Hilbert-Siegel moduli spaces in positive characteristic

Jeffrey D. Achter
achter@math.columbia.edu

April 19, 2001

Hilbert-Siegel varieties are moduli spaces for abelian varieties equipped with an action by an order OK
in a fixed, totally real field K. As such, they include both the Siegel moduli spaces (use K = Q and the
action is the standard one) and Hilbert-Blumenthal varieties (where the dimension of K is the same as
that of the abelian varieties in question). In this paper we study certain phenomena associated to Hilbert-
Siegel varieties in positive characteristic. Specifically, we show that ordinary points are dense in moduli
spaces of mildly inseparably polarized abelian varieties with action by a given totally real field. Moreover,
we introduce a combinatorial invariant of the first cohomology of an abelian variety which allows us to
compute and explain the singularities of such a moduli space.
The problem considered here arises in two distinct but closely related lines of inquiry. On one hand, recall
that if X is an abelian variety over a field k of characteristic p, then its p-torsion is described by X[p](k) ∼
=
(Z/ pZ)ρ for some ρ. This integer ρ, the p-rank, is between zero and dim X. When ρ is maximal, the abelian
variety is said to be ordinary. Deuring shows that the generic elliptic curve is ordinary [4]. Mumford
announces [12], and Norman and Oort prove [14], the obvious generalization of this statement to higher
dimension: ordinary points are dense in the moduli space of polarized abelian varieties. Wedhorn has
recently obtained similar results [18] for families of principally polarized abelian varieties with given ring
of endomorphisms.
On the other hand, moduli spaces of PEL type – those parametrizing abelian varieties with certain polariza-
tion, endomorphism and level-structure data – are important spaces in their own right. Roughly speaking,
when the characteristic of the ground field is relatively prime to the moduli problem, the resulting space is
smooth. When the characteristic resonates with the moduli functor, things get interesting and the spaces
get singular. The singularities of such spaces have attracted considerable attention. In particular, the pro-
gram established in [17] studies singularities arising from ramification of the endomorphism ring or the
level structure at p. The present work is complementary to these efforts, in that it addresses bad-reduction
behavior coming from inseparable polarizations. (The final paragraph of this paper observes that a weak
form of theorem 3.2.2 proves a special case of a conjecture of [17].)
The first section gives the precise definition of the moduli stacks in question. The second section collects
a number of results on the deformation theory of polarized abelian varieties equipped with an action by
a ring. Since it requires little extra notation and no extra work, we state and prove many of these results
in a slightly broader context than that required for Hilbert-Siegel varieties. We conclude by applying these
techniques to deduce the density of the ordinary locus in certain cases and to compute the structure of local
rings on these spaces.
Much of this paper has already appeared as the author’s doctoral dissertation. I would like to thank Ching-
Li Chai for his support and guidance. I similarly thank Chia-Fu Yu for both general discussions and tech-

1
nical remarks. They, along with Eyal Goren and Frans Oort, read preliminary versions of this work; the
paper has certainly benefited from their comments. Finally, I would like to thank the referee, particularly
for inspiring the discussion in remark 3.2.3.

1 Moduli spaces
The moduli spaces under consideration are defined in the following way. Let B a finite-dimensional Q-
algebra with positive involution ∗, and let OB ⊂ B be an order stable under ∗ and maximal at a rational
prime p.
Let OB be an order in a finite-dimensional Q-algebra B with positive involution ∗. Let ∆ be the product of
all primes which ramify in B. Finally, let g and d be natural numbers.

Definition 1.1 e OB the category of triples (X / S, ι, λ) where

We denote by A g,d

i. X → S → Spec Z[ ∆1 ] is an abelian scheme of relative dimension g.

ι
ii. OB ,→ End(X) is a ring homomorphism taking 1 to id X , so that Lie(X) is a locally free OB ⊗ OS -module.
λ
iii. X → X∨ is a polarization of degree d2 , taking the given involution on OB to the Rosati involution of
End(X) ⊗ Q.

Fix an algebraically closed field Z[ ∆1 ] → k of characteristic p > 0. Denote the reduction of the global moduli
space modulo p by

AgO,Bd def
=A e OB ×
g,d Spec Z[ ∆1 ] Spec k .

We will often use the locution “polarized OB -abelian variety” to denote a point (X /k, ι, λ) ∈ AgO,Bd (k).

Remark 1.2 The demanded compatibilities in (ii) and (iii) are quite reasonable requests of our moduli
space. The freeness constraint in (ii) expresses one instance of Kottwitz’s “determinantal condition” [8].
While modifying this condition still yields a reasonable moduli space, other such conditions may forbid the
existence of ordinary points. Moreover, if d is invertible on S and the center of B is fixed by ∗, then (ii) is
automatic. (If d is not invertible, however, then Lie(X) is not necessarily locally free over OB ⊗ OS ; (ii) is not
a vacuous condition.)
It may be worth making (iii)’s meaning explicit, too. An ample line bundle L on an abelian variety X over
def
a field k induces an isogeny φL : X → X∨ = Pic0 (X), x 7→ L ⊗ Tx∗ L −1 . An isogeny arising in this way is
called a polarization of X /k. If X is an abelian scheme over S, then a polarization of X is a map λ : X → X∨
which is a polarization of abelian varieties at every geometric point of S. The degree of a polarization is
simply its degree as an isogeny, that is, the rank of its kernel. Any polarization induces a Rosati involution
on End(X) ⊗ Q, defined by α† = λ−1 ◦ α∨ ◦ λ. We insist that, for any b ∈ OB , ι(b∗ ) = ι(b)† .
e OB as a fibered category over Sch 1 .
The functor (X / S, ι, λ) 7→ S clearly reveals A g,d Z[ ]
∆

2
Theorem 1.3 e OB is an algebraic stack over Z[ 1 ] in the sense of [3].
The category A g,d ∆

Proof The sketch in Théorème 1.20 of [16], which treats the case where OB is a totally real field of dimen-
sion g, is an exegesis of Artin’s method which works for general B. Alternatively, one can consider the
φ
e OB →
forgetful functor A Ae g,d to the moduli space of abelian schemes of relative dimension g equipped with
g,d
a polarization of degree d2 . The target space is well-known to be an algebraic stack, and it suffices to show
that φ is relatively representable. This last property, in turn, comes down to the algebraicity of condition
(ii) above. Since the freeness of Lie(X /k) is a statement about the mutiplicities of various representations
χ : OB → Aut Lie(X), and since these multiplicities are clearly constructible functions, φ is representable.
(In fact, a class-number argument shows that φ is quasifinite, and a rigidity statement on homomorphisms
shows φ is even finite; but we will not need this here.)♥

2 Deformation theory

For the most part, we will attempt to understand AgO,Bd by studying its local behavior. Indeed, let Art p (k)
be the category of Artin local k-algebras with residue field k and let (X /k, ι, λ) ∈ AgO,Bd (k) be any k-point.
e / R, φ), where X
Then Def(X) is the covariant functor Art p (k) → Set taking R to the set of all pairs ( X e/R
φ
is an abelian scheme and X e ×Spec R Spec k → X is an isomorphism. The closed subfunctors Def(X , λ) and
b OB
Def(X , ι, λ) are defined analogously. Tautologically, O pro-represents Def(X , ι, λ). We describe here
Ag,d ,(X,ι,λ)
three different approaches to the deformation theory of a polarized OB -abelian variety.
As a preliminary step we remark that Def(X /k, ι, λ) depends not on the global arithmetic of B but on its
def
p-adic behavior. To make this precise, let X[p∞ ] = lim→n X[pn ] be the p-divisible (or Barsotti-Tate) group of
X. Denote by bι an action of OB ⊗Z Z p on a p-divisible group.

Lemma 2.1 The natural map Def(X , ι, λ) → Def(X[p∞ ], bι, λ) is an isomorphism of functors.

Proof The Serre-Tate theorem (first announced in section 6 of [9], although the reader might profitably
consult V.2.3 of [11] or 1.2.1 of [7]) implies that Def(X , ι, λ) → Def(X[p∞ ], ι, λ) is an isomorphism. The only
novelty here lies in passing from an OB -action ι to an OB ⊗ Z p -action bι. However, any p-divisible group
has a canonical structure of Z p -module. Thus, one may indifferently place an OB - or OB ⊗ Z p -structure on
X[p∞ ].♥

Remark 2.2 Since B is by hypothesis unramified and maximal at p, OB ⊗ Z p ∼ = ⊕ j Mats j W(F p f j ). Idem-
potents of OB ⊗ Z p will often give a (noncanonical) decomposition (X[p ], λ) = ⊕ j (X[p∞ ] j , λ j ), where each
∞

(X[p∞ ] j , λ j ) is a polarized p-divisible group equipped with an action by a certain ring W(F p j ). In order to
show that (X , ι, λ) admits a deformation to an ordinary abelian variety, it thus suffices to produce an ordi-
ιj
nary deformation of each triple (X[p∞ ] j , W(F p j ) ,→ End(X[p∞ ] j ), λ j ). Thus, if desired, 2.1 would often let
us assume that OB ⊗ Z p ∼
= W(F p f ), i.e., that B has the same local structure as a number field inert at p.

3
2.1 Dieudonné theory

Reducing a question of abelian varieties to one of formal groups would not be much of an improvement,
were it not for Dieudonné theory. There are several different functors which come under the rubric of
Dieudonné theory. They all associate to a formal or p-divisible group a σ -linear algebraic object. We will
make heavy use of the covariant Dieudonné theory, efficiently documented in [19], which has the distinct
advantage of working over an arbitrary base ring.
Let G be a formal p-divisible group over a ring R of characteristic p. Associated to it is its Dieudonné
def b , G) ⊂ G(R[[T]]), the group of p-typical curves on G. This group is a module over
module D∗ (G) = Hom(W
def b opp (see [19] 4.17). When R is a perfect field k, the local Cartier ring
the local Cartier ring Cart p (R) = (End W)
is

W(k)[F][[V]]
Cart p (k) =
(FV − p, VaF − aτ , Fa − aσ F, Vaσ − aV)

where σ and τ are the Frobenius and Verschiebung of W(k). For general R there is always an embedding
W(R) ,→ Cart p (R), and σ - and τ -linear elements F and V, respectively, of Cart p (R).
A Dieudonné module over R is a V-adically separated and complete Cart p (R)-module M such that V : M →
M is injective and M/ VM is a locally free, finite R-module. Over a perfect field k, a Dieudonné module may
then be thought of as a free W(k)-module of rank height(G), equipped with σ - and τ -linear operators F and
V satisfying certain identities; and M/ VM is canonically the tangent space of G. The fundamental theorem
of Dieudonné theory says that there is an equivalence between the category of formal groups over R and
the category of Dieudonné modules over R ([19] 4.23).
For simplicity of exposition, let X /k be an abelian variety with p-rank zero. By the Dieudonné module M
of X we mean D∗ (X[p∞ ]). It is a free, rank 2g = 2 dim X module over W(k). By functoriality, a polarization
λ λ
X → X∨ induces a homomorphism of Dieudonné modules D∗ (X) →∗ D∗ (X∨ ). Moreover D∗ (X∨ ) is, up to
Tate twist, the W(k)-linear dual of D∗ (X). Carefully following through dualities (as in [12] or [13]; see also
section 5.1 of [1]) shows λ induces a W(k)-linear pairing h·, ·i : M × M → W(k) such that hm, m0 i = −hm0 , mi
σ
and h Fm, m0 i = hm, Vm0 i . This is the Dieudonné-theoretic analogue of the Riemann form of a polarized
complex abelian variety.
Roughly speaking, twisting the action of F on M by a nilpotent endomorphism gives a family of deforma-
tions of X. A special case of this idea is made precise in the following lemma.

Construction 2.1.1 Let (X , ι, λ) be a polarized OB -abelian variety, and let M be its Dieudonné module.
To any nilpotent endomorphism ν : M → M corresponds a deformation M f this
f/k[[]]. If h·, ·i extends to M,
e /k[[]], λ). If ν commutes with OB , this construction gives a deformation
construction gives a deformation ( X
e
( X /k[[]], ι, λ).

Proof This is the main theorem of section 1 of [13], although our coordinate-free formulation more closely
follows section 4 of [2]. Let  be the Teichmüller lift of  to W(k[[]]). Set µ = id +ν : M f → M,f where
f e
M = M ⊗Cart p (k) Cart p (k[[]]). Set F = µ ◦ F, a twisted form of the original Frobenius. One can adapt the
Verschiebung as well so that ( M e is a Dieudonné module. Now recall the Serre-Tate theorem (lemma 2.1)
f, F)
to get the full statement.♥

4
Suppose X is equipped with an action by OK , where K is a totally real field inert at p. The action of OK on M
becomes particularly easy to describe. There is a canonical structure of W(k)-module on M, and OK acts on
M via embeddings OK ,→ W(k). For convenience’s sake, identify Hom(OK , W(k)) with Z/ f Z by fixing one
σ0
such map OK ⊗ Z p ∼
= W(F f ) ,→ W(k). Let σ be the Frobenius of W(F f ) and set σi = σ0 ◦ σ i : W(F f ) ,→ W(k)
p p p
for 1 ≤ i ≤ f − 1. Let Mi be the eigenspace where OK acts via σi . There is a decomposition of M as a
W(k)-module,

M = ⊕i∈Z/ f Z Mi .

This is not a direct sum of Dieudonné modules. Indeed, the Frobenius and Verschiebung operators inter-
weave the Mi . For any element mi of Mi ,

F(αmi ) = F(σi (α)mi )

= σi (α)σ Fmi
= σi+1 (α)Fmi .

Thus, FMi ⊆ Mi+1 with the expected arithmetic for indices. Similarly, VMi ⊆ VMi−1 .
Let h·, ·i be the nondegenerate alternating form on M induced by the polarization λ. It turns out that
h·, ·i| Mi is nondegenerate for each i. Recall that, as K is totally real, the Rosati involution is actually trivial on
elements of K. For any α ∈ OK and mi ∈ Mi , m j ∈ M j , on one hand hαmi , m j i = hσi (α)mi , m j i = σi (α)hmi , m j i;
on the other, hαmi , m j i = hmi , αm j i = hmi , σ j (α)m j i = σ j (α)hmi , m j i. If i 6= j, then choosing any α with
σi (α) 6= σ j (α) shows that hmi , m j i = 0.
It is possible – and, for the explicit deformation theory which follows, desirable – to choose bases for M
which clearly expose the behavior of F, V and h·, ·i.

Lemma 2.1.2 Let (M, ι, h·, ·i) be a quasipolarized Dieudonné module equipped with an action by OK . Let
Mi be the eigenspace corresponding to σi as above. There are W(k)-bases {e1i , · · · , e2r
i
} and { f 1i , · · · , f 2ri } for
i
M such that

• Feij ∈ { f ji+1 , p f ji+1 }.

• If W(k)(Feri + j ) is a direct summand then so is W(k)(Feij ).

0
• heij , eij0 i 6= 0 ⇐⇒ i = i 0 , | j − j0 | = r.
i
• heij , eri + j i = pδ j for some δ ij ∈ Z≥0 .

Notationally, let f li = ∑rj=1 aijl eij . Then (aijl ) is an automorphism of h·, ·i| Mi , and in particular is invertible over
W(k).

5
Proof In view of the previous computations, the proof is essentially a careful meditation on the elemen-
tary divisors lemma. It may be worth pointing out that, in the absence of an OK -action, this recovers the
“displayed module” of [13].
Fix i ∈ Z/ f Z. By the freeness hypothesis, (M/ FM)i = ∼ kr . By, say, the elementary divisors lemma
∼ Lie(X∨ )i =
there are bases e1i j and f 1i j so that Fe1i j ∈ { f 1i j , p f 1i j }. Let’s agree to say that F acts unimodularly, or with index
zero, on e1i j if W(k)Fe1i j is a direct summand; and with index one if Fe1i j = p f 1i j . (In general, the index of an
element x ∈ M is the largest n ∈ Z with x ∈ pn M; and if T is a [σ ±1 -]linear operator on M, declare that T acts
with index (ind Tx − ind x) on x.) The idea is simply to diagonalize this basis with respect to the symplectic
form.
i
Order the e1i j and f 1i j so that ord p he11 i
, e1i ,r+1 i is minimal among all ord p he1i j , e1l i. We may further choose
i
these first elements so that ind Fe11 + ind Fe1,r+1 is maximal over all i, j with ord p he1i j , e1i j i minimal. Start
i

orthogonalizing, by setting

 e1i j j = 1, r + 1
e2i j = he1i j ,e1i ,r+1 i i he1i j ,e11
i
i i .
 e1i j + i ,ei
he11
e11 + i , ei
he11
e1,r+1 j 6= 1, r + 1
1 , r +1 i 1,r+1 i

Note that, by the minimality of ord p he11 i

, e1i ,r+1 i, the division is permissible over W(k). Moreover, for j 6=
i
1, r + 1, he2i j , e21 i = he2i j , e2i ,r+1 i = 0. Indeed,

i
he1i j , e11 i
i
he21 , e2i j i = he11
i
, e1i j i + i , ei
i
he11 , e1i ,r+1 i
he11 1,r+1 i
= 0

and the same computation works for he2i j , e2i ,r+1 i.

i
The only issue is whether e21 , · · · , e2i ,2r is still a good basis for describing the action of F. Fix a j =
6 1, r + 1.
If Fe1i j is unimodular, then Fe2i j is clearly such, too. If Fe1i j has index one, however, there is still a small
amount of verification to be done. Ideally, Fe2i j should have index one as well.
i
i
If ord p he1i j , e11 i and ord p he1i j , e1i ,r+1 i are strictly greater than ord p he11 , e1i ,r+1 i, then there’s no problem, as e2i j −
e1i j ∈ pM.

The situation to worry about is the following: Fe1i j = p f 1i j , Fe11

i i
= f 11 i
, ord p he1i j , e1i ,r+1 i = ord p he11 , e1i ,r+1 i. But
then ind Fe1 j + ind Fe1,r+1 > ind Fe11 + ind Fe1,r+1 , contradicting the second assumption on e11 and e1i ,r+1 .
i i i i i

Now iterate this procedure, ultimately constructing eij = eri j , to finish; and the f ji are determined by the eij .♥
Call any such choice of bases a normal form for (M, ι, λ∗ ). Empirically, it is much easier to write down
deformations of Dieudonné modules which enjoy the following property:

There is a normal form such that, for each i ∈ Z/ f Z and 1 ≤ j ≤ r,

(*) Fei = f i+1 and Fei = p f i+1 .
j j r+ j r+ j

6
Suppose such exists. Define certain direct summands of M in terms of the given normal form.

Qi = ⊕rj=1 W(k)eij Q = ⊕i∈Z/ f Z Qi

Pi = ⊕rj=1 W(k)eri + j P = ⊕i∈Z/ f Z Pi

By the definition of normal form, these summands satisfy the following conditions:

i. Mi = Pi ⊕ Qi , hence M = P ⊕ Q.
ii. h P, Pi = h Q, Qi = (0) ⊂ W(k).
iii. P mod pM = VM/ pM.

In fact, such summands characterize the sort of Dieudonné module we’re after:

Lemma 2.1.3 M satisfies (*) if and only if there are Pi , Qi ⊆ M satisfying (i) through (iii).

Proof If M satisfies (*), then the Pi and Qi obviously satisfy (i) through (iii), by the definition of normal
form.
Conversely, suppose we are given such Pi and Qi . Start with arbitrary W(k)-bases for Pi and Qi , and
diagonalize as in the proof of lemma 2.1.2. Since h P, Pi = h Q, Qi = 0, the algorithm will merely produce
new bases for P and Q. By (iii), F acts with index one on P. So it must act with index zero on Q, and (*) is
satisfied.♥

Definition 2.1.4 Call a polarized abelian variety whose quasipolarized Dieudonné module satisfies (*)
nice. With a slight abuse of nomenclature, say the W(k)-summand Mi is nice if there are Pi and Qi as above.
With a somewhat more serious abuse, we will sometimes say X and M are nice if the polarization is clear
from context.
In view of the commutative diagram associated to any such decomposition,

0 - P - M= P⊕Q - Q - 0

? ? ?
0 - VM/ pM - M/ pM - M/ VM - 0

= = =
? ? ?
0 - H 1 (X , OX )∨ - H1dR (X) - Lie(X) - 0

this condition may be reasonably paraphrased as demanding that the Hodge filtration admit an isotropic
lifting.

7
There is a nice rank n(X) = (n0 , · · · , n f −1 ) which measures the defect of (X , ι, λ) from nice. Set

+1
ni = max #{ j|1 ≤ j ≤ r, Feij = f ji+1 , Feri + j = p f ri+ j },

where the maximum is taken over all possible normal forms for M. Note that X is nice if and only if each
ni = r.

Remark 2.1.5 If X is separably polarized, then D∗ (X) is nice. For suppose not. Then there are i and j so
σ
that F acts with index zero on eij and eri + j . On one hand, h Feij , Feri + j i = heij , VFeri + j i = p; on the other hand,
h Feij , Feri + j i = heij+1 , eri++1j i. Thus, p|d, contradicting the hypothesis of separability.♥

Remark 2.1.6 The following remarks, while not logically necessary in the sequel, may help give the reader
some idea of the lay of the land.
i. One might reasonably ask whether it is necessary to consider all possible normal forms for M to determine
its nice rank; it is certainly distasteful. In the special case where all elementary divisors of M are 1 or p, it
suffices to consider a single normal form. Indeed, suppose M is such and that there is some normal form
which is not visibly nice. Then there are i ∈ Z/ f Z and 1 ≤ j ≤ r with heij , eri + j i = 1, F{eij , eri + j } = { f ji , f ri+ j }.
Define Pi and Qi as above, although h Pi , Qi i ) (0). Any apparent improvement to the nice rank must come
from finding x, y ∈ pPi + Qi , not both in Qi , so that heij + y, eri + j + xi = 0, i.e., hx, yi = 1. Given the constraints
on the elementary divisors, this is impossible.
A similar argument shows the same claim when the relative dimension r is two. Unfortunately, it fails for
arbitrary polarized OK -abelian varieties; this may help explain why we only use this notion in studying
mildly inseparable polarizations.
ii. In contrast with the p-rank, the nice rank depends on the integral structure of (X , ι, λ); it is not preserved
by isogenies. Still, this rank induces a reasonable stratification on AgO,Kd . Nice is an open condition; we
sketch a proof. Suppose (X , ι, λ) is nice. This is equivalent to the existence of W(k)-submodules Q, R ⊂
M so that h Q, Qi = h R, Ri = (0); dimk FQ mod pM = rkW(k) Q = g; and dimk VR mod pM = rkW(k) R = g
[simply take R = V −1 P]. Consider any deformation of this polarized Dieudonné module. Working only
with the polarization, there are submodules Q e , R,
e lifting Q and R, so that h Qe, Qei = hR e, Rei = (0). Since
“having full rank under F mod p or V mod p” is an open condition, the generic lifts Q and R e e still have
e
dimk F Q mod pM = dimk V R mod pM = g. Now, the deformation of M also changes the action of Fe and
e
e but if F Q
V; e mod pM has full rank, then so must the generic FeQ e mod pM. This argument works for any
suitable summands, not just ones of full rank. Thus, if N f is equipped with the product partial order, then
the function (X , ι, λ) 7→ n(X) is a lower semicontinuous function on AgO,Kd . In fact, we will see in 3.2 that, in
certain cases, the nice stratification recovers the stratfication by singularity.

Remark 2.1.7 It is not hard to adapt these notions to handle totally imaginary rings of endomorphisms,
too. However, insofar as the smooth case is adequately addressed in [18], and the ramified case seems to
require different techniques, we forgo the immediate temptation to generalize.

8
2.2 Crystalline Cohomology
Crystalline cohomology is another tool well-adapted to the study of p-divisible groups and abelian varieties
in positive characteristic. The classical theory is ably documented in chapters IV and V of [6] and I of [1].
π def
1 def
To an abelian scheme X → S = Spec k we associate its Dieudonné crystal D∗ (X) = Hcris (X) = R1 π∗,crisOX , a
1 ∼ 1
sheaf of crystals on Scris . The Hodge filtration on HdR (X) = Hcris (X)(k) extends to a filtration of the actual
Dieudonné crystal.
It is easy to formulate the crystalline solution to our deformation problem.

Lemma 2.2.1 Let S0 be a divided power extension of S, and let (X , ι, λ) be a polarized OB -abelian variety.
To give a deformation of (X / S, ι, λ) to S0 is to give Fil(S0 ) ⊂ D∗ (X)(S0 ) so that

i. Fil(S0 ) is a locally direct summand of D∗ (X)(S0 ) as a OS0 ⊗ OB -module.

ii. Fil(S0 ) ⊂ D∗ (X)(S0 ) lifts the Hodge filtration.
iii. Fil(S0 ) is isotropic with respect to the induced bilinear form h·, ·iλ .

Proof This is essentially the Grothendieck-Messing theory of admissible filtrations (see, e.g., V.4 of [6]).
The elucidation of the OB -structure is clear in view of 2.1.♥
We will often prefer to work with the linear dual H1cris (X) and its (dual) Hodge filtration, as this exposes the
connection between the crystalline theory and the covariant Dieudonné theory. Indeed, let M = D∗ (X[p∞ ]).
There are canonical isomorphisms M/ pM = H1dR (X) = H1cris (X)(k) and M/ VM ∼ = Lie(X) (see [1]). Dualizing
the Hodge filtration yields

0 - Lie(X∨ )∨ - H1dR (X) - Lie(X) - 0

= = =
? ? ?
0 - Fil(M/ pM) = VM/ pM - M/ pM - M/ VM - 0

Let M∗ = D∗ (X∨ ); up to a Tate twist, it is the W(k)-linear dual of the free W(k)-module M. Clearly, Fil(M∗ / pM∗ )
may be computed in the same way. Alternately, observe that Fil(M∗ / pM∗ ) = Lie(X∨ ∨ )∨ = Hom(M/ VM, k) =
{e∗ ∈ M∗ : (VM, e∗ ) = (0)}.

2.3 Kodaira-Spencer theory

A third approach, which actually works well in any characteristic, is Kodaira-Spencer theory. We refer to
[15] for a careful exposition of the algebraic formulation of this technique, essentially due to Mumford and
•
Grothendieck. Instead, we content ourselves with recalling that Def(X) = \ (Te X∨ ⊗k Te X), and that
∼ Symm
the obstruction to lifting a polarization lives in H 2 (X , OX ) ∼
= H 1 (X , OX ) ∧k H 1 (X , OX ).
A modest variant of these classical results lets us study Def(X , ι, λ). The first-order deformations are now
parametrized by Te X ⊗k⊗OB Te X∨ , and the obstruction to lifting a polarization lives in H 1 (X , OX ) ∧k⊗OB
H 1 (X , OX ).

9
Theorem 2.3.1 Let s = dimk Te X ⊗k⊗OB Te X∨ and c = dimk (Te X∨ ∧k⊗OB Te X∨ ). Then there are power series
a1 , · · · , ac so that

k[[t1 , · · · , ts ]]
Def(X , ι, λ) ∼
= .
(a1 , · · · , ac )

Proof The proof is quite similar to that of theorem 2.3.3 of [15], which proves the analogous result for
Def(X , λ). Clearly, Def(X , ι) is a smooth, pro-representable subfunctor of Def(X). Moreover, either using
general arguments from Kodaira-Spencer theory or (the dual of) the Dieudonné-theoretic description of
∼ Homk⊗O ((Te X)∨ , Te X∨ ) = Te X ⊗k⊗O Te X∨ , and
Def(X) in [13], we see that Def(X , ι)(k[]/(2 )) = B B

•
Def(X , ι) ∼ \ (Te X ⊗k⊗O Te X∨ )
= Symm k B

∼ def
= k[[t1 , · · · , ts ]] = D

It remains to compute the closed subfunctor Def(X , ι, λ) of Def(X , ι), necessarily represented by D /a for
some ideal a. Let m be the maximal ideal of D . By the Artin-Rees lemma, there is an n > 0 so that mn ∩ a =
mn ∩ ma, and thus

- a - D - D - 0
0
ma ma + mn a + mn

k k k

0 - I - R - R0 - 0

is exact. Note that I is an ideal with I 2 = mI = (0). The canonical surjections

D -
- D /a

? ?
?
?
R -
- R0

give an OB -abelian scheme (X / R, ι), a polarized OB -abelian scheme (X 0 / R0 , ι0 , λ0 ), and an isomorphism

∼ (X 0 , ι0 ). We work with the first de Rham homology H dR (X) and H dR (X 0 ) in order to apply
(X , ι) ⊗ R R0 = 1 1
the discussion of endomorphisms in section 2.1. The former is a free filtered R-module Fil(X) = (Te X∨ )∨ ⊂
H1dR (X) equipped with an eigenspace decomposition Fil(X) = ⊕ Fili (X), H1dR (X) = ⊕ H1dR (X)i , and H1dR (X 0 )
is an analogous object over R0 . Of course, ⊕ Fili (X) ⊂ ⊕ H1dR (X)i lifts ⊕ Fili (X 0 ) ⊂ H1dR (X 0 )i . The polarization
λ0 induces a bilinear form on H1dR (X 0 ), for which Fil(X 0 ) is a Lagrangian subspace. The polarization lifts
to X / R if and only if Fil(X) is isotropic under the induced form. Choose a basis {bij } for each eigenspace
H1dR (X)i , and set

10
 e
bijl = hbij , bil i

if the center of B, acting on Fili , is fixed by the involution of B, and e

bijl = hbij , bil i otherwise.

Since hFil(X 0 ), Fil(X 0 )i = (0) ⊂ R0 , e

bijl ∈ I. For each i, j, l let bijl be a lift of e
bijl to a; these are the a1 , · · · , ac
promised in the theorem. Finally, let b ⊂ D be the ideal generated by all bijl .
D
In fact, b = a. To see this, let R00 = b+ma +mn . By the construction of R , hFil(X ), Fil(X )i = (0) ⊂ R .
00 00 00 00
n
Consequently the polarization λ lifts to R and there is a map D /a → R . Thus, a ⊂ b + ma + m , and
0 00 00

b = a.♥
Note that this gives a quick lower bound on the dimension of each component of AgO,Bd . The following simple
observation will be of decisive importance later.

g
Corollary 2.3.2 Let K be a totally real field of dimension f = [K : Q], and let r = f . Then the dimension of
each component of AgO,Kd is at least f · r(r+1)
2 .

Proof Indeed, dimOK ⊗k Te X = dimOK ⊗k Te X∨ = r, so s = dimk (Te X ⊗OK ⊗k Te X∨ ) = f r2 ; and c = dimk (Te X∨ ∧k⊗OK
1) r(r+1)
Te X∨ ) = f r(r−
2 . This gives a lower bound of s − c = f 2 on the dimension of the local ring at any point
OK
of Ag,d , and thus on the dimension of each component.♥

Corollary 2.3.3 Let Z be an irreducible component of AgO,Kd containing an ordinary point. Then every
1)
point of Z has a lift to a ring of characteristic zero; dimk Z = f r(r+
2 ; and Z is (everywhere) a local complete
intersection.

Proof Observe that, as the Serre-Tate theory produces a lift of an ordinary abelian variety along with all
its endomorphisms, any ordinary point of Z lifts. Now, by hypothesis, every point of Z is a specialization
of an ordinary abelian scheme, and thus of a liftable point. Since liftability is a closed property – indeed,
the “liftable locus” of a scheme M /W(k) is precisely the intersection of M × Spec k and the closure of
M × Spec(Frac W(k)) inside M – the first claim follows.
As for the second claim, this is the dimension predicted from characteristic zero; but as every point lifts, the
prediction must come true.
The third claim is now an immediate consequence of the second claim and the description of formal neigh-
borhoods in 2.3.1.♥

3 Density of the ordinary locus

Armed with the deformation theory of section two, it makes sense to approach the objects defined in 1.1.
Subsection 3.1 studies the deformation theory of the nice points defined in 2.1.4, and immediately deduces
the density of the ordinary locus in the smooth case. The final subsection introduces singularities by allow-
ing inseparability in the polarization. In cases of mild inseparability we compute the structure of the local
rings, and again conclude that ordinary points are dense.

11
3.1 The ordinary locus of smooth moduli spaces

The condition introduced in 2.1.4 is a convenient hypothesis for the following result.

Lemma 3.1.1 Let K be a totally real field. Suppose (X , ι, λ) ∈ AgO,Kd (k) is nice but not ordinary. Then (X , ι, λ)
admits an infinitesimal deformation to a polarized OK -abelian variety with strictly bigger p-rank.

Proof We use the covariant Dieudonné theory described in 2.1.1. The Serre-Tate theory assures us we
may work directly with the p-divisible group X[p∞ ] = X[p∞ ]0 ⊕ X[p∞ ]tor ⊕ X[p∞ ]ét , where X[p∞ ]0 is the
local-local part of X[p∞ ] which keeps X from being ordinary. By, say, the classification of p-divisible groups
[10], this decomposition is stable under the OK -action, so we may study X[p∞ ]0 and its Dieudonné module
M. As explained in remark 2.2, we may and do assume that K is actually inert at p.
We will produce a nontrivial deformation of X[p∞ ] to a family of p-divisible groups over k[[]]. The
quasipolarization will be preserved; by the Serre-Tate theory, this gives a polarized formal abelian scheme
e ). By [EGA] III.5.4.5, this algebraizes to an honest abelian scheme.
e /k[[]], eι, λ
(X
Choose a normal basis for M as in lemma 2.1.2. Define a nilpotent endomorphism ν of M by
š
0 1≤ j≤r
ν (eij ) =
eij−r r + 1 ≤ j ≤ 2r

and apply construction 2.1.1. As ν preserves the blocks Mi , and hµ(x), µ(y)i = hx, yi for all x, y ∈ M =
M ⊗ 1 ⊂ M, f Hence, this gives a deformation ( X
f ι and λ extend to M. e ) of (X , ι, λ). It is worth remarking
e , eι, λ
that it is exactly the nice condition which makes it so easy to produce deformations which preserve the
quasipolarization.
In order to show that the p-rank has increased under our deformation, it’s certainly enough to produce
some x ∈ Mf and l ∈ N with Fel x = γ x, γ ∈ W(k(()))× . (This is, of course, the same as showing that Fe is not
f/V
nilpotent on M e M.)
f It is in fact slightly more convenient, and for the purposes of computing the p-rank
harmless, to verify this for a geometric generic point of the formal deformation. So let ek = k(())perf , and base
change to e
k.
Consider, say, e11 ; F acts unimodularly on it, so

e1
Fe = µ( f 12 )
1
r 2r
= ∑ a2j1 e2j + ∑ (e2j1 + e2j−r,1 )
j=1 j=r+1
r 2r
= ∑ (a2j1 + a2r+ j,1 )e2j + ∑ a2j1 e2j .
j=1 j=r+1

Now, there is some 1 ≤ j ≤ r such that a2j1 + a2r+ j,1 ∈ W(e k)× ; otherwise, p|a2j1 for all j, and (a2jk ) would
e 1 ∈ W(e
be singular. So Fe k)× e2j ⊕ W(e
k)e2l l= e 2
1 6 j , and F acts unimodularly on e j . Continuing in this way, and
remembering that there are only finitely many ei , we produce some ei on which Fe does not act nilpotently.♥
j j

12
3.2 Mildly inseparable polarizations

When a low power of p divides d, the moduli spaces tend to be singular but not unmanageably so. We
consider here a class of such spaces.
As always, let K be a totally real field unramified at p of degree f = [K : Q]. Throughout this section assume
d = p f m with m prime to p. Moreover, since we make vital use of crystalline techniques, assume throughout
this section that p > 3. Crystalline cohomology supplies a good description of the local geometry of AgO,Kd . We
start by computing an infinitesimal neighborhood of (X /k, ι, λ) in AgO,Kd .

We have seen in 2.2.1 that to deform X /k to a PD extension R of k is to give an admissible filtration of

e if the filtration is OK -linear, in the sense that
H1cris (X)(R). An action ι extends to X

Fil(X)(R) = ⊕i∈Z/ f Z Fil(X)(R)i ⊂ ⊕i∈Z/ f Z H1cris (X)(R)i = H1cris (X)(R).

Because of this, we may compute deformations to PD rings by examining each eigenspace H1cris (X)i sepa-
rately. This program is carried out below. The reader is invited to perform these computations for herself,
possibly after glancing briefly at the exposition given here.
Some notation is necessary to state the result of this calculation. As usual, let M = D∗ (X) = ⊕ Mi . For
i ∈ Z/ f Z, let Ri = k[[αijl ]]1≤ j,l≤r . For 1 ≤ j < l ≤ r define certain power series f jli in the following way.

If Mi is nice, set
(
αijl − αil j 1≤ j < l ≤r−1
f jli = .
αil j 1 ≤ j ≤ r − 1, l = r

If Mi is not nice, set

 i i i i
 α12 α21 − α11 α22 j = 1, l = 2
i i
f jli =
i
α α − α α − αi i
1≤l<3≤ j≤r .
 ij1 l2 i ij2 l1 i il j
αl j − α j1 αl2 + α j2 αl1 − αijl 3≤l< j≤r

Let mi ⊂ Ri be the maximal ideal, and let I i be the ideal generated by the f jli .

Lemma 3.2.1

p ∼b Ri
ObA OK ,(X,ι,λ) /m(X ,ι,λ) = ⊗i∈Z/ f Z .
g,d (I i , (mi ) p )

Proof As promised, the proof is an involved computation in crystalline cohomology. Recall that there is a
canonical isomorphism (VM/ pM ⊂ M/ pM) ∼ = Fil(X)(k) ⊂ H1cris (X)(k)).

Mi nice Suppose Mi is nice. Using 2.1.2, we may choose a normal form for Mi ; Mi = W(k){x1i , · · · , xri ,
y1i , · · · , yri }, where

13
 Fxij ∈ Mi+1 − pMi+1 
Fyij ∈ pMi+1  1 j=l<r
i i
h x , y i = p j=l=r
hxij , xil i = 0 j l

0 j 6= l
h yij , yil i = 0

The filtration on Mi is given by Fil(Mi / pMi ) = (VM/ pM)i = k{ y1i , · · · , yri }. According to identifications in
∨
section 2.2, Fil(Mi∗ / pMi∗ ) = (M/VM)i = k{x1i∗ , · · · , xri∗ }.
Up to order p − 1, the formal moduli space for the filtered vector space Fil(Mi / pMi ) ⊂ Mi / pMi is Ri =
k[[αijl ]]1≤ j,l≤r /(αijl ) p . The universal filtration, of course, is

r
Fil(M j ∑ jl l
f / pM)i = spanh yi + αi xi i.
l =1

Similarly, the local moduli space for the filtration on the first homology of the dual abelian variety is
k[[β jl ]]/(β jl ) p , and the filtration which lives over it is

r
Fil(M j ∑ l
f ∗ / pM∗ )i = spanhxi∗ + β jl yi∗ i.
l =1

In the present setting these two moduli spaces should be somehow linked; to any algebraic deformation of
X corresponds a deformation of X∨ , so that X∨ truly does remain the dual abelian scheme. The important
condition is that

f / pM)i , Fil(M
hFil(M f ∗ / pM∗ )i i = (0).

This imposes certain relations, e.g.,

0 = h yij + ∑ αijl xil , xij∗0 + ∑ β j0 l 0 yil∗0 i

l0

= β j0 j + αij j0
β j0 j = −αij j0

So make these identifications systematically and write

r
Fil(M j ∑ lj l
f ∗ / pM∗ )i = spanhxi∗ − αi yi∗ i.
l =1

λ λ λ
The polarization X → X∨ induces M →∗ M∗ and H1cris (X) →∗ H1cris (X∨ ). If the deformation is algebraic, it
must be a map of filtered crystals; λ∗ (Fil(X)i ) ⊆ Fil(X∨ )i .

14
 −1 i i∗ f
For 1 ≤ j ≤ r − 1, λ∗ (yij + ∑ αijl xil ) = −xij∗ + ∑rl= i i∗ i
1 α jl yl + p α jr yr . If this is to lie in Fil(X) , then

r−1 r
−xij∗ + ∑ αijl yil∗ + pαijr yri∗ = −xij∗ + ∑ αil j yil∗ .
l =1 l =1

Equate coefficients of yil∗ to find that

αijl = αil j 1≤ j < l ≤r−1

αri j = 0 1≤ j ≤r−1

−1 i i∗
Similarly, λ∗ (yri + ∑ αrli xil ) = ∑rl= i
1 αrl yl mod p, which again forces αrl = 0 for 1 ≤ l ≤ r − 1.

This gives the leading terms of certain local equations for the moduli space at (X /k, ι, λ). We will see shortly
that these represent all the equations.

Mi not nice Not surprisingly, a similar methodology computes local equations for the non-nice eigenspaces.
This time, choose a normal form for Mi = W(k){x1i , · · · , xri , y1i , · · · , yri }, where

Fxij ∈ Mi+1 − pMi+1

Fyij ∈ pMi+1
hx1i , x2i i = 1
h y1i , y2i i = p
hxij , yij i = 1 (3 ≤ j ≤ r)

and all other elements pair to zero. Again, the universal filtrations are given by

r
f
Fil(X) i
= spanh yij + ∑ αijl xil i
l =1
r
f ∨ )i
Fil(X = spanhxij∗ − ∑ αil j yil∗ i
l =1

f
and we must find the conditions ensuring that λ∗ (Fil(X) i f ∨ )i . We find that
) ⊆ Fil(X

r r
λ∗ (y1i + ∑ α1l
i i
xl ) = py2i∗ + α11
i i∗ i
x2 − α12 x1i∗ + ∑ α1l
i i∗
yl
l =1 l =3
r r
i
= α11 (x2i∗ − ∑ αil2 yil∗ ) − α12
i
(x1i∗ − ∑ αil1 yil∗ )
l =1 l =1
r
i
= −α12 x1i∗ + α11
i
x2i∗ + ∑ (α12
i
αil1 − α11
i
αil2 )yil∗
l =1

15
No relation comes from the coefficient of y1i∗ , but

i i i i
α12 α21 − α11 α22 =0 l=2
i i i
α1l = α12 αl1 − α11 i
αil2 3≤l≤r

Already, we see that the defect from nice introduces singularities to the moduli space. Similarly, examining
λ∗ (y2i + ∑rl=1 α2l
i i
xl ) yields the additional relation

i
i
α2l = α22 αil1 − α21
i
αil2 3≤l≤r

When 3 ≤ j ≤ r, the natural constraint is

r r
λ∗ (yij + ∑ αijl xil ) = −xij∗ + αij1 x2i∗ − αij2 x1i∗ + ∑ αijl yil∗
l =1 l =3
r r r
= −(xij∗ + ∑ αil j yil∗ ) + αij1 (x2i∗ − ∑ αil2 yil∗ ) − αij2 (x1i∗ − ∑ αil1 yil∗ )
l =1 l =1 l =1

Thus, we get local equations

αil j = αij1 αil2 − αij2 αil1 1≤l<3≤ j≤r

αijl = αil j − αij1 αil2 + αij2 αil1 3≤l< j≤r

We resume the general discussion. Note that, whether or not Mi is nice, the crystalline cohomology sees
r(r−1)
2 local equations for each i ∈ Z/ f Z. A priori, it is possible that there are other equations of sufficiently
high leading degree that we cannot detect them using crystalline techniques. However, in view of the lower
bound 2.3.2, we know that we have seen the avatars of all equations for the local moduli space.♥

Theorem 3.2.2 Assume p f ||d. The moduli space AgO,Kd is a local complete intersection, and the smooth
locus is the nice locus. Ordinary points are dense in AgO,Kd .

Proof The lemma shows that any nice point is smooth. Conversely, let J be the formal local ring of (AgO,Kd )red
at a non-nice point (X , ι, λ). Lemma 3.2.1 gives the initial forms of elements f ji ∈ k[[αijl ]] presenting J, one
of which has the form

i i i i
α12 α21 − α11 α22 + higher order terms = 0.

It is conceivable that there are additional relations in rad(J); but if they were linear or quadratic, then the
dimension of OA OK ,(X,ι,λ) would drop below the lower bound guaranteed by 2.3.2. Thus, any non-nice point
g,d

is singular in (AgO,Kd )red , and a fortiori in AgO,Kd .

16
The leading terms computed above tell us a little bit more about the (formal) local structure of AgO,Kd . At any
1)
fixed k-point, lemma 3.2.1 provides the initial forms of f r(r− 2 equations. The tangent cone is the spectrum
i i
of a quotient of k[[α jl ]]/(I ), the algebra defined by these initial forms. However, since this ring already has
the minimum allowed dimension, it must actually be the ring of functions of the tangent cone. In particular,
the local ring is a local complete intersection, and it is an integral domain since the tangent cone is one, too.
(Still more particularly, the local ring is reduced, and AgO,Kd is a reduced scheme.)
Finally, given the density of the smooth – and thus the nice – locus, lemma 3.1.1 lets us deduce the density
of the ordinary locus.♥

Remark 3.2.3 As mentioned in the introduction, this result gives evidence for a much more general (albeit
somewhat less precise) conjecture of Rapoport and Zink. For a broad class of Shimura varieties of PEL type,
Rapoport and Zink conjecture [17, p.95] the existence of a flat local model over OE , the ring of integers of the
completion of the reflex field at p. This has been verified in many cases, see especially [5] for a discussion
of the Hilbert-Siegel case.
Rapoport and Zink consider local models for moduli spaces of chains of lattices equipped with a skew-
symmetric pairing; grosso modo, these correspond to Dieudonné modules of p-divisible groups of abelian
varieties, and the elementary divisors of one lattice in the next define a parahoric level structure.
Let (X , ι, λ) be a point in AgO,Kd . There is a self-dual lattice inside D∗ (X[p∞ ]); the relation between it and the
Dieudonné module of X depends on the structure of the polarization. In this way, the local deformation
space of (X , ι, λ) may be modelled on one of the local PEL problems of [17].
Now, Görtz proves the conjecture of Rapoport and Zink for a class of examples which essentially subsumes
the situation of 3.2.2. Still, it may be worth noting that the present, more precise description of the local
e OK is reduced.
rings of AgO,Kd implies the conjecture. As discussed in the proof of 3.2.2, the special fiber of A g,d
Moreover, by 2.3.3, the closure of the generic fiber A e OK is the entire moduli space. Thus, A e OK is flat over
g,d g,d
W(k), as predicted by [17].

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18