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CUBO A Mathematical Journal
Vol.16, No¯ 03, (67–85). October 2014

On the supersingular loci of quaternionic Siegel space

Oliver Bültel

Mathematische Fakultät,

Universität Duisburg-Essen,

Thea-Leymann Strasse 9, 45127 Essen, Germany.

oliver.bueltel@uni-due.de

ABSTRACT

The paper studies the supersingular locus of the characteristic p moduli space of prin-

cipally polarized abelian 8-folds that are equipped with an action of a maximal order

in a quaternion algebra, that is non-split at ∞ but split at p. The main result is that

its irreducible components are Fermat surfaces of degree p + 1.

RESUMEN

El art́ıculo estudia el lugar supersingular del espacio módulo caracteŕıstico p de abeliano

polarizado 8-veces principal que son equipados con una acción de un orden maximal

en un álgebra quaterniona, que es no-divisible en ∞, pero se divide en p. El principal

resultados es que los componentes irreducibles son superficies de Fermat de grado p+1.

Keywords and Phrases: supersingular abelian varieties, Shimura varieties, orthogonal groups.

2010 AMS Mathematics Subject Classification: 14L05, 14K10.



68 Oliver Bültel CUBO
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1 Introduction

Let p be a prime number. In [9] Oort and Li give a description of the supersingular locus Sg,1 of

Ag,1 × Fp, the fibre over p of the Siegel modular variety of principally polarized abelian g-folds.

Among their results are that Sg,1 has Hg(p,1) irreducible components if g is odd and Hg(1,p) if

g is even, and all of these components have dimension [g2/4].

In this paper we study the supersingular locus of certain PEL-moduli spaces SK of type D
H
4 , see

body of text for a more precise explanation. These moduli spaces are associated to groups G that

are twists of GO(8). In the complex analytic context there exist uniformisations by quaternionic

Siegel half-spaces, these are tube domains of the shape

h = {X + iY|X,Y ∈ Mat2(H),X
t,ι = X,Yt,ι = Y > 0}, (1)

where H is the non-split quaternion algebra over R, and ι is the standard involution.

In the algebraic context SK is a 6-dimensional variety parameterizing abelian 8-folds with a par-

ticular kind of additional structure, and on a mild assumption on the level structure this variety is

smooth. For every prime of good reduction we introduce the usual integral model for this Shimura

variety, and we move on to exhibit the geometry of each individual irreducible component of the

supersingular locus of the mod p-reduction. Our main result says that these components are Fer-

mat surfaces. This comes as a surprise, because for a more general Shimura variety, the structure

of the supersingular locus is usually quite complicated and might not even be smooth, for example

this happens in the case of S3,1 the 2-dimensional space of supersingular principally polarized

abelian 3-folds, cf. [12, Paragraph(4)], [14, Proposition 2.4], [9, Example(9.4)] or [15] for a

very precise exposition. We round off the discussion by turning to the non-supersingular points

also, we prove that their p-divisible groups do not have parameters, which is somewhat the exact

opposite to their behaviour on the supersingular locus. This too seems unusual, as one sees from

the non-supersingular principally polarized abelian 3-folds. These results were already applied in

[2] to obtain an Eichler-Shimura congruence relation for SK and for its Shimura divisors.

I am indebted to the referee, and there remains the pleasant task of thanking C.Kaiser, Prof.

M.Rapoport, and Prof. T.Wedhorn for remarks on the topic and especially Prof. R.Taylor for

much good advice and Prof. F.Oort for some email exchanges.

The paper is organized as follows: Section 2 focuses on local aspects, section 3 on global ones.

In subsections 2.1/ 3.1 we sum up definitions and conventions. In subsections 2.2 we explain

techniques needed to understand supersingular Dieudonné modules. We apply these techniques

to supersingular Dieudonné modules with the particular additional structure under consideration

in the subsections 2.3- 2.4. In subsection 2.5 a result on the non-supersingular locus is obtained

(Corollary 2.7).



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On the supersingular loci of quaternionic Siegel space 69

2 Structure Theorems on Dieudonné Modules with Pairing

2.1 Notions and Notations

We continue to fix a prime p. If k is a perfect field of characteristic p, then one denotes by W(k)

and K(k) the Witt ring and fraction field thereof. Unless otherwise said, k will be assumed to be

algebraically closed. The absolute Frobenius x 7→ xp induces automorphisms x 7→ Fx on W(k)

and K(k) which again will be referred to as absolute Frobenii. Recall that a Dieudonné module

is a finitely generated, torsion free W(k)-module together with a F-linear endomorphism F and a
F−1-linear endomorphism V that satisfies FV = VF = p. If one tensorizes with Q one obtains the

isocrystal of M, this is a finite dimensional K(k)-vector space together with a F-linear bijection F.

Dieudonné modules are called isogenous if they give rise to isomorphic isocrystals.

By a pairing on a Dieudonné module M one understands a W(k)-bilinear map φ : M × M →

W(k) which satisfies φ(x,Fy) = Fφ(Vx,y). When thinking of M as the co-variant Dieudonné

module of a p-divisible group A over k, this means that φ gives rise to a morphism from A to

the Serre-dual of A. Dieudonné modules with pairings (M,φ) and (M′,φ′) are called isometric

if there exists an isomorphism from M to M′ taking φ to φ′. The dimension, dimk(M/VM)

of a Dieudonné module with non-degenerate pairing is equal to the codimension dimk(M/FM),

the rank of M is necessarily even. A pairing is called antisymmetric if φ(x,y) = −φ(y,x) and

symmetric if φ(x,y) = φ(y,x). In either of these cases we denote {x ∈ M ⊗ Q|φ(M,x) ⊂ W(k)}

by Mt, if M = Mt we say that φ is perfect.

In [13, Definition(3.1)] the crucial notion of crystalline discriminant of a non-degenerate symmetric

pairing is introduced: Say the underlying Dieudonné module has rank 2n, and choose a K(k)-basis

x1, . . . ,x2n with the additional property Fx1 ∧ · · · ∧ Fx2n = p
nx1 ∧ · · · ∧ x2n. The determinant

det(φ(xi,xj)), regarded as an element in K(Fp)
×/(K(Fp)

×)2, can be checked to be independent

of the choice of basis and is called the crystalline discriminant crisdisc(M,φ). It only depends

on the isogeny class of M allowing one to also write crisdisc(M ⊗ Q,φ) for crisdisc(M,φ). When

fixing once and for all an element t ∈ W(Fp2)
× with tσ = −t, the target group K(Fp)

×/(K(Fp)
×)2

can be identified with {1,t2,p,pt2} if p is odd and with {±1,±t2,±p,±pt2} if p = 2, notice that

the kernel of the forgetful map from K(Fp)
×/(K(Fp)

×)2 to K(k)×/(K(k)×)2 consists of {1,t2}.

Notice also that the image of crisdisc(M,φ) in the group K(k)×/(K(k)×)2 is the discriminant of a

symmetric pairing in the usual sense of linear algebra, hence is independent of the structure of M

as a Dieudonné module. The following characterization of crystalline discriminants within {1,t2}

will be useful, see [13, Corollary(3.5)] for a proof:

Fact 1 (Ogus). Assume that φ is a non-degenerate symmetric pairing on a Dieudonné module M

of rank 2n. Assume also that there exists a φ-isotropic K(k)-subspace A ⊂ M ⊗ Q of dimension

n. Then

crisdisc(M,φ) = (−1)nt2 dimK(k)(A+FA/A),

in particular, if A is an isocrystal, then crisdisc(M,φ) = (−1)n.



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Recall that the Grassmannian of n-dimensional isotropic subspaces of the K(k)-vector space

M ⊗ Q has two connected components. Two such spaces A1 and A2 lie in the same component

if and only if the integer dimK(k)(A1 + A2/A1) is even, we will say that A1 and A2 have the

same parity if this is the case. Thus by the above fact (−1)n crisdisc(M,φ) is the trivial element

of K(Fp)
×/(K(Fp)

×)2 if and only if the bijection F does not change the parity of the maximal

isotropic subspaces. If p is odd and if φ is a perfect form on M, then one can deduce a further

formulation: Pick a maximal isotropic k-subspace A ⊂ M = M/pM complementary to VM. Lift

it to W(k) to obtain a maximal isotropic W(k) submodule A of M (the Grassmannian is smooth).

Observe that FA = FM to conclude that (mod 2):

dimk(A + FA/A)

≡ dimk(A + VM/A) + dimk(VM + FM/VM)

≡ dimk(M/VM + FM) (mod 2).

The integer dimk(M/VM + FM) is called the Oort invariant of M and denoted a(M). Thus,

we have derived the consequence, implicitely stated in [11, Section 5.3]:

Fact 2 (Moonen). Assume that φ is a perfect symmetric pairing on the Dieudonné module M of

rank 2n. If p is odd, then crisdisc(M,φ) = (−1)nt2a(M).

The Dieudonné module M is called superspecial if it satisfies FM = VM, i.e. if rankW(k)(M) =

2a(M). Superspecial Dieudonné modules may conveniently be described in terms of their skele-

tons, these are the W(Fp2)-submodules defined by M̃ = {x ∈ M|Fx = Vx}. We write OB for the ring

extension of W(Fp2), obtained by adjoining an indeterminate σ subject to the relations σ
2 = p

and σa = Faσ, it operates in a self-explanatory way on M̃. As remarked in [8] the assignment

M 7→ M̃ sets up an equivalence of the category of superspecial Dieudonné modules with the cat-

egory of finitely generated torsion free OB-modules. We also write B for OB ⊗ Q, it is the unique

non-split quaternion algebra over K(Fp). Observe that OB is the maximal order of B. We let mB

be the maximal ideal of OB, one has OB/mB ∼= Fp2.

We need to put pairings into the picture as follows. If φ is a pairing on a superspecial

Dieudonné module M, then one considers a OB-valued pairing on M̃ defined by:

Φ(x,y) = φ(x,σty) − φ(x,y)σt (2)

This is OB-sesquilinear, i.e. satisfies Φ(ux,vy) = uΦ(x,y)v
ι, for u,v ∈ OB. The involution ι

is the standard one, mapping a+bσ to Fa−bσ. Conversely any OB-sesquilinear form arises from

a pairing on M in the way described, φ is non-degenerate/perfect if and only if Φ is.

Unless otherwise said we assume from now on that φ is symmetric, in terms of Φ this means



CUBO
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On the supersingular loci of quaternionic Siegel space 71

Φ(y,x)ι = −Φ(x,y) for all x,y ∈ M̃. The OB-module M̃ with form Φ is called hyperbolic if on a

suitable OB-basis e1, . . . ,en/2,f1, . . . ,fn/2 of M̃ one has

Φ(ei,ej) = Φ(fi,fj) = 0 , Φ(ei,fj) = wδi,j (3)

for some non-zero w ∈ OB, uniquely determined only up to multiplication by O
×

B
. It turns out

that w = −σrt is a very convenient choice as the values of the corresponding form φ will then

read:

φ(ei,fj) =

{
0 r ≡ 0 (mod 2)

pr−1/2δi,j r ≡ 1 (mod 2)

and

φ(ei,Ffj) =

{
pr/2δi,j r ≡ 0 (mod 2)

0 r ≡ 1 (mod 2)

and φ(ei,ej) = φ(fi,fj) = 0. Equivalently, (M,φ) is hyperbolic if and only if the Dieudonné

module M allows a decomposition into a direct sum of Dieudonné modules A and B with φ(A,A) =

φ(B,B) = 0, and Mt = F−rM, so that φ identifies the dual of A with F−rB.

2.2 Results of Oort and Li

A Dieudonné module is called supersingular if it is isogenous to a superspecial one, or equivalently

if all its Newton slopes are equal to 1/2. This section is primarily concerned with supersingu-

lar Dieudonné modules, so recall some of the techniques which are usually applied to them: If

M is supersingular it has a biggest superspecial sub-module S0(M) which one can construct as

S0(M) = M̃ ⊗W(F
p2

) W(k). Dually there is S
0(M), the smallest superspecial module containing

M, see [10, Chapter III.2] for proofs of this.

The following facts on the relation of the lattices S0(M) ⊂ M ⊂ S
0(M) are basic to the study of

supersingular Dieudonné modules. The first of them can be found in [8, Corollary(1.7)], along

with more information on the functors S0 and S
0. For the other two facts we refer the reader to

[8, Lemma(1.5/1.6)] (or [9, Fact(5.8)]) and [8, 1.10(i)] (or [9, Chapter(12.2)]):

Fact 3 (Li). Let M be a supersingular Dieudonné module of rank 2g over W(k). Then one has

Fg−1S0(M) =
∑

i+j=g−1 F
iVjM. It follows that Fg−1S0(M) ⊂ S0(M), in particular the length

of the W(k)-module S0(M)/S0(M) is bounded by g(g − 1) and equality is acquired if and only if

a(M) = 1.

Fact 4 (Li). Let N be a superspecial Dieudonné module of rank 2g over W(k). Let x be an element

of N. Then one has S0(W(k)[F,V]x) = N if and only if the elements

Fg−1x,Fg−2Vx,. . . ,FVg−2x,Vg−1x

form a basis of the k-vector space Fg−1N/FgN. Moreover, an element with this property exists.



72 Oliver Bültel CUBO
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Fact 5 (Li). Let M be a supersingular Dieudonné module of rank 2g over W(k). For a non-negative

integer i let

si = dimk(M ∩ F
iS0(M)/M ∩ Fi+1S0(M)).

Then one has si ≤ si+1 and equality holds if and only if si = g.

The work [9] studies supersingular Dieudonné modules which are equipped with a perfect

anti-symmetric form ψ. Following their method we notice that we have to incorporate additional

structure which by the Morita-equivalence of subsection 3.2 leads to Dieudonné modules M of

rank 8 equipped with a symmetric form φ.

Analogous to [9, Proposition(6.1)] we need to analyze the restriction of φ to N = S0(M), or more

generally, a classification of non-degenerate symmetric forms on superspecial Dieudonné modules:

Theorem 2.1. Let k be an algebraically closed field of characteristic p 6= 2 and let N be a superspe-

cial Dieudonné module of rank 2n over W(k), which is equipped with a non-degenerate symmetric

pairing φ. Then N contains Dieudonné modules Ni of rank 2, with φ(Ni,Nj) = 0, for i 6= j, and

N =
⊕n

i=1 Ni. Moreover, each Ni has a W(k)-basis consisting of elements xi,Fxi = Vxi = yi

such that one of the two cases:

(i) φ(xi,xi) = φ(yi,yi) = 0, and φ(xi,yi) = p
ni,

(ii) φ(xi,yi) = 0, φ(xi,xi) = ǫip
ni, and φ(yi,yi) = ǫ

σ
i p

ni+1,

holds for some integers ni and some elements ǫi ∈ W(Fp2)
× which are unique up to multiplication

by elements in (W(Fp2)
×)2. Moreover, the cristalline discriminant can be computed from this

decomposition as

crisdisc(Ni,φ) =

{
−t2 (Ni,φ) of type (i)

pt2ǫiǫ
σ
i (Ni,φ) is of type (ii)

,

and crisdisc(N,φ) =
∏n

i=1 crisdisc(Ni,φ).

Proof. The skeleton construction descends N to a W(Fp2)-Dieudonné module Ñ which at the same

time is a OB-module. As in (2) we consider the OB-valued sesquilinear form Φ and diagonalize it

as follows: Let x0 ∈ Ñ be an element with Φ(x0,x0) of mB-adic valuation as small as possible, i.e.

such that Φ(x,x) ∈ mrB = OBΦ(x0,x0) for all x ∈ Ñ. By the usual polarization process it follows

that Φ(x,y)−Φ(x,y)ι ∈ mrB, and also Φ(x,y)+Φ(x,y)
ι ∈ mrB by replacing tx for x. Consequently

Φ(Ñ,Ñ) ⊂ mrB. Therefore we obtain an orthogonal direct sum Ñ = (Ñ ∩ (Bx0)
⊥) ⊕ OBx0, as

any x ∈ Ñ has Φ(x,x0)Φ(x0,x0)
−1 = α ∈ OB which allows to write x as a sum of αx0 ∈ Ñ and

x − αx0 ∈ Ñ ∩ (Bx0)
⊥.

Having obtained a decomposition Ñ =
⊕n

i=1 Ñi we search for basis elements x̃i ∈ Ñi with Φ(x̃i, x̃i)

manageable: In Ñi ⊗ Q one can certainly find elements x̃i with Φ(x̃i, x̃i) ∈ W(Fp2)
× ∪ FW(Fp2)

×

for example by [16, Chapter 10, Theorem(3.6.(i))]. Observe that the mB-adic valuation of Φ(x̃i, x̃i)

must be congruent modulo 2 to ri = lengthOB Ñ
t
i/Ñi. Hence after adjusting the x̃i’s by multiplying



CUBO
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On the supersingular loci of quaternionic Siegel space 73

them by Fri/2, if ri is even, and by F
(ri−1)/2, if ri is odd, one gets generators of the OB-modules

Ñi on which the sesquilinear form takes values in F
riW(Fp2)

×.

It is clear how to obtain the desired basis x1, . . . ,xn,y1, . . . ,yn from these generators. If ri is

even Ni will be of type (i) with ni = ri/2, and if ri is odd then Ni will be of the type (ii) with

ni = (ri − 1)/2.

Remark 2.2. Suppose N is a superspecial Dieudonné module of rank 2 with a symmetric form

φ. Then one checks from the above classification that (N,φ) is isometric to (N,−φ). It follows

that N⊕2, the orthogonal direct sum of two copies of N, is hyperbolic. One checks this by using

the sesquilinear form (2) as Φ((u1 + u2,u1 − u2),(v1 + v2,v1 − v2)) = (u1 + u2)w(v1 + v2)
ι −

(u1 − u2)w(v1 − v2)
ι = (2u1w)v

ι
2 − u2(2v1w)

ι. (cf. [9, Remark(6.1)] for the analog in the

anti-symmetric setting)

For later use we note an immediate corollary:

Corollary 2.3. Let (N1,φ1) and (N2,φ2) be supersingular Dieudonné modules of rank two,

equipped with symmetric pairings. There exists an isometry between them if and only if the follow-

ing holds:

lengthW(k) N
t
1/N1 = lengthW(k) N

t
2/N2

crisdisc(N1,φ1) = crisdisc(N2,φ2).

Consequently for any non negative integer n, there is only one isometry class of rank two supersin-

gular Dieudonné modules with pairing (N,φ) where lengthW(k) N
t/N = 2n. There are two such

classes of modules with pairing where lengthW(k) N
t/N = 2n + 1.

2.3 Classification of symmetric Dieudonné modules

This section is the core of the work, we give a classification of Dieudonné modules with the

additional structure of interest.

Theorem 2.4. Let M be a supersingular Dieudonné module over W(k) with perfect symmetric

pairing φ. Assume that:

rankW(k) M = 8

crisdisc(M,φ) = 1.

Consider S0(M) = N, the smallest superspecial Dieudonné lattice in M ⊗ Q, which contains M.

Choose a decomposition N =
⊕4

i=1 Ni with properties as granted by Theorem 2.1, and with

S0(M) = N
t =

⊕4
i=1 F

riNi for integers r1 ≤ r2 ≤ r3 ≤ r4. Then (r1,r2,r3,r4) is one of

(i) (0,0,0,0)

(ii) (1,1,1,1)



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(iii) (0,2,2,2)

(iv) (2,2,2,2),

moreover, there exists a superspecial Dieudonné lattice Q, which contains FM and satisfies

(a) Qt = Q

(b) dimk(M/M ∩ Q) = dimk(Q/M ∩ Q) = 1.

If M is of the form (iii) or (iv), then the superspecial Dieudonné lattice Q, satisfying (a) and (b)

is unique.

Proof. For the proof we need two auxiliary lemmas:

Lemma 2.5. Let the assumptions on M be as in the above theorem, then there exist two different

indices i1 and i2 such that Ni1 and Ni2 are isometric.

Proof. If an even integer r occurs twice amongst the various ri’s one is done, and if an odd integer

r occurs three times one is done as well, use the pigeon hole principle and Corollary 2.3. The

condition on the discriminant forces the number of indices i with ri odd to be even. This means

that one is left with checking the lemma for the ri-quadruples (0,1,2,3), (0,2,3,3), (0,1,1,2), and

(1,1,3,3).

The three quadruples with r1 = 0 do not arise, because otherwise M would be an orthogonal direct

sum of N1 and some supersingular Dieudonné module M
′ of rank 6 and equipped with a perfect

symmetric form φ′. Applying Fact 2 to M′ would give that M′ has Oort invariant 1 or 3, as

crisdisc(M′) = crisdisc(N1) = −t
2. Fact 3 applied to M′ would further imply that the elementary

divisors of S0(M′)/S0(M
′) are either all 0 or all equal to 2. Hence the elementary divisors of

S0(M)/S0(M) would be (0,0,0,0) or (0,2,2,2).

It remains to do the (r1,r2,r3,r4) = (1,1,3,3)-case. Assume that no two of the N
′
is were isometric.

This would lead to a basis xi,Fxi = Vxi = yi with

φ(x1,x1) = p
−1, φ(y1,y1) = 1

φ(x2,x2) = ǫp
−1, φ(y2,y2) = ǫ

σ

φ(x3,x3) = p
−2, φ(y3,y3) = p

−1

φ(x4,x4) = ǫp
−2, φ(y4,y4) = ǫ

σp−1,

other products = 0,

and with ǫ some non-square in W(Fp2)
×. The module M has to contain an element of the form

α1x1 +α2x2 + α3x3 + α4x4 + β3y3 +β4y4 such that βi,αi ∈ W(k) but not both of α3 and α4 in

pW(k). As

φ(x,x) = p−1(α21 + ǫα
2
2 + β

2
3 + ǫ

σβ24) + p
−2(α23 + ǫα

2
4)



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On the supersingular loci of quaternionic Siegel space 75

one has α23 + ǫα
2
4 ≡ 0 (mod p), but as

φ(x,F2x) = α1α
σ2

1 + ǫα2α
σ2

2 + β3β
σ2

3 + ǫ
σβ4β

σ2

4 + p
−1(α3α

σ2

3 + ǫα4α
σ2

4 )

one has α
p2+1
3 +ǫα

p2+1
4 ≡ 0 (mod p) as well. As ǫ is a non-square in W(Fp2)

×, one has ǫ
p2−1

2 ≡

−1 (mod p), so that we derive the contradiction

α
p2+1
3 ≡ (−ǫα

2
4)

p2+1
2 ≡ ǫα

p2+1
4 (mod p).

Lemma 2.6. With the same notation as in the theorem ri ≤ 2 for all indices i.

Proof. Observe that the lemma would be immediate if one of the ri was zero. So we can assume

0 < ri for all indices i. Pick two indices i 6= j with ri = rj = r and crisdisc(Ni) = crisdisc(Nj),

according to the previous lemma such indices will exist. Say (i, j) = (1,2) after relabeling, and

write according to Remark 2.2 N1⊕N2 = A⊕B, with φ(A,A) = φ(B,B) = 0 and A×F
rB → W(k)

a perfect pairing. Consider along the lines of [9, Proposition(6.3)] a W(k)-module M′ which is the

image of (B⊕N3⊕N4)∩M under the projection map B⊕N3⊕N4 → N3⊕N4. M
′ inherits a perfect

form and is indeed canonically isomorphic to the sub-quotient (B⊥ ∩ M)/(B ∩ M) of M. One has

crisdisc(M′) = 1 because M′ is isogenous to N3 ⊕N4. By Fact 2 it follows that M
′ is superspecial.

Furthermore the proof of [9, Proposition(6.3)] shows that FN3 ⊕ FN4 ⊂ M
′ ⊂ N3 ⊕ N4. For

convenience of the reader we reproduce the argument in loc.cit.: Pick an element in M of the form

x = e + f + n3 + n4 with e ∈ Ã, f ∈ B, n3 ∈ N3, n4 ∈ N4 and

S0(M) = S0(W(k)[F,V]x),

it exists due to Fact 4. The elements F3x, F2Vx, FV2x, V3x will then form a basis of the k-vector

space F3N/F4N so that F3x−F2Vx, F2Vx−FV2x, FV2x−V3x is a basis of F3(B⊕N3 ⊕N4)/F
4(B⊕

N3 ⊕ N4). It follows that

S0(W(k)[F,V](F − V)x) = F(B ⊕ N3 ⊕ N4),

but (F − V)x ∈ M ∩ (B ⊕ N3 ⊕ N4) which projects surjectively onto M
′. As S0 is a functor in

supersingular Dieudonné modules FN3 ⊕FN4 will be contained in S
0(M′) = M′, and consequently

FN3 ⊕ FN4 ⊂ M
′ = M′t ⊂ F−1Nt3 ⊕ F

−1Nt4 = F
r3−1N3 ⊕ F

r4−1N4

i.e. r3,r4 ≤ 2. However, r3 ≡ r4 (mod 2), as crisdisc(N3) = crisdisc(N4). Therefore r3 = r4, as

r3,r4 ∈ {1,2}. Now, note that this does indeed imply that N3 is isometric to N4.

In order to find that r1,r2 ≤ 2 also, we redo the whole argument, with the roles of N1 and N2

being replaced by N3 and N4.

Return to proof of theorem



76 Oliver Bültel CUBO
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We move on to investigate the set of possible quadruples (r1,r2,r3,r4). If one of the numbers

in that sequence is 0, then Fact 2 shows that we must have either (0,0,0,0) or (0,2,2,2). For

the remaining cases (2,2,2,2), (1,1,1,1) and (1,1,2,2) are conceivable. We show that (1,1,2,2)

can not arise: Assume we had a Dieudonné module M with (r1,r2,r3,r4) = (1,1,2,2). It would

follow that one had crisdisc(N3) = crisdisc(N4) by Corollary 2.3, and so would crisdisc(N1) =

crisdisc(N2). By applying Remark 2.2 to both N1 ⊕ N2 and N3 ⊕ N4 one obtains a basis of N

consisting of say e1, e2, f1, f2, Fe1 = Ve1, Fe2 = Ve2, Ff1 = Vf1, Ff2 = Vf2 and with the only

non-zero products being given by

φ(Fe1,Ff1) = 1

φ(e1,f1) = φ(e2,Ff2) = φ(f2,Fe2) = p
−1.

As F−1Nt is superspecial one has M 6⊂ F−1Nt, so that M contains an element of the form x =

α1e1 + β1f1 + α2e2 + β2f2 + α3Fe2 + β3Ff2, with all α1, . . . ,β3 ∈ W(k) and at least one of α2

and β2 a unit. From Fx ∈ α
σ
2Fe2 + β

σ
2Ff2 + N

t and φ(M,M) ⊂ W(k) one infers φ(x,Fx) ∈

p−1(ασ2β2 + β
σ
2α2) + W(k), which means that α

σ
2β2 + β

σ
2α2 ≡ 0 (mod p). As we may alter the

elements α1, . . . ,β3 by any element in pW(k) we can actually assume that α
σ
2β2 + β

σ
2α2 = 0, but

then the Dieudonné module

W(k)Fx + Nt = W(k)(ασ2Fe2 + β
σ
2Ff2) + N

t

is superspecial contradicting S0(M) = N
t.

Having done the first assertion of the theorem we now focus on the existence of Q. If M is of the

form (i), then use Remark 2.2 to write N1 ⊕N2 as direct sum of two isotropic Dieudonné modules

A and B, between which there is the duality that is induced from the pairing on N. Then one finds

that Q = F−1A ⊕ FB ⊕ N3 ⊕ N4 is a superspecial Dieudonné lattice that does the job. Similarly

for the (ii)-case: Write N = A1 ⊕ A2 ⊕ B1 ⊕ B2 with isotropic Ai and Bi, this time equipped with

a canonical isomorphism Ati
∼= FBi. The superspecial lattices

FA1 ⊕ FA2 ⊕ B1 ⊕ B2

A1 ⊕ FA2 ⊕ FB1 ⊕ B2

both satisfy Qt = Q, and one of them satisfies property (b) as well.

In the (iii)-case property (a) forces to look at Q = N1 ⊕
⊕4

i=2 FNi, whereas Q = FN in the

(iv)-case. We have to show that this module does indeed satisfy (b), to this end observe that the

numbers dimk M/M ∩ Q and dimk M ∩ Q/M ∩ FQ are nonzero and sum up to 4, it thus suffices

to see that the first of them is strictly smaller than the second. In the (iv)-case this is the content

of Fact 5. In the (iii)-case apply Fact 5 to the orthogonal complement of N1 in M, which is a

Dieudonné module of rank 6 with perfect symmetric form.

2.4 Moduli of symmetric Dieudonné modules

We consider the graded Fp-algebra R := Fp[A1,A2,B1,B2]/(
∑2

i=1 AiB
p
i +BiA

p
i ), and its associated

projective variety X1 := ProjR, which is smooth of relative dimension 2. Let Y1 denote the affine



CUBO
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On the supersingular loci of quaternionic Siegel space 77

chart determined by A1 6= 0, it is the spectrum of R(A1)
∼= Fp[a2,b1,b2]/(b1 +b

p
1 +a2b

p
2 +b2a

p
2),

where a2 :=
A2
A1

, b1 :=
B1
A1

, and b2 :=
B2
A1

. Let α2,β1,β2 ∈ W(R(A1)) be lifts of a2,b1,b2 with

β1 +
Fβ1 + α2

Fβ2 + β2
Fα2 = 0. Let T(A1) be the W(R(A1))-module

⊕4
i=1 W(R(A1))ti, L(A1) be

the W(R(A1))-module
⊕4

i=1 W(R(A1))li, and M(A1) be L(A1) ⊕ T(A1). Putting:

F(t1) = l1

F(t2) = l2 + (β2 −
F2β2)t1

F(t3) = l3 + (
F
2

α2 − α2)t2 + (
F
2

β2 − β2)t4

F(t4) = l4 + (α2 −
F2α2)t1

V−1(l1) = t1

V−1(l2) = t2

V−1(l3) = t3

V−1(l4) = t4

and using the formula V−1(Vαx) = αF(x) defines the structure of a display ([17]) on M(A1), which

moreover has the normal decomposition L(A1) ⊕T(A1). One checks that a pairing is given on M(A1)
by φ(li, lj) = φ(ti,tj) = 0, φ(li,tj) = δ|i−j|,2. Let also N = LN ⊕TN be the display obtained from

the formulas F(ti) = li, V
−1(li) = ti and with pairing defined analogously. Putting:

ǫ(t1) = pt3

ǫ(t2) = l2 −
Fβ2l3

ǫ(t3) = t1 +
Fα2t2 +

Fβ1t3 +
Fβ2t4

ǫ(t4) = −
Fα2l3 + l4

ǫ(l1) = pl3

ǫ(l2) = pt2 − pβ2t3

ǫ(l3) = l1 + α2l2 + β1l3 + β2l4

ǫ(l4) = −pα2t3 + pt4

defines an embedding of displays ǫ(A1) : M(A1) →֒ N ×Fp Y1, satisfying pφ(x,y) = φ(ǫ(x),ǫ(y)).

Neither M(A1) nor ǫ(A1) depend on the choice of the lifts α2,β1,β2, which can be checked upon

passage to the perfection R
perf
(A1)

(here notice that R(A1) → R
perf
(A1)

is flat, because R(A1) is regular).

Moreover, the natural action of the Kleinian group on X1 gives rise to analogous subdisplays of the

constant display N regarded over each of the translates {A2 6= 0}, {B1, 6= 0}, and {B2 6= 0}, which

in turn gives rise to an inclusion

ǫ : M →֒ N ×Fp X1,

of sheaves of displays with respect to the Zariski topology of X1. This is because the closed points

can be used to check the cocycle condition. However, notice that there does not exist a global

normal decomposition for M.



78 Oliver Bültel CUBO
16, 3 (2014)

2.5 Miscellaneous

The study of families of Dieudonné modules with our additional structure within a given isogeny

class is meaningful not just for the supersingular one. Recall that every isogeny class of Dieudonné

modules can be written as a direct sum of certain simple ones. These are parameterized by pairs of

coprime non-negative integers a and b and denoted by Ga,b, see [10] for details. The isogeny class

Ga,b contains usually more than one Dieudonné module except if a or b is equal to 1, in which

case we are allowed to speak of “the” Dieudonné module of type Ga,b. We have the following result:

Corollary 2.7. Let M be a non-supersingular Dieudonné module over W(k) that is equipped with

a perfect symmetric pairing φ. Assume that:

rankW(k) M = 8

crisdisc(M,φ) = 1.

Then M is an orthogonal direct sum
⊕

i Mi where for each of the (Mi,φ) one of the following

alternatives hold:

(i.n) (Mi,φ) can be written as A ⊕ B with mutually dual isotropic Dieudonné modules A and B,

which lie in the isogeny classes G1,n and Gn,1 for some n ∈ {0,1,2,3}.

(ii) (Mi,φ) is supersingular of rank 2 and the perfect pairing thereon is the one described by part

(i) of theorem 2.1.

(iii) (Mi,φ) is supersingular of rank 4, and the pairing is such that S0(Mi) decomposes into the

two Dieudonné modules with pairings described by part (ii) of theorem 2.1.

Moreover, the only combinations which occur are:

• 4 × (i.0)

• 2 × (i.0) ⊕ (i.1)

• (i.0) ⊕ (ii) ⊕ (iii)

• (i.0) ⊕ (i.2)

• (i.3)

Proof. We consider the canonical decomposition of M = M0 ⊕M
′ ⊕M1 into the étale-local, local-

local, local-étale parts. The assertion of the corollary has solely something to do with M′ which

is of some even rank equal to 8 − 2f and has crisdisc(M′) = (−1)f, here f is the p-rank of M. As

M′ is also self-dual it can have only one of the following isogeny types:



CUBO
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On the supersingular loci of quaternionic Siegel space 79

(1) 3 × G1,1

(2) 2 × G1,1

(3) G1,2 ⊕ G2,1

(4) G1,3 ⊕ G3,1

If M′ has the above isogeny types 3., or 4. we deduce from [7, Paragraph(16), Satz(3)] and

a(M′) = 2 that M′ is a direct sum of two Dieudonné modules A and B, each with Oort invariant

equal to one. The assertion on the pairing is then immediate as neither A nor B is selfdual.

If M′ has isogeny type 2G1,1 it must be superspecial. Then use theorem 2.1 in conjunction with

remark 2.2 to check that M′ has the shape A ⊕ B with isotropic A and B.

In the case in which the isogeny type of M′ is 3G1,1, we have to work a bit harder: First consider a

diagonalization of S0(M′) = N =
⊕3

i=1 Ni with S0(M
′) = Nt = FriNi. An analysis as in the proof

of lemma 2.5 yields that (r1,r2,r3) = (0,1,1), therefore the orthogonal direct summand (N1,φ)

has a complement with perfect form, say M′′, its Oort invariant is 1. Therefore crisdisc(M′′) = t2.

As r2 = r3 = 1 this implies that crisdisc(N2,φ), and crisdisc(N3,φ), are the two numbers p, and

pt2, which is what we wanted.

3 The Shimura variety SKp

3.1 Further Notation

Before we proceed we want to introduce the input data for our PEL-moduli problem: Fix once

and for all a quaternion algebra B over Q and write R for the set of places at which B is non-split.

Assume that ∞ ∈ R, i.e. that BR is definite. Let p be a prime which is not in {2} ∪ R and choose

a maximal Z(p)-order OB ⊂ B, together with an isomorphism κp : Zp ⊗ OB
∼= Mat2(Zp). The

standard involution b 7→ bι = tr(b) − b preserves OB and is positive.

Let V be a left B-module of rank 4 with non-degenerate alternating pairing satisfying (bv,w) =

(v,bιw). For simplicity we require that the skew-Hermitian B-module V is hyperbolic in the

following sense: We want it to have a B-basis e1,e2,f1,f2 such that (
∑2

i=1 aiei +bifi,
∑2

i=1 a
′
iei +

b′ifi) = trB/Q(
∑2

i=1 aib
′ι
i − bia

′ι
i ) for all ai,bi,a

′
i,b

′
i ∈ B. Set further Λ0 =

⊕2
i=1 OBei ⊕ OBfi,

it is a self-dual OB-invariant Z(p)-lattice in V.

Let G/Q be the reductive group of all B-linear symplectic similitudes of V. This group is a form

of GO(8). Write Kp ⊂ G(Qp) for the hyperspecial subgroup consisting of group elements that

preserve Λ0 and let K
p ⊂ G(A∞,p) be an arbitrary compact open subgroup.

Finally we specify a particular ∗-homomorphism h0 : C → EndB(VR) by the rule h0(i)(
∑2

i=1 aiei+

bifi) =
∑2

i=1 biei − aifi, and R-linear extension. The reflex field of (G,h0) is equal to Q.

Now, for every connected scheme SKp/Z(p) with a geometric base point s we consider the set of

Z(p)-isogeny classes of quadruples (A,λ, ı,η) with:



80 Oliver Bültel CUBO
16, 3 (2014)

(M1) A is a 8-dimensional abelian scheme over S up to prime-to-p isogeny

(M2) λ : A → At is a Z×
(p)

-class of prime-to-p polarizations of A

(M3) ı : OB → End(A) ⊗ Z(p) is a homomorphism satisfying ı(b
ι) = ı(b)∗, here ∗ is the Rosati

involution associated to λ

(M4) η is a π1(S,s)-invariant K
p-orbit of OB-linear isomorphisms η : V ⊗ A

∞,p ∼= H1(As,A
∞,p)

which are compatible with the alternating form up to scalars.

By geometric invariant theory this functor is representable by a quasi-projective Z(p)-scheme

SKp . Moreover, the deformation theory of Grothendieck-Messing shows that S is smooth of relative

dimension 6 over Z(p), cf. [6, Chapter 5]. See also [6, Chapter 8] for the complex uniformizations

of SKp(C).

Finally, let us write SsiKp (resp. S
sp
Kp ) for the subsets SKp × F

ac
p whose sets of geometric points

consist of those quadruples (A,λ, ı,η) where D(A[p∞]) is supersingular (resp. superspecial), here

D(G) denotes the (covariant) Dieudonné module of a p-divisible group G over a perfect field. Notice

that we always have crisdisc(D(G),φ) = 1, by [1].

3.2 Morita equivalence

Let us write G∗ for the Serre-dual of a p-divisible group G =
⋃

l G[p
l] over some base scheme S.

We will say that G is polarized (resp. anti-polarized) if it is endowed with an isomorphism φ to its

dual which satisfies φ = −φ∗ (resp. φ = φ∗). In particular, consider the anti-polarized p-divisible

groups G1 := BT (M) and G0 := BT (N), where M and N are as in section 2.4. The emdedding

ǫ : M →֒ N ×Fp X1 gives rise to a canonical isogeny ǫ : G1 → G0 ×Fp X1 satisfying ǫ
∗ ◦ ǫ = p idG1

and ǫ ◦ ǫ∗ = p id
G0×Fp X1

, notice also that ker(ǫ) ⊂ G1[p] and ker(ǫ
∗) ⊂ G0[p] ×Fp X1 are finite,

flat, maximal isotropic subgroup schemes of order p4.

If an isomorphism Zp ⊗OB
κp
→ Mat2(Zp) is fixed once and for all, one obtains a Morita-equivalence

(G,φ) 7→ (G⊕2,

(

0 φ

−φ 0

)

)

from the category of anti-polarized p-divisible groups to the category of polarized p-divisible groups

with Rosati-invariant OB-action. In this manner one obtains an anti-polarized p-divisible group

(G,φ) from every S-valued point on SKp, say represented by (A,λ, ı,η), by the requirement

(A[p∞],ψλ) ∼= (G
⊕2,

(

0 φ

−φ 0

)

),

where ψλ : A[p
∞] → A[p∞]∗ is the p-adic Weil-pairing, which is induced from the polarization λ :

A → At. If S is the spectrum of a perfect field of characteric p, we always have crisdisc(D(G),φ) =



CUBO
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On the supersingular loci of quaternionic Siegel space 81

1, by [1]. We next want to define a family of morphisms

cx,ηp : X1 × F
ac
p → SKp × F

ac
p (4)

which are indexed by superspecial Facp -points x = (A,λ, ı,η), equipped with the following additional

datum: By a frame for x we mean an isomorphism ηp : G0 ×Fp F
ac
p → G, where (G,φ) corresponds

to x ∈ SKp(F
ac
p ) by the above Morita-equivalence while (G0,φ0) is the previously exhibited anti-

polarized p-divisible group. Let us consider the abelian variety which is defined by the exact

sequence:

0 → ηp(ker(ǫ
∗))⊕2 → A

et

→ A1 → 0,

the isotropicity and the OB-invariance of ηp(ker(ǫ
∗))⊕2 give rise to a canonical Z×

(p)
-class of

prime-to-p polarizations λ1 : A1 → A
t
1, together with a Rosati-invariant operation ı1 : OB →

End(A1) ⊗ Z(p) and level structure η1, each gotten by transport of structure. Finally one sees

that the quadruple x1 = (A1,λ1, ı1,η1) thus obtained constitutes a X1 × F
ac
p -valued point, whose

classifying morphism we define to be (4). It is easy to see that the image of cx,ηp is a closed subset,

whose geometric points consist of exactly those quadruples (A1,λ1, ı1,η1) which allow an OB-linear

isogeny e : A1 → A, wich is compatible with the level structure and satisfies pλ1 = e
t ◦ λ ◦ e.

Remark 3.1. Fix (A,λ, ı,η) = x ∈ S
sp
Kp (F

ac
p ). Notice, that we have just shown, that the Zariski-

closed subset cx,ηp(X1 × F
ac
p ) := S

sp
x,Kp does not dependent on the choice of frame.

3.3 Description of SsiKp

Now, we would like to investigate whether or not cx,ηp is a closed immersion, the next lemma is a

step towards this direction:

Lemma 3.2. Let x and ηp be as above, then cx,ηp induces an injection on the tangentspaces to

each geometric point u ∈ X1(k), where k is an arbitrary algebraically closed field of characteristic

p.

Proof. Recall that every k-display P of dimension d and codimension c allows structural equations:

F(tj) =

d∑

i=1

ui,jti +

c∑

i=1

ui+d,jli

V−1(lj) =

d∑

i=1

ui,j+dti +

c∑

i=1

ui+d,j+dli

for some display-matrix









u1,1 . . . u1,c+d
...

...
...

uc+d,1 . . . uc+d,c+d









= U ∈ GL(c + d,W(k)),



82 Oliver Bültel CUBO
16, 3 (2014)

where t1, . . . ,td, l1, . . . , lc ∈ P, and t1 +Q,.. . ,td +Q ∈ P/Q are bases. Let L and T be the W(k)-

submodules of P that are generated by l1, . . . , lc and t1, . . . ,td, and write J := HomW(k)(L,T).

Due to the technique of Norman-Oort the isomorphism classes of infinitesimal deformations of P

over the ring of dual numbers kD := k[s]/(s
2) are parameterized by the elements in J ⊗W(k) k =

Homk(Q/pP,P/Q), in fact each deformation may be described explicitly as follows: Pick a tangent

direction N ∈ J ⊗W(k) k, say with d × c-matrix representation









n1,1 . . . n1,c
...

...
...

nd,1 . . . nd,c









(with respect to the two bases above). Write W(skD) for the kernel of the natural map from

W(kD) to W(k), and choose elements ñi,j ∈ W(skD) whose 0-th Witt coordinate is equal to the

dual number sni,j. Then

Ũ :=



























1 . . . 0 ñ1,1 . . . ñ1,c
...

...
...

...
...

...

0 . . . 1 ñd,1 . . . ñd,c

0 . . . 0 1 . . . 0
...

...
...

...
...

...

0 . . . 0 0 . . . 1



























U ∈ GL(c + d,W(kD))

displays an infinitesimal deformation of P, that corresponds to the tangent direction N, in particular

it is the trivial deformation if and only of N = 0.

Now let (X1 : X2 : Y1 : Y2) be the homogeneous coordinates of u ∈ X1(k), and fix one of its

non-zero tangent directions u′ ∈ X1(kD). To finish the proof of the lemma we only have to show

that the associated kD-display Mu′ is a non-trivial infinitesimal deformation (of Mu, i.e. the

special fiber of Mu′). Of course we can assume (X1 : X2 : Y1 : Y2) = (1 : x2 : y1 : y2) from the

start, so let (1 : x2 + sa : y1 − s(ay
p
2 − bx

p
2) : y2 + sb) be the homogeneous coordinates of u

′,

where (a,b) ∈ k2 − {(0,0)}. Now recall from section 2.4 that the restriction of M to the affine

chart Spec Fp[a2,b1,b2]/(b1 +b
p
1 +a2b

p
2 +b2a

p
2) ⊂ X1 has already a normal decomposition and is

explicitly displayed in an extremely convenient way, namely by means of the matrix U =

(

H E

E 0

)

,

where E denotes the identity matrix, and where the (so-called ‘Hasse-Witt’) matrix H is given by:












0 β2 −
F2β2 0 α2 −

F2α2

0 0 F
2

α2 − α2 0

0 0 0 0

0 0 F
2

β2 − β2 0













,

for certain α2,β1,β2 ∈ W(Fp[a2,b1,b2]/(b1 + b
p
1 + a2b

p
2 + b2a

p
2)). Now consider the skD-

valued Witt-vectors α := u′(α2) − u(α2) and β := u
′(β2) − u(β2), in fact it is easy to see that



CUBO
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On the supersingular loci of quaternionic Siegel space 83

u′(β1)−u(β1) = −(αu(β2)
σ +βu(α2)

σ), because α and β are killed by F. Moreover, the 0th Witt-

coordinates of α and β are just sa and sb. It follows immediately that u′(U) =

(

E Ñ

0 E

)

u(U),

with Ñ being the deformation matrix:












0 β 0 α

0 0 −α 0

0 0 0 0

0 0 −β 0













,

whose matrix of 0th Witt-components is clearly nonvanishing.

As a consequence of theorem 2.4 we have:

SsiKp =
⋃

x∈S
sp

Kp

Ssix,Kp,

and S
sp
Kp is a finite set of closed points. It follows from this (or from Grothendieck’s specialization

theorem [3, p.149]), that SsiKp is Zariski closed. Our aim is to describe S
si
Kp together with its induced

reduced subscheme structure. Let us fix x ∈ S
sp
Kp, which classifies some quadruple (A,λ, ı,η), and

let ∗ denote the Rosati-involution on the Q-algebra End0B(A). Let us write Ix/Q for the group

scheme which represents the functor

C 7→ {g ∈ (End0B(A) ⊗ C)
×|ggt ∈ C×}. (5)

Every full level structure η : V ⊗ A∞,p ∼= H1(A,A
∞,p) yields an isomorphism

I × A∞,p
∼=
→ G × A∞,p;γ 7→ η−1γη.

Notice that the preimage of Kp under the above isomorphism depends only on the Kp-orbit of η,

and hence we can define K
p
x := ηK

pη−1 for any η ∈ η, this is again a compact open subgroup of

Ix(A
∞,p). Consider the compact set K̃p := {γ ∈ I(Qp)|γ,γ

−1 ∈ p−1Zp ⊗ EndB(A)}, and let us say

that Kp is superneat for x if and only if Ix(Q) ∩ K̃p × K
p
x = {1}. The left-hand side is always a

finite group, because Ix is anisotropic. In particular K
p will always contain some a compact open

subgroup which is superneat for every x ∈ SsiKp

Lemma 3.3. If Kp is superneat for x, then (4) is a closed immersion.

Proof. A morphism from a proper Facp -variety to a separated one is a closed immersion if and only if

it radicial and injective on the tangent spaces to all Facp -valued points, this is elementary and can be

proved along the lines of [4, Lemma 7.4.]. In view of lemma 3.2 it suffices to check that (4) is indeed

injective on geometric points. Suppose it wasn’t. Then there existed SKp (k) ∋ x1 = (A1,λ1, ı1,η1)

which lies in the image of (4) in two different ways. According to the thoughts at the end of

subsection 3.2, this means that there existed two degree-p8-isogenies e,e′ : A1 → A each of which

induce the additional structures λ1, ı1, η1 from the additional structures λ, ı, η on A. It follows

immediately that idA 6= e
′ ◦ e−1 is in contradiction to Kp being superneat for x.



84 Oliver Bültel CUBO
16, 3 (2014)

Received: October 2012. Accepted: March 2013.

References

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On the supersingular loci of quaternionic Siegel space 85

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	Introduction
	Structure Theorems on Dieudonné Modules with Pairing
	Notions and Notations
	Results of Oort and Li
	Classification of symmetric Dieudonné modules
	Moduli of symmetric Dieudonné modules
	Miscellaneous

	The Shimura variety SKp
	Further Notation
	Morita equivalence
	Description of SKpsi