Remarks on Ekedahl-Oort stratifications

0.19 MB·

by  Chao Zhang

Remarks on Ekedahl-Oort stratifications
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Chao Zhang
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Abstract: This short paper is a continuation of the author’s Ph.D thesis, where Ekedahl-Oort
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J strata are defined and studied for Shimura varieties of Hodge type. The main results here are as

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follows.
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1. The Ekedahl-Oort stratification is independent of the choices of symplectic embeddings.

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G 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl-Oort strati-
A fications with respect to morphisms of Shimura varieties.
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[ 0 Introduction

1
v
Let (G,X) be a Shimuradatum of Hodge type, and Sh (G,X) be the Shimura variety attached to
2 K
3 a compact open subgroup K ⊆ G(Af). We assume that K = KpKp, where Kp is hyperspecial, i.e.
6 there is a reductive group G/Z such that G = G and that K = G(Z ). By works of Deligne,
6 p Qp Qp p p
1. ShK(G,X) is defined over a number field E. Let v be a place of E over p, then Kisin proved in
0 [2] that ShK(G,X) has a smooth model SK(G,X) over OE,(v). Moreover, SK(G,X) is uniquely
4 determined by the Shimura datum in the sense that lim S (G,X) satisfies a certern extension
1 ←−Kp K
: property (see [2] 2.3.7 for a precise statement).
v
i Ekedahl-Oortstratifications forgoodreductionsofShimuravarietiesofHodgetypeweredefined
X
and studied in [14] using [2] and [8]. Let κ = O /(v) and S (resp. G ) be the special fiber
E,(v) 0 0
r
a of S (G,X) (resp. G). The Shimura datum determines a cocharacter µ : G → G which
K m,κ 0,κ
is unique up to G (κ)-conjugacy. We constructed in [14] a morphism ζ : S → G -Zipµ, where
0 0 0 κ
G -Zipµ isthestackofG -zipsoftypeµ(see[8]or§1.2ofthispaper). Fibersofζ areEkedahl-Oort
0 κ 0
strata. Note that to construct ζ, we need to fix a symplectic embedding.
There are many basic questions that were not solved in [14]. Here we mention two of them.
1. Whether the Ekedahl-Oort stratification (namely, the morphism ζ) is independent of the
choices of symplectic embeddings.
2. How to study behavior of stratifications under morphisms of Shimura varieties.
The motivation for the first question is the observation that both the reduction of the Shimura
variety and the “list” of Ekedahl-Oort strata are independent of choices of symplectic embeddings.
While the second one is a question that can not be more natural.
The first question is solved by the following theorem.
1
Theorem 0.1. The morphism ζ is uniquely determined by (G,X) and µ, and hence independent
of choices of symplectic embeddings.
The first section is devoted to a proof of the above result.
Thesecondquestionistoogeneralandtooinexplicittostudy,soweraisethefollowingquestion.
Let f : (G,X) → (G′,X′) be a morphism of Shimura data of Hodge type. Let E and E′ be their
reflex fields. Then E ⊇ E′. Let K ⊆ G(A ) and K′ ⊆ G′(A ) be such that K and K′ are
f f p p
hyperspecial. Assume that f(K)⊆ K′, then there is a morphism f : ShK(G,X) → ShK′(G′,X′)E.
Let v′ be a place of E′ over p with residue field κ′ and v be a place of E over v′ with residue
field κ, then there is a morphism SK(G,X) → SK′(G′,X′)OE,(v) extending f. Still write f for the
morphism on special fibersS0,K(G,X) → S0,K′(G′,X′)κ. Let G0 (resp. G′0) bethe reduction of G
(resp. G′),andletµ(resp. µ′)bethecocharacteruniqueuptoconjugacy. Thenthereisamorphism
ζ : S0,K(G,X) → G0-Zipµκ (resp. ζ′ : S0,K′(G′,X′) → G′0-Zipµκ′′) giving the Ekedahl-Oort strata
on S0,K(G,X) (resp. S0,K′(G′,X′)).
The question is, whether there is any compatibility between f, ζ and ζ′. We have the following
result.
Theorem 0.2. Assume that f extends to a morphism of reductive group schemes over Z , then
Qp p
there is a canonical commutative diagram
S0,K(G,X) f //S0,K′(G′,X′)κ
ζ ζ′⊗κ
(cid:15)(cid:15) (cid:15)(cid:15)
G −zipµ α // G′−zipµ′ ⊗κ.
0 κ 0 κ′
The proof to the above result will be given in the second section.
As we have seen, the second question is not yet totally solved. More questions will be raised
and studied in the author’s future research.
1 Independence of symplectic embeddings
Let (G,X) be a Shimura datum of Hodge type with good reduction at a prime p > 2. Let E be
the reflex field and v be a place of E over p. The residue field at v will be denoted by κ. Let
K ⊆ G(Q ) be a hyperspecial subgroup, and Kp ⊆ G(Ap) be a compact open subgroup which is
p p f
small enough. Let K be K ×Kp. Then by [2], the Shimura varieties Sh (G,X) has an integral
p K
canonical model S (G,X) which is smooth over O .
K E,(v)
LetS bethespecialfiberofS (G,X). Bythemainresultsof[14],thereisatheoryofEkedahl-
0 K
Oortstratification on S . Todefinethestratification, weneedto fixasymplectic embedding,while
0
the variety S is independent of symplectic embeddings. A natural question is whether different
0
symplectic embeddings give the same stratification.
Theabove question is not yet precise enough to work with. Let usfirstrecall how Ekedahl-Oort
stratifications are defined and raise precise questions.
2
1.1 Integral canonical models
Let K and G be as above, then by [9] Proposition 3.1.2.1 c) and e), G extends uniquely to a
p Qp
reductive group G/Z such that K =G(Z ). More precisely, for any two extensions e : G → G
b p p b p 1 Qp b1
ande :G → G suchthate (K )= G (Z )ande (K ) = G (Z ),thereisanuniqueisomorphism
2 Qp b2 1 p b1 p 2 p b2 p
f :G → G such that f ◦e = e and that f(G (Z )) =G (Z ).
1 2 1 2 1 p 2 p
b b b b
Let i : (G,X) ֒→ (GSp(V,ψ),X′) be a symplectic embedding. Then by [2] Lemma 2.3.1, there
exists a Z -lattice L ⊆ V , such that i : G ⊆ GL(V ) extends uniquely to a closed em-
p Zp Qp Qp Qp Qp
bedding G ֒→ GL(L ). So there is a Z-lattice L ⊆ V such that G, the Zariski closure of G in
Z
b p
GL(L ), is reductive, as the base change to Z of G is G. Moreover, we can assume L is such
Z p
(p) b
that L∨ ⊇ L. Let d = |L∨/L| and g = 1dim(V), then the integral canonical model S (G,X)
2 K
of Sh (G,X) is constructed as follows. We can choose K′ ⊆ GSp(V,ψ)(A ) small enough such
K f
that K′ ⊇ K and that ShK′(GSp(V,ψ),X) affords a moduli interpretation. There is a finite
morphism f : ShK(G,X) → ShK′(GSp(V,ψ),X)E. Let Ag,d,K′/Z be the moduli scheme of
(p)
abelian schemes over Z -schemes with a polarization of degree d and level K′ structure. Then
(p)
Ag,d,K′/Z ⊗Q = ShK′(GSp(V,ψ),X). The integral canonical model SK(G,X) is the normaliza-
(p)
tion of the Zariski closure of ShK(G,X) in Ag,d,K′/Z ⊗OE,(v). Here the word “normalization”
(p)
make sense. As ShK(G,X) is regular, and on each open affine, OSK(G,X) is obtain by taking
elements in OShK(G,X) that is integral over OAg,d,K′/OE,(v).
Notethatwedidn’tassumethatK′ issuchthatthemorphismf isaclosedembedding. Because
if we take K′′ ⊆K′ small enough such that the induced morphism
g : ShK(G,X) → ShK′′(GSp(V,ψ),X)E
is a closed embedding, then f factors through g. The natural morphism Ag,d,K′′/Z → Ag,d,K′/Z
(p) (p)
is finite, so the normalization gives the same S (G,X). The special fiber of S (G,X) will be
K K
denoted by S .
0
1.2 G-zips
Let G (resp. L ) be the special fiber of G (resp. L ). We remark that G is uniquely determined
0 0 Zp 0
by (G,K ), as it is also the special fiber G which is uniquely determined by (G,K ). But L is not
p p 0
b
unique, there might bemany choices. By [14], the Shimuradatum (G,X) determines a cocharacter
µ : G → G which is unique up to G(W(κ))-conjugacy. The special fiber of µ will still be
m,W(κ) bW(κ) b
denoted by µ.
Setting 1.3. We start with G and µ : G → G . For an F -scheme S, let σ : S → S be
0 m,κ 0,κ p
the absolute Frobenius. For an S-scheme T, we will write T(p) for the pull back of T via σ. In
particular, we will write µ(p) for the pull back via Frobenius of µ. Note that it is a cocharacter of
G .
0,κ
Let P+ (resp. P−) be the unique parabolic subgroup of G0,κ such that its Lie algebra is the
sum of spaces with non-negative weights (resp. non-positive weights) in Lie(G ) under Ad◦µ.
0,κ
Let U+ (resp. U−) be the unipotent radical of P+ (resp. P−), and L be the common Levi
subgroup of P+ and P−. Note that L is also the centralizer of µ.
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Definition 1.4. Let S be a scheme over κ. A G0-zip of type µ over S is a tuple I = (I,I+,I−,ι)
(p)
consisting of a right Gκ-torsor I over S, a right P+-torsor I+ ⊆ I, a right P− -torsor I− ⊆ I, and
an isomorphism of L(p)-torsors ι: I+(p)/U+(p) → I−/U−(p).
A morphism (I,I+,I−,ι) → (I′,I+′ ,I−′ ,ι′) of G0-zips of type µ over S consists of equivariant
morphisms I → I′ and I± → I±′ that are compatible with inclusions and the isomorphisms ι and ι′.
The category of G -zips of type µ over a κ-scheme S will be denoted by G -Zipµ(S). They
0 0 κ
form a fibered category G -Zipµ over the category of κ-schemes if we only consider isomorphisms
0 κ
as morphisms.
Pink, Wedhorn and Ziegler proved the following result.
Theorem 1.5. The fibered category G -Zipµ is a smooth algebraic stack over κ of dimension 0.
0 κ
Proof. This is [8] Corollary 3.12.
1.6 Ekedahl-Oort strata
Now we will explain how to construct Ekedahl-Oort stratification follow [14]. Note that we will
NOTfollow [14]strictly, asitseemsmorenaturaltocompareL∨ withcohomologies, seealso[3]and
[13]. Our theory of Ekedahl-Oort stratification is base on the theory of G -zips of type µ defined
0
and studied by Pink, Wedhorn and Ziegler in [8].
Let A be the pull back to SK(G,X) of the universal abelian scheme on Ag,d,K′/Z , and V be
(p)
H1 (A/S (G,X)). Let L ⊆ V and G be as in 1.1. Then by [2] Proposition 1.3.2, there is a tensor
dR K
⊗ ⊗
s ∈ L defining G ⊆ GL(L ). Corollary 2.3.9 of [2] implies that the tensor s ∈ L induces a
Z(p) Z(p) Z(p)
section s ∈ V⊗. By [14] Lemma 2.3.2 1), the scheme
dR
I = IsomSK(G,X)(cid:0)(L∨Z(p),s)⊗OSK(G,X),(V,sdR)(cid:1)
is a right G-torsor.
The first main result of [14] is as follows.
Setting 1.7. Still write V, s, s and I the reduction mod p of V, s, s and I. Let F : V(p) → V
dR dR
and V : V → V(p) be the Frobenius and Verschiebung on V respectively. Let δ : V → V(p) be the
semi-linear map sending v to v ⊗1. Then we have a semi-linear map F ◦δ : V → V. There is a
descending filtration V ⊇ ker(F ◦δ) ⊇ 0 and an ascending filtration 0 ⊆ im(F) ⊆ V. The morphism
V induces an isomorphism V/im(F) → ker(F) whose inverse will be denoted by V−1. Then F and
V−1 induce isomorphisms
ϕ : (V/ker(F ◦δ))(p) → ker(F)
0
and
ϕ : (ker(F ◦δ))(p) → V/(im(F)).
1
Setting 1.8. Let L be the special fiber of L . The cocharacter
0 Z
(p)
µ : G → G ⊆ GL(L ) ∼= GL(L∨ )
m,κ 0,κ 0,κ 0,κ
4
induces an F-zip structure on L∨ as follows. Let (L∨ )0 (resp. (L∨ )1) be the subspace of L∨ of
0,κ 0,κ 0,κ 0,κ
weight 0 (resp. 1) with respect to µ, and (L∨ ) (resp. (L∨ ) ) be the subspace of L∨ of weight
0,κ 0 0,κ 1 0,κ
0 (resp. 1) with respect to µ(p). Then we have a descending filtration L∨ ⊇ (L∨ )1 ⊇ 0 and
0,κ 0,κ
an ascending filtration 0 ⊆ (L∨ ) ⊆ L∨ . Let ξ : L∨ → (L∨ )(p) be the isomorphism given by
0,κ 0 0,κ 0,κ 0,κ
l⊗k 7→ l⊗1⊗k, ∀ l ∈ L∨, ∀ k ∈ κ. Then ξ induces isomorphisms
0
φ :(L∨ )(p)/((L∨ )1)(p) p→r2 ((L∨ )0)(p) −ξ−→1 (L∨ )
0 0,κ 0,κ 0,κ 0,κ 0
and
φ : ((L∨ )1)(p) −ξ−→1 ((L∨ ) ≃ L∨ /(L∨ ) .
1 0,κ 0,κ 1 0,κ 0,κ 0
Theorem 1.9.
1) Let I ⊆ I be the closed subscheme
+
I+ := IsomS0(cid:0)(L∨0,κ,s,(L∨0,κ)1)⊗OS0, (V,sdR,ker(F ◦δ))(cid:1).
Then I is a P -torsor over S .
+ + 0
2) Let I− ⊆ I be the closed subscheme
I− := IsomS0(cid:0)(L∨0,κ,s,(L∨0,κ)0)⊗OS0, (V,sdR,im(F))(cid:1).
Then I− is a P−(p)-torsor over S0.
(p) (p) (p)
3) Let ι :I+ /U+ → I−/U− be the morphism induced by
(p) (p)
I+ → I−/U−
f 7→ (ϕ ⊕ϕ )◦gr(f)◦(φ−1⊕φ−1),∀ S/S and ∀ f ∈ I(p)(S).
0 1 0 1 0 +
Then ι is an isomorphism of L(p)-torsors.
Hence the tuple (I,I+,I−,ι) is a G0-zip of type µ over S0.
Proof. This is [14] Theorem 2.4.1.
The G -zip of type µ over S constructed above induces a morphism ζ : S → G -Zipµ. As
0 0 0 0 κ
we have seen, to construct ζ, we need to choose a Z -model G of G, a symplectic embedding
(p)
i : (G,X) ֒→ (GSp(V,ψ),X′), a Z -lattice L ⊆ V, and a tensor s ∈ L defining G. So, by
(p) Z(p) Z(p)
independenceof symplectic embeddings, we mean that ζ is independent of the choices of G, i, L
Z
(p)
and s.
1.10 Uniqueness of G
The Z -model G of G we obtained is actually unique.
(p)
Proposition 1.11. Let V and V be two finite dimensional Q-vector spaces. Let i : G → GL(V )
1 2 1 1
and i : G → GL(V ) be two closed embeddings of reductive groups. Assume that there is a Z -
2 2 (p)
lattice L ⊆ V (resp. L ⊆ V ) such that the Zariski closure G (resp. G ) of G in GL(L ) (resp.
1 1 2 2 1 2 1
GL(L )) is reductive. Then G ∼=G .
2 1 2
5
Proof. Let V = V ⊕V be the direct sum of the two representations i and i , let L = L ⊕L .
1 2 1 2 1 2
The we have a sequence of closed embeddings G ×G ⊆ GL(L )×GL(L ) ⊆GL(L). Let G/Z be
1 2 1 2 p
b
the reductive model of G and G be the Zariski closure of G in GL(L). Then G = G = G .
Qp 3 1,Zp b 2,Zp
Flat base-change implies that G is the diagonal subgroup of G ×G = G ×G . So G is
3,Zp b b 1,Zp 2,Zp 3
reductive. Note that the morphism G → G ×G −p→1 G is an isomorphism, so G ∼= G . Similarly,
3 1 2 1 3 1
∼
G = G .
3 2
Let G/Z be a reductive model of G. Then there exists a free Z -module M of finite rank
(p) (p)
such that there is a closed embedding G ֒→ GL(M). The generic fiber of this embedding satisfies
the condition of the above proposition. Sotwo reductive models over Z of Gmust beisomorphic.
(p)
1.12 Comparing G -zips (I)
0
We will first show that the morphism ζ does not depend on the choices of s, once G, i and L
Z
(p)
are fixed. Let us recall our notations and constructions in 1.1.
For the symplectic embedding i : (G,X) ⊆ (GSp(V,ψ),X′) and a the chosen reductive model
G/Z ofG,thereisaZ-latticeL ⊆ V suchthattheZariskiclosureofGinGL(L )isG andthatG
(p) Z(p)
isdefinedbyatensors∈ L⊗ . OnecanchooseLsuchthatL∨ ⊇ L. Letd = |L∨/L|,g = 1dim(V),
Z(p) 2
K = G(Z ) and K = K Kp with Kp ⊆ G(Ap) small enough. Then the integral canonical model
p p p f
S (G,X) of Sh (G,X) is constructed as follows. We can choose K′ ⊆ GSp(V,ψ)(A ) small
K K f
enough such that K′ ⊇ K and that ShK′(GSp(V,ψ),X) affords a moduli interpretation. There
is a finite morphism f : ShK(G,X) → ShK′(GSp(V,ψ),X)E. Let Ag,d,K′/Z be the moduli
(p)
scheme of abelian schemes over Z -schemes with a polarization of degree d and level K′ structure.
(p)
Then Ag,d,K′/Z ⊗Q = ShK′(GSp(V,ψ),X), and the integral canonical model SK(G,X) is the
(p)
normalization of the Zariski closure of ShK(G,X) in Ag,d,K′/Z ⊗OE,(v).
(p)
Let A be the pull back to SK(G,X) of the universal abelian scheme on Ag,d,K′/Z , and V be
(p)
H1 (A/S (G,X)). Then the tensor s ∈ L⊗ induces a section s ∈ V⊗. For a different choice
dR K Z dR
(p)
of s′ ∈ L⊗ , we have another section s′ ∈ V⊗. We have two G-torsors
Z dR
(p)
I = IsomSK(G,X)(cid:0)(L∨Z(p),s)⊗OSK(G,X),(V,sdR)(cid:1)
and I = IsomSK(G,X)(cid:0)(L∨Z(p),s′)⊗OSK(G,X),(V,s′dR)(cid:1).
Lemma 1.13. The two G-torsors I and I′ are canonically isomorphic.
Proof. WewillshowthatI andI′arethesameclosedsubschemeofIsomSK(G,X)(L∨⊗OSK(G,X),V).
Let
I′′ := IsomSK(G,X)(cid:0)(L∨Z(p),s,s′)⊗OSK(G,X),(V,sdR,s′dR)(cid:1).
Then it is a closed subscheme of both I and I′. But I′′ is also a G-torsor over S (G,X), so
K
I = I′′ = I′.
Let S (G,X) (resp. G ) be the special fiber of S (G,X) (resp. G). The construction in 1.6,
0,K 0 K
especially Theorem 1.9, gives a G0-zip of type µ (I,I+,I−,ι) on S0,K(G,X), using L∨Z ,s,V,sdR
(p)
and the F-zip structure on V. Similarly, there is a G -zip (I′,I′ ,I′ ,ι′) attached to L∨ ,s′,V,s′ .
0 + − Z dR
(p)
6
Corollary 1.14. The two G0-zips of type µ (I,I+,I−,ι) and (I′,I+′ ,I−′ ,ι′) on S0,K(G,X) are
canonically isomorphic.
Proof. By Lemma 1.13, the torsors I and I′ are canonically isomorphic. Noting that (I+,I−,ι) and
(I′ ,I′ ,ι′) are constructed usingFrobenius and Verschiebung on V, the two G -zips are canonically
+ − 0
isomorphic.
1.15 Symplectic embeddings
Let i : (G,X) ֒→ (GSp(V ,ψ ),X ) and i : (G,X) ֒→ (GSp(V ,ψ ),X ) be two symplectic
1 1 1 1 2 2 2 2
embeddings. We can construct another symplectic embedding as follows.
By definition of symplectic similitude groups, there is a character χ : GSp(V ,ψ ) → G , such
1 1 1 m
that GSp(V ,ψ ) acts on ψ via χ . Note that changing χ to a power of it will not change the
1 1 1 1 1
symplectic similitude group. Similarly, we have χ : GSp(V ,ψ ) → G . Let w : G → G be the
2 2 2 m m
weight cocharacter of G. Then χ ◦w and χ ◦w are two characters G → G of weights m and
1 2 m m 1
m respectively. After changing χ to χm2 and χ to χm1, we see that G acts on ψ and ψ via the
2 1 1 2 2 1 2
same character.
Let V =V ⊕V and ψ : V ×V → Q be such that
1 2
′ ′ ′ ′ ′ ′
ψ(cid:0)(v1,v2),(v1,v2)(cid:1) = ψ1(v1,v1)+ψ2(v2,v2), ∀ v1,v1 ∈ V1 and ∀ v2,v2 ∈ V2.
Then G ⊆ GSp(V,ψ), and this embedding induces an embedding of Shimura data
′
(G,X) ⊆ (GSp(V,ψ),X ).
1.16 Comparing G -zips (II)
0
Now we will show that the morphism ζ : S (G,X) → G −zipµ is independent of choices of
0,K 0 κ
symplectic embeddings, reductive models and lattices. Note that ζ is independent of K. More
precisely, for K ⊆ K′, there is a commutative diagram
S0,K(G,X) // S0,K′(G,X) ,
PPPPPζPPPPPPP(( (cid:15)(cid:15)ζ′
G −zipµ
0 κ
inducing a G(Ap)-equivariant morphism
f
S (G,X) = limS (G,X) −→ G −zipµ.
0,Kp ←− KpKp 0 κ
Kp
Here the G(Ap)-action on G −zipµ is the trivial action, and that on S (G,X) is the unique one
f 0 κ 0,Kp
induced by the action on Sh (G,X). So, we can shrink Kp if necessary.
Kp
Let i : (G,X) ֒→ (GSp(V ,ψ ),X ) and i : (G,X) ֒→ (GSp(V ,ψ ),X ) be two symplectic
1 1 1 1 2 2 2 2
embeddings. Let G be the reductive model of G over Z such that G(Z )= K .
(p) p p
7
There are lattices L ⊆ V , t = 1,2, such that
t t
1. ψ takes integral value on L .
t t
2. The Zariski closure of G in GL(L ) is G.
t,Z
(p)
Let d = |L∨/L |, g = 1dim(V ), and n ≥ 3 be an integer such that (n,p) = 1. Let A
t t t t 2 t gt,dt,n/Z(p)
be the moduli scheme of abelian schemes over Z -schemes of relative dimension g with a po-
(p) t
larization λ of degree d and a level n structure τ . We write (A ,λ ,τ ) for the universal fam-
t t t t t t
ily on A . Let Kp ⊆ G(Ap) be small enough such that there are natural morphisms
gt,dt,n/Z(p) f
Sh (G,X) → A and Sh (G,X) → A . Then by the construction of the inte-
K g1,d1,n/E K g2,d2,n/E
gral canonical model, there are natural finite morphisms i : S (G,X) → A and
1 K g1,d1,n/OE,(v)
i : S (G,X) → A .
2 K g2,d2,n/OE,(v)
Let V be V ⊕V , L be L ⊕L and ψ : V ×V → Q be such that
1 2 1 2
′ ′ ′ ′ ′ ′
ψ(cid:0)(v1,v2),(v1,v2)(cid:1) = ψ1(v1,v1)+ψ2(v2,v2), ∀ v1,v1 ∈ V1 and ∀ v2,v2 ∈ V2.
Then by 1.15, there is an embedding of Shimura data i : (G,X) → (GSp(V,ψ),Y). Consider the
diagram
A ×A .
uu❦❦❦❦❦❦❦❦❦p❦1❦g1❦,❦d❦1❦,n/Z(p) g2,d2❚,❚n❚/❚Z❚(p❚p2❚)❚❚❚❚❚❚❚❚))
A A
g1,d1,n/Z(p) g2,d2,n/Z(p)
Theabelian schemep∗(A ,λ ,τ )×p∗(A ,λ ,τ )onA ×A isanabelian scheme
1 1 1 1 2 2 2 2 g1,d1,n/Z(p) g2,d2,n/Z(p)
of dimension g +g with a polarization of degree d d and level n structure. There is a unique
1 2 1 2
morphism
i′ :A ×A −→ A
g1,d1,n/Z(p) g2,d2,n/Z(p) g1+g2,d1d2,n/Z(p)
such that
′∗ ∗ ∗
i (A,λ,τ) = p (A ,λ ,τ )×p (A ,λ ,τ ),
1 1 1 1 2 2 2 2
where (A,λ,τ) is the universal family on A .
g1+g2,d1d2,n/Z(p)
By the construction of S (G,X), we have a commutative diagram
K
S (G,X) i1×i2 // A ×A i′ //A
K ❄❄❄❖❄❖❄❄❖❄❖❄❖❄❖❄❖❄i❄❖1i❄2❖❄❖A❄❖❄❄❖g❄❖1❄'',❄d❄1(cid:31)(cid:31) ,n/OyytEt,t(tt✐tv)t✐t✐t✐t✐t✐t✐t✐ptp✐t21✐tt✐gt1✐t,✐dt✐1t✐,tn✐t/tOttEt,(v) g2,d2,n/OE,(v) g1+g2,d1d2,n/OE,(v)
A
g2,d2,n/OE,(v)
such that the generic fiber of i′◦(i ×i ) is induced by i. We will write i for i′◦(i ×i ). The pull
1 2 1 2
back via i of the universal family on A is precisely i∗(A ,λ ,τ )×i∗(A ,λ ,τ ).
g1+g2,d1d2,n/OE,(v) 1 1 1 1 2 2 2 2
LetA ,A ,AbethepullbacktoS (G,X)oftheuniversalabelianschemesonA ,A
1 2 K g1,d1,n g2,d2,n
and A respectively. Then A = A × A . Let V = H1 (A /S (G,X)), t = 1,2, and
g1+g2,d1d2,n 1 2 t dR t K
V = H1 (A/S (G,X)). Then V = V ⊕V .
dR K 1 2
8
⊗
Let L = L ⊕ L and V = V ⊕ V be as before. Let s ∈ L be a tensor defining
1 2 1 2 t t,Z
(p)
G ⊆ GL(L ), for t = 1,2. By our construction, we have a sequence of closed embeddings
t,Z
(p)
(1.17) G ⊆ G ×G ⊆ GL(L )×GL(L ) ⊆ GL(L ).
1,Z 2,Z Z
(p) (p) (p)
Here the first embedding is the diagonal embedding.
⊗
By [2] Proposition 1.3.2, G ⊆ GL(L ) is defined by a tensor s ∈ L . We need some explicit
Z(p) Z(p)
conditionsthatcutoutG×G ⊆ GL(L ). FirstnotethatGL(L )×GL(L )isthesubgroup
Z 1,Z 2,Z
(p) (p) (p)
of GL(L ) respecting the splitting L = L ⊕L . So the group scheme G ×G is such
Z Z 1,Z 2,Z
(p) (p) (p) (p)
that for any Z -algebra R,
(p)
G ×G(R) = {g ∈ GL(L )(R) |g(L ⊗R)= L ⊗R and g(s ⊗1) = s ⊗1 for t = 1,2}.
Z t t t t
(p)
But then G will be the group scheme such that for any Z -algebra R,
(p)
G(R) = {g ∈ GL(L )(R) | g(L ⊗R)= L ⊗R and g(s ⊗1) = s ⊗1 for t = 1,2,g(s⊗1) = s⊗1}.
Z t t t t
(p)
Clearly, if we remove the conditions on L and s , we get the same group scheme.
2 2
Let s ∈ V⊗ (resp. s ∈ V⊗) be the section corresponding to s (resp. s). Let
1,dR 1 dR 1
I = IsomSK(G,X)(cid:0)(L∨Z(p),L∨1,Z(p),s1,s)⊗OSK(G,X), (V,V1,s1,dR,sdR)(cid:1).
where L∨ ⊆ L∨ is given by taking dual of the surjection p : L ։ L .
1,Z(p) Z(p) 1 Z(p) 1,Z(p)
Lemma 1.18. The scheme I is a right G-torsor over S (G,X).
K
Proof. By [2], the direct summand L∨ ⊆ L∨ induces a direct summand L∨ ⊆ V. To prove
1,Z Z 1,dR
(p) (p)
the lemma, it suffices to prove that L∨ = V . But then it suffices to prove that
1,dR 1
∨
L | = V | .
1,dR ShK(G,X) 1 ShK(G,X)
As if we denote by Grass2g1 the S (G,X)-scheme of locally direct summands of V with rank 2g .
V K 1
Then Grass2g1 is proper over S (G,X). The sub-bundles L∨ ⊆ V and V ⊆ V induce an
V K 1,dR 1
S (G,X)-morphism i : S (G,X) → Grass2g1 ×Grass2g1. That L∨ | = V |
K K V V 1,dR ShK(G,X) 1 ShK(G,X)
means that the restriction to Sh (G,X) of i factors through the diagonal. But the diagonal is
K
closed and S (G,X) is reduced, so i factors through the diagonal, which means that L∨ = V .
K 1,dR 1
But L∨ | = V | follows from the construction of these two bundles. We will
1,dR ShK(G,X) 1 ShK(G,X)
follow [2] 2.2 and work with L | and V∨| . They are both closed subschemes of
1,dR ShK(G,X) 1 ShK(G,X)
V∨| , so to prove that they are equal, it suffices to prove that
ShK(G,X)
∨ ∨
L | = L | +V | = V | .
1,dR ShK(G,X) 1,dR ShK(G,X) 1 ShK(G,X) 1 ShK(G,X)
But then one can pass to Sh (G,X) and use descent. Let V∨|^ be the pull back to
K C 1 ShK(G,X)
X × G(A )/K of V∨| . Then by the de Rham isomorphism, V∨|^ equals to the
f 1 ShK(G,X)C 1 ShK(G,X)
vectorbundleattached tothevariationofHodgestructuresgivenbyX andG → GL(V ). Notethat
1
the quotient by G(Q) of these two bundles give L | and V∨| on Sh (G,X)
1,dR ShK(G,X)C 1 ShK(G,X)C K C
respectively, so L | = V∨| .
1,dR ShK(G,X)C 1 ShK(G,X)C
9
We write i ,i ,p ,p ,i for the morphisms of the special fibers. To prove that the Ekedahl-
1 2 1 2
Oort stratifications are independent of choices of symplectic embedding, it suffices to prove that
the stratifications induced by i and i coincide. By Corollary 1.14 and the proof of Lemma 1.18,
1
the special fiber of I is precisely the G -torsor in the G -zip over S (G,X) constructed using
0 0 0,K
i. Let us write I for this special fiber and (I,I+,I−,ι) for the G0-zip constructed using i. Let
(I1,I1,+,I1,−,ι1) be the G0-zip over S0,K(G,X) constructed using i1.
There is a natural morphism ǫ :I → I given by
1
f ∈ I(S) 7→ f|L∨1,κ⊗OS ∈ I1(S), for all S0,K(G,X)-scheme S.
Theorem 1.19. The morphism ǫ induces an isomorphism (I,I+,I−,ι) → (I1,I1,+,I1,−,ι1) of G0-
zips. In particular, i and i give the same Ekedahl-Oort stratification.
1
Proof. Themorphismǫ :I → I is clearly G -equivariant, and hencean isomorphism of G -torsors.
1 0 0
For any S/S (G,X), and any f ∈ I (S) ⊆ I(S), f maps the weight 1 subspace of L∨ ⊗O to
0,K + κ S
ker(F), where F is the Frobenius on V. Let F be the Frobenius on V , then ker(F )= ker(F)∩V ,
1 1 1 1
as V ⊆ V is induced by a morphism of abelian schemes and hence compatible with Frobenius. So
1
ǫ(f)= f|L∨1,κ⊗OS maps the weight 1 subspace of L∨1,κ⊗OS to ker(F)∩V1 = ker(F1), and hence lies
in I (S). But then ǫ| will automatically be an isomorphism of P -torsors. Similarly, ǫ| is an
1,+ I+ + I−
(p)
isomorphism of P -torsors.
−
Now we check the compatibility between ι and ι . We first recall how ι and ι′ are defined
1
in Theorem 1.9 3). Let ϕ , ϕ be as in Setting 1.7, and φ , φ be as in Setting 1.8. Then
0 1 0 1
(p) (p) (p)
ι: I+ /U+ → I−/U− is the morphism induced by
(p) (p)
I+ → I−/U−
f 7→ (ϕ ⊕ϕ )◦gr(f)◦(φ−1⊕φ−1),∀ S/S and ∀ f ∈I(p)(S).
0 1 0 1 0 +
We apply the constructions in Setting 1.7 and Setting 1.8 to V and L∨ respectively, and
1 1,κ
denote the obtained morphisms by ϕ′, ϕ′, φ′ and φ′. Let (L∨ )0 (resp. (L∨ )1) be the subspace
0 1 0 1 1,κ 1,κ
of L∨ of weight 0 (resp. 1) with respect to µ, and (L∨ ) (resp. (L∨ ) ) be the subspace of L∨
1,κ 1,κ 0 1,κ 1 1,κ
of weight 0 (resp. 1) with respect to µ(p). Then φ′ and φ′ are compatible with φ and φ , in the
0 1 0 1
sense that
φ0|(L∨ )(p)/((L∨ )1)(p) =φ′0 : (L∨1,κ)(p)/((L∨1,κ)1)(p) −→ (L∨1,κ)0
1,κ 1,κ
and
φ1|((L∨ )1)(p) = φ′1 : ((L∨1,κ)1)(p) −→ L∨0,κ/(L∨1,κ)0.
1,κ
Let F′ :V(p) → V and V′ :V → V(p) be the Frobenius and Verschiebung on V respectively. Then
1 1 1 1 1
V and F are compatible with V′ and F′. This implies that
ϕ | = ϕ′ : (V /ker(F′◦δ))(p) → ker(F′)
0 (V1/ker(F′◦δ))(p) 0 1
and
ϕ | = ϕ′ : (ker(F′◦δ))(p) → V /(im(F′)).
1 ker(F′◦δ))(p) 1 1
10