Estimating the Newton polygon of a p-divisible group from its p …tetsushi/workshop200911/... · 2009-12-01 · Estimating the Newton polygon of a p-divisible group from its p-kernel

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Estimating the Newton polygon of ap-divisible group from its p-kernel

Shushi Harashita

Department of Mathematics, Graduate School of Science, Kobe University

Workshop on Arithmetic Geometry in Kanazawa

Shushi Harashita (Kobe) Estimate the NP of a p-divisible group 24 November 2009 1 / 33

The moduli space of abelian varieties

Let Ag be the moduli space (over Z) of principallypolarized abelian varieties.

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Fact

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Ag(C) = Sp2g(Z)\H,

where H is the Siegel upper half space

H = Z ∈ Mg(C) | Z = tZ, Im Z > 0.

What can we say about Ag ⊗ Fp?From now on we write Ag for Ag ⊗ Fp.


An expectation on Ag over Fp

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1 There exists a natural decomposition of Ag intofinitely many locally closed subschemes:

Ag =∐ν

Tν

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2 Each Tν can be beautifully described.

Here

a natural decomp. = decomp. by natural invariants

of the p-divisible groups

of abelian varieties


Two invariants

For a principally polarized abelian variety A, we have twomain invariants (combinatorial data):

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. .1 N (A) : Newton polygon

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2 E(A) : Final element of Wg the Weyl group of Sp2g

Newton polygons = p-divisible groups A[p∞]/isog.

Final elements = p-kernels A[p]/isom.

Here A[n] is the kernel of the n-multiplicationn : A → A and

A[p∞] =⋃i

A[pi]


Let A be a principally polarized abelian variety. It wouldbe nice if we could estimate N (A) := isog. cl. of A[p∞]from E(A) := isom. cl. of A[p].

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Result (polarized case)

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There exists a concrete combinatorial algorithmdetermining the Newton polygon ξ(w) satisfying

∀A, E(A) = w =⇒ N (A) ≺ ξ(w),

∃A, E(A) = w and N (A) = ξ(w).

We also have an unpolarized analogue, where ‘abelianvariety’ is replaced by ‘p-divisible group’.


Two stratifications

Ag =∐

ξ

W0ξ : Newton polygon stratification,

Ag =∐w

Sw : Ekedahl-Oort stratification,

where we define

W0ξ := A ∈ Ag | N (A) = ξ,

Sw := A ∈ Ag | E(A) = w.


Problems

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Open problem

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. . 1 When W0ξ ∩ Sw 6= ∅?

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2 Can W0ξ ∩ Sw be beautifully described?

Remark that W0ξ has singularities in general. If g ≤ 3, all

W0ξ ∩ Sw are regular.

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Today’s aim

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Determining the Newton polygon of any generic point ofSw, equivalently when Sw ⊂ Wξ(= W0

ξ ).

We shall also give an unpolarized analogue.


Minimal p-divisible groups

• K : a perfect field.

• W (K) : the ring of Witt vectors with coord. in K.

• AK : the p-adic completion of the ring

W (K)[F ,V ]/(Fa−aσF ,Vaσ−aV ,FV−p,VF−p),

where σ is the Frobenius map W (K) → W (K).

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Definition

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A Dieudonne module (DM) over K is a left AK-modulewhich is finitely generated as W (K)-module.


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Theorem

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. .1 p-div. groups/K D' DM/Kfree as W (K)-mod

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2 fin. p-group sch./K D' DM of fin. length/K

Let m,n be non-negative integers with gcd(m,n) = 1.Define a minimal simple p-div. group Hm,n over Fp by

D(Hm,n) =m+n−1⊕

i=0

Zpei

with Fei = ei+n, Vei = ei+m and ei+m+n = pei.


A Newton polygon (NP) is a formal sum of pairs ofnon-negative coprime integers

ξ =t∑

i=1

(mi, ni).

The i-th slope is defined to be λi = mi/(mi + ni), andthey are supposed to satisfy λ1 ≤ λ2 ≤ · · · ≤ λt−1 ≤ λt.

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1 Symmetric NP : λi + λt+1−i = 1,

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2 NP of dimension g :∑t

i=1 mi = g.

ζ ≺ ξ ⇐⇒ ∀point of ζ is above or on ξ

H(ξ) =t⊕

i=1

Hmi,niminimal p-divisible group


Let k be an algebraically closed field of characteristic p.

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Theorem (Dieudonne-Manin classification)

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For any p-divisible group G over k, there exists a uniqueNewton polygon ξ such that there exists an isogeny

G −−→ H(ξ).

If G = A[p∞] for an abelian variety A, then the Newtonpolygon ξ of G is symmetric. Moreover

A[p∞] | A ∈ Ag(k)/isog.N' sym. Newton polygon


The Newton polygon stratification

Let ξ be a symmetric Newton polygon. The Newtonpolygon stratum of ξ is defined by

W0ξ := A ∈ Ag | N (A) = ξ,

which is a locally closed subset of Ag; we consider it asa locally closed subscheme by giving it the inducedreduced structure.In this way, we obtain the Newton polygon stratification:

Ag =∐

ξ

W0ξ


Final elements of Wg

Let Wg denote the Weyl group of Sp2g:

Wg = w ∈ Aut1, . . . , 2g | w(i)+w(2g+1−i) = 2g+1

Let

J Wg = w ∈ Wg | w−1(1) < · · · < w−1(g).

This is the set of minimal length representatives ofSg\Wg.We call J Wg the set of final elements.


Classification of polarized BT1

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Theorem (Kraft, Oort, Moonen-Wedhorn)

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E : A[p] | A ∈ Ag(k)/isom.∼−−→ J Wg .

To each element w of J Wg, we associate a sequence ψw

defined by

ψw(i) = ]a ∈ 1, . . . , i | w(a) > g. (1)

Let G be a polarized BT1 over k with E(G) = w. Wecan express the Dieudonne module N = D(G) as follows:


N =

2g⊕i=1

kbi

with the operators F and V defined by

F(bi) :=

bψw(i) if w(i) > g,

0 otherwise,

V(bj) :=

±bi if j = g + i − ψw(i) with w(i) ≤ g,

0 otherwise

and the polarization 〈 , 〉 defined by

〈bi, b2g+1−j〉 = ±δij.


The Ekedahl-Oort stratification

For each w ∈ J Wg, the Ekedahl-Oort stratum of w isdefined to be

Sw := A ∈ Ag | E(A) = w,

which is a locally closed subset of Ag; we consider this asa locally closed subscheme by giving it the inducedreduced structure. Thus we have

Ag =∐w

Sw.

This is called the Ekedahl-Oort stratification.


Open problem : When W0ξ ∩ Sw 6= ∅?

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Conjecture (Oort)

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W0ξ ∩ Sw 6= ∅ ⇒ Zξ ⊂ Sw

Here Zξ, the central stream of ξ, is defined to be

Zξ = A ∈ Ag | A[p∞]Ω ' H(ξ)Ω for some Ω = Ω,

which is a closed subset of W0ξ ; we consider it as a

closed subscheme of W0ξ by giving it the induced reduced

structure.


Irreducibility of Ekedahl-Oort strata

Oort conjectured:

(i) Sw is irreducible if Sw 6⊂ Wσ.(ii) Sw is reducible for sufficient large p if Sw ⊂ Wσ.

Oort showed

Sw ⊂ Wσ ⇔ w(i) = i for all i ≤ g − [g/2].

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Theorem (Ekedahl - van der Geer)

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Conjecture (i) is true.

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Theorem (H.)

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Conjecture (ii) is true.


Main theorem (polarized case)

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Definition

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The generic newton polygon of Sw is defined to be

ξ(w) = Newton polygon of a (every) generic point of Sw

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Theorem (H., to appear in Ann. Inst. Fourier)

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For any final element w ∈ J Wg, we have

Zξ(w) ⊂ Sw


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Corollaries

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(1) Oort’s conjecture is true: W0ξ ∩Sw 6= ∅ ⇒ Zξ ⊂ Sw.

(2) ξ(w) = max≺

ξ | Zξ ⊂ Sw.

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Remark

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1 Cor. (2) gives a combinatorial algorithm determining thegeneric Newton polygon ξ(w) of Sw:

∀A, isog. cl. of A[p] = w =⇒ N (A) ≺ ξ(w)

∃A, isog. cl. of A[p] = w & N (A) = ξ(w)

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2 For Hilbert modular varieties over inert primes, Goren andOort have proved an analogous result.


Ingredients

For the main theorem,

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1 we use the result on the first slope of the genericNewton polygon ξ(w),

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2 we find an induction step (based on an idea byOort).

For the corollaries, we use

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1 Grothendieck-Katz: Newton polygon goes down(w.r.t. ≺) or stays under specialization,

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2 the result on the configuration of central streams.


The first slope

Let w ∈ J Wg. We define a map

Ψw : 1, . . . , 2g → 1, . . . , 2g

by Ψw(i) = g + i if w(j) ≤ g for ∀j ≤ i andΨw(i) = ]a ≤ i | w(a) > g otherwise. LetD =

⋂∞k=1 Im(Ψw)k and C = D ∩ g + 1, . . . , 2g. Set

λw := ]C/]D.

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Theorem (H., J. Algebraic Geom. (2007))

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The first slope of ξ(w) equals λw.

An unpol. analogue: see J. Pure and Applied Algebra (2009).


Induction

This step is based on an idea by Oort. In order to getthe main theorem, i.e., Zξ(w) ⊂ Sw, it suffices to show

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Proposition

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Assume w is not minimal. There exists a non-constantfamily of principally polarized p-divisible groups whosep-kernels are geometrically constant of type w andNewton polygons are geometrically constant ξ(w).

The following induction works: Zξ(w) ⊂ Sw is reduced to

Zξ(w′) ⊂ Sw′ for Sw′ ⊂ Sw with ξ(w′) = ξ(w) & w′ 6= w.Outline of the proof: Take a geometric pointx ∈ Sw ∩W0

ξ(w). Let X := Ax[p∞].


Step 1: The first-slope theory shows that there exists aself-dual complex:

0 −−→ Hm,ngt0−−→ X

g0−−→ Hn,m −−→ 0.

Taking these p-kernels:

0 −−→ Hm,n[p]f t0−−→ X[p]

f0−−→ Hn,m[p] −−→ 0.

Step 2: We construct a non-constant family over S:

0 −−→ Hm,n[p]Sf t

−−→ X[p]Sf−−→ Hn,m[p]S −−→ 0.

Step 3: We extend this to a complex over S ′ (finite/S):

0 −−→ (Hm,n)S′gt

−−→ X g−−→ (Hn,m)S′ −−→ 0,

so that we have a non-constant family X → S ′.Shushi Harashita (Kobe) Estimate the NP of a p-divisible group 24 November 2009 24 / 33

Configuration of central streams

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Theorem (H., Asian J. Math. (2009))

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(1) Zζ ⊂ Zξ ⇔ ζ ≺ ξ.

(2) codimZξZζ = 2

∑gi=1 ζ(i) − ξ(i).

The dimension formula of Zξ has been obtained by Oortand by Chai-Oort. Viehmann also get the formula inanother way.Oort’s conjecture follows from (1) and the main theorem:


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Proof of Corollary (1).

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Assume W0ξ ∩ Sw 6= ∅. By Grothendieck-Katz, we have

ξ ≺ ξ(w). By Zζ ⊂ Zξ ⇔ ζ ≺ ξ we have Zξ ⊂ Zξ(w).

Then the main theorem Zξ(w) ⊂ Sw implies

Zξ ⊂ Sw.

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Proof of Corollary (2).

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Put Ξw = ξ | Zξ ⊂ Sw. The main theorem is nothingbut ξ(w) ∈ Ξw. For any ξ ∈ Ξw, by Grothendieck-Katzwe have ξ ≺ ξ(w). These mean

ξ(w) = max≺

ξ | Zξ ⊂ Sw


The case of Sw ⊂ Wσ, i.e., ξ(w) = σ

For a non-negative integer c ≤ g, we put

J W[c]g =

w ∈ J Wg

∣∣ w(i) = i, ∀ i ≤ g − c

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and set J W(c)g = J W[c]

g −J W[c−1]g . We define a map

r : J W(c)g −−→ Sc\Wc /Sc

by sending w to w′ characterized byw′(i) = w(g − c + i) − (g − c) for all 1 ≤ i ≤ c. Put

W′c := Im r.


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Theorem (H., to appear in J. Algebraic Geom.)

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Assume c ≤ [g/2]. For each w′ ∈ W′c, there exists a

finite surjective morphism

G(Q)\X(w′) × G(Af)/K →⋃

r(w)=w′

Sw,

which is bijective on geometric points.

Here X(w′) is the (generalized) Deligne-Lusztig variety:

P ∈ Sp2c / P0 | h P = P0,hFr(P) = w′

P0 for ∃h ∈ Sp2c,

and G is a certain quaternion unitary group over Q.


Conjecture (ii) follows from this theorem and theestimate of the class number by the mass:

]G(Q)\G(Af)/K ≥ 2mg,c.

Here mg,c is computed as

g∏i=1

(2i − 1)!ζ(2i)

(2π)2i·(

g

2c

)p2

·g−2c∏i=1

(pi+(−1)i)c∏

i=1

(p4i−2−1),

where ζ(s) is the Riemann zeta function and(g

r

)q

:=

∏gi=1(q

i − 1)∏ri=1(q

i − 1)∏g−r

i=1 (qi − 1)∈ Z[q].


The unpolarized case

We replace ‘abelian variety of dim. g’ by ‘p-divisiblegroups of dim. d and codim. c’. Put r = c + d. Let Wr

be the Weyl group of GLr. Let JWr be the set of theminimal length representatives of Wd × Wc\Wr.

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Result (unpolarized case)

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For each w ∈ JWr, there exists the Newton polygonξ(w) satisfying

∀X, E(X) = w =⇒ N (X) ≺ ξ(w),

∃X, E(X) = w and N (X) = ξ(w).

There exists an algorithm determining ξ(w).


The existence of the optimal upper bound ξ(w) followsfrom the existence of an irreducible catalogue ofp-divisible groups with a given type:

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Lemma

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Let m ∈ N. Let w be a pm-kernel type. There exists ap-divisible group X over an irreducible scheme S of fin.type over k s.t.

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1 every geometric fiber Xs is of pm-kernel type u;

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2 For any p-divisible group X with pm-kernel type u,there exists a geometric point s ∈ S such thatX ' Xs.


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Theorem (H.)

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Let w ∈ JWGLr. Let µ(ζ) be the p-kernel type of H(ξ).

ξ(w) = ζ | µ(ζ) ⊂ w.

This gives a combinatorial algorithm determining ξ(w).

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Theorem (H.)

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µ(ζ) ⊂ µ(ξ) ⇔ ζ ≺ ξ.

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Corollary (unpolarized Oort conjecture)

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If there exists a p-divisible group X with Newtonpolygon ζ and p-kernel type w, then we have µ(ζ) ⊂ w.

The proof is quite similar to that in the polarized case.Shushi Harashita (Kobe) Estimate the NP of a p-divisible group 24 November 2009 32 / 33

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Definition

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We say w′ ⊂ w if there is an F -zip over an irreduciblescheme S such that its special fiber is of type w′ and itsgeneric fiber is of type w.

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Definition (Moonen-Wedhorn)

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An F -zip over S is a quintuple Z = (N,C,D, ϕ, ϕ)consisting of locally free OS-module N andOS-submodules C,D of N which are locally directsummands of N , and isomorphisms ϕ : (N/C)(p) → Dand ϕ : C(p) → N/D.

If S = Spec(K) with a perfect field K, then

BT1 ’s over K ∼−−→ F -zips over K.Shushi Harashita (Kobe) Estimate the NP of a p-divisible group 24 November 2009 33 / 33

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Estimating the Newton polygon of a p-divisible group from its p …tetsushi/workshop200911/... · 2009-12-01 · Estimating the Newton polygon of a p-divisible group from its p-kernel