Foliations in moduli spaces of abelian varieties

0.39 MB·English

by  Frans Oort

Foliations in moduli spaces of abelian varieties
Frans Oort
7
Version 21-XI-2003
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This paper appeared: J. Amer. Math. Soc. 17 (2004), 267-296.
a
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7
Introduction
1

] In this paper we study abelian varieties and p-divisible groups in characteristic p.
G
Even though a non-trivial deformation of an abelian variety can produce a non-trivial
A
Galois-representation, say on the Tate-ℓ-group of the generic fiber (in any characteristic), the
.
h geometric generic fiber has a “constant” Tate-ℓ-group. The same phenomenon we encounter
t
a for the p-structure in positive characteristic: any two ordinary abelian varieties of the same
m
dimension over an algebraically closed field have isomorphic p-divisible groups.
[ However, for non-ordinary abelian varieties this seems to break down. The fascinating

3 structure which comes out of this is that the maximal locus where a given geometric isomor-
v phism class of a p-divisible group is realized (e.g. in a family of abelian varieties) is a locally
0
closed set. The locus defined by the geometric isomorphism type of a p-divisible group will be
5
0 called a “central leaf”, see (2.1), and (3.4). This gives riseto a “foliation” of the open stratum
7 attached to a Newton polygon ξ; the dimension of any “leaf” in the same Newton polygon
0
stratum solely depends on ξ. In extreme cases
2
0 either the leaf is the whole stratum, as in the ordinary case,
/
h or in the “almost ordinary case” (the p-rank equals g−1),
t or a leaf is zero-dimensional as in the supersingular case;
a
m in intermediate cases a leaf can be a proper subset and still be positive dimensional: we have
: worked out the example of g = 4 in (8.1).
v
i
X
Between two central leaves in the same Newton polygon stratum there is a correspondence by
r
a iterated αp-isogenies. This leads naturally to the study of “Hecke-orbits” by such isogenies.
In the case of moduli spaces of abelian varieties we obtain “isogeny leaves”, see (4.2). The
beautyofthesetwostructuresisthattheyarealmosttransversal,thatbothfeellikeafoliation,
and that they give, up to a finite morphism, a natural product structure on every irreducible
component W of an open Newton polygon stratum,
We prove the following theorems:
(1.3) - a finitelevel of a geometrically fiberwiseconstant family of p-divisiblegroups becomes
constant over an appropriate finite cover of the base;
(2.2), (3.3) - a central leaf is closed in its open Newton polygon stratum;
1
(3.13) - central leaves are smooth over the base field, and the dimension of a central leaf
depends only on the Newton polygon;
(4.2) - for polarized abelian varieties, the union of all irreducible iterated α -Hecke-orbits
p
through one point is a closed subset;
(5.3) - every component of a Newton polygon stratum is up to a finite morphism isomorphic
with the product of any of the isogeny leaves with a finite cover of any of the central
leaves.
Motivation / explanation. In the moduli space of abelian varieties in characteristic p we
consider Hecke orbits related to isogenies of degree prime to p, and Hecke orbits related to
iterated α -isogenies. The first “moves” points in a central leaf: under such isogenies the
p
geometric p-divisible group does not change; the second moves points in an isogeny leaf. We
can expect that these two natural foliations describethese two “transversal” actions: see (6.1)
and (6.2).
EO-strata and central leaves. In [18]we described thestratification of the modulispace of
∼
abelianvarietiesgivenbyp-kernels: (A,λ)and(B,µ)areinthesameEO-stratumif(A,λ)[p] =
(B,µ)[p], the isomorphism over an algebraically closed field. It is natural to consider more
generally the relation given by the isomorphism type of (A,λ)[pi] for some i ∈ Z≥1; for every i
this definesnaturally subsets of themodulispace, butweshouldnotexpect only finitely many
of such subspaces. The central leaves studied in this paper are the ones obtained by choosing
i large enough (depending on g), which is the same, see (1.7), as considering the isomorphism
type of (A,λ)[p∞].
Isogeny correspondences. Such correspondences are finite-to-finite in characteristic zero,
or, more generally if only isogenies of degree prime to the characteristic are considered. In
positive characteristic such correspondences in general blow up and down. However it turns
out that restricted to central leaves all correspondences are finite-to-finite, see (3.16).
Acknowledgments. I thank Thomas Zink for his ideas; suggestions by him have been incorporated in
this paper; especially his definition of completely slope divisible p-divisible groups, his paper [25] and our
cooperations on [22] have been of value for me in preparing this paper. I thank Johan de Jong for discussions
whilepreparingthispaper,andforcarefulreadingofthemanuscript. Someaspects ofthispaperalready have
beenincorporatedbyElenaMantovaninherHarvardPhD-thesis,see[13],afterdiscussionswehadintheFall
of 2000 and in theSpringof 2002. I am grateful to MIT, where part of this paper was written.
Some terminology and notations used in this paper are brought together in Section 7 and
Section 8. In particular a field denoted by k will be supposed algebraically closed, k = k ⊃ Fp.
1 Preliminaries on p-divisible groups and on finite group
schemes
(1.1) Definition. Let S be a scheme, and let X → S be a p-divisible group. We say that
X/S is geometrically fiberwise constant if there exist a field K, a p-divisible group X over
0
2
K, a morphism S → Spec(K), and for every s ∈S an algebraically closed field k ⊃ κ(s) ⊃ K
containing the residue class field of s and an isomorphism X ⊗k ∼= X ⊗k.
0 k s
The analogous terminology will be used for (polarized) abelian schemes.
(1.2) Definition. Let T be a scheme and let G → T be a group scheme. We say that G is
constant over T if there exist a field L, a group scheme G over L, a morphism T → Spec(L)
0
and an isomorphism
G × T ∼= G.
0 Spec(L) T
In this section we show:
(1.3) Theorem. Let S be a scheme which satisfies condition (N), see (7.2); let X → S be a
p-divisible group; let n ∈ Z≥0. Suppose that X → S is geometrically fiberwise constant. Then
there exists a finite surjective morphism T = T → S, such that X[pn]× T is constant over
n S
T.
Remark. The choice of T → S can be made in such a way that this equals a composition of
finite morphisms T → T′ → S′ → S, where S′ → S factors the normalization, S˜ → S′ → S,
and T′ → S′ is purely inseparable and T → T′ is ´etale.
For the proof of (1.3) we need some preliminaries.
We say that the condition ∗(Z/S,n) is satisfied, where Z → S is a p-divisible group, and
n ∈ Z≥0, if there exists a finite surjective morphism T → S such that Z[pn]×S T is constant.
(1.4) Lemma. For any completely slope divisible Y → S, where S is noetherian and
integral, the property ∗(Z/S,N) is satisfied for every N ∈ Z≥0.
Proof. At first we claim that there exist a purely inseparable, finite, surjective morphism
T = T → S such that Y[pN] = ⊕ ((Y /Y )[pN]) , in the notation of (7.6).
1 T i i i−1 T
In fact, for large r ∈ Z, over T′ = S(p−rs) the homomorphism
(cid:18)ψ := pFs−sti(cid:19)r :((Yi/Yi−1)[pN])(Tp′−rs) −→ Yi[pN]T′
splits off this subquotient, because ψ is an isomorphism on Y /Y and nilpotent on Y [pN].
i i−1 i−1
Note that S is integral and noetherian, and Y [pN] → S is finite; hence there exists a factor-
i
ization T′ = S(p−rs) → T → S, where T → S is finite, and for which the claim holds.
1 1
By [22], 1.10 we see that all G = (Y /Y [pN] become constant, for 1 ≤ i ≤ m, under a
i i i−1)
(separable) finite, surjective morphism T → S. Any T → S dominating T → S and T → S
2 1 2
has the desired property. This proves the lemma. (cid:3)
For p-divisible groups X,Y over a field and for N ≥ n ≥ 0 we denote by
Φ :Hom(X,Y)→ Hom(X[n],Y[n]) and ΦN :Hom(X[N],Y[N]) → Hom(X[n],Y[n])
n n
the natural restriction maps.
3
(1.5) Lemma. For p-divisible groups X and Y over k, there exists for every n ∈ Z≥0 an
integer N(X,Y,n) such that for every N ≥ N(X,Y,n) we have
Im(Φ ) = Im(ΦN).
n n
Proof. For every m ∈ Z≥0 consider the functor T 7→ Hom(X[pm]T,Y[pm]T) on k-schemes.
As is easily seen this functoris representable by a groupscheme of finitetype over k, notation:
G = Hom(X[pm],Y[pm]). For m ≥ n the natural restriction map yields a homomorphism,
m
of algebraic groups ρm : G → G . We write G′ = (G ) . Note that the image of
n m n m m red
a homomorphism between algebraic groups is a (closed) subgroup; consider the finite set
Φ (Hom(X,Y)) ⊂ G′ (k) as an algebraic subgroup of G′ . We obtain a descending chain of
n n n
algebraic groups
G′ ⊃ ρn+1(G′ ) ⊃ ··· ⊃ ρn+i(G′ )⊃ ··· ⊃ Φ (Hom(X,Y)).
n n n+1 n n+i n
The sequence of subgroups schemes {ρnn+i(G′n+i)| i∈ Z≥0} stabilizes on the one hand, and it
is equal to Φ (Hom(X,Y)) on the other hand. (cid:3)
n
We are going to show that n 7→ N(X,Y,n) can be chosen uniformly depending only on the
heights of the p-divisible groups in consideration.
(1.6) Proposition. For every h ∈ Z>0 there exists an integer N(h′,h′′,n) such that for
p-divisible groups X and Y over an algebraically closed field k of height h′ =height(X) respec-
tively h′′ =height(Y) and for every N ≥N(h′,h′′,n) the following images are equal:
Im(Φ ) = Im ΦN .
n n
(cid:0) (cid:1)
Proof. We split up the proof in various steps.
(1)Given h there exists an integer d(h) such that for every p-divisible group X of height h and
with Newton polygon N(X) =β there is an isogeny ρ:H(β) → X of degree deg(ρ) = d(h).
This follows from [12], (3.4) and (3.5) on page 44, and from the property that H(β) admits
endomorphisms of degree p. (cid:3)(1)
(2) In order to prove the proposition is suffices to show:
For every h ≥ 0 there exists N(h) such that for every p-divisible group Z of height h and for
every N ≥ N(h) we have:
Im(Φ :End(Z) → End(Z[n])) = Im(ΦN :End(Z(N)) → End(Z[n])).
n n
Indeed, apply this result for Z = X ⊕Y. (cid:3)(2)
(3) Let H and Z be p-divisible groups and let ρ: H → Z be an isogeny of degree deg(ρ) = ps.
Suppose that the property mentioned in (2) holds for H with the function b 7→ N (b); then
H
that property holds for Z with the function n 7→ N (n):= N (n+s)+s.
Z H
Proof of (3). We consider Dieudonn´e modules and induced maps:
D(ρ: H → Z) =(D(H)=: Q ֒→ M := D(Z)).
For n ≥ 0 and N ≥ N (n+s)+s we consider the inclusions:
H
pNM ⊂ pNH(n+s)Q ⊂ pn+sQ ⊂ pn+sM ⊂ pnQ ⊂ Q ⊂ M ⊂ p−sQ.
4
Suppose ϕ ∈ End(Z[N]) restricts to ϕ = ΦN(ϕ ); we claim that in this case ϕ can be
N n n N n
lifted to ϕ ∈ End(Z). Indeed, consider ψ ∈ End(Z[pn+s] defined by ψ = psϕ . We
n+s n+s n
denote the endomorphisms induced on the Dieudonn´e modules by the same symbols; note
that ψ :M/pn+sM → M/pn+sM restricts to ψ′ :Q/pn+sM → Q/pn+sM by:
n+s n+s
Q M psϕ psM Q
֒→ −→ ֒→ .
pn+sM pn+sM pn+sM pn+sM
We see that ϕ restricts to ψ′ ∈ End(H[pNH(n+s)]) = End(Q/pNH(n+s)Q); hence the
N NH(n+s)
restriction to End(H[pn+s]) can be lifted to H. This shows that ψ′ ∈ End(Q/pn+sM) can
n+s
be lifted to Q. From this it follows that ϕ ∈ End(M) can be lifted to End(M). (cid:3)(3)
n
(4) For a given H = H(β) there is a function N such that (2) is satisfied for H and this
H
function. Hence, for all minimal groups of height h there is a function n 7→ N (h,n)
min
satisfying (2) for all minimal groups of height h.
Thefirststatementisaconsequenceof(1.5). Thesecondfollowsbecausethereareonlyfinitely
many Newton polygons of a given height. (cid:3)(4)
Remark (not used in this paper). An easy and explicit consideration shows that for any
Newton Polygon β the function N (n) := n+1 satisfies the property in (4).
H(β)
We have seen in (2) that it suffices to show the proposition for endomorphisms. For a given
h ∈ Z>0 we define N(h,n) = Nmin(h,n +s)+s, with with s := d(h) as in (1). By (1) we
see that for every Z of height h there exists an isogeny H = H(N(Z)) → Z of degree d(h).
By (4) we know that property (2) holds for the function N (m)= N (h,m). By (3) we see
H min
that property (2) holds for Z with the function N (n) = N (h,n+s)+s. (cid:3)(1.6)
Z min
Notethatthedetourviaminimalgroupsisnotnecessary: thepropositionfollows,usingspecial
Dieudonn´e submodules as in [12], with a less explicit bound, from: [12], (3.4) and (3.5) on
page 44, (1.5) and (2) and (3). Note however that the number of completely slope divisible
p-divisible groups, up to isomorphism over k, of given height, in general is not finite; for that
reason we dit not use these p-divisible groups in the previous proof.
The following result will be used several times in this paper. Probably this fact is known to
all experts in the field.
(1.7) Corollary. For any h ∈ Z≥0 there exists N(h) ∈ Z≥0 with the property that for p-
divisible groups X , and X of height h over an algebraically closed field, and any n ≥ N(h),
1 2
then:
X [pn] ∼= X [pn] ⇒ X ∼= X .
1 2 1 2
Apply (1.6) with n = 1. (cid:3)
(1.8) Corollary. Any completely slope divisible Y → S, over a noetherian and integral K-
scheme S with S(K) 6= ∅ is geometrically fiberwise constant.
This follows from (1.4) and (1.7). (cid:3)
5
(1.9) Lemma. Suppose given X → S, a p-divisible group, and i ∈Z≥0, the functor
T 7→ Gr(T) := {G ֒→ X | rk(G/T) = pi}
T
is representable by a proper scheme Gr = Gr → S, and a universal G → Gr.
X/S,i
Proof. There is a locally free sheaf B of algebras on S giving X[pi] → S. Any G ֒→ X as
T
considered is included in X[pi] , and it is given by an ideal sheaf I in B ⊗O . Hence the
T T
functor is a subfunctor of the related Grassmannian of B/S. The properties that B → B/I
with G= Spec(B/I) is in fact a morphismof bialgebras is a closed condition on points in that
Grassmannian. (cid:3)
(1.10) Lemma. Let Z and Z be p-divisible groups over an algebraically closed field k,
1 2
and let i ∈ Z≥0. We write
Ψ = Ψ(Z ,Z ;i) := {G ⊂Z | rk(G) = pi, ∃ψ : Z → Z , Ker(ψ) = G};
1 2 1 1 2
then
#(Ψ) < ∞.
Proof. The module Γ := Hom(Z1,Z2) is free of finite rank over Zp. Hence Γ/piΓ is finite.
Let Φ′ ⊂ Γ and its image Φ′′ ⊂ Γ/piΓ be defined as:
Φ′ := {ψ ∈ Γ | rk(Ker(ψ)) = pi}; Φ′ mod piΓ =:Φ′′ ⊂ Γ/piΓ.
If ψ ∈ Φ′, and ρ ∈ Φ′, and ψ +pi·ρ =: ψ ∈ Φ′, then Ker(ψ ) = Ker(ψ ). The map Φ′ ։ Φ
1 1 2 1 2
defined by ψ 7→ Ker(ψ) induces a surjective map Φ′′ ։ Ψ; hence #(Ψ)< ∞. (cid:3)
(1.11) Proof ofTheorem(1.3). LetS′ → S bethenormalizationmorphism,andconsider
X′ = XS′ → S′; by [22], Theorem (2.1), or [22], Proposition (2.7), we know there exists an
isogeny ϕ : Y → X′ over S′, where Y/S′ is completely slope divisible. Let pi := deg(ϕ). We
are going to show ∗(XS′,n). We choose N ∈ Z≥0 such that N ≥ n+i. As Y → S′ satisfies
∗(Y,N), see (1.4), we can choose a finite T → S′, with T connected, a field L, a morphism
T → Spec(L), a p-divisible group Y over L , and an isomorphism
0
Y0[pN]×T −∼→ Y[pN]×S′ T;
from now on we fix this isomorphism and write it as an equality. We write X = (X ) , and
0 T 0
we have the morphism ϕ : Y → X . Because N ≥ i we see that Ker(ϕ) ⊂ Y[pi] ⊂ Y[pN];
T,0 0 0
hence Gr := Gr = Gr ×T. Consider the finite set Ψ = Ψ(Z = Y ,Z = X ;i) as in
Y,i Y0,i 1 0 2 0
(1.10).
The morphism ϕ : Y → X defines G := Ker(ϕ )⊂ Y[pN] = Y ×T; this is equivalent
T T T T T 0
with a section
←σ−
Gr −→ T.
NotethatY /T iscompletelyslopedivisible;henceforeverygeometricpointtofT thereexists
T
some isomorphism Y ∼= Y ⊗k, see (1.8); also we have some isomorphism X ∼= X ⊗k. Hence
t 0 t 0
for every geometric point t of T we have σ(t) ∈ Ψ = Ψ(Y ,X ;i); note that this statement
0 0
does not depend on the choice of the isomorphisms Y ∼= Y ⊗k and X ∼= X ⊗k: note that
t 0 t 0
Aut(Y ) acts from the right on Ψ, and Aut(X ) acts from the left on Ψ. As T is connected,
0 0
6
and σ(t) ∈ Ψ for every t, this shows σ : T → Gr ×T to be a constant map; hence there
Y0,i
exists G ⊂ Y such that
0 0
Ker((ϕ ) )= G ×T ⊂ Y[pN] = Y ×T.
T 0 0 T 0
As N ≥n+i we see that
ϕ (Y[pN] )⊃ X[pn] , hence Y[pN] ⊃ H := ϕ−1(X[pn] )= Ker(pn·ϕ ).
T T T T T T T
We have
X[pn] =H/G, and G ⊂ H ⊂ Y .
0 0 0
We see that H ⊂ Y[pN] = Y ×T defines a section Σ of Gr ×T, with j = hn+i. Every
T 0 Y0,j
geometric fiber H is the kernel of pn·ϕ : Y ∼= Y → X ∼= X ; hence H defines a section
t t 0 t t 0
Σ : T → Gr = Gr ×T; this section has the property that for every geometric point t we
Y0,j
have Σ(t)∈ Ψ(Y ,X ;j); hence, by the same argument as before,
0 0
H = H ×T ⊂ Y[pN] = Y ×T, with G ⊂ H giving (G ⊂ H)= (G ⊂ H )×T.
0 T 0 0 0
This proves that
X[pn] = H/G = (H /G )×T.
T 0 0
This proves ∗(X,n) and we have concluded the proof of the theorem. 2(1.3)
2 Central leaves
(2.1) Notation. Let K ⊃ Fp be a field, and let X be a p-divisible group over K. Let
S → Spec(K) be a noetherian scheme over K, and let Y → S be a p-divisible group over S.
We write:
C (S) := {s ∈ S |∃k = k ⊃ κ(s), ∃ ∼=: Y ⊗ k ∼= X ⊗ k}.
X s κ(s) K
Note that Y ⊗ k ∼= X⊗ k as in the definition exists iff there is a field extension κ(s) ⊂ L
s κ(s) K
and an isomorphism Y ⊗ L ∼= X ⊗ L.
s κ(s) K
Remark. We will write C (Y → S) if confusion might be possible. The isomorphism in
X
the definition exists iff such an isomorphisms exists over an algebraic closure of κ(s). We
have defined C (S) ⊂ S as a subset. Once (2.2) has been proved, working over a perfect
X
field, we consider C (S) ⊂ S as a closed subscheme of W0 (S) with the induced reduced
X N(X)
scheme structure. Once the theorem has been proved, in such situations the formation C (−)
X
commutes with base change.
We remind the reader of the following notation. Let β be a Newton polygon, see (7.4), and
let Y → S be a p-divisible group with the same d, and h = d+c. We write:
W0(S) = {s |N(Y ) = β},
β s
and
W (S) = {s |N(Y ) ≺ β}.
β s
By Grothendieck-Katz, see [10], 2.3.1 and 2.3.2, we know that W (S) ⊂ S is closed and
β
W0(S) ⊂ S is locally closed; these sets are given the induced, reduced scheme structure. Note
β
that C (S) ⊂ W0 (S).
X N(X)
In this section we will show:
7
(2.2) Theorem. Let K be a field, let X be a p-divisible group over K. Let S → Spec(K)
where S satiefies is an excellent scheme over a field K. Let Y → S be a p-divisible group.
Then
C (S) ⊂ W0 (S)
X N(X)
is a closed subset.
(2.3) Definition. Let S be a scheme. Let G → S and G → S be finite, locally free group
1 2
schemes over S. Define the “isom - functor” I by:
∀T → S, I(T)= Isom(G × T,G × T).
1 S 2 S
(2.4) Lemma. This functor is representable by a scheme I(G ,G ) = I → S of finite type
1 2
over S.
Proof. Let B , respectively B be the O -algebra defining G → S, respectively G → S.
1 2 S 1 2
An isomorphism as considered defines an O -isomorphism B ⊗O → B ⊗O ; the functor
T 2 T 1 T
of such maps is represented by a scheme of finite type over S. The condition that such maps
moreover define an isomorphism of bialgebras (defining an isomorphism of group schemes) is
a closed condition in this scheme. (cid:3)
(2.5) Corollary. With the notation as in (2.2), by (2.4) we see that C (S) ⊂ S is a
X
constructible subset. Hence it contains a dense subset U ⊂ C (S), open in the closure of
X
C (S).
X
Indeed, use (1.7). (cid:3)
(2.6) Proof of (2.2). Let S′ = Zar C (S) ⊂ W0 be the Zariski-closure. Let
X N(X)(S)
(cid:16) (cid:17)
S′′ → S′ be the normalization, and X′′ = X × S′′. Note that S′′ = W0 (S′′): the Newton
S N(X)
polygon of X′′ → S′′ is constant. By [22], Theorem 2.1 there exists ϕ : Y → X′′ such that
Y → S′′ is completely slope divisible. Let deg(ϕ) = pi. Note that Y → S′′ is geometrically
fiberwiseconstant, see (1.8). Consider N(h) as in (1.7), where h is the height of X/S. Choose
N ∈ Z>0 such that N ≥ N(h)+ i. As Y → S′′ satisfies ∗(Y,N), see (1.4), we can choose
a finite surjective T ։ S′′, a field L, a morphism T → Spec(L), a point 0 ∈ T(L), and an
isomorphism
Y0[pN]×T −∼→ Y[pN]×S′′ T;
we fix this isomorphism, and write it as an equality.
The morphism ϕ : Y → X defines G := Ker(ϕ ) ⊂ Y[pN] ⊂ Y × T, and this is
T T T T T 0
equivalent with a section
←σ−
Gr = Gr ×T −→ T.
Y,i Y0,i
Choose U ⊂ CX(S) as in (2.5); as X ×S (U ×S T) and Y ×S′′ (U ×S T) are geometrically
fiberwise constant we conclude that σ is a constant map over U × T; hence it is constant
S
over T, hence there exists G ⊂ Y such that
0 0
Ker((ϕ ) )= G ×T ⊂ Y[pN] = Y ×T.
T 0 0 T 0
As N ≥N(h)+i, defining H := ϕ−1(X[pN(h)]), we see, as in the proof of (1.3), that
T T
for every geometric point t ∈ T(k) we have X [pN(h)]∼= X [pN(h)]⊗k.
t 0
8
By (1.7) this proves that X ∼= X ⊗k for every geometric point t ∈ T(k). Hence X → T
t 0 T
is geometrically fiberwise constant; this proves that C (S) = S′, which finishes the proof of
X
Theorem (2.2). 2
(2.7) Proposition / Notation cu(−). (i) Let ϕ : Y → X be an isogeny of p-divisible
0 0 0
groups over a field k; let R be a complete local noetherian domain with residue class field k
and let Y → T := Spec(R) be a geometrically fiberwise constant p-divisible group. Consider
C := C (Def(X )) and the restriction X → C of the universal family. There exists a corre-
X0 0
spondence T ← Γ → C = C (Def(X)) and an isogeny ϕ :Y → X extending ϕ with T ← Γ
X Γ Γ 0
a finite, surjective morphism.
(ii)For every Newton polygon β there exists an integer cu(β) ∈Z≥0 (c =central leaf, u=un-
polarized)such that for every p-divisible group X with N(X) = β the closed locus C (Def(X))
X
is pure of dimension cu(β).
The first part follows from (1.3); for more details see the proof of (3.14). The second part is
an immediate consequence of the first. (cid:3)
(2.8) Remark. In [1] we prove the following formula. Let β be a Newton polygon of
dimension d and height h, given by the slopes 0 ≤ λ ≤ ··· ≤ λ ≤ 1. Let β∗ be the (upper
1 h
convex) polygon given by λ ,···,λ . We have cu(β) = (β∗(j)−β(j)).
h 1 0<j<h
P
3 Central leaves: the polarized case
(3.1) Let A → S be an abelian scheme. Let λ : A → At be a polarization on A → S; this
isogeny is symmetric, i.e. λ = λt under the identification A −∼→ Att.
Let Y → S be a p-divisible group. Let Yt → S be its Serre-dual. We say that an isogeny
λ : Y → Yt is a quasi-polarization on Y/S if
(λt :Ytt → Yt) = −(Ytt = Y −λ→ Yt).
Note that a polarization (a symmetric isogeny) on an abelian scheme A induces a quasi-
polarization (an anti-symmetric isogeny) on its p-divisible group X = A[p∞], see [15], Propo-
sition 1.12; a polarization on A and the induced quasi-polarization on X = A[p∞] will be
denoted by the same symbol, unless there might be danger of confusion.
(3.2) Notation. Let K ⊃ Fp be a field, and let (X,λ) be a quasi-polarized p-divisible group
over K. Let S → Spec(K) be a scheme over K, and let (Y,µ) → S be a quasi-polarized
p-divisible group over S. We write:
C (S) := {s ∈ S |∃k = k ⊃ κ(s), ∃ ∼=: (Y ,µ )⊗ k ∼= (X,λ)⊗ k}.
(X,λ) s s κ(s) K
In case K is perfect we consider C (S) as a scheme with the induced reduced scheme
(X,λ)
structure.
(3.3) Theorem. Let K be a field, let (X,λ) be a quasi-polarized p-divisible group over K.
Let S → Spec(K) where S is an excelent scheme over K. Let (Y,µ) → S be a quasi-polarized
p-divisible group over S. Then
C (S) ⊂ W0 (S)
(X,λ) N(X)
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is a closed subset, which is a union of connected components of C (S).
X
By (2.2) we know that C (S) ⊂ W0 (S) is closed. Take the union of those irreducible
X N(X)
componentsC ⊂ C (S)forwhichforthegenericpointη ∈ C anisomorphism(Y ,µ )⊗ k ∼=
X η η κ(η)
(X,λ) ⊗ k exists; using Theorem (1.3) and Corollary (1.7) we see that this union equals
K
C (S). (cid:3)
(X,λ)
(3.4) Central leaves. For any point [(A,λ)] = x ∈ Ag,d ⊗ Fp we choose a field K over
which (X := A[p∞],λ) is defined, and we consider:
C(x) := C (A ⊗K).
(X,λ) g,d
SupposeK isperfect; ageometrically irreduciblecomponentofC(x)withtheinducedreduced
scheme structure will be called a central leaf. We will see that for x∈ A ⊗k with n ≥ 3
g,d,n
prime to p there is exactly one irreducible component of C(x) containing x; this component
will be denoted by C ⊂ A ⊗k. See (6.5).
x g,d,n
(3.5) Notation. Over Fp2 we consider two series of quasi-polarized superspecial p-divisible
groups (see [11], 6.1):
Ir: (S = G1,1,τr), where on M = D(S) = Wx⊕W·Fx, with Fx = Vx the pairing τr is given
by:
ǫ ∈ W −pW, ǫσ = −ǫ, < x,Fx >=pr·ǫ, deg(τr)= p2r r ∈ Z≥0;
IIr: (T = ((G1,1)2,νr) , where on M = D(T) = Wx⊕W·Fx⊕Wy ⊕W·Fy the pairing is
given by < x,y >= pr, < Fx,Fy >= pr+1 (and all other pairs of base vectors pair to zero);
here deg(νr) = p4r+2, and r ∈ Z≥0.
In [11], 6.1 it is proved that over an algebraically closed field a superspecial quasi-polarized
p-divisible group is isomorphic to a direct sum of summands each of type I or II.
(3.6) Notation. Choose coprime, non-negative integers m,n ∈ Z≥0, with m > n. For any
r ∈ Z≥0 we consider over Fp a quasi-polarized p-divisible group (U,ζ), sometimes written as
U = U(m,n), ζ = ζ (m,n) by: U(m,n)= H ⊕H , and
r m,n n,m
0 −πr
ζ = : H ⊕H −→ (H ⊕H )t,
(cid:18) πr 0 (cid:19) m,n n,m m,n n,m
where π ∈ End(Hm,n ⊗Fp) is a uniformizer in End(Hm,n ⊗Fp), see [7], 5.3, and π = πm,n :
Hm,n → Hm,n = (Hn,m)t, and π = πn,m ∈ End(Hn,m⊗Fp) is a uniformizer. Note that πt = π
under the identification (H )t =H .
m,n n,m
We assume gcd(m ,n )= 1, and m > n , and n /(m +n )< n /(m +n ) for j > i.
i i i i i i i j j j
(3.7) Proposition (elementary divisors). In this proposition we work over an algebraically
closed field k. Suppose ξ is a symmetric Newton polygon, and let (H,ζ) be a quasi-polarized
p-divisible group with
H ∼= H(ξ) = (⊕ (H )ri) (G )s (⊕ (H )ri),
i mi,ni 1,1 i ni,mi
M M
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