) is the tangent bundle of A/S.

Definition A.6.4. The p-divisible group of A/S is

A[p∞ ] = lim A[pn ]
−→
n

where the maps of the directed system are given by inclusion.

Definition A.6.5. A full level N structure on A/S consists of a collection of sections σi :
S → A where 1 ≤ i ≤ 2N such that
(i). for all geometric points s of S, the images σi (s) form a basis for As [N ]; and
(ii). [N ] ◦ σi = ε where [N ] : A → A is multiplication by N .

We now seek to define a polarization of an abelian scheme. Let L be an invertible sheaf on
A and consider the invertible sheaf on A ×S A given by
µ∗ (L) ⊗ p∗1 (L)−1 ⊗ p∗2 (L)−1

p2

where A ×S A ⇒ A.
p1

Regarding A×S A as a scheme over A via p1 , this sheaf defines an A-valued point Λ(L) : A →

160

Pic(A/S). Now Λ(L) ◦ ε : S → Pic(A/S) is the identity and thus ψ is a homomorphism.
Also, since the geometric fibers of A over S are connected, we have Λ(L) factors through
A∨ = Pic0 (A/S) → Pic(A/S).

Definition A.6.6. A polarization of an abelian scheme A/S is an S-homomorphism λ :
A → A∨ such that for all geometric points s of S, the induced homomorphism λs : As → A∨s
is of the form Λ(Ls ) for some ample invertible sheaf Ls on As . Such a polarization is said to
be principal if λ is an isomorphism.

Theorem A.6.7. [MFK, Theorem 7.9] Let An,1,N denote the moduli functor of abelian
schemes of relative dimension n, equipped with a principal polarization and level N structure.
If N ≥ 3, then a fine moduli scheme for An,1,N exists.

We now describe the Rosati involution. Let A be an abelian variety over an algebraically
closed field k equipped with a polarization λ : A → A∨ . Let End0 (A) = End(A) ⊗ Q.
Definition A.6.8. The Rosati involution on End0 (A) with respect to λ is defined by

ϕ → ϕ = λ−1 ◦ ϕ∨ ◦ λ

for ϕ ∈ End0 (A) where ϕ∨ ∈ End0 (A∨ ) is the dual isogeny of ϕ.

Proposition A.6.9. [Mum, pg. 189-190] The Rosati involution satisfies the following properties.
(i). The map (·) : End0 (A) → End0 (A) is a Q-algebra homomorphism.
(ii). eλ (ϕx, y) = eλ (x, ϕ y), where eλ is the Weil pairing induced by λ. As eλ is a non-

161

degenerate bilinear form, this immediately implies that ϕ → ϕ is an involution.
(iii). The Rosati involution is positive.

Definition A.6.10. Suppose R is a subring of Q. An R-isogeny f : A → A between two
abelian S-schemes is an isomorphism in the category of whose objects are abelian schemes
over S and whose morphisms consist of Hom(A, A ) ⊗Z R.

Let A/k be an abelian variety of dimension n with k algebraically closed and let = char(k)
be a rational prime. Then A[ i ] ∼
= (Z/li Z)n . We have surjective homomorphisms [ ] :
A[

i+1

] → A[ i ] given by multiplication by . These homomorphisms are compatible in the

sense that they form an inverse system.

Definition A.6.11. Let A/k be an abelian variety with k algebraically closed. The -adic
Tate module of A/k is defined to be the inverse limit

T (A) = lim A[ i ].
←−
i

With A/k still an abelian variety over an algebraically closed field, suppose A is equipped
ˆ Then the polarization induces the Weil pairing
with a principal polarization λ : A → A.

A[ i ] × A[ i ] → µ i

where µ i is the group of

i

roots of unity in k. Define the -adic Tate module Z (1) = limi µ i ,
←−
taking the inverse limit over the Weil pairing gives

T (A) × T (A) → Z (1).

162

∼
With Z (1) ∼
→ Z and thus have
= Z noncanonically, we may choose an isomorphism Z (1) −

the Weil pairing on the -adic Tate modules take values in Z . As the choice of such an
isomorphism is up to some Z× -multiple, T × T → Z is well-defined up to a Z× multiple.

Proposition A.6.12. If f : A → A is an isogeny with kernel N , there is an exact sequence
T (f )

0 → T (A) −−−→ T (A ) → N → 0

where N is the pro- part of N .

Proposition A.6.13. Let f : A → A be an isogeny that is also a Z(p) -isogeny. Then f
is an isomorphism if and only if for all primes

= p, the induced homomorphism T (f ) :

T (A) → T (A ) is an isomorphism.

The paring on T (A) induces a pairing on the rational Tate module V := T (A) ⊗Z Q which
we also call the Weil pairing.

Proposition A.6.14. If f : A → A is a Z(p) -isogeny, then the induced homomorphism
V (f ) : V (A) → V (A ) is an isomorphism.

Define
ˆ (p) ) =
H1 (A, Z

T (A)

and

=p

H1 (A, Apf ) =

T (A) ⊗ Q.
=p

We again have the Weil pairing on H1 (A, Apf ) taking values in Apf (1) :=

=p

Z (1) ⊗ Q.

Proposition A.6.15. If f : A → A is a Z(p) -isogeny, then f induces an isomorphism

163

∼

→ H1 (A , Apf ).
H1 (f ) : H1 (A, Apf ) −

164

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165

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