### On Hyper-Symmetric Abelian Varieties

### Ying Zong

### A Dissertation

### in

### Mathematics

### Presented to the Faculties of the University of Pennsylvania in Partial

### Fulfillment of the Requirements for the Degree of Doctor of Philosophy

### 2008

### Advisor’s Name

### Supervisor of Dissertation

## 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.

ABSTRACT

On Hyper-Symmetric Abelian Varieties

Ying Zong Ching-Li Chai, Advisor

## 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 Dieudonn´e’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

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

## 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 _{F}p. 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 integerg ≥1 and a prime numberp. Consider the Siegel modular variety

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 pis Γ-hyper-symmetric, if the natural map

End0_{Γ}(Y)⊗_{Q} _{Q}p →EndΓ(H1(Y))

is a bijection.

For simplicity we denote byH1_{(Y}_{) the isocrystal}_{H}1

crys(Y /W(k))⊗ZQ. The goal of this paper is to answer the following question:

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. LetY =Y0⊗_{Fpa}_{F}p be a Γ-simple hyper-symmetric

abelian variety over _{F}p, where Y0 is a Γ-simple abelian variety over a finite field

Fpa. By the theory of Honda-Tate, up to isogeny, Y0 is completely characterized
by its Frobenius endomorphism πY0. Let F be the center of Γ. Assume that _{F}_{p}a
is sufficiently large. We show in (3.3.1) that Y is Γ-hyper-symmetric if and only if
the extension F(πY0)/F is totally split everywhere above p, that is,

F(πY0)⊗_{F} F_{v} 'F_{v}× · · · ×F_{v},

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(πY0) :F] indexed by `∈T_{Γ}. For each `∈T_{Γ}, P_{`} is given by

P`(`0) = [F(πY0)_{`}0 :F_{`}]

with `0 ranging over the places of F(πY0) above `. The pair (H1(Y), P) is the

partitioned isocrystal attached toY. In particular, we denote bysΓthe pair attached

to the unique Γ-simple supersingular abelian variety up to isogeny over _{F}p, see

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 End0_{F}(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`)`∈TΓ is

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, `)`∈TΓ given by

PK/F, `(`0) := [K`0 :F_{`}], ∀ `0 | `

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 Y0 be the

unique abelian variety up to Γ-isogeny corresponding to π. For some integer e,
(Y0)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, Dieudonn´e’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 _{F}p (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_{(Y}

## 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. Letv1,· · · , vt be the places ofF above p.

We have

Γ⊗_{Q}_{Q}p = Γv1 × · · · ×Γvt.
Let TΓ denote the following set

### 2.2

### Brauer invariants

Recall the computation of Brauer invariants. Let K be a finite extension of _{Q}p.

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 elementu∈A, the vectors
1, u,· · · , ud−1 _{form an} _{L-basis of} _{A, and}

ua=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

Ifk is a perfect field of characteristicp, we denote byW(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

p

1,· · ·)

### 2.4

### Isocrystals

An isocrystal over k is a finite dimensional K(k)-vector space M equipped with a σ-linear automorphism Φ. A morphismf : (M,Φ)→(M0,Φ0) is aK(k)-linear map f :M →M0 such that fΦ = Φ0f. Isocrystals over k form an abelian category.

### 2.5

### Dieudonn´

### e’s classfication of isocrystals

Let k be an algebraic closure of k, a perfect field of characteristic p. We have the fundamental theorem of Dieudonn´e, cf. Kottwitz [8]:

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

(2) A set of representatives of simple objects _{E}r can be given as follows,

Er = (K(k)[T]/(Tb−pa), T)

wherer =a/b is a rational number with (a, b) = 1, b >0. The endomorphism
ring of _{E}r is a central division algebra over Qp with Brauer invariant −r ∈

Q/Z.

(3) Every isocrystal M overk admits a unique decomposition

M =M

r∈Q M(r)

for an integer mr.

The rational numbers occurred in the decomposition M = L

r∈QM(r) are called the slopes of M. If all slopes are non-negative, the isocrystal iseffective.

### 2.6

### Polarization

Apolarization of weight1 or simply apolarization of an isocrystalM 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 Dieudonn´e’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

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

IfM is a Γ-linear isocrystal, andM =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 atv 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 andv collected together, arranged in increasing order, are symmetric with respect to 1/2.

### 2.9

### Theory of Honda-Tate

The relative Frobenius morphism

FX/k :X →X(p)

is an isogeny. We call πX = FX/ka 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 πsuch 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) Ifa ∈2_{Z}, and π=pa/2_{, then} _{X}

π 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}−2_{Z}, and π = pa/2_{, then} _{X}

π ⊗kk0 is isogenous to the product

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

away from p. For a place wof _{Q}(π) abovep, 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 polarizationλ. 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 Y1e1 × · · · ×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 Xiei. We say Yi is of type Xi.

### 2.11

### A dimension relation

LetY be a Γ-simple abelian variety of typeX, i.e. Y ∼isog Xe, for an integere. Let

relation [8],

e.[End0(X) :Z0]1/2[Z0 :Q] = [Γ :F]1/2[End0Γ(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
End0_{Γ}(Y).

### 2.12

### A variant of the Honda-Tate theory

Let k = _{F}pa 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 ofpa_{-Weil numbers.}

(2) The endomorphism algebra End0_{Γ}(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 typeXπ.

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

## 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 Y1e1 × · · · ×Yrer be the Γ-isotypic decomposition

of Y, cf. (2.10). For the rest, H1_{(Y}_{) stands for the first crystalline cohomology of}

Y, H1

crys(Y /W(k))⊗ZQ.

### 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

### 3.2

### Rigidity

Proposition 3.2.1. If Y is Γ-hyper-symmetric, there exists a Γ-hyper-symmetric
abelian variety Y0 over_{F}p such that Y0⊗_{Fp}k is Γ-isogenous to Y.

We first prove a weaker result.

Corollary 3.2.2. There is a Γ-hyper-symmetric abelian variety Y0 over _{F}p such

that the isocrystal H1(Y0⊗_{Fp}k) is isomorphic to H1(Y).

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

subfield K such that YK⊗Kk 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_{(}_{Y}

x) is constructible. Let S0 be the

open subset consisting of points x with the generic Newton polygon, i.e. the same
Newton polygon with that ofH1(Y). AsS0 is regular, the canonical homomorphism
End(YS0) → End(Y_{K}) is an isomorphism. So there is a well defined specialization

map sp : End(YK) → End(Yt) for any point t ∈ S0. By the rigidity lemma 6.1

[9], sp is injective. Let t be a closed point of S0 and Y_{t} = Yt ⊗k(t)k(t). As Y is

Thus the composite map

End0_{Γ}(YK)⊗QQp ,→End

0

Γ(Yt)⊗QQp ,→EndΓ(H

1_{(}_{Y}

t))

is bijective. It follows that Y_{t} is a desired Γ-hyper-symmetric abelian variety over
k(t)'_{F}p.

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 _{F}p 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 End0_{Γ}(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[End0_{Γ}(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 Y0 be an abelian variety over _{F}p as in Corollary (3.2.2). Similarly, the

dimension r0 of any maximal ´etale sub-algebra of End0(Y0) is equal to

where E0 is the center of EndΓ(H1(Y0)).

By the choice of Y0, r and r0 are equal. As any abelian variety over _{F}p has

sufficiently many complex multiplication (2.9), we have r = r0 = 2.dim(Y0). 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 End0_{Γ}(Y).

Proposition 3.3.1. A Γ-linear polarized abelian variety Y over _{F}p 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 Y0 be a Γ-linear polarized abelian variety over a finite field _{F}pa, such
that Y0⊗_{Fpa} _{F}p 'Y and End(Y0) = End(Y). The center Z can be identified with

F(π), the sub-algebra generated by the Frobenius endomorphism of Y0. By Tate
[11], over _{F}pa, the map

End0(Y0)⊗_{Q}_{Q}p →End(H1(Y0))

is bijective.

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

Let M0 := H1_{(Y}0_{), and} _{M}0 _{=} L

v|pM 0

v be the decomposition defined in (2.7).

The isocrystal M_{v}0 is Γv-linear and has a decomposition into isotypic components,

M_{v}0 =M

r∈_{Q}

M_{v}0(r).

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

EndΓv(M

0

v(r)) = EndΓv(M

0

v(r)⊗K(Fpa)K(Fp)),

for any v|p, and r ∈_{Q}.

On the left hand side, the center of EndΓv(M

0

v(r)) is Fv(πv,r), where πv,r stands

for the endomorphism π|M_{v}0(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 Mv0(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−1_{Φ}a_{, i.e.} _{H}1_{(Y}0_{). Hence} _{f} _{∈}_{End}

Γ(H1(Y0)). This implies thatY is

## Chapter 4

## Partitions and Partitioned

## Isocrystals

### 4.1

### Partitions

Definition 4.1.1. LetN be a positive integer. A partition of N with support in a
finite set I is a function P :I →_{Z}>0, such that

P

i∈IP(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 eachs ∈S, P |f−1(s) is a partition of N with

support in f−1_{(s).}

X P //

f

Z>0

Definition 4.1.3. Let P be an S-partition of N with structural map f :X → S. For any map g :S0 →S, the pull-back partition g∗(P) = P ◦p is an S0-partition of N, where p:X×SS0 →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, andf|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

ThisPK/F defines anS-partition ofN. The most interesting case isK =F(πY), the

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

abelian varietyY over a finite fieldk (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 aTΓ-partition

{P` : [1, d`]→Z>0| `∈TΓ} of N = [K :F], which satisfies the following property

P`(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(`)| w6=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 property

P`(2i−1) = P`(2i), if `=`, i∈[1, c1(`)]

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

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 degreeN overF.

Partitions of CM-type can be characterized as follows.

Proposition 4.1.9. ATΓ-partitionP ={P`;`∈TΓ}of an integerN 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. LetDbe 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,

I R //

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. HereDt denotes a local field, the completion ofDwith respect to the t-adic

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

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

a

i∈u−1_{(}_{t}_{)}

Xi,

for t ∈T.

(a) Assume first that F is a totally real number field. We define aTΓ-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` =

a

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 degreeN/2, such that X` is isomorphic to the spectrum of OE⊗OF OF`. Define a scheme Y` over X`,

Y` :=

a

j∈[1,c1(`)]

(Xj

a

Xj)

a

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 ringOK⊗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 ofR of the same integerN as

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

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 '

Y

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 '

Y

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`, forj ∈[c1(`) + 1, d`−c1(`)].

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

is equivalent to P.

### 4.2

### Partitioned isocrystals

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

I P //

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 `0 ∈I,n(x).invf(`0_{)}(Γ)P(`0) = 0 in _{Q}/_{Z}.

We shorten Γ-linear polarized simply partitioned isocrystal tosimply partitioned isocrystal if this causes no confusion. We call M the underlying isocrystal, P the defining partition of x= (M, P). The dimension, slopes, multiplicityn(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 4.2.4. There is a partially defined sum operation on the set of sim-ply partitioned isocrystals. Suppose that the simsim-ply 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 thesum 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, thenx2+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.

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

component ofx. The direct sum of the underlying isocrystals ofxa,M =La∈AMa,

is called the underlying isocrystal of x.

Definition 4.2.6. Two partitioned isocrystalsx={xa;a∈A}andy ={yb;b∈B}

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

to yu(a).

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

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

Definition 4.2.7. Letx={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 isdivisible if x=h.yfor some integerh >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 paira∈A,b ∈B, and for each place v of F abovep,xa and yb have

no common slopes at v.

we define their joint, s=xW

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 bexa.

(ii) If bothxand yhave 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

### s

ΓDefinition 4.3.1. We definesΓ to be the simply partitioned isocrystal (H1(A), P)

associated to the unique Γ-simple super-singular abelian variety A up to isogeny
over _{F}p. The partition P is the unique TΓ-partition of 1, i.e. P(`) = 1, for any

` ∈TΓ.

TΓ

P _{/}_{/}

id

Z>0

TΓ

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

### 4.4

### Partitioned isocrystals with (S)-Restriction

Definition 4.4.1. A partitioned isocrystalx={xa;a∈A}is said to satisfy the

su-persingular restriction (S) if there exist an integerh≥0 and a partitioned isocrystal y ={yb;b∈B} such that

(S1) x=h.sΓ

W

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

## Chapter 5

## Main Theorem and Examples

For the rest of the paper, all abelian varieties and isocrystals are defined over _{F}p.

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 Dieudonn´e 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 themultiplicity

### 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) +

X

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), atv1

1.(0) + 1.(1), atv2

admit no hyper-symmetric point.

The isocrystalsF is isotypic of slope 1/2 at every placev ofF. 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.

Letzbe one of theyi’s , and let{λv,1−λv}be the two slopes of zatv. 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, anF-linear polarized isocrytalM of dimension 2doverK(_{F}p)

is realizable by anF-hyper-symmetric abelian variety over_{F}pif 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

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

y=y1 +· · ·+yn,

where eachyi 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, atv

1−µv, 1−νv, atv

with λv, λv ∈[0,1/2), µv 6=νv ∈[0,1].

The partitioned isocrystalsΓis isotypic of slope 1/2 with multiplicityn(sΓ) = 2,

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

Ify is (S)-restricted, then by (S3), its partition is of the following form

P`(2i−1) =P`(2i), i∈[1, c1(`)]

P`(i) is even, i∈[2c1(`) + 1, d`]

for some integer c1(`)∈Z≥0.

Now letM 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 thatN(y) is an even integer because M is polarized and has no slope 1/2 component.

With this choice of partitionPl, the simply partitioned isocrystaly decomposes

as a finite sum

y =y1+· · ·+ym,

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

For example, let us work out the slopes and multiplicities of all (S)-restricted
partitioned isocrystals of dimension 12 over K(_{F}p). 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 atv1and a finite prime-to-pplace`,` =`; invv1(Γ) = 1/3,

The Brauer classc= [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 ofy 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 partitionPl 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 admitsno hyper-symmetric point. ξ=

1.(0) + 1.(1), atv1

1.(0) + 1.(1), atv2

Note that if M has ξ as Newton polygon, then

dim_{K}_{(}_{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(_{F}p). By (S1), we can write x = h.sΓ

W

y. Note that

the dimension of sΓ is 72. One has eitherx=sΓ orx=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}]X

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 =X

b∈B

N(yb)n(yb).

(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:

## 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}

such that the nilponent part n=f(π). The image of`n, for a sufficiently divisible integer `, is a proper sub-abelian variety ofX, 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 Y1e1 × · · · ×Yrer

be the Γ-isotypic decomposition. By (3.1.1), for the only-if part, we only need
to show that each H1_{(Y}

i) 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}

W

· · ·W

{er.xr}.

From now on, we assume that Y is Γ-simple. Let q =pa _{and} _{Y}

Fq be a Γ-linear
polarized abelian variety over _{F}q such that YFq ⊗Fq Fp ' Y. Suppose that a is
sufficiently divisible. The abelian variety Y_{Fq} is Γ-simple, therefore, Y_{Fq} ∼isog X_{Fq}s ,

for some X_{Fq} simple over _{F}q. Let π denote the Frobenius endomorphism of YFq as
well as that of X_{Fq}. 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,∞⊗QF]−[Γ] in Br(F).

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

[End0_{Γ}(Y)] in Br(K).

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

C := End0_{Γ}(Y_{Fq}) and Lv :=Fv⊗QpK(Fq). Decompose
H1(Y_{Fq}) = M

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) ofπ as anLv-linear transformation of

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

fv(T) =

Y

w|v

(T −ιw(π))nw

is a product of linear polynomials, whereιw :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(Y_{Fq})) of theK(_{F}q)-linear endomorphismπ can be factored as

Y

v

NormLv/K(Fq)fv(T) =

Y

v

Y

w

NormFv/Qp(T −ιw(π))

nw_{.}

Since the_{Q}-embeddingsιu of F(π) intoQp 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 _{Q}p-linear homomorphismτ :Fv ,→Qp, we can rewrite f(T) as

f(T) = Y

u

(T −ιu(π))nw.

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

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 `0 of K above a place `∈ TΓ, the local invariant of C

at `0 is

inv`0(C) = −inv_{`}(Γ)[K_{`}0 :F_{`}].

It certainly follows that order([C])inv`(Γ)P(`0) = 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,X_{Fq} is a super-singular elliptic curve. The isocrystal
H1_{(Y}_{) underlies the simply partitioned isocrystal} _{s}

Γ (4.3.1). At every placev of F

above p, sΓ is isotypic of slope 1/2.

## Chapter 7

## Proof of the “if ” part of (5.1.1)

Letx=h.sΓWybe 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.

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

only one prime wabove v. In the first case, letaw ∈ OK be a generator of the ideal

wh_{; in the latter case, let}_{a}

v ∈ OE be a generator of vh, where h is the ideal class

number of K. Consider the factorization

pOK =

Y

v

(ww)e(v|p)Y

v

ve(v|p),

where the first product counts thosev split inK/E, the second counts thosev inert or ramified in K/E. Raising to the h-th power, one has

ph =Y

v

(awaw)e(v|p)

Y

v

ae_{v}(v|p).u.

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

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

algebraic integer π as

π =Y

v

(amv

w a nv

w )

e(v|p)Y

v

ace_{v} (v|p)/2.uc/2.

One checks easily that ππ=phc _{and} _{π} _{is the desired}_{p}hc_{-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

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 containingSn−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 thatYK :=Y×SSpec(K)is geometrically irreducible andXK :=X×SSpec(K)

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

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

Remark 7.2.2. LetX be a scheme of finite type over a number field K. Recall that a subset E ofX(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 ofKv.

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_{`}0 be a
finite ´etale algebra overK` of rankn. Then there is a totally real extensionK0/K of

degree n, such that its normal hull has a Galois group isomorphic to the symmetric group Sn of n letters, and K0⊗K K` 'K`0, for all ` ∈Σ.

Proof. We consider the following situation. LetS = Spec(OK),X0 =S[a1,· · · , an],

an S-affine space with coordinates a1,· · · , an. Let Y0 be the hyper-surface in X0[t]

defined by the equation

f =tn+a1tn−1+· · ·+an.

Let R be the resultant of f and its derivative f0. We denote by X the complement of {R = 0} in X0 and by Y := Y0 ×X0 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 Y_{K} :=YK⊗KK

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

that Γ(OY_{K}) 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 pointsxwhereYx is connected, i.e. Yx is the spectrum of a field extensionK0ofKof

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

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. LetK be a factorial domain, A=K[a1,· · ·, an] a polynomial

alge-bra over K. The “generic” polynomial f =tn_{+}_{a}

1tn−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 inB[t] with respect to the prime an.

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

W

y. For proving

the “if” part, it suffices to show that each component ofyis 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 extensionB/F such thatP is equivalent to PB/F|TΓ. LetB0 be the maximal totally

real subfield ofB. We also letN be the common number of isotypic components of y at all places v of F abovep.

### 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 ofF.

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 placev ofF abovep, 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)/2

v , if v =v, N odd

FvN/2, if v =v, N even

It follows from Proposition (7.2.3) that there is a totally real extensionE/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,

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

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

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 apa_{-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. IfN >1,π is not an element ofF, because, otherwise, we would have ordw1(π) = ordw2(π), for any two placesw1, 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 Y0

Fq defined over Fq, q = p

a_{, corresponding to the} _{p}a_{-Weil}

number π. We assume that a is chosen to be sufficiently divisible so that Y_{Fq}0 is
absolutely Γ-simple. Let Y0 := Y_{Fq}0 ⊗_{Fq} _{F}p. Kottwitz [8] proved that there exists

a Γ-linear _{Q}-polarization on Y0. Since the center F(π) of End0_{Γ}(Y0) is totally split
at every place v|p of F, the abelian variety Y0 is therefore Γ-hyper-symmetric, cf.
(3.3.1).

The pairy0 = (H1_{(Y}0_{), P}

K/F|TΓ) is a simply partitioned isocrystal satisfying the

supersingular restriction (S) by (6.2.1). By construction, y0 and y have the same slopes at every place v of F above p.

of [End0_{Γ}(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`0 :F_{`}] = 0 in_{Q}/_{Z}. Look at the local Brauer

invariants of C := End0_{Γ}(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(y0) divides n(y). Let e be the integer such that n(y) =e.n(y0).

It remains to prove that the underlying isocrystals ofyande.y0 are isomorphic as

polarized Γ-linear isocrystals. Indeed, we can modify the polarization on Y :=Y0e
so that e.y0 with this modified polarization is isomorphic toy. For a proof, letS be
the_{Q}-vector space of the symmetric elements in Hom0_{Γ}(Y, Y∗), whereY∗denotes the
dual abelian variety of Y. AsY is Γ-linear hyper-symmetric,S⊗_{Q}_{Q}p is isomorphic

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

S⊗_{Q}_{Q}p, 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

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|pof F, E⊗F Fv 'F N/2

v ,

(2) for each `∈TΓ, there is an F`-isomorphismf` :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 extensionK/E such that

(i) for each place ν of E above p,K ⊗EEν '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 yatv as{λw;w|v} withw 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

for all places w of K above p.

We assume that a is sufficiently divisible. The unique Γ-simple abelian variety
Y_{Fpa}0 up to isogeny corresponding toπadmits a Γ-linear _{Q}-polarization by Kottwitz
[8]. Let Y0 :=Y0

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