On Hyper-Symmetric Abelian Varieties

Ying Zong

A Dissertation

in

Mathematics

Presented to the Faculties of the University of Pennsylvania in PartialFulfillment of the Requirements for the Degree of Doctor of Philosophy

2008

Advisor’s NameSupervisor of Dissertation

Graduate Chair’s NameGraduate Group Chairperson

Acknowledgments

The five years I spent in the graduate study has changed me a lot. Suddenly I feel

that I am no longer asleep in my dearest dream. Burdens and responsibilities drop

on my shoulders. Had no care and help from my wife Lei, I would not know where

to go. I dedicate this thesis to her.

This thesis is finished under the supervision of my advisor Ching-Li Chai. I

admire his pure spirit and I thank heartily for his patient and constant support.

I have been a dear student of all the mathematicians of the Univerisity of Penn-

sylvania, to whom I thank from the bottom of my heart. I thank in particular the

encouragement and support of Ted Chinburg.

I am grateful to Professors Paula Tretkoff and Steven Zucker; they gave me as

much support as they can when I met difficulties.

ii

ABSTRACT

On Hyper-Symmetric Abelian Varieties

Ying Zong

Ching-Li Chai, Advisor

Motivated by Oort’s Hecke-orbit conjecture, Chai introduced hyper-symmetric

points in the study of fine structures of modular varieties in positive characteristics.

We prove a necessary and sufficient condition to determine which Newton polygon

stratum of PEL-type contains at least one such point.

iii

Contents

1 Introduction 1

2 Notations and Generalities 6

2.1 The positive simple algebra Γ . . . . . . . . . . . . . . . . . . . . . 6

2.2 Brauer invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Isocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Dieudonne’s classfication of isocrystals . . . . . . . . . . . . . . . . 8

2.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.7 Isocrystals with extra structure . . . . . . . . . . . . . . . . . . . . 9

2.8 Γ-linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.9 Theory of Honda-Tate . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.10 Γ-linear abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . 12

2.11 A dimension relation . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.12 A variant of the Honda-Tate theory . . . . . . . . . . . . . . . . . . 13

iv

3 A Criterion of Hyper-Symmetry 14

3.1 A lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 A criterion of hyper-symmetry . . . . . . . . . . . . . . . . . . . . . 17

4 Partitions and Partitioned Isocrystals 19

4.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Partitioned isocrystals . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 A simply partitioned isocrystal sΓ . . . . . . . . . . . . . . . . . . . 28

4.4 Partitioned isocrystals with (S)-Restriction . . . . . . . . . . . . . . 29

5 Main Theorem and Examples 30

5.1 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . 30

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Proof of the “only-if” part of (5.1.1) 39

6.1 Semi-simplicity of the Frobenius action . . . . . . . . . . . . . . . . 39

6.2 Proof of the only-if part . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Proof of the “if” part of (5.1.1) 43

7.1 Weil numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.2 Hilbert irreducibility theorem . . . . . . . . . . . . . . . . . . . . . 45

7.3 If F is a CM field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.4 If F is a totally real field . . . . . . . . . . . . . . . . . . . . . . . . 50

v

Chapter 1

Introduction

This work is to extend the study of hyper-symmetric abelian varieties initiated by

Chai-Oort [1]. The notion is motivated by the Hecke-orbit conjecture.

For the reduction of a PEL-type Shimura variety, the conjecture claims that

every orbit under the Hecke correspondences is Zariski dense in the leaf containing

it. In positive characterisitic p, the decomposition of a Shimura variety into leaves

is a refinement of the decomposition into disjoint union of Newton polygon strata.

A leaf is a smooth quasi-affine scheme over Fp. Its completion at a closed point is

a successive fibration whose fibres are torsors under certain Barsotti-Tate groups.

The resulting canonical coordinates, a terminology of Chai, provides the basic tool

for understanding its structure.

Fix an integer g ≥ 1 and a prime number p. Consider the Siegel modular variety

Ag in characteristic p. Denote by C(x) the leaf passing through a closed point x. By

1

applying the local stabilizer principle at a hyper-symmetric point x, Chai [3] first

gave a very simple proof that the p-adic monodromy of C(x) is big. Later, in their

solution of the Hecke-orbit conjecture for Ag, Chai and Oort used the technique of

hyper-symmetric points to deduce the irreducibility of a non-supersingular leaf from

the irreducibility of a non-supersingular Newton polygon stratum, see [2]. Note that

although hyper-symmetric points distribute scarcely, at least one such point exists

in every leaf [1].

Here we are mainly interested in the existence of hyper-symmetric points of

PEL-type. Let us fix a positive simple algebra (Γ, ∗), finite dimensional over Q.

Following Chai-Oort [1], we have the definition:

Definition 1.0.1. A Γ-linear polarized abelian variety (Y, λ) over an algebraically

closed field k of characteristic p is Γ-hyper-symmetric, if the natural map

End0Γ(Y )⊗Q Qp → EndΓ(H1(Y ))

is a bijection.

For simplicity we denote by H1(Y ) the isocrystal H1crys(Y/W (k))⊗Z Q. The goal

of this paper is to answer the following question:

Question. Does every Newton polygon stratum contain a hyper-symmetric point?

The answer to the question in general is no; a Newton polygon stratum must

satisfy certain conditions to contain a Γ-hyper-symmetric point. See (5.2.2) for an

example when Γ is a real quadratic field split at p, and (5.2.6) when Γ is a division

algebra over a CM-field and the Γ-linear isocrystal M only has slopes 0, 1.

2

In the main theorem (5.1.1), we characterize isocrystals of the form H1(Y ) for

Γ-hyper-symmetric abelian varieties Y as the underlying isocrystals of partitioned

isocrystals with supersingular restriction (S).

Consider a typical situation. Let Y = Y ′⊗Fpa Fp be a Γ-simple hyper-symmetric

abelian variety over Fp, where Y ′ is a Γ-simple abelian variety over a finite field

Fpa . By the theory of Honda-Tate, up to isogeny, Y ′ is completely characterized

by its Frobenius endomorphism πY ′ . Let F be the center of Γ. Assume that Fpa

is sufficiently large. We show in (3.3.1) that Y is Γ-hyper-symmetric if and only if

the extension F (πY ′)/F is totally split everywhere above p, that is,

F (πY ′)⊗F Fv ' Fv × · · · × Fv,

for every prime v of F above p. Thus Y is Γ-hyper-symmetric if and only if it is

F -hyper-symmetric.

Denote by TΓ the set of finite prime-to-p places ` of F where Γ is ramified. To

Y , one can associate its isocrystal H1(Y ) as well as a family of partitions P = (P`)

of the integer N = [F (πY ′) : F ] indexed by ` ∈ TΓ. For each ` ∈ TΓ, P` is given by

P`(`′) = [F (πY ′)`′ : F`]

with `′ ranging over the places of F (πY ′) above `. The pair (H1(Y ), P ) is the

partitioned isocrystal attached to Y . In particular, we denote by sΓ the pair attached

to the unique Γ-simple supersingular abelian variety up to isogeny over Fp, see

(4.3.1).

3

To study the pair (H1(Y ), P ), it is more convenient to consider Y as an F -

linear abelian variety equipped with a Γ-action. Write ρ : Γ → EndF (H1(Y ))

for the ring homomorphism defining the Γ-action induced by functoriality on its

isocrystal H1(Y ). In essence, the definition (4.2.1) of partitioned isocrystals is a

purely combinatorial formulation of the conditions that Y is F -hyper-symmetric

and ρ factors through the endomorphism algebra End0F (Y ) of the F -linear abelian

variety Y .

The introduction of supersingular restriction (S) (4.4.1) has its origin in the

following example. Assume that F is a totally real number field. If a Γ-linear

isocrystal M contains a slope 1/2 component at some place v of F above p, but

not all, then there is no Γ-hyper-symmetric abelian variety Y such that H1(Y )

is isomorphic to M . In the proof of the main theorem (5.1.1), we treat specially

supersingular abelian varieties and isocrystals containing slope 1/2 components.

Given any pair y = (M,P ) satisfying the supersingular restriction (S) and con-

taining no sΓ component, the construction of a Γ-hyper-symmetric abelian variety

Y realizing y goes as follows. Let N be the integer such that P = (P`)`∈TΓis

a family of partitions of N . The Hilbert irreducibility theorem [4] enables us to

find a suitable CM extension K/F of degree N , so that the family of partitions

(PK/F, `)`∈TΓgiven by

PK/F, `(`′) := [K`′ : F`], ∀ `′ | `

concide with (P`). Then a simple formula (7.1.1) gives directly a pa-Weil number

4

π for a certain integer a ≥ 1, such that K = F (π) and the slopes of M at a

place v of F above p are equal to λw = ordw(π)/ordw(pa), for w|v. Let Y ′ be the

unique abelian variety up to Γ-isogeny corresponding to π. For some integer e,

(Y ′)e ⊗Fpa Fp equipped with a suitable polarization is a desired Γ-hyper-symmetric

abelian variety.

The organization of this thesis is as follows. In chapter 2 we set up the nota-

tions and review the fundamentals of isocrystals with extra structures, Dieudonne’s

theorem on the classification of isocrystals and the Honda-Tate theory. In chapter

3, we show that every Γ-hyper-symmetric abelian variety is isogenous to an abelian

variety defined over Fp (3.2.1). Then we prove a criterion of hyper-symmetry in

terms of endomorphism algebras (3.3.1). In the next chapter, we define partitions

and partitioned isocrystals. The main theorem (5.1.1) is stated in chapter 5. Sev-

eral examples are provided to illustrate how to determine which data of slopes are

realizable by hyper-symmetric abelian varieties. The proof of (5.1.1) is divided into

two parts. The “only-if” part, in chapter 6, shows that to every Γ-hyper-symmetric

abelian variety Y , one can associate a partitioned isocrystal y. We prove that y

satisfies the supersingular restriction (S). A key ingredient of the proof is that the

characteristic polynomial of the Frobenius endomorphism of H1(YFpa ) has rational

coefficients. In chapter 7 we prove the inverse, the “if” part.

5

Chapter 2

Notations and Generalities

Let p be a prime number fixed once and for all.

2.1 The positive simple algebra Γ

Let Γ be a positive simple algebra, finite dimensional over the field of rational

numbers. We fix a positive involution ∗ on Γ. Let F be the center of Γ; F is either

a totally real number field or a CM field. Let v1, · · · , vt be the places of F above p.

We have

Γ⊗Q Qp = Γv1 × · · · × Γvt .

Let TΓ denote the following set

TΓ = {` ∈ Spec(OF )| ` - p, ` 6= (0), inv`(Γ) 6= 0}.

6

2.2 Brauer invariants

Recall the computation of Brauer invariants. Let K be a finite extension of Qp.

Let A be a central simple K-algebra of dimension d2. By Hasse, A contains a d-

dimensional unramified extension L/K such that for an element u ∈ A, the vectors

1, u, · · · , ud−1 form an L-basis of A, andua = σ(a)u, ∀a ∈ L

ud = α ∈ L

where σ ∈ Gal(L/K) is the Frobenius automorphism of L/K. Then we define the

Brauer invariant invK(A) ∈ Br(K) ' Q/Z as

invK(A) = −ordL(α)/d,

where ordL is the normalized valuation of L, i.e. ordL(π) = 1, for a uniformizer

π ∈ OL.

2.3 Witt vectors

If k is a perfect field of characteristic p, we denote by W (k) the ring of Witt vectors

of k. Let K(k) be the fraction field of W (k). The Frobenius automorphism of k

induces by functoriality an automorphism σ of W (k), namely,

σ(a0, a1, · · · ) = (ap0, a

p1, · · · )

for all a0, a1, · · · ∈ k.

7

2.4 Isocrystals

An isocrystal over k is a finite dimensional K(k)-vector space M equipped with a

σ-linear automorphism Φ. A morphism f : (M,Φ) → (M ′,Φ′) is a K(k)-linear map

f : M →M ′ such that fΦ = Φ′f . Isocrystals over k form an abelian category.

2.5 Dieudonne’s classfication of isocrystals

Let k be an algebraic closure of k, a perfect field of characteristic p. We have the

fundamental theorem of Dieudonne, cf. Kottwitz [8]:

(1) The category of isocrystals over k is semi-simple.

(2) A set of representatives of simple objects Er can be given as follows,

Er = (K(k)[T ]/(T b − pa), T )

where r = a/b is a rational number with (a, b) = 1, b > 0. The endomorphism

ring of Er is a central division algebra over Qp with Brauer invariant −r ∈

Q/Z.

(3) Every isocrystal M over k admits a unique decomposition

M =⊕r∈Q

M(r)

where M(r) is the largest sub-isocrystal of slope r, i.e.

M(r)⊗K(k) K(k) ' Emrr

8

for an integer mr.

The rational numbers occurred in the decomposition M =⊕

r∈QM(r) are called

the slopes of M . If all slopes are non-negative, the isocrystal is effective.

2.6 Polarization

A polarization of weight 1 or simply a polarization of an isocrystal M is a symplectic

form ψ : M ×M → K(k) such that

ψ(Φx,Φy) = pσ(ψ(x, y))

for all x, y ∈M . The slopes of a polarized isocrystal, arranged in increasing order,

are symmetric with respect to 1/2.

2.7 Isocrystals with extra structure

Let Γ be as in (2.1). A Γ-linear isocrystal over k is an isocrystal (M,Φ) over k

together with a ring homomorphism i : Γ → End(M,Φ). The following variant of

Dieudonne’s theorem is proven in Kottwitz [8],

(1) The category of Γ-linear isocrystals over k is semi-simple. It is equivalent to

the direct product of Cv, the Γv-linear isocrystals over k.

(2) For each place v of F above p, the simple objects of Cv are parametrized by

r ∈ Q, whose endomorphism ring is a central division algebra over Fv, with

9

Hasse invariant −[Fv : Qp]r − invv(Γ) in the Brauer group Br(Fv).

If M is a Γ-linear isocrystal, and M = Mv1 ×· · ·×Mvt is the decomposition defined

in (1), we call the slopes of Mv the slopes of M at v and define the multiplicity of

a slope r at v by

multMv(r) = dimK(k)Mv(r)/([Fv : Qp][Γ : F ]1/2)

2.8 Γ-linear polarization

A Γ-linear polarized isocrystal is a quadruple (M,Φ, i, ψ), where (M,Φ) is an

isocrystal, i : Γ → End(M,Φ) is a ring homomorphism, and ψ is a polarization

on M such that

ψ(γx, y) = ψ(x, γ∗y)

for all γ ∈ Γ, x, y ∈ M . If F is a totally real number field, the slopes of M at each

place v of F above p, arranged in increasing order, are symmetric about 1/2. If F

is a CM field, the slopes at v and v collected together, arranged in increasing order,

are symmetric with respect to 1/2.

2.9 Theory of Honda-Tate

Recall that a morphism of abelian varieties f : X → X ′ is an isogeny if it is

surjective with a finite kernel. Let X be an abelian variety over a finite field k = Fpa .

10

The relative Frobenius morphism

FX/k : X → X(p)

is an isogeny. We call πX = F aX/k the Frobenius endomorphism of X. If X is a

simple abelian variety, the Frobenius endomorphism πX is a pa-Weil number, that

is, an algebraic integer π such that for every complex imbedding ι : Q(π) ↪→ C, one

has

| ι(π) |= pa/2.

Here is a basic result, due to Honda-Tate [11]:

(1) The mapX 7→ πX defines a bijection from the isogeny classes of simple abelian

varieties over k to the conjugacy classes of pa-Weil numbers.

(2) The endomorphism algebra End0(Xπ) of a simple abelian variety Xπ corre-

sponding to π is a central division algebra over Q(π). One has

2.dim(Xπ) = [Q(π) : Q][End0(Xπ) : Q(π)]1/2.

(a) If a ∈ 2Z, and π = pa/2, then Xπ is a supersingular elliptic curve, whose

endomorphism algebra is Dp,∞, the quaternion division algebra over Q,

ramified exactly at p and the infinity.

(b) If a ∈ Z − 2Z, and π = pa/2, then Xπ ⊗k k′ is isogenous to the product

of two supersingular elliptic curves, where k′ is the unique quadratic

extension of k.

11

(c) If π is totally imaginary, the division algebraD = End0(Xπ) is unramified

away from p. For a place w of Q(π) above p, the local invariant of D at

w is

invw(D) = −ordw(π)/ordw(pa).

2.10 Γ-linear abelian varieties

A Γ-linear polarized abelian variety is a triple (Y, λ, i) consisting of a polarized

abelian variety (Y, λ) and a ring homomorphim i : Γ → End0(Y ). We require

that i is compatible with the involution ∗ and the Rosati involution on End0(Y )

associated to the polarization λ. The category of Γ-linear polarized abelian varieties

up to isogeny is semi-simple. In particular, any such abelian variety Y admits a

Γ-isotypic decomposition,

Y ∼Γ-isog Ye11 × · · · × Y er

r

where each Yi is Γ-simple and for different i, j, Yi and Yj are not Γ-isogenous.

For each i, there exist a simple abelian variety Xi and an integer ei, such that

Yi ∼isog Xeii . We say Yi is of type Xi.

2.11 A dimension relation

Let Y be a Γ-simple abelian variety of type X, i.e. Y ∼isog Xe, for an integer e. Let

Z0, Z be the center of End0(X) and End0Γ(Y ), respectively. There is the following

12

relation [8],

e.[End0(X) : Z0]1/2[Z0 : Q] = [Γ : F ]1/2[End0

Γ(Y ) : Z]1/2[Z : Q].

One deduces that the Q-dimension of any maximal etale sub-algebra of End0(Y )

is equal to [Γ : F ]1/2 times the Q-dimension of any maximal etale sub-algebra of

End0Γ(Y ).

2.12 A variant of the Honda-Tate theory

Let k = Fpa be a finite field. Kottwitz [8] proved a variant of the theorem of

Honda-Tate:

(1) The map Y 7→ πY is a bijection from the set of isogeny classes of Γ-simple

abelian varieties over k to the F -conjugacy classes of pa-Weil numbers.

(2) The endomorphism algebra End0Γ(Yπ) of a Γ-simple abelian variety Yπ corre-

sponding to π is a central division algebra over F (π). Let Xπ be a simple

abelian variety up to isogeny corresponding to π as in (2.9); Yπ is of type Xπ.

Let D = End0(Xπ), C = End0Γ(Yπ). Then one has the equality

[C] = [D ⊗Q(π) F (π)]− [Γ⊗F F (π)]

in the Brauer group of F (π), and

2.dim(Yπ) = [F (π) : Q][Γ : F ]1/2[C : F (π)]1/2.

13

Chapter 3

A Criterion of Hyper-Symmetry

Let Y be a Γ-linear polarized abelian variety over an algebraically closed field k of

characteristic p, and let Y ∼Γ-isog Ye11 × · · · × Y er

r be the Γ-isotypic decomposition

of Y , cf. (2.10). For the rest, H1(Y ) stands for the first crystalline cohomology of

Y , H1crys(Y/W (k))⊗Z Q.

3.1 A lemma

Lemma 3.1.1. The abelian variety Y is Γ-hyper-symmetric if and only if each Yi

is Γ-hyper-symmetric and for any place v of F above p, for different i, j, Yi and Yj

have no common slopes at v.

Proof. This is clear.

14

3.2 Rigidity

Proposition 3.2.1. If Y is Γ-hyper-symmetric, there exists a Γ-hyper-symmetric

abelian variety Y ′ over Fp such that Y ′ ⊗Fpk is Γ-isogenous to Y .

We first prove a weaker result.

Corollary 3.2.2. There is a Γ-hyper-symmetric abelian variety Y ′ over Fp such

that the isocrystal H1(Y ′ ⊗Fpk) is isomorphic to H1(Y ).

Proof. There is a Γ-linear polarized abelian variety YK over a finitely generated

subfield K such that YK ⊗K k is isomorphic to Y and End(YK) = End(Y ).

Choose a scheme S, irreducible, smooth, of finite type over the prime field, so

that, if η denotes the generic point of S, k(η) = K. We may and do assume that

YK extends to an abelian scheme Y over S.

By a theorem of Grothendieck-Katz [6], the function assigning any point x of

S the Newton polygon of the isocrystal H1(Yx) is constructible. Let S ′ be the

open subset consisting of points x with the generic Newton polygon, i.e. the same

Newton polygon with that of H1(Y ). As S ′ is regular, the canonical homomorphism

End(YS′) → End(YK) is an isomorphism. So there is a well defined specialization

map sp : End(YK) → End(Yt) for any point t ∈ S ′. By the rigidity lemma 6.1

[9], sp is injective. Let t be a closed point of S ′ and Yt = Yt ⊗k(t) k(t). As Y is

Γ-hyper-symmetric, End0Γ(YK)⊗Q Qp and EndΓ(H1(Yt)) have the same dimension.

15

Thus the composite map

End0Γ(YK)⊗Q Qp ↪→ End0

Γ(Yt)⊗Q Qp ↪→ EndΓ(H1(Yt))

is bijective. It follows that Yt is a desired Γ-hyper-symmetric abelian variety over

k(t) ' Fp.

Proof. of (3.2.1). Recall that by Grothendieck [10], an abelian variety Y over an

algebraically closed field k of characteristic p is isogenous to an abelian variety

defined over Fp if and only if Y has sufficiently many complex multiplication, i.e.

any maximal etale sub-algebra of End0(Y ) has dimension 2.dim(Y ) over Q.

We only need to show that Y has sufficiently many complex multiplication.

Without loss of generality we assume that Y is Γ-simple of type X, namely, X is

simple and Y ∼isog Xe for an integer e. Let Z0, Z denote respectively the center

of End0(X) and End0Γ(Y ). The dimension r of any maximal etale sub-algebra of

End0(Y ) is

e.[End0(X) : Z0]1/2[Z0 : Q],

thus by (2.11), is equal to

[Γ : F ]1/2[End0Γ(Y ) : Z]1/2[Z : Q] = [Γ : F ]1/2[EndΓ(H1(Y )) : E]1/2[E : Qp],

since Y is Γ-hyper-symmetric. In the above, E denotes the center of EndΓ(H1(Y )).

Let Y ′ be an abelian variety over Fp as in Corollary (3.2.2). Similarly, the

dimension r′ of any maximal etale sub-algebra of End0(Y ′) is equal to

[Γ : F ]1/2[EndΓ(H1(Y ′)) : E ′]1/2[E ′ : Qp],

16

where E ′ is the center of EndΓ(H1(Y ′)).

By the choice of Y ′, r and r′ are equal. As any abelian variety over Fp has

sufficiently many complex multiplication (2.9), we have r = r′ = 2.dim(Y ′). This

finishes the proof.

3.3 A criterion of hyper-symmetry

In the following we prove a criterion of Γ-hyper-symmetry in terms of the center Z

of End0Γ(Y ).

Proposition 3.3.1. A Γ-linear polarized abelian variety Y over Fp is Γ-hyper-

symmetric if and only if the Fv-algebra Z ⊗F Fv is completely decomposed, i.e.,

Z ⊗F Fv ' Fv × · · · × Fv, for every place v of F above p.

Proof. Let Y ′ be a Γ-linear polarized abelian variety over a finite field Fpa , such

that Y ′ ⊗Fpa Fp ' Y and End(Y ′) = End(Y ). The center Z can be identified with

F (π), the sub-algebra generated by the Frobenius endomorphism of Y ′. By Tate

[11], over Fpa , the map

End0(Y ′)⊗Q Qp → End(H1(Y ′))

is bijective.

Hence, the condition for Y to be Γ-hyper-symmetric is equivalent to

EndΓ(H1(Y ′)) = EndΓ(H1(Y )).

17

Let M ′ := H1(Y ′), and M ′ =⊕

v|pM′v be the decomposition defined in (2.7).

The isocrystal M ′v is Γv-linear and has a decomposition into isotypic components,

M ′v =

⊕r∈Q

M ′v(r).

With these decompositions, the condition for Y to be Γ-hyper-symmetric is equiv-

alent to

EndΓv(M′v(r)) = EndΓv(M

′v(r)⊗K(Fpa ) K(Fp)),

for any v|p, and r ∈ Q.

On the left hand side, the center of EndΓv(M′v(r)) is Fv(πv,r), where πv,r stands

for the endomorphism π|M ′v(r). On the right hand side, the center is isomorphic to

a direct product Fv × · · · × Fv with the number of factors equal to the number of

Γv-simple components of M ′v(r)⊗K(Fpa ) K(Fp).

Therefore, if Y is Γ-hyper-symmetric, the F -algebra Z = F (π) is completely

decomposed at every place v of F above p. Conversely, if Z/F is completely

decomposed everywhere above p, any Γ-linear endomorphism f of the isocrystal

(H1(Y ),Φ) commutes with the operator π−1Φa, and thus stabilizes the invariant

sub-space of π−1Φa, i.e. H1(Y ′). Hence f ∈ EndΓ(H1(Y ′)). This implies that Y is

Γ-hyper-symmetric.

18

Chapter 4

Partitions and Partitioned

Isocrystals

4.1 Partitions

Definition 4.1.1. Let N be a positive integer. A partition of N with support in a

finite set I is a function P : I → Z>0, such that∑

i∈I P (i) = N .

Definition 4.1.2. Let f : X → S be a surjective map of sets such that for all

s ∈ S, f−1(s) is finite. An S-partition of N with support in the fibres of f is a

function P : X → Z>0 such that for each s ∈ S, P | f−1(s) is a partition of N with

support in f−1(s).

XP //

f

��

Z>0

S

19

Definition 4.1.3. Let P be an S-partition of N with structural map f : X → S.

For any map g : S ′ → S, the pull-back partition g∗(P ) = P ◦ p is an S ′-partition of

N , where p : X ×S S′ → X is the projection.

Definition 4.1.4. Let Pi be an Si-partition of N , i = 1, 2. We say that P1 is

equivalent to P2 if there exist a bijection u : S1 → S2 and a u-isomorphism g :

X1 → X2 such that P1 = P2 ◦ g.

Definition 4.1.5. Consider S-partitions Pi of Ni, i = 1, 2. Let fi : Xi → Z>0 be

the structural maps. The sum P1 ⊕ P2 is the following S-partition P of N1 +N2,

X1

∐X2

P //

f

��

Z>0

S

where P |Xi = Pi, and f |Xi = fi, i = 1, 2.

Example 4.1.6. Let S be a scheme, f : X → S a finite etale cover of rank N . We

define an S-partition P : X → Z>0 of N associated to f by

P (x) = [k(x) : k(f(x))], ∀x ∈ X.

Example 4.1.7. Let F be a number field, K/F a finite field extension of degree

N . Let S = Spec(OF ), I = Spec(OK), and f : I → S the structural morphism.

Consider the function PK/F : I → Z>0 defined as

PK/F (w) =

[Kw : Ff(w)], if w is a finite prime

N, if w = (0)

20

This PK/F defines an S-partition of N . The most interesting case is K = F (πY ), the

field generated by the Frobenius endomorphism πY of a Γ-simple non-super-singular

abelian variety Y over a finite field k (2.12). We study this example in more detail.

(a) F is totally real, K is a CM extension.

One has [Kw : Ff(w)] = [Kw : Ff(w)], and [Kw : Ff(w)] is an even integer if

w = w. Recall that TΓ (2.1) denotes the set of finite prime-to-p places ` of F

where Γ is ramified. The restriction PK/F |TΓ (4.1.3) is equivalent to a TΓ-partition

{P` : [1, d`] → Z>0| ` ∈ TΓ} of N = [K : F ], which satisfies the following propertyP`(2i− 1) = P`(2i), for i ∈ [1, c1(`)]

P`(i) is even, for i ∈ [2c1(`) + 1, d`]

where d` = Card(f−1(`)), 2c1(`) = Card({w ∈ f−1(`)| w 6= w}).

(b) F is a CM field, K is a CM extension.

One has [Kw : Ff(w)] = [Kw : Ff(w)]. The restriction PK/F |TΓ is equivalent to

{P` : [1, d`] → Z>0| ` ∈ TΓ}

which satisfies the propertyP`(2i− 1) = P`(2i), if ` = `, i ∈ [1, c1(`)]

P`(i) = P`(i), if ` - `

where d` = Card(f−1(`)). If ` = `, 2c1(`) := Card({w ∈ f−1(`)| w 6= w}).

21

Definition 4.1.8. A TΓ-partition P of an integer N is said to be of CM-type or

a CM-type partition if it is equivalent to the pull-back partition PK/F |TΓ for a CM

field K of degree N over F .

Partitions of CM-type can be characterized as follows.

Proposition 4.1.9. A TΓ-partition P = {P`; ` ∈ TΓ} of an integer N is of CM-type

if and only if it satisfies the properties in (4.1.7) (a) or (b).

For a proof, we need the following lemma.

Lemma 4.1.10. Let D be a number field, T a set of maximal ideals in OD. For any

T -partition R : I → Z>0 of an integer N with support in the fibres of u : I → T ,

IR //

u

��

Z>0

T

there is a finite etale cover ft : Xt → Spec(ODt) of rank N , such that the partition

associated to ft restricted to {t} is equivalent to R|u−1(t), for every t ∈ T .

Proof. Here Dt denotes a local field, the completion of D with respect to the t-adic

absolute value. For each i ∈ I, t = u(i), let Xi be the unique connected etale cover

of Spec(ODt) of rank R(i). The desired scheme Xt can be chosen as

Xt =∐

i∈u−1(t)

Xi,

for t ∈ T .

Proof. of (4.1.9). It remains to prove the “if”-part of the Proposition (4.1.9). Let

P be a given TΓ-partition of N satisfying the conditions of (4.1.7) (a) or (b).

22

(a) Assume first that F is a totally real number field. We define a TΓ-partition R

of the integer N/2,

R`(j) =

P`(2j), j ∈ [1, c1(`)]

P`(j + c1(`))/2, j ∈ [c1(`) + 1, d` − c1(`)].

For each ` ∈ TΓ, let

X` =∐

j∈[1,d`]

Xj

be the etale cover of Spec(OF`) constructed in Lemma (4.1.10) corresponding to

the partition R. Then by Proposition (7.2.3), there exists a totally real extension

E of F of degree N/2, such that X` is isomorphic to the spectrum of OE ⊗OFOF`

.

Define a scheme Y` over X`,

Y` :=∐

j∈[1,c1(`)]

(Xj

∐Xj)

∐j∈[c1(`)+1,d`−c1(`)]

Yj

where, for j ∈ [c1(`) + 1, d` − c1(`)], Yj denotes the unique connected etale cover of

Xj of rank 2. We apply weak approximation to get a CM quadratic extension K of

E, so that for each ` ∈ TΓ, Y` is isomorphic to the spectrum of the ring OK⊗OFOF`

.

One verifies that K is a desired solution.

(b) Next assume that F is totally imaginary. Let F0 be its maximal totally real

subfield, and T0 be the image of TΓ under the morphism Spec(OF ) → Spec(OF0).

From the TΓ-partition P we construct a T0-partition of R of the same integer N as

follows. If `0 = `` is split in F ,

R`0(j) := P`(j), j ∈ [1, d`].

23

If `0 is inert or ramified in F , `0 = `|F0, ` ∈ TΓ,

R`0(j) :=

2.P`(2j), j ∈ [1, c1(`)]

P`(j + c1(`)), j ∈ [c1(`) + 1, d` − c1(`)]

By Proposition (7.2.3), for a suitable totally real extension E/F0 of degree N , one

has

(i) if `0 = `` is split,

E ⊗F0 (F0)`0 '∏

j∈[1,d`]

Ej,

where Ej is the unique unramified extension of (F0)`0 of degree R`0(j).

(ii) if `0 = `|F0 is inert or ramified in F ,

E ⊗F0 (F0)`0 '∏

j∈[1,d`−c1(`)]

Ej

where Ej is the unique unramified extension of F` of degree R`0(j)/2, for

j ∈ [1, c1(`)], and is an extension of (F0)`0 of degree R`0(j) linearly disjoint

with F`, for j ∈ [c1(`) + 1, d` − c1(`)].

Form the tensor product K := E ⊗F0 F . One checks that the TΓ-partition PK/F |TΓ

is equivalent to P .

4.2 Partitioned isocrystals

Definition 4.2.1. A Γ-linear polarized simply partitioned isocrystal x is a pair

(M,P ) consisting of a polarized Γ-linear isocrystal M and a TΓ-partition of an

24

integer N(x), P : I → Z>0, with support in the fibres of f : I → TΓ,

IP //

f

��

Z>0

TΓ

which satisfies the following conditions:

(SPI1) There exists a constant n(x) such that for every place v of F above p, the

Γv-linear isocrystal Mv has N(x) isotypic components, and the multiplicity

(2.7) of each component is equal to n(x).

(SPI2) For every `′ ∈ I, n(x).invf(`′)(Γ)P (`′) = 0 in Q/Z.

We shorten Γ-linear polarized simply partitioned isocrystal to simply partitioned

isocrystal if this causes no confusion. We call M the underlying isocrystal, P the

defining partition of x = (M,P ). The dimension, slopes, multiplicity n(x), Newton

polygon, and polarization of x will be understood to be those of M .

Definition 4.2.2. Two simply partitioned isocrystals x, y are said to be equivalent

if their isocrystals are isomorphic and their partitions are equivalent (4.1.4).

Definition 4.2.3. Let x = (M,P ) be a simply partitioned isocrystal. For any

non-negative integer a, we define the scalar multiple a.x to be (Ma, P ); a.x is a

simply partitioned isocrystal. If a ≥ 1, then N(a.x) = N(x), n(a.x) = a.n(x). If

there exist an integer a > 1 and a simply partitioned isocrystal y such that x = a.y,

then x is called divisible.

25

Definition 4.2.4. There is a partially defined sum operation on the set of sim-

ply partitioned isocrystals. Suppose that the simply partitioned isocrystals xi =

(Mi, Pi), i = 1, 2, satisfy the following assumptions:

(1) Their multiplicities are equal n(x1) = n(x2).

(2) For any place v of F above p, (M1)v and (M2)v have no common slopes.

Then we define the sum x1 + x2 to be the pair (M1 ⊕M2, P1 ⊕ P2), see (4.1.5);

x1 + x2 is a simply partitioned isocrystal.

One verifies that if x1 + x2 is defined, then x2 + x1 is also defined and

x1 + x2 = x2 + x1.

If x1 + x2 and (x1 + x2) + x3 are both defined, then x2 + x3 and x1 + (x2 + x3) are

also defined, and the associativity holds, i.e.

(x1 + x2) + x3 = x1 + (x2 + x3).

Definition 4.2.5. A Γ-linear polarized partitioned isocrystal is a finite collection of

simply partitioned isocrystals x = {xa; a ∈ A}, such that the following conditions

are satisfied.

(PI1) For each pair a, b ∈ A, and each place v of F above p, (xa)v and (xb)v have

no common slopes.

(PI2) The multiplicities n(xa) are distinct, for a ∈ A.

26

We call x a partitioned isocrystal if no confusion arises. Each xa is called a

component of x. The direct sum of the underlying isocrystals of xa, M =⊕

a∈AMa,

is called the underlying isocrystal of x.

Definition 4.2.6. Two partitioned isocrystals x = {xa; a ∈ A} and y = {yb; b ∈ B}

are equivalent if there exists a bijection u : A→ B such that each xa is equivalent

to yu(a).

Up to equivalence, every partitioned isocrystal x = {xa; a ∈ A} can be naturally

indexed by the multiplicities of its simple components, cf. (PI2) (4.2.5).

Definition 4.2.7. Let x = {xa; a ∈ A} be a partitioned isocrystal (4.2.5). For any

non-negative integer h, we define the scalar multiple h.x to be {h.xa; a ∈ A}. A

partitioned isocrystal is divisible if x = h.y for some integer h > 1 and a partitioned

isocrystal y, cf. (4.2.3).

Definition 4.2.8. The sum operation defined for simply partitioned isocrystals

can be extended to partitioned isocrystals. Given two partitioned isocrystals x =

{xa; a ∈ A}, y = {yb; b ∈ B} satisfying the following restriction,

(N) For each pair a ∈ A, b ∈ B, and for each place v of F above p, xa and yb have

no common slopes at v.

we define their joint, s = x∨y, another partitioned isocrystal, as follows. Let C be

the finite set of positive integers c such that either x or y or both has a component

27

whose multiplicity is c. This set C will parametrize the components of s. In other

words, we have

s = {sc; c ∈ C}

(i) If exactly one of the x, y has a component with multiplicity c, say n(xa) = c,

one defines sc to be xa.

(ii) If both x and y have components, say xa, yb, such that n(xa) = n(yb) = c, one

defines sc to be the sum xa + yb (4.2.4).

Whenever it is defined, the joint operation is clearly commutative and associative

up to canonical equivalence.

4.3 A simply partitioned isocrystal sΓ

Definition 4.3.1. We define sΓ to be the simply partitioned isocrystal (H1(A), P )

associated to the unique Γ-simple super-singular abelian variety A up to isogeny

over Fp. The partition P is the unique TΓ-partition of 1, i.e. P (`) = 1, for any

` ∈ TΓ.

TΓP //

id��

Z>0

TΓ

At every place v of F above p, sΓ is isotypic of slope 1/2 and its multiplicity n(sΓ)

is equal to the order eΓ of the class [Dp,∞ ⊗Q F ]− [Γ] in Br(F ), see (6.2.1).

28

4.4 Partitioned isocrystals with (S)-Restriction

Definition 4.4.1. A partitioned isocrystal x = {xa; a ∈ A} is said to satisfy the su-

persingular restriction (S) if there exist an integer h ≥ 0 and a partitioned isocrystal

y = {yb; b ∈ B} such that

(S1) x = h.sΓ

∨y,

(S2) if F is totally real, y contains no slope 1/2 part,

(S3) the partition Pb of each component yb = (Mb, Pb) is of CM-type (4.1.8).

For simplicity we call x an (S)-restricted partitioned isocrystal.

Remarks 4.4.2. (a). When h ≥ 1, the condition (S1) implies that for every place v

of F above p, y has no slope 1/2 component at v, see (4.2.8).

(b). The condition (S3) is a purely combinatorial condition, see the characteri-

zation of CM-type partitions in (4.1.9).

29

Chapter 5

Main Theorem and Examples

For the rest of the paper, all abelian varieties and isocrystals are defined over Fp.

Now we formulate our criterion for a Γ-linear polarized isocrystal to be realizable

by a Γ-hyper-symmetric abelian variety.

5.1 Statement of the main theorem

Theorem 5.1.1. An effective Γ-linear polarized isocrystal M is isomorphic to the

Dieudonne isocrystal H1(Y ) of a Γ-hyper-symmetric abelian variety Y if and only

if M underlies an (S)-restricted partitioned isocrystal.

The theorem will be proven in the next two sections. Here we apply it to some

examples of simple algebras Γ for which we work out explicitly the slopes and

multiplicities of the Γ-hyper-symmetric abelian varieties. Note that the multiplicity

is defined in (2.7).

30

5.2 Examples

Example 5.2.1. (Siegel) Γ = Q. As TΓ is empty, the supersingular restriction (S)

is reduced to (S1) and (S2). A non-divisible simply partitioned isocrystal without

slope 1/2 component is called balanced in the terminology of Chai-Oort [1]. In

general, any simply partitioned isocrystal x can be expressed uniquely as

x = h.sΓ +m.y

with integers h,m ≥ 0 and a balanced isocrystal y. One deduces that any Newton

polygon of the form

ρ0.(1/2) +∑

i∈[1,t]

(ρi.(λi) + ρi.(1− λi))

can be realized by a hyper-symmetric abelian variety, where λi ∈ [0, 1/2) are pair-

wise distinct slopes, ρ0 = mult(1/2), ρi = mult(λi) are multiplicities. This example

recovers the Proposition (2.5) of Chai-Oort [1].

Example 5.2.2. Let F be a real quadratic field split at p, p = v1v2. The following

slope data 2.(1/2), at v1

1.(0) + 1.(1), at v2

admit no hyper-symmetric point.

Example 5.2.3. Let Γ = F be a totally real field of degree d over Q. The restriction

(S) is reduced to (S1) and (S2).

31

The isocrystal sF is isotypic of slope 1/2 at every place v of F . The multiplicity

is n(sF ) = eF , the order of the class [Dp,∞ ⊗Q F ] in the Brauer group of F , cf.

(6.2.1).

Any simply partitioned isocrystal y without slope 1/2 component can be de-

composed as a finite sum

y = y1 + · · ·+ yn,

where each yi has two isotypic components at every place v|p.

Let z be one of the yi’s , and let {λv, 1−λv} be the two slopes of z at v. Then the

multiplicity n(z) is a common multiple of the denominators of [Fv : Qp]λv, where v

runs over the places of F above p.

As a consequence, an F -linear polarized isocrytal M of dimension 2d over K(Fp)

is realizable by an F -hyper-symmetric abelian variety over Fp if and only if the slopes

of M has exactly one of the following two patterns:

(i) At every place v|p, there is only one slope 1/2 with multiplicity 2.

(ii) At every place v|p, there are two slopes {λv, 1 − λv}, each of multiplicity 1.

These λv are such that [Fv : Qp]λv ∈ Z.

Example 5.2.4. Let Γ = F be a CM field, [F : Q] = 2d. The restriction (S) is

reduced to (S1).

The isocrystal sF is isotypic of slope 1/2 at every place v of F above p. The

multiplicity is n(sF ) = eF , the order of the class [Dp,∞ ⊗Q F ] in the Brauer group

of F .

32

Any (S)-restricted simply partitioned isocrystal y is decomposed as a finite sum

y = y1 + · · ·+ yn,

where each yi has either one or two isotypic components. More explicitly, for a fixed

z = yi,

(i) if z has one isotypic component at every place v|p, the slopes are such that

λv + λv = 1. In particular, λv = 1/2, if v = v. The multiplicity n(z) is a

multiple of the common denominator of [Fv : Qp]λv, for v|p.

(ii) if z has two isotypic components at every place v|p,

(a) if v = v, the slopes are {λv, 1− λv}, with λv ∈ [0, 1/2).

(b) if v 6= v, the slopes are either

λv, 1− λv, at v

λv, 1− λv, at v

or µv, νv, at v

1− µv, 1− νv, at v

with λv, λv ∈ [0, 1/2), µv 6= νv ∈ [0, 1].

Example 5.2.5. Let Γ be a definite quaternion division algebra over Q. We assume

that Γ is ramified exactly at the infinity and a prime ` different from p. Hence

TΓ = {`} and inv`(Γ) = 1/2.

33

The partitioned isocrystal sΓ is isotypic of slope 1/2 with multiplicity n(sΓ) = 2,

because the order eΓ of the class [Dp,∞]− [Γ] in the Brauer group of Q is 2.

Let y be a simply partitioned isocrystal without slope 1/2 component. Let

P` : [1, d`] → Z>0

be the defining partition of y. The condition (SPI2) says that

n(y).P`(i).1/2 ∈ Z, for all i ∈ [1, d`].

If y is (S)-restricted, then by (S3), its partition is of the following formP`(2i− 1) = P`(2i), i ∈ [1, c1(`)]

P`(i) is even, i ∈ [2c1(`) + 1, d`]

for some integer c1(`) ∈ Z≥0.

Now let M be any effective Γ-linear polarized isocrystal satisfying the condition

(SPI1) and without slope 1/2 component. We claim that M underlies an (S)-

restricted simply partitioned isocrystal y. In fact, one can choose y = (M,Pl),

where d` = 1, P`(1) = N(y), and N(y) is the number of isotypic components of

M . Note that N(y) is an even integer because M is polarized and has no slope 1/2

component.

With this choice of partition Pl, the simply partitioned isocrystal y decomposes

as a finite sum

y = y1 + · · ·+ ym,

where each yi has exactly two isotypic components with slopes {λi, 1 − λi}. The

multiplicity n(y) is a multiple of the common denominator of the λi’s.

34

For example, let us work out the slopes and multiplicities of all (S)-restricted

partitioned isocrystals of dimension 12 over K(Fp). There are exactly five Newton

polygons which are realizable by 6-dimensional Γ-hyper-symmetric abelian varieties:

a. 3.(1/2).

b. 1.(0) + 1.(1) + 2.(1/2).

c. 2.(0) + 2.(1) + 1.(1/2).

d. 3.(0) + 3.(1).

e. 1.(1/3) + 1.(2/3).

The above notation, for example, 1.(0) + 1.(1) + 2.(1/2) means that the slopes are

{0, 1, 1/2}, with multiplicities {1, 1, 2}, respectively.

Example 5.2.6. Let F be a CM field, and Γ be a positive central division algebra

over F . We make the following assumptions on Γ,

(i) [F : Q] = 4; [Fv1 : Qp] = 2, [Fv2 : Qp] = [Fv2 : Qp] = 1, v1, v2, v2 are above p.

(ii) Γ is ramified exactly at v1 and a finite prime-to-p place `, ` = `; invv1(Γ) = 1/3,

inv`(Γ) = 2/3.

35

The Brauer class c = [Dp,∞ ⊗Q F ]− [Γ] ∈ Br(F ) has local invariants

invν(c) =

−1/3, if ν = v1

−1/2, if ν = v2, v2

−2/3, if ν = `

0, otherwise

Hence the order of c, as well as the multiplicity n(sΓ), is equal to 6.

Let y be a simply partitioned isocrystal. Let N(y) be the number of isotypic

components, n(y) the multiplicity of y at each place v ∈ {v1, v2, v2}. Denote by P`

the defining partition of y

P` : [1, d`] → Z>0.

In this case, the condition (SPI2) says that

n(y)P`(i).2/3 ∈ Z, for all i ∈ [1, d`].

If y is (S)-restricted, then by (4.1.9), its partition Pl satisfies the condition

P`(2i− 1) = P`(2i), ∀ i ∈ [1, c1(`)],

for some integer c1(`), with 0 ≤ 2c1(`) ≤ d`.

We give another example of Newton polygon which admits no hyper-symmetric

point.

ξ =

1.(0) + 1.(1), at v1

1.(0) + 1.(1), at v2

1.(0) + 1.(1), at v2

36

Note that if M has ξ as Newton polygon, then

dimK(Fp)(Mv1) = 12, dimK(Fp)(Mv2) = dimK(Fp)(Mv2) = 6.

At each place v ∈ {v1, v2, v2}, M has N = 2 isotypic components, the multiplicity

of every isotypic component is n = 1. But there is no partition P` of N = 2, such

that n.P`(i).2/3 ∈ Z.

Now we compute the Newton polygons of all (S)-restricted partitioned isocrys-

tals of dimension 72 over K(Fp). By (S1), we can write x = h.sΓ

∨y. Note that

the dimension of sΓ is 72. One has either x = sΓ or x = y. Consider the case x = y

and write

y = {yb; b ∈ B},

where yb are the simple components of y. Comparing the dimension of yb and y,

one has

72 = [Γ : F ]1/2[F : Q]∑b∈B

N(yb)n(yb),

where N(yb) denotes the number of isotypic components, n(yb) the multiplicity, of

yb at each place of F above p. Since [Γ : F ]1/2 = 3, [F : Q] = 4, this equation is

reduced to

6 =∑b∈B

N(yb)n(yb).

One verifies that this condition forces that y is simply partitioned, N(y) = 2, and

n(y) = 3. Here we list all the realizable Newton polygons as follows.

37

(i) The slopes at v1 are one of: 0, 1

1/3, 2/3

1/6, 5/6

(ii) The slopes at v2, v2, in this order, are one of:

0, 1; 1/3, 2/3

0, 1/3; 1, 2/3

0, 2/3; 1, 1/3

1, 1/3; 0, 2/3

1, 2/3; 0, 1/3

1/3, 2/3; 0, 1

38

Chapter 6

Proof of the “only-if” part of

(5.1.1)

6.1 Semi-simplicity of the Frobenius action

The following lemma is certainly well known and an analogous statement for `-adic

cohomology can be found in Mumford’s book on abelian varieties.

Lemma 6.1.1. If X is an abelian variety over a finite field k, the Frobenius endo-

morphism π acts in a semi-simple way on the isocrystal H1(X).

Proof. We may and do assume that X is a simple abelian variety. Let π = s + n

be the Jordan decomposition of π considered as a linear endomorphism of H1(X).

By Katz-Messing [7], the characterisic polynomial det(T − π|H1(X)) has rational

coefficients. Hence we can find a polynomial f(T ) ∈ Q[T ] without constant term,

39

such that the nilponent part n = f(π). The image of `n, for a sufficiently divisible

integer `, is a proper sub-abelian variety of X, thus equal to 0.

6.2 Proof of the only-if part

Given a Γ-hyper-symmetric abelian variety Y , we let Y ∼Γ-isog Y e11 × · · · × Y er

r

be the Γ-isotypic decomposition. By (3.1.1), for the only-if part, we only need

to show that each H1(Yi) underlies an (S)-restricted partitioned isocrystal xi. In-

deed, if this is proved, H1(Y ) is isomorphic to the underlying isocrystal of x =

{e1.x1}∨· · ·

∨{er.xr}.

From now on, we assume that Y is Γ-simple. Let q = pa and YFq be a Γ-linear

polarized abelian variety over Fq such that YFq ⊗Fq Fp ' Y . Suppose that a is

sufficiently divisible. The abelian variety YFq is Γ-simple, therefore, YFq ∼isog XsFq

,

for some XFq simple over Fq. Let π denote the Frobenius endomorphism of YFq as

well as that of XFq . Let K = F (π).

Proposition 6.2.1. The pair x = (H1(Y ), PK/F |TΓ) associated to the Γ-simple

hyper-symmetric abelian variety Y is a simply partitioned isocrystal satisfying the

supersingular restriction (S). More explicitly,

(a) if π is totally real, then x = sΓ is isotypic of slope 1/2 with multiplicity n(sΓ)

equal to the order of the Brauer class [Dp,∞ ⊗Q F ]− [Γ] in Br(F ).

(b) if π is totally imaginary, then x has N(x) = [K : F ] isotypic components

40

at every place v of F above p, the multiplicity n(x) is the order of the class

[End0Γ(Y )] in Br(K).

Proof. Let N = [F (π) : F ] and denote by P the TΓ-partition PK/F |TΓ of N . Let

C := End0Γ(YFq) and Lv := Fv ⊗Qp K(Fq). Decompose

H1(YFq) =⊕v|p

Mv

as in (2.7). Each Mv is a free Lv-module, by (6.1.1) and Lemma 11.5 [8]. We

consider the characterisitic polynomial fv(T ) of π as an Lv-linear transformation of

Mv. Since Y is Γ-hyper-symmetric, by (3.3.1),

fv(T ) =∏w|v

(T − ιw(π))nw

is a product of linear polynomials, where ιw : F (π) ↪→ Fv denote the F -embeddings

of F (π) into Fv indexed by the places w. Thus the characterisitic polynomial

f(T ) = det(T −π|H1(YFq)) of the K(Fq)-linear endomorphism π can be factored as

∏v

NormLv/K(Fq)fv(T ) =∏

v

∏w

NormFv/Qp(T − ιw(π))nw .

Since the Q-embeddings ιu of F (π) into Qp are one-to-one correspondence with the

set of triples u = (v, w, τ) consisting of a place v of F above p, a place w of F (π)

above v, and a Qp-linear homomorphism τ : Fv ↪→ Qp, we can rewrite f(T ) as

f(T ) =∏u

(T − ιu(π))nw .

By Katz-Messing [7], the polynomial f(T ) ∈ Z[T ], so nw = n is independent of the

place w, and thus, is equal to 2.dim(Y )/[F (π) : Q]. Because fv(T ) has N different

41

irreducible factors, i.e. T − ιw(π), H1(Y ) has N isotypic components at every place

v of F above p [8]. By the dimension formula in (2.12), the multiplicity of each

isotypic component is equal to

[Lv : K(Fq)]n/([Γ : F ]1/2[Fv : Qp]) = order([C]).

Observe that for every place `′ of K above a place ` ∈ TΓ, the local invariant of C

at `′ is

inv`′(C) = −inv`(Γ)[K`′ : F`].

It certainly follows that order([C])inv`(Γ)P (`′) = 0 in Q/Z.

If now π = q1/2 is a totally real algebraic number, then, since we have assumed

that a is sufficiently divisible, XFq is a super-singular elliptic curve. The isocrystal

H1(Y ) underlies the simply partitioned isocrystal sΓ (4.3.1). At every place v of F

above p, sΓ is isotypic of slope 1/2.

If π is totally imaginary, the field K = F (π) is a CM extension of F ; so the

condition (S3) is a priori satisfied. In case that F is a totally real number field,

the slopes of H1(Y ) at a place v of F above p, if arranged in increasing order, are

symmetric with respect to 1/2. As there are N = [K : F ] of them, and N is even,

H1(Y ) contains no slope 1/2 component. The proof is now complete.

42

Chapter 7

Proof of the “if” part of (5.1.1)

Let x = h.sΓ

∨y be an (S)-restricted partitioned isocrystal. This section is devoted

to showing that x is realizable by a Γ-hyper-symmetric abelian variety. Here is the

first step towards proving the existence theorem.

7.1 Weil numbers

Proposition 7.1.1. Let K be a CM field, {λw;w|p} a set of rational numbers

contained in the interval [0, 1] and indexed by the places w of K above p. Assume

that λw + λw = 1. Then there exist an integer a ≥ 1 and a pa-Weil number π such

that

ordw(π)/ordw(pa) = λw,

for all w|p.

Proof. Let E be the maximal totally real subfield of K. For any place v of E above

43

p, we define λv := min{λw, λw}, v = w|E. Either v is split, v = ww, or there is

only one prime w above v. In the first case, let aw ∈ OK be a generator of the ideal

wh; in the latter case, let av ∈ OE be a generator of vh, where h is the ideal class

number of K. Consider the factorization

pOK =∏

v

(ww)e(v|p)∏

v

ve(v|p),

where the first product counts those v split in K/E, the second counts those v inert

or ramified in K/E. Raising to the h-th power, one has

ph =∏

v

(awaw)e(v|p)∏

v

ae(v|p)v .u.

The element u is a unit of OE. Now choose a sufficiently divisible positive integer

c, and write λv = mv/(mv + nv), with c = mv + nv, mv, nv ∈ Z. We then define an

algebraic integer π as

π =∏

v

(amvw anv

w )e(v|p)∏

v

ace(v|p)/2v .uc/2.

One checks easily that ππ = phc and π is the desired phc-Weil number.

In case that K is an extension of F , it is important to know when the Weil

number we have just constructed generates K over F .

Proposition 7.1.2. Let F be a field, and K/F be a separable field extension of

degree n. Assume that the normal hull L of K/F has a Galois group isomorphic to

the symmetric group Sn of n letters. Then K/F has no sub-extensions other than

F and itself.

44

Proof. This is equivalent to the assertion that the stabilizer subgroup Sn−1 of the

letter 1 ∈ {1, · · · , n} is a maximal subgroup of Sn. It suffices to show that any

subgroup H properly containing Sn−1 acts transitively on the letters {1, · · · , n}. If

n = 1, 2, this is clear. Assume that n ≥ 3. Let τ be an element of H, τ(1) = i,

i 6= 1. For any j ∈ {1, · · · , n}, different from 1 and i, the permutation σ := (ij)τ

in H sends 1 to j.

7.2 Hilbert irreducibility theorem

Proposition 7.2.1. (Ekedahl) Let K be a number field, and OK its ring of inte-

gers. Let S be a dense open sub-scheme of Spec(OK). Let X, Y be two schemes of

finite type over S, and let g : Y → X be a finite etale surjective S-morphism. Sup-

pose that YK := Y ×S Spec(K) is geometrically irreducible and XK := X×S Spec(K)

satisfies the property of weak approximation. Then the set of K-rational points x of

X such that g−1(x) is connected satisfies also the property of weak approximation.

Remark 7.2.2. Let X be a scheme of finite type over a number field K. Recall that

a subset E of X(K) is said to satisfy the property of weak approximation, if for any

finite number of places {v1, · · · , vr} of K, E is dense in the product

X(Kv1)× · · · ×X(Kvr)

under the diagonal embedding. The topology on X(Kv) is induced from that of Kv.

In particular, theK-schemeX is said to satisfy the property of weak approximation,

45

if X(K) does.

Proposition 7.2.3. Let n be a positive integer, and K a totally real number field.

Let Σ be a finite set of non-archimedean places of K. For each ` ∈ Σ let K ′` be a

finite etale algebra over K` of rank n. Then there is a totally real extension K ′/K of

degree n, such that its normal hull has a Galois group isomorphic to the symmetric

group Sn of n letters, and K ′ ⊗K K` ' K ′`, for all ` ∈ Σ.

Proof. We consider the following situation. Let S = Spec(OK), X ′ = S[a1, · · · , an],

an S-affine space with coordinates a1, · · · , an. Let Y ′ be the hyper-surface in X ′[t]

defined by the equation

f = tn + a1tn−1 + · · ·+ an.

Let R be the resultant of f and its derivative f ′. We denote by X the complement

of {R = 0} in X ′ and by Y := Y ′ ×X′ X; Y is an etale cover of X of rank n.

The scheme XK , being a non-empty open sub-scheme of an affine space, clearly

satisfies the property of weak approximation. The geometric fibre YK := YK ⊗K K

is affine of ring Γ(OYK) = (K[a1, · · · , an, t]/(f))R. We will prove in the next lemma

that Γ(OYK) is an integral domain. Now it is ready to apply Ekedahl’s Hilbert

irreducibility theorem (7.2.1) according to which, the subset M of the K-rational

points x where Yx is connected, i.e. Yx is the spectrum of a field extension K ′ of K of

degree n, satisfies the property of weak approximation. Requiring the Kl-algebras

K ′⊗K Kl to be isomorphic to some given etale algebras at finitely many places l of

46

K imposes a weak approximation question on the parameters a1, · · · , an ∈ K. The

condition on the Galois group of the normal hull is a weak approximation property,

cf. [5]. The proposition follows by modifying a little the content but not the proof

of Ekedahl’s theorem [4].

Lemma 7.2.4. Let K be a factorial domain, A = K[a1, · · · , an] a polynomial alge-

bra over K. The “generic” polynomial f = tn + a1tn−1 + · · · + an is irreducible in

A[t].

Proof. Let B = K[b1, · · · , bn], where bi = ai/an, for 1 ≤ i ≤ n − 1, and bn = an.

As A is a subring of B, it suffices to prove that f is irreducible in B[t]. This is so

because f is an Eisenstein polynomial in B[t] with respect to the prime an.

Now consider an (S)-restricted partitioned isocrystal x = h.sΓ

∨y. For proving

the “if” part, it suffices to show that each component of y is realizable by a Γ-isotypic

hyper-symmetric abelian variety. From now on, we assume that y = (M,P ) is a

simply partitioned isocrystal. By the supersingular restriction (S), there is a CM

extension B/F such that P is equivalent to PB/F |TΓ. Let B0 be the maximal totally

real subfield of B. We also let N be the common number of isotypic components of

y at all places v of F above p.

These reductions and hypothesis are in force for the rest.

47

7.3 If F is a CM field

Let us now finish the proof of the main theorem (5.1.1). First, assume that F is a

CM field. Let F0 be the maximal totally real subfield of F .

Proposition 7.3.1. Assume that F is a CM field. Suppose that y = (M,P ) is an

(S)-restricted simply partitioned isocrystal. Then there exists a Γ-isotypic hyper-

symmetric abelian variety Y such that M is Γ-isomorphic to H1(Y ).

Proof. For each place v of F above p, we define an (F0)v|F0-algebra Tv|F0 of rank N :

Tv|F0 =

(F0)

Nv|F0

, if v 6= v

(F0)v|F0 × F(N−1)/2v , if v = v,N odd

FN/2v , if v = v,N even

It follows from Proposition (7.2.3) that there is a totally real extension E/F0 of

relative degree N such that its normal hull has a Galois group isomorphic to SN

and that

(1) for each v|p, E ⊗F0 (F0)v|F0 ' Tv|F0 ,

(2) for every ` ∈ TΓ, E ⊗F0 (F0)`|F0 ' B0 ⊗F0 (F0)`|F0 .

Consider the CM field K := E ⊗F0 F . One has

(i) the normal hull of K/F has a Galois group isomorphic to SN ,

(ii) for each ` ∈ TΓ, K ⊗F F` ' B ⊗F F`,

48

(iii) for each place v of F above p, K ⊗F Fv ' FNv is totally split.

The property (iii) allows us to index the slopes of y at v as {λw;w|v}, where w

runs over the places of K above v. One can even arrange that λw + λw = 1, since

the underlying isocrystal M of y is polarized, cf. (2.7). We apply (7.1.1) to get an

integer a ≥ 1 and a pa-Weil number π ∈ K, so that

ordw(π)/ordw(pa) = λw, for all w|p

Note that the field F (π) must be equal to K. Indeed, if N = 1, this is clear

because F = F (π) = K. If N > 1, π is not an element of F , because, otherwise, we

would have ordw1(π) = ordw2(π), for any two places w1, w2 above v. This is absurd

in view of the choice of π. By (7.1.2) and (i), we have F (π) = K.

According to the theorem of Honda-Tate (2.12), up to isogeny there is a unique

Γ-simple abelian variety Y ′Fq

defined over Fq, q = pa, corresponding to the pa-Weil

number π. We assume that a is chosen to be sufficiently divisible so that Y ′Fq

is

absolutely Γ-simple. Let Y ′ := Y ′Fq⊗Fq Fp. Kottwitz [8] proved that there exists

a Γ-linear Q-polarization on Y ′. Since the center F (π) of End0Γ(Y ′) is totally split

at every place v|p of F , the abelian variety Y ′ is therefore Γ-hyper-symmetric, cf.

(3.3.1).

The pair y′ = (H1(Y ′), PK/F |TΓ) is a simply partitioned isocrystal satisfying the

supersingular restriction (S) by (6.2.1). By construction, y′ and y have the same

slopes at every place v of F above p.

Now we prove that the multiplicity n(y′) divides n(y). In fact, n(y′) is the order

49

of [End0Γ(Y )] in the Brauer group of K, cf. (6.2.1). Since y satisfies the condition

(SPI2) (4.2.1), one has n(y).inv`(Γ)[K`′ : F`] = 0 in Q/Z. Look at the local Brauer

invariants of C := End0Γ(Y )

invν(C) =

−[Fv : Qp]λν − invv(Γ), if ν | v

−[Kν : F`].inv`(Γ), if ν - p.

By Kottwitz 11.5 [8], n(y).invw(C) = 0 in Q/Z, for all w above p. These two

equations together show that n(y′) divides n(y). Let e be the integer such that

n(y) = e.n(y′).

It remains to prove that the underlying isocrystals of y and e.y′ are isomorphic as

polarized Γ-linear isocrystals. Indeed, we can modify the polarization on Y := Y ′e

so that e.y′ with this modified polarization is isomorphic to y. For a proof, let S be

the Q-vector space of the symmetric elements in Hom0Γ(Y, Y ∗), where Y ∗ denotes the

dual abelian variety of Y . As Y is Γ-linear hyper-symmetric, S⊗Q Qp is isomorphic

to the symmetric elements of HomΓ(H1(Y ∗), H1(Y )). The space S being dense in

S⊗Q Qp, our claim is clearly justified and the proof in the case that F is a CM field

is now complete.

7.4 If F is a totally real field

Proposition 7.4.1. Assume that F is a totally real number field. And suppose that

y = (M,P ) is an (S)-restricted simply partitioned isocrystal. Then there exists a

Γ-isotypic hyper-symmetric abelian variety Y such that M ' H1(Y ).

50

Proof. As y is Γ-linearly polarized and contains no slope 1/2 part by (S2), N is an

even integer. By Proposition (7.2.3), there is a totally real extension E/F of degree

N/2 such that

(1) for each place v|p of F , E ⊗F Fv ' FN/2v ,

(2) for each ` ∈ TΓ, there is an F`-isomorphism f` : E ⊗F F` ' B0 ⊗F F`,

(3) the normal hull of E/F has a Galois group isomorphic to SN/2.

By the lemma 5.7 [1], there exists a totally imaginary quadratic extension K/E

such that

(i) for each place ν of E above p, K ⊗E Eν ' Eν × Eν ,

(ii) for each ` ∈ TΓ, there is an isomorphism g` : K ⊗F F` ' B ⊗F F` compatible

with f`,

(iii) the field K contains no proper CM sub-extension of F .

The properties (1) and (i) show that K/F is totally split everywhere above v.

Thus we can index the slopes of y at v as {λw;w|v} with w running over the places

of K above v. Moreover, as y is Γ-linearly polarized, one can even arrange that

λw + λw = 1, cf. (2.7). Similarly as in the preceding proposition, there is a pa-Weil

number π, for a suitable integer a ≥ 1, such that F (π) = K, and

ordw(π)/ordw(pa) = λw,

51

for all places w of K above p.

We assume that a is sufficiently divisible. The unique Γ-simple abelian variety

Y ′Fpa up to isogeny corresponding to π admits a Γ-linear Q-polarization by Kottwitz

[8]. Let Y ′ := Y ′Fpa ⊗Fpa Fp, which is by construction Γ-hyper-symmetric. We then

modify, if necessary, the polarization on Y ′ so that a copy Y := Y ′e realizes y. The

argument is the same as that in (7.3.1). We have proved the Proposition (7.4.1).

52

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