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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

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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.

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ABSTRACT

On Hyper-Symmetric Abelian Varieties

Ying Zong Ching-Li Chai, Advisor

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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

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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

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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 integerg ≥1 and a prime numberp. Consider the Siegel modular variety

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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

End0Γ(Y)⊗Q Qp →EndΓ(H1(Y))

is a bijection.

For simplicity we denote byH1(Y) the isocrystalH1

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

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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⊗FpaFp be a Γ-simple hyper-symmetric

abelian variety over Fp, 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 Fpa 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 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 partitionsP = (P

`)

of the integer N = [F(πY0) :F] indexed by `∈TΓ. For each `∈TΓ, 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 Fp, see

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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`)`∈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 | `

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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 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(Y

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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⊗QQp = Γv1 × · · · ×Γvt. Let TΓ denote the following set

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

   

  

ua=σ(a)u, ∀a∈L ud=αL

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,· · ·)

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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 Er 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 Er 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)

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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

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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

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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 ∈2Z, 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−2Z, and π = pa/2, then X

π ⊗kk0 is isogenous to the product

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(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

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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 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 ofpa-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 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

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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

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3.2

Rigidity

Proposition 3.2.1. If Y is Γ-hyper-symmetric, there exists a Γ-hyper-symmetric abelian variety Y0 overFp such that Y0⊗Fpk 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 Fp such

that the isocrystal H1(Y0⊗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⊗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(YK) 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 Yt = Yt ⊗k(t)k(t). As Y is

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Thus the composite map

End0Γ(YK)⊗QQp ,→End

0

Γ(Yt)⊗QQp ,→EndΓ(H

1(Y

t))

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 Y0 be an abelian variety over Fp as in Corollary (3.2.2). Similarly, the

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

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

By the choice of Y0, r and r0 are equal. As any abelian variety over Fp 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 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 Y0 be a Γ-linear polarized abelian variety over a finite field Fpa, such that Y0⊗Fpa Fp '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 Fpa, the map

End0(Y0)⊗QQp →End(H1(Y0))

is bijective.

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

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Let M0 := H1(Y0), and M0 = L

v|pM 0

v be the decomposition defined in (2.7).

The isocrystal Mv0 is Γv-linear and has a decomposition into isotypic components,

Mv0 =M

r∈Q

Mv0(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|Mv0(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−1Φa, i.e. H1(Y0). Hence f End

Γ(H1(Y0)). This implies thatY is

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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

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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

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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 `-`

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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.

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(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,

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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

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

I P //

f

Z>0

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).

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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.

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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

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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

Γ

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

` ∈TΓ.

P //

id

Z>0

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

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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).

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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 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

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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.

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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(Fp)

is realizable by anF-hyper-symmetric abelian variety overFpif 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

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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].

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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

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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 atv1and a finite prime-to-pplace`,` =`; invv1(Γ) = 1/3,

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The Brauer classc= [Dp,∞⊗QF]−[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

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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

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).

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(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:

                                      

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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

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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 Fq such that YFq ⊗Fq Fp ' Y. Suppose that a is sufficiently divisible. The abelian variety YFq is Γ-simple, therefore, YFq ∼isog XFqs ,

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,∞⊗QF]−[Γ] in Br(F).

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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⊗QpK(Fq). Decompose H1(YFq) = 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(YFq)) of theK(Fq)-linear endomorphismπ can be factored as

Y

v

NormLv/K(Fq)fv(T) =

Y

v

Y

w

NormFv/Qp(T −ιw(π))

nw.

Since theQ-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 Qp-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

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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,XFq 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.

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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.

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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, leta

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

aev(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

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

One checks easily that ππ=phc and π is the desiredphc-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

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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.

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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 YK :=YK⊗KK

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 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

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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.

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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,

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(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 pa-Weil

number π. We assume that a is chosen to be sufficiently divisible so that YFq0 is absolutely Γ-simple. Let Y0 := YFq0 ⊗Fq Fp. Kottwitz [8] proved that there exists

a Γ-linear Q-polarization on Y0. Since the center F(π) of End0Γ(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(Y0), 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.

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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`0 :F`] = 0 inQ/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(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 theQ-vector space of the symmetric elements in Hom0Γ(Y, Y∗), whereY∗denotes the dual abelian variety of Y. AsY is Γ-linear hyper-symmetric,S⊗QQp is isomorphic

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

S⊗QQp, 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

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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

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for all places w of K above p.

We assume that a is sufficiently divisible. The unique Γ-simple abelian variety YFpa0 up to isogeny corresponding toπadmits a Γ-linear Q-polarization by Kottwitz [8]. Let Y0 :=Y0

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Bibliography

[1] C. L. Chai, F. Oort. Hypersymmetric Abelian Varieties. Pure and Applied Mathematics Quarterly.2 (2006), 1-27.

[2] C. L. Chai. Hecke orbits on Siegel modular varieties. Progress in Mathematics, Birkhauser. 235 (2004), 71-107.

[3] C. L. Chai. Methods for p-adic Monodromy. J.Inst.Math.Jessieu, to appear. [4] T. Ekedahl. An effective version of Hilbert’s irreduciblity theorem. S´eminaire

de Th´eorie des Nombres, Paris 1988-89. Progress in Math. Birkhauser. 91

(1990), 241-249.

[5] J. Ellenberg, A. Venkatesh. The number of extensions of a number field with fixed degree and bounded discriminant. Ann.of.Math. 163 (2006), 723-741. [6] N. Katz. Slope filtration of F-crystals. Journ.G´eom.Alg.Rennes., Ast´erisque.

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[7] N. Katz, W. Messing. Some consequences of the Riemann Hypothesis for Varieties over Finite Fields. Invent.Math. 23 (1974), 73-77.

[8] R. E. Kottwitz. Points on some Shimura varieties over finite fields.

J.Amer.Math.Soc.2 (1992), 373-444.

[9] D. Mumford. Geometric Invariant Theory. Springer Verlag, 1965.

[10] F. Oort. The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field. Journ.Pure.Appl.Algebra. 3 (1973), 399-408. [11] J. Tate. Classes d’isog´enie de vari´et´es ab´eliennes sur un corps fini (d’apres

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