ON HYPER-SYMMETRIC ABELIAN VARIETIES

YING ZONG

Abstract. Motivated by Oort’s Hecke-orbit conjecture, Chai introduced hyper-symmetric points in the study of fine structures of modular varieties in positivecharacteristics. We prove a necessary and sufficient condition to determine whichNewton polygon stratum of PEL-type contains at least one such point.

1. Introduction

This work is to extend the study of hyper-symmetric abelian varieties initiatedby Chai-Oort [1]. The notion is motivated by the Hecke-orbit conjecture.

For the reduction of a PEL-type Shimura variety, the conjecture claims thatevery orbit under the Hecke correspondences is Zariski dense in the leaf containingit. In positive characterisitic p, the decomposition of a Shimura variety into leavesis 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 isa successive fibration whose fibres are torsors under certain Barsotti-Tate groups.The resulting canonical coordinates, a terminology of Chai, provides the basic toolfor understanding its structure.

Fix an integer g ≥ 1 and a prime number p. Consider the Siegel modular varietyAg in characteristic p. Denote by C(x) the leaf passing through a closed point x. Byapplying the local stabilizer principle at a hyper-symmetric point x, Chai [3] firstgave a very simple proof that the p-adic monodromy of C(x) is big. Later, in theirsolution of the Hecke-orbit conjecture for Ag, Chai and Oort used the techniqueof hyper-symmetric points to deduce the irreducibility of a non-supersingular leaffrom the irreducibility of a non-supersingular Newton polygon stratum, see [2].Note that although hyper-symmetric points distribute scarcely, at least one suchpoint exists in every leaf [1].

Here we are mainly interested in the existence of hyper-symmetric points ofPEL-type. Let us fix a positive simple algebra (Γ, ∗), finite dimensional over Q.Following Chai-Oort [1], we have the definition:

Definition 1.1. A Γ-linear polarized abelian variety (Y,λ) over an algebraicallyclosed field k of characteristic p is Γ-hyper-symmetric, if the natural map

End0Γ(Y )⊗Q Qp → EndΓ(H

1(Y ))

is a bijection.1

2 YING ZONG

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

of this paper is to answer the following question:

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

The answer to the question in general is no; a Newton polygon stratum mustsatisfy certain conditions to contain a Γ-hyper-symmetric point. See (5.3) for anexample when Γ is a real quadratic field split at p, and (5.7) when Γ is a divisionalgebra over a CM-field and the Γ-linear isocrystal M only has slopes 0, 1.

In the main theorem (5.1), we characterize isocrystals of the form H1(Y ) forΓ-hyper-symmetric abelian varieties Y as the underlying isocrystals of partitionedisocrystals with supersingular restriction (S).

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

abelian variety over Fp, where Y ′ is a Γ-simple abelian variety over a finite fieldFpa. By the theory of Honda-Tate, up to isogeny, Y ′ is completely characterizedby its Frobenius endomorphism πY ′. Let F be the center of Γ. Assume that Fpa

is sufficiently large. We show in (3.4) that Y is Γ-hyper-symmetric if and only ifthe extension F (πY ′)/F is totally split everywhere above p, that is,

F (πY ′)⊗F Fv ≃ Fv × · · ·× Fv,

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

Denote by TΓ the set of finite prime-to-p places ℓ of F where Γ is ramified. ToY , one can associate its isocrystal H1(Y ) as well as a family of partitions P = (Pℓ)of the integer N = [F (πY ′) : F ] indexed by ℓ ∈ TΓ. For each ℓ ∈ TΓ, Pℓ is given by

Pℓ(ℓ′) = [F (πY ′)ℓ′ : Fℓ]

with ℓ′ ranging over the places of F (πY ′) above ℓ. The pair (H1(Y ), P ) is thepartitioned isocrystal attached to Y . In particular, we denote by sΓ the pairattached to the unique Γ-simple supersingular abelian variety up to isogeny overFp, see (4.19).

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 itsisocrystal H1(Y ). In essence, the definition (4.11) of partitioned isocrystals is apurely combinatorial formulation of the conditions that Y is F -hyper-symmetricand ρ factors through the endomorphism algebra End0

F (Y ) of the F -linear abelianvariety Y .

The introduction of supersingular restriction (S) (4.20) has its origin in thefollowing example. Assume that F is a totally real number field. If a Γ-linearisocrystal M contains a slope 1/2 component at some place v of F above p, butnot all, then there is no Γ-hyper-symmetric abelian variety Y such that H1(Y )

ON HYPER-SYMMETRIC ABELIAN VARIETIES 3

is isomorphic to M . In the proof of the main theorem (5.1), we treat speciallysupersingular 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 varietyY realizing y goes as follows. Let N be the integer such that P = (Pℓ)ℓ∈TΓ isa family of partitions of N . The Hilbert irreducibility theorem [4] enables us tofind a suitable CM extension K/F of degree N , so that the family of partitions(PK/F, ℓ)ℓ∈TΓ given by

PK/F, ℓ(ℓ′) := [Kℓ′ : Fℓ], ∀ ℓ′ | ℓ

concide with (Pℓ). Then a simple formula (7.1) gives directly a pa-Weil numberπ for a certain integer a ≥ 1, such that K = F (π) and the slopes of M at aplace v of F above p are equal to λw = ordw(π)/ordw(pa), for w|v. Let Y ′ be theunique abelian variety up to Γ-isogeny corresponding to π. For some integer e,(Y ′)e⊗Fpa

Fp equipped with a suitable polarization is a desired Γ-hyper-symmetricabelian variety.

The organization of this paper is as follows. In section 2 we set up the notationsand review the fundamentals of isocrystals with extra structures, Dieudonne’stheorem on the classification of isocrystals and the Honda-Tate theory. In section3, we show that every Γ-hyper-symmetric abelian variety is isogenous to an abelianvariety defined over Fp (3.2). Then we prove a criterion of hyper-symmetry interms of endomorphism algebras (3.4). In the next section, we define partitionsand partitioned isocrystals. The main theorem (5.1) is stated in section 5. Severalexamples are provided to illustrate how to determine which data of slopes arerealizable by hyper-symmetric abelian varieties. The proof of (5.1) is divided intotwo parts. The “only-if” part, in section 6, shows that to every Γ-hyper-symmetricabelian variety Y , one can associate a partitioned isocrystal y. We prove that ysatisfies the supersingular restriction (S). A key ingredient of the proof is that thecharacteristic polynomial of the Frobenius endomorphism of H1(YFpa

) has rationalcoefficients. In section 7 we prove the inverse, the “if” part.

Acknowledgement. This thesis is under the supervision of my advisor Ching-LiChai, to whom I thank for his constant and patient support.

2. Notations and Generalities

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

2.1. Let Γ be a positive simple algebra, finite dimensional over the field of rationalnumbers. We fix a positive involution ∗ on Γ. Let F be the center of Γ; F is eithera totally real number field or a CM field. Let v1, · · · , vt be the places of F abovep. We have

Γ⊗Q Qp = Γv1 × · · ·× Γvt .

4 YING ZONG

Let TΓ denote the following set

TΓ = {ℓ ∈ Spec(OF )| ℓ ! p, ℓ = (0), invℓ(Γ) = 0}.

2.2. Recall the computation of Brauer invariants. Let K be a finite extension ofQp. Let A be a central simple K-algebra of dimension d2. By Hasse, A containsa d-dimensional unramified extension L/K such that for an element u ∈ A, thevectors 1, u, · · · , ud−1 form an L-basis of A, and

{ua = σ(a)u, ∀a ∈ Lud = α ∈ L

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

2.3. If k is a perfect field of characteristic p, we denote by W (k) the ring of Wittvectors of k. Let K(k) be the fraction field of W (k). The Frobenius automorphismof k induces by functoriality an automorphism σ of W (k), namely,

σ(a0, a1, · · · ) = (ap0, ap1, · · · )

for all a0, a1, · · · ∈ k.

2.4. An isocrystal over k is a finite dimensional K(k)-vector space M equippedwith a σ-linear automorphism Φ. A morphism f : (M,Φ) → (M ′,Φ′) is a K(k)-linear map f : M → M ′ such that fΦ = Φ′f . Isocrystals over k form an abeliancategory.

2.5. Let k be an algebraic closure of k, a perfect field of characteristic p. We havethe fundamental theorem of Dieudonne, cf. Kottwitz [8]:

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

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

Er = (K(k)[T ]/(T b − pa), T )

where r = a/b is a rational number with (a, b) = 1, b > 0. The endomor-phism ring of Er is a central division algebra over Qp with Brauer invariant−r ∈ Q/Z.

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

M =⊕

r∈QM(r)

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

M(r)⊗K(k) K(k) ≃ Emrr

ON HYPER-SYMMETRIC ABELIAN VARIETIES 5

for an integer mr.

The rational numbers occurred in the decomposition M =⊕

r∈QM(r) are calledthe slopes of M . If all slopes are non-negative, the isocrystal is effective.

2.6. A polarization of weight 1 or simply a polarization of an isocrystal M is asymplectic 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. Let Γ be as in (2.1). A Γ-linear isocrystal over k is an isocrystal (M,Φ) overk together with a ring homomorphism i : Γ → End(M,Φ). The following variantof Dieudonne’s theorem is proven in Kottwitz [8],

(1) The category of Γ-linear isocrystals over k is semi-simple. It is equivalentto 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 parametrizedby r ∈ Q, whose endomorphism ring is a central division algebra over Fv,with Hasse invariant −[Fv : Qp]r − invv(Γ) in the Brauer group Br(Fv).

If M is a Γ-linear isocrystal, and M = Mv1× · · ·×Mvt is the decomposition definedin (1), we call the slopes of Mv the slopes of M at v and define the multiplicity ofa slope r at v by

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

2.8. A Γ-linear polarized isocrystal is a quadruple (M,Φ, i,ψ), where (M,Φ) is anisocrystal, i : Γ → End(M,Φ) is a ring homomorphism, and ψ is a polarization onM 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 eachplace v of F above p, arranged in increasing order, are symmetric about 1/2. IfF is a CM field, the slopes at v and v collected together, arranged in increasingorder, are symmetric with respect to 1/2.

2.9. Recall that a morphism of abelian varieties f : X → X ′ is an isogeny if itis surjective with a finite kernel. Let X be an abelian variety over a finite fieldk = Fpa. The relative Frobenius morphism

FX/k : X → X(p)

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

simple abelian variety, the Frobenius endomorphism πX is a pa-Weil number, thatis, an algebraic integer π such that for every complex imbedding ι : Q(π) ↪→ C,one has

| ι(π) |= pa/2.

6 YING ZONG

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

(1) The map X +→ πX defines a bijection from the isogeny classes of simpleabelian varieties over k to the conjugacy classes of pa-Weil numbers.

(2) The endomorphism algebra End0(Xπ) of a simple abelian variety Xπ cor-responding to π is a central division algebra over Q(π). One has

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

(a) If a ∈ 2Z, and π = pa/2, then Xπ is a supersingular elliptic curve,whose endomorphism algebra is Dp,∞, the quaternion division algebraover Q, ramified exactly at p and the infinity.

(b) If a ∈ Z− 2Z, and π = pa/2, then Xπ ⊗k k′ is isogenous to the productof two supersingular elliptic curves, where k′ is the unique quadraticextension of k.

(c) If π is totally imaginary, the division algebra D = End0(Xπ) is unram-ified away from p. For a place w of Q(π) above p, the local invariantof D at w is

invw(D) = −ordw(π)/ordw(pa).

2.10. A Γ-linear polarized abelian variety is a triple (Y,λ, i) consisting of a polar-ized abelian variety (Y,λ) and a ring homomorphim i : Γ → End0(Y ). We requirethat i is compatible with the involution ∗ and the Rosati involution on End0(Y )associated to the polarization λ. The category of Γ-linear polarized abelian vari-eties up to isogeny is semi-simple. In particular, any such abelian variety Y admitsa Γ-isotypic decomposition,

Y ∼Γ-isog Ye11 × · · ·× Y er

r

where each Yi is Γ-simple and for i = j, Yi and Yj are not Γ-isogenous. For eachi, there exist a simple abelian variety Xi and an integer ei, such that Yi ∼isog X

eii .

We say Yi is of type Xi.

2.11. Let Y be a Γ-simple abelian variety of type X , i.e. Y ∼isog Xe, for aninteger e. Let Z0, Z be the center of End0(X) and End0

Γ(Y ), respectively. Thereis the following relation [8],

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

Γ(Y ) : Z]1/2[Z : Q].

One deduces that the Q-dimension of any maximal etale sub-algebra of End0(Y )is equal to [Γ : F ]1/2 times the Q-dimension of any maximal etale sub-algebra ofEnd0

Γ(Y ).

ON HYPER-SYMMETRIC ABELIAN VARIETIES 7

2.12. Let k = Fpa be a finite field. Kottwitz [8] proved a variant of the theoremof Honda-Tate:

(1) The map Y +→ πY is a bijection from the set of isogeny classes of Γ-simpleabelian varieties over k to the F -conjugacy classes of pa-Weil numbers.

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

responding to π is a central division algebra over F (π). Let Xπ be a simpleabelian variety up to isogeny corresponding to π as in (2.9); Yπ is of typeXπ. Let D = End0(Xπ), C = End0

Γ(Yπ). Then one has the equality

[C] = [D ⊗Q(π) F (π)]− [Γ⊗F F (π)]

in the Brauer group of F (π), and

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

3. A Criterion of Hyper-Symmetry

Let Y be a Γ-linear polarized abelian variety over an algebraically closed field kof characteristic p, and let Y ∼Γ-isog Y

e11 ×· · ·×Y er

r be the Γ-isotypic decompositionof Y , cf. (2.10). For the rest, H1(Y ) stands for the first crystalline cohomology ofY , H1

crys(Y/W (k))⊗Z Q.

Lemma 3.1. The abelian variety Y is Γ-hyper-symmetric if and only if each Yi isΓ-hyper-symmetric and for any place v of F above p, for different i, j, Yi and Yj

have no common slopes at v.

Proof. This is clear. !Proposition 3.2. If Y is Γ-hyper-symmetric, there exists a Γ-hyper-symmetricabelian variety Y ′ over Fp such that Y ′ ⊗Fp

k is Γ-isogenous to Y .

We first prove a weaker result.

Corollary 3.3. There is a Γ-hyper-symmetric abelian variety Y ′ over Fp such thatthe isocrystal H1(Y ′ ⊗Fp

k) is isomorphic to H1(Y ).

Proof. There is a Γ-linear polarized abelian variety YK over a finitely generatedsubfield K such that YK ⊗K k is isomorphic to Y and End(YK) = End(Y ).

Choose a scheme S, irreducible, smooth, of finite type over the prime field, sothat, if η denotes the generic point of S, k(η) = K. We may and do assume thatYK extends to an abelian scheme Y over S.

By a theorem of Grothendieck-Katz [6], the function assigning any point x of Sthe Newton polygon of the isocrystal H1(Yx) is constructible. Let S ′ be the opensubset consisting of points x with the generic Newton polygon, i.e. the same New-ton polygon with that of H1(Y ). As S ′ is regular, the canonical homomorphismEnd(YS′) → End(YK) is an isomorphism. So there is a well defined specializationmap sp : End(YK) → End(Yt) for any point t ∈ S ′. By the rigidity lemma 6.1

8 YING ZONG

[9], sp is injective. Let t be a closed point of S ′ and Yt = Yt ⊗k(t) k(t). As Y isΓ-hyper-symmetric, End0

Γ(YK)⊗QQp and EndΓ(H1(Yt)) have the same dimension.Thus the composite map

End0Γ(YK)⊗Q Qp ↪→ End0

Γ(Yt)⊗Q Qp ↪→ EndΓ(H1(Yt))

is bijective. It follows that Yt is a desired Γ-hyper-symmetric abelian variety overk(t) ≃ Fp. !Proof. of (3.2). Recall that by Grothendieck [10], an abelian variety Y over analgebraically closed field k of characteristic p is isogenous to an abelian varietydefined 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 issimple and Y ∼isog Xe for an integer e. Let Z0, Z denote respectively the centerof End0(X) and End0

Γ(Y ). The dimension r of any maximal etale sub-algebra ofEnd0(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(H

1(Y )) : E]1/2[E : Qp],

since Y is Γ-hyper-symmetric. In the above, E denotes the center of EndΓ(H1(Y )).Let Y ′ be an abelian variety over Fp as in Corollary (3.3). Similarly, the dimen-

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

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

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

By the choice of Y ′, r and r′ are equal. As any abelian variety over Fp hassufficiently many complex multiplication (2.9), we have r = r′ = 2.dim(Y ′). Thisfinishes the proof. !

In the following we prove a criterion of Γ-hyper-symmetry in terms of the centerZ of End0

Γ(Y ).

Proposition 3.4. A Γ-linear polarized abelian variety Y over Fp is Γ-hyper-symmetric if and only if the Fv-algebra Z ⊗F Fv is completely decomposed, i.e.,Z ⊗F Fv ≃ Fv × · · ·× Fv, for every place v of F above p.

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

Fp ≃ Y and End(Y ′) = End(Y ). The center Z can be identified withF (π), the sub-algebra generated by the Frobenius endomorphism of Y ′. By Tate[11], over Fpa, the map

End0(Y ′)⊗Q Qp → End(H1(Y ′))

ON HYPER-SYMMETRIC ABELIAN VARIETIES 9

is bijective.

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

EndΓ(H1(Y ′)) = EndΓ(H

1(Y )).

Let M ′ := H1(Y ′), and M ′ =⊕

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

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

M ′v =

⊕

r∈QM ′

v(r).

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

EndΓv(M′v(r)) = EndΓv(M

′v(r)⊗K(Fpa ) K(Fp)),

for any v|p, and r ∈ Q.

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

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

to a direct product Fv × · · ·× Fv with the number of factors equal to the numberof Γv-simple components of M ′

v(r)⊗K(Fpa ) K(Fp).

Therefore, if Y is Γ-hyper-symmetric, the F -algebra Z = F (π) is completelydecomposed at every place v of F above p. Conversely, if Z/F is completelydecomposed everywhere above p, any Γ-linear endomorphism f of the isocrystal(H1(Y ),Φ) commutes with the operator π−1Φa, and thus stabilizes the invariantsub-space of π−1Φa, i.e. H1(Y ′). Hence f ∈ EndΓ(H1(Y ′)). This implies that Yis Γ-hyper-symmetric. !

4. Partitions and Partitioned Isocrystals

Definition 4.1. Let N be a positive integer. A partition of N with support in afinite set I is a function P : I → Z>0, such that

∑i∈I P (i) = N .

Definition 4.2. 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 functionP : X → Z>0 such that for each s ∈ S, P | f−1(s) is a partition of N with supportin f−1(s).

XP

f

Z>0

S

Definition 4.3. Let P be an S-partition of N with structural map f : X → S.For any map g : S ′ → S, the pull-back partition g∗(P ) = P ◦ p is an S ′-partitionof N , where p : X ×S S ′ → X is the projection.

10 YING ZONG

Definition 4.4. Let Pi be an Si-partition of N , i = 1, 2. We say that P1 isequivalent to P2 if there exist a bijection u : S1 → S2 and a u-isomorphismg : X1 → X2 such that P1 = P2 ◦ g.Definition 4.5. Consider S-partitions Pi of Ni, i = 1, 2. Let fi : Xi → Z>0 bethe structural maps. The sum P1 ⊕ P2 is the following S-partition P of N1 +N2,

X1

∐X2

P

f

Z>0

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

Example 4.6. Let S be a scheme, f : X → S a finite etale cover of rank N . Wedefine an S-partition P : X → Z>0 of N associated to f by

P (x) = [k(x) : k(f(x))], ∀x ∈ X.

Example 4.7. Let F be a number field, K/F a finite field extension of degreeN . 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 primeN, if w = (0)

This PK/F defines an S-partition of N . The most interesting case is K = F (πY ),the field generated by the Frobenius endomorphism πY of a Γ-simple non-super-singular abelian variety Y over a finite field k (2.12). We study this example inmore 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 ifw = w. Recall that TΓ (2.1) denotes the set of finite prime-to-p places ℓ of Fwhere Γ is ramified. The restriction PK/F |TΓ (4.3) is equivalent to a TΓ-partition{Pℓ : [1, dℓ] → Z>0| ℓ ∈ TΓ} of N = [K : F ], which satisfies the following property

{Pℓ(2i− 1) = Pℓ(2i), for i ∈ [1, c1(ℓ)]Pℓ(i) is even, for i ∈ [2c1(ℓ) + 1, dℓ]

where dℓ = Card(f−1(ℓ)), 2c1(ℓ) = Card({w ∈ f−1(ℓ)| w = 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

{Pℓ : [1, dℓ] → Z>0| ℓ ∈ TΓ}which satisfies the property

{Pℓ(2i− 1) = Pℓ(2i), if ℓ = ℓ, i ∈ [1, c1(ℓ)]Pℓ(i) = Pℓ(i), if ℓ ! ℓ

ON HYPER-SYMMETRIC ABELIAN VARIETIES 11

where dℓ = Card(f−1(ℓ)). If ℓ = ℓ, 2c1(ℓ) := Card({w ∈ f−1(ℓ)| w = w}).

Definition 4.8. A TΓ-partition P of an integer N is said to be of CM-type or aCM-type partition if it is equivalent to the pull-back partition PK/F |TΓ for a CMfield K of degree N over F .

Partitions of CM-type can be characterized as follows.

Proposition 4.9. A TΓ-partition P = {Pℓ; ℓ ∈ TΓ} of an integer N is of CM-typeif and only if it satisfies the properties in (4.7) (a) or (b).

For a proof, we need the following lemma.

Lemma 4.10. Let D be a number field, T a set of maximal ideals in OD. For anyT -partition R : I → Z>0 of an integer N with support in the fibres of u : I → T ,

IR

u

Z>0

T

there is a finite etale cover ft : Xt → Spec(ODt) of rank N , such that the partitionassociated to ft restricted to {t} is equivalent to R|u−1(t), for every t ∈ T .

Proof. Here Dt denotes a local field, the completion of D with respect to the t-adicabsolute value. For each i ∈ I, t = u(i), let Xi be the unique connected etale coverof Spec(ODt) of rank R(i). The desired scheme Xt can be chosen as

Xt =∐

i∈u−1(t)

Xi,

for t ∈ T . !

Proof. of (4.9). It remains to prove the “if”-part of the Proposition (4.9). Let Pbe a given TΓ-partition of N satisfying the conditions of (4.7) (a) or (b).

(a). Assume first that F is a totally real number field. We define a TΓ-partitionR of the integer N/2,

Rℓ(j) =

{Pℓ(2j), j ∈ [1, c1(ℓ)]Pℓ(j + c1(ℓ))/2, j ∈ [c1(ℓ) + 1, dℓ − c1(ℓ)].

For each ℓ ∈ TΓ, let

Xℓ =∐

j∈[1,dℓ]

Xj

be the etale cover of Spec(OFℓ) constructed in Lemma (4.10) corresponding to the

partition R. Then by Proposition (7.5), there exists a totally real extension E

12 YING ZONG

of F of degree N/2, such that Xℓ is isomorphic to the spectrum of OE ⊗OF OFℓ.

Define a scheme Yℓ over Xℓ,

Yℓ :=∐

j∈[1,c1(ℓ)]

(Xj

∐Xj)

∐

j∈[c1(ℓ)+1,dℓ−c1(ℓ)]

Yj

where, for j ∈ [c1(ℓ) + 1, dℓ − c1(ℓ)], Yj denotes the unique connected etale coverof Xj of rank 2. We apply weak approximation to get a CM quadratic extensionK of E, so that for each ℓ ∈ TΓ, Yℓ is isomorphic to the spectrum of the ringOK ⊗OF OFℓ

. One verifies that K is a desired solution.

(b). Next assume that F is totally imaginary. Let F0 be its maximal totally realsubfield, and T0 be the image of TΓ under the morphism Spec(OF ) → Spec(OF0).From the TΓ-partition P we construct a T0-partition of R of the same integer Nas follows. If ℓ0 = ℓℓ is split in F ,

Rℓ0(j) := Pℓ(j), j ∈ [1, dℓ].

If ℓ0 is inert or ramified in F , ℓ0 = ℓ|F0, ℓ ∈ TΓ,

Rℓ0(j) :=

{2.Pℓ(2j), j ∈ [1, c1(ℓ)]Pℓ(j + c1(ℓ)), j ∈ [c1(ℓ) + 1, dℓ − c1(ℓ)]

By Proposition (7.5), for a suitable totally real extension E/F0 of degree N , onehas

(i) if ℓ0 = ℓℓ is split,

E ⊗F0 (F0)ℓ0 ≃∏

j∈[1,dℓ]

Ej ,

where Ej is the unique unramified extension of (F0)ℓ0 of degree Rℓ0(j).

(ii) if ℓ0 = ℓ|F0 is inert or ramified in F ,

E ⊗F0 (F0)ℓ0 ≃∏

j∈[1,dℓ−c1(ℓ)]

Ej

where Ej is the unique unramified extension of Fℓ of degree Rℓ0(j)/2, forj ∈ [1, c1(ℓ)], and is an extension of (F0)ℓ0 of degree Rℓ0(j) linearly disjointwith Fℓ, for j ∈ [c1(ℓ) + 1, dℓ − c1(ℓ)].

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

is equivalent to P . !

Definition 4.11. A Γ-linear polarized simply partitioned isocrystal x is a pair(M,P ) consisting of a polarized Γ-linear isocrystal M and a TΓ-partition of an

ON HYPER-SYMMETRIC ABELIAN VARIETIES 13

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

IP

f

Z>0

TΓ

which satisfies the following conditions:

(SPI1) There exists a constant n(x) such that for every place v of F above p, theΓv-linear isocrystal Mv has N(x) isotypic components, and the multiplicity(2.7) of each component is equal to n(x).

(SPI2) For every ℓ′ ∈ I, n(x).invf(ℓ′)(Γ)P (ℓ′) = 0 in Q/Z.We shorten Γ-linear polarized simply partitioned isocrystal to simply partitioned

isocrystal if this causes no confusion. We call M the underlying isocrystal, P thedefining partition of x = (M,P ). The dimension, slopes, multiplicity n(x), Newtonpolygons, and polarizations of x will be understood to be those of M .

Definition 4.12. Two simply partitioned isocrystals x, y are said to be equivalentif their isocrystals are isomorphic and their partitions are equivalent (4.4).

Definition 4.13. Let x = (M,P ) be a simply partitioned isocrystal. For anynon-negative integer a, we define the scalar multiple a.x to be (Ma, P ); a.x is asimply partitioned isocrystal. If a ≥ 1, then N(a.x) = N(x), n(a.x) = a.n(x).If there exist an integer a > 1 and a simply partitioned isocrystal y such thatx = a.y, then x is called divisible.

Definition 4.14. There is a partially defined sum operation on the set of sim-ply partitioned isocrystals. Suppose that the simply partitioned isocrystals xi =(Mi, Pi), i = 1, 2, satisfy the following assumptions:

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

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

Then we define the sum x1+x2 to be the pair (M1⊕M2, P1⊕P2), see (4.5); x1+x2

is a simply partitioned isocrystal.

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

x1 + x2 = x2 + x1.

If x1 + x2 and (x1 + x2) + x3 are both defined, then x2 + x3 and x1 + (x2 + x3) arealso defined, and the associativity holds, i.e.

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

Definition 4.15. A Γ-linear polarized partitioned isocrystal is a finite collection ofsimply partitioned isocrystals x = {xa; a ∈ A}, such that the following conditionsare satisfied.

14 YING ZONG

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

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

We call x a partitioned isocrystal if no confusion arises. Each xa is called a com-ponent of x. The direct sum of the underlying isocrystals of xa, M =

⊕a∈A Ma,

is called the underlying isocrystal of x.

Definition 4.16. Two partitioned isocrystals x = {xa; a ∈ A} and y = {yb; b ∈ B}are equivalent if there exists a bijection u : A → B such that each xa is equivalentto yu(a).

Up to equivalence, every partitioned isocrystal x = {xa; a ∈ A} can be naturallyindexed by the multiplicities of its simple components, cf. (PI2) (4.15).

Definition 4.17. Let x = {xa; a ∈ A} be a partitioned isocrystal (4.15). For anynon-negative integer h, we define the scalar multiple h.x to be {h.xa; a ∈ A}. Apartitioned isocrystal is divisible if x = h.y for some integer h > 1 and a partitionedisocrystal y, cf. (4.13).

Definition 4.18. The sum operation defined for simply partitioned isocrystalscan be extended to partitioned isocrystals. Given two partitioned isocrystals x ={xa; a ∈ A}, y = {yb; b ∈ B} satisfying the following restriction,

(N) For each pair a ∈ A, b ∈ B, and for each place v of F above p, xa and ybhave no common slopes at v.

we define their joint, s = x∨

y, another partitioned isocrystal, as follows. LetC be the finite set of positive integers c such that either x or y or both has acomponent whose multiplicity is c. This set C will parametrize the components ofs. In other words, we have

s = {sc; c ∈ C}

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

(ii) If both x and y have components, say xa, yb, such that n(xa) = n(yb) = c,one defines sc to be the sum xa + yb (4.14).

Whenever it is defined, the joint operation is clearly commutative and associativeup to canonical equivalence.

Definition 4.19. A simply partitioned isocrystal sΓ. We define sΓ to be thesimply 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

ON HYPER-SYMMETRIC ABELIAN VARIETIES 15

TΓ-partition of 1, i.e. P (ℓ) = 1, for any ℓ ∈ TΓ.

TΓP

id

Z>0

TΓ

At every place v of F above p, sΓ is isotypic of slope 1/2 and its multiplicity n(sΓ)is equal to the order eΓ of the class [Dp,∞ ⊗Q F ]− [Γ] in Br(F ), see (6.1).

Definition 4.20. Partitioned Isocrystal with (S)-Restriction. A partitionedisocrystal x = {xa; a ∈ A} is said to satisfy the supersingular restriction (S) if thereexist an integer h ≥ 0 and a partitioned isocrystal y = {yb; b ∈ B} such that

(S1) x = h.sΓ∨y,

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

(S3) the partition Pb of each component yb = (Mb, Pb) is of CM-type (4.8).

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

Remarks 4.21. (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 (4.18).

(b). The condition (S3) is a purely combinatorial condition, see the characteri-zation of CM-type partitions in (4.9).

5. Main Theorem and Examples

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

Now we formulate our criterion for a Γ-linear polarized isocrystal to be realizableby a Γ-hyper-symmetric abelian variety.

Theorem 5.1. An effective Γ-linear polarized isocrystal M is isomorphic to theDieudonne isocrystal H1(Y ) of a Γ-hyper-symmetric abelian variety Y if and onlyif M underlies an (S)-restricted partitioned isocrystal.

The theorem will be proven in the next two sections. Here we apply it to someexamples of simple algebras Γ for which we work out explicitly the slopes andmultiplicities of the Γ-hyper-symmetric abelian varieties. Note that themultiplicityis defined in (2.7).

Example 5.2. (Siegel) Γ = Q. As TΓ is empty, the supersingular restriction (S)is reduced to (S1) and (S2). A non-divisible simply partitioned isocrystal withoutslope 1/2 component is called balanced in the terminology of Chai-Oort [1]. Ingeneral, any simply partitioned isocrystal x can be expressed uniquely as

x = h.sΓ +m.y

16 YING ZONG

with integers h,m ≥ 0 and a balanced isocrystal y. One deduces that any Newtonpolygon of the form

ρ0.(1/2) +∑

i∈[1,t]

(ρi.(λi) + ρi.(1− λi))

can be realized by a hyper-symmetric abelian variety, where λi ∈ [0, 1/2) arepairwise distinct slopes, ρ0 = mult(1/2), ρi = mult(λi) are multiplicities. Thisexample recovers the Proposition (2.5) of Chai-Oort [1].

Example 5.3. Let F be a real quadratic field split at p, p = v1v2. The followingslope data {

2.(1/2), at v11.(0) + 1.(1), at v2

admit no hyper-symmetric point.

Example 5.4. Let Γ = F be a totally real field of degree d over Q. The restriction(S) is reduced to (S1) and (S2).

The isocrystal sF is isotypic of slope 1/2 at every place v of F . The multiplicityis n(sF ) = eF , the order of the class [Dp,∞ ⊗Q F ] in the Brauer group of F , cf.(6.1).

Any simply partitioned isocrystal y without slope 1/2 component can be de-composed as a finite sum

y = y1 + · · ·+ yn,

where each yi has two isotypic components at every place v|p.Let z be one of the yi’s , and let {λv, 1− λv} be the two slopes of z at v. Then

the multiplicity n(z) is a common multiple of the denominators of [Fv : Qp]λv,where v runs over the places of F above p.

As a consequence, an F -linear polarized isocrytal M of dimension 2d over K(Fp)is realizable by an F -hyper-symmetric abelian variety over Fp if and only if theslopes 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 multiplicity1. These λv are such that [Fv : Qp]λv ∈ Z.

Example 5.5. Let Γ = F be a CM field, [F : Q] = 2d. The restriction (S) isreduced to (S1).

The isocrystal sF is isotypic of slope 1/2 at every place v of F above p. Themultiplicity is n(sF ) = eF , the order of the class [Dp,∞ ⊗Q F ] in the Brauer groupof F .

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

y = y1 + · · ·+ yn,

ON HYPER-SYMMETRIC ABELIAN VARIETIES 17

where each yi has either one or two isotypic components. More explicitly, for afixed z = yi,

(i) if z has one isotypic component at every place v|p, the slopes are such thatλv + λv = 1. In particular, λv = 1/2, if v = v. The multiplicity n(z) is amultiple 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 = v, the slopes are either

{λv, 1− λv, at vλv, 1− λv, at v

or {µv, νv, at v1− µv, 1− νv, at v

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

Example 5.6. Let Γ be a definite quaternion division algebra over Q. We assumethat Γ is ramified exactly at the infinity and a prime ℓ different from p. HenceTΓ = {ℓ} and invℓ(Γ) = 1/2.

The partitioned isocrystal sΓ is isotypic of slope 1/2 with multiplicity n(sΓ) = 2,because the order eΓ of the class [Dp,∞]− [Γ] in the Brauer group of Q is 2.

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

Pℓ : [1, dℓ] → Z>0

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

n(y).Pℓ(i).1/2 ∈ Z, for all i ∈ [1, dℓ].

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

Pℓ(2i− 1) = Pℓ(2i), i ∈ [1, c1(ℓ)]Pℓ(i) is even, i ∈ [2c1(ℓ) + 1, dℓ]

for some integer c1(ℓ) ∈ Z≥0.

Now let M be any effective Γ-linear polarized isocrystal satisfying the condition(SPI1) and without slope 1/2 component. We claim that M underlies an (S)-restricted simply partitioned isocrystal y. In fact, one can choose y = (M,Pl),where dℓ = 1, Pℓ(1) = N(y), and N(y) is the number of isotypic components ofM . Note that N(y) is an even integer because M is polarized and has no slope1/2 component.

With this choice of partition Pl, the simply partitioned isocrystal y decomposesas a finite sum

y = y1 + · · ·+ ym,

18 YING ZONG

where each yi has exactly two isotypic components with slopes {λi, 1 − λi}. Themultiplicity n(y) is a multiple of the common denominator of the λi’s.

For example, let us work out the slopes and multiplicities of all (S)-restrictedpartitioned isocrystals of dimension 12 over K(Fp). There are exactly five Newtonpolygons which are realizable by 6-dimensional Γ-hyper-symmetric abelian vari-eties:

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 5.7. Let F be a CM field, and Γ be a positive central division algebraover F . We make the following assumptions on Γ,

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

(ii) Γ is ramified exactly at v1 and a finite prime-to-p place ℓ, ℓ = ℓ; invv1(Γ) =1/3, invℓ(Γ) = 2/3.

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

invν(c) =

⎧⎪⎪⎨

⎪⎪⎩

−1/3, if ν = v1−1/2, if ν = v2, v2−2/3, if ν = ℓ0, 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 isotypiccomponents, n(y) the multiplicity of y at each place v ∈ {v1, v2, v2}. Denote by Pℓ

the defining partition of y

Pℓ : [1, dℓ] → Z>0.

In this case, the condition (SPI2) says that

n(y)Pℓ(i).2/3 ∈ Z, for all i ∈ [1, dℓ].

If y is (S)-restricted, then by (4.9), its partition Pl satisfies the condition

Pℓ(2i− 1) = Pℓ(2i), ∀ i ∈ [1, c1(ℓ)],

for some integer c1(ℓ), with 0 ≤ 2c1(ℓ) ≤ dℓ.

ON HYPER-SYMMETRIC ABELIAN VARIETIES 19

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

ξ =

⎧⎨

⎩

1.(0) + 1.(1), at v11.(0) + 1.(1), at v21.(0) + 1.(1), at v2

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 multiplicityof every isotypic component is n = 1. But there is no partition Pℓ of N = 2, suchthat n.Pℓ(i).2/3 ∈ Z.

Now we compute the Newton polygons of all (S)-restricted partitioned isocrys-tals of dimension 72 over K(Fp). By (S1), we can write x = h.sΓ

∨y. Note that

the dimension of sΓ is 72. One has either x = sΓ or x = y. Consider the casex = y and write

y = {yb; b ∈ B},where yb are the simple components of y. Comparing the dimensions of yb and y,one has

72 = [Γ : F ]1/2[F : Q]∑

b∈B

N(yb)n(yb),

where N(yb) denotes the number of isotypic components, n(yb) the multiplicity, ofyb at each place of F above p. Since [Γ : F ]1/2 = 3, [F : Q] = 4, this equation isreduced to

6 =∑

b∈B

N(yb)n(yb).

One verifies that this condition forces that y is simply partitioned, N(y) = 2, andn(y) = 3. Here we list all the realizable Newton polygons as follows.

(i) The slopes at v1 are one of:⎧⎨

⎩

0, 11/3, 2/31/6, 5/6

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

⎪⎪⎪⎪⎪⎩

0, 1; 1/3, 2/30, 1/3; 1, 2/30, 2/3; 1, 1/31, 1/3; 0, 2/31, 2/3; 0, 1/31/3, 2/3; 0, 1

20 YING ZONG

6. Proof of the “only-if” part of (5.1)

Given a Γ-hyper-symmetric abelian variety Y , we let Y ∼Γ-isog Ye11 × · · ·× Y er

r

be the Γ-isotypic decomposition. By (3.1), for the only-if part, we only needto show that each H1(Yi) underlies an (S)-restricted partitioned isocrystal xi.Indeed, if this is proved, H1(Y ) is isomorphic to the underlying isocrystal ofx = {e1.x1}

∨· · ·

∨{er.xr}.

From now on, we assume that Y is Γ-simple. Let q = pa and YFq be a Γ-linearpolarized abelian variety over Fq such that YFq ⊗Fq Fp ≃ Y . Suppose that a issufficiently divisible. The abelian variety YFq is Γ-simple, therefore, YFq ∼isog Xs

Fq,

for some XFq simple over Fq. Let π denote the Frobenius endomorphism of YFq aswell as that of XFq . Let K = F (π).

Proposition 6.1. The pair x = (H1(Y ), PK/F |TΓ) associated to the Γ-simplehyper-symmetric abelian variety Y is a simply partitioned isocrystal satisfying thesupersingular restriction (S). More explicitly,

(a) if π is totally real, then x = sΓ is isotypic of slope 1/2 with multiplicityn(sΓ) equal to the order of Brauer class [Dp,∞ ⊗Q F ]− [Γ] in Br(F ).

(b) if π is totally imaginary, then x has N(x) = [K : F ] isotypic componentsat 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 . LetC := End0

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

H1(YFq) =⊕

v|p

Mv

as in (2.7). Each Mv is a free Lv-module, by the next lemma and 11.5 [8]. Weconsider the characterisitic polynomial fv(T ) of π as an Lv-linear transformationof Mv. Since Y is Γ-hyper-symmetric, by (3.4),

fv(T ) =∏

w|v

(T − ιw(π))nw

is a product of linear polynomials, where ιw : F (π) ↪→ Fv denote the F -embeddingsof F (π) into Fv indexed by the places w. Thus the characterisitic polynomialf(T ) = det(T − π|H1(YFq)) of the K(Fq)-linear endomorphism π can be factoredas ∏

v

NormLv/K(Fq)fv(T ) =∏

v

∏

w

NormFv/Qp(T − ιw(π))nw .

Since the Q-embeddings ιu of F (π) into Qp are one-to-one correspondence withthe set of triples u = (v, w, τ) consisting of a place v of F above p, a place w of

ON HYPER-SYMMETRIC ABELIAN VARIETIES 21

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

f(T ) =∏

u

(T − ιu(π))nw .

By Katz-Messing [7], the polynomial f(T ) ∈ Z[T ], so nw = n is independent of theplace w, and thus, is equal to 2.dim(Y )/[F (π) : Q]. Because fv(T ) has N differentirreducible factors, i.e. T−ιw(π), H1(Y ) has N isotypic components at every placev of F above p [8]. By the dimension formula in (2.12), the multiplicity of eachisotypic component is equal to

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

Observe that for every place ℓ′ of K above a place ℓ ∈ TΓ, the local invariant of Cat ℓ′ is

invℓ′(C) = −invℓ(Γ)[Kℓ′ : Fℓ].

It certainly follows that order([C])invℓ(Γ)P (ℓ′) = 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 isocrystalH1(Y ) underlies the simply partitioned isocrystal sΓ (4.19). At every place v of Fabove p, sΓ is isotypic of slope 1/2.

If π is totally imaginary, the field K = F (π) is a CM extension of F ; so thecondition (S3) is a priori satisfied. In case that F is a totally real number field,the slopes of H1(Y ) at a place v of F above p, if arranged in increasing order, aresymmetric with respect to 1/2. As there are N = [K : F ] of them, and N is even,H1(Y ) contains no slope 1/2 component. The proof is now complete. !

The following lemma is certainly well known and an analogous statement forℓ-adic cohomology can be found in Mumford’s book on abelian varieties.

Lemma 6.2. 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 + nbe 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 rationalcoefficients. Hence we can find a polynomial f(T ) ∈ Q[T ] without constant term,such that the nilponent part n = f(π). The image of ℓn, for a sufficiently divisibleinteger ℓ, is a proper sub-abelian variety of X , thus equal to 0. !

7. Proof of the “if” part of (5.1)

Let x = h.sΓ∨

y be an (S)-restricted partitioned isocrystal. This section isdevoted to showing that x is realizable by a Γ-hyper-symmetric abelian variety.Here is the first step towards proving the existence theorem.

22 YING ZONG

Proposition 7.1. Let K be a CM field, {λw;w|p} a set of rational numbers con-tained in the interval [0, 1] and indexed by the places w of K above p. Assume thatλw + λw = 1. Then there exist an integer a ≥ 1 and a pa-Weil number π such that

ordw(π)/ordw(pa) = λw,

for all w|p.

Proof. Let E be the maximal totally real subfield ofK. For any place v of E abovep, we define λv := min{λw,λw}, v = w|E. Either v is split, v = ww, or there isonly one prime w above v. In the first case, let aw ∈ OK be a generator of theideal wh; in the latter case, let av ∈ OE be a generator of vh, where h is the idealclass number of K. Consider the factorization

pOK =∏

v

(ww)e(v|p)∏

v

ve(v|p),

where the first product counts those v split in K/E, the second counts those vinert or ramified in K/E. Raising to the h-th power, one has

ph =∏

v

(awaw)e(v|p)

∏

v

ae(v|p)v .u.

The element u is a unit of OE . Now choose a sufficiently divisible positive integerc, and write λv = mv/(mv + nv), with c = mv + nv, mv, nv ∈ Z. We then definean algebraic integer π as

π =∏

v

(amvw anv

w )e(v|p)∏

v

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

One checks easily that ππ = phc and π is the desired phc-Weil number. !In case that K is an extension of F , it is important to know when the Weil

number we have just constructed generates K over F .

Proposition 7.2. Let F be a field, and K/F be a separable field extension ofdegree n. Assume that the normal hull L of K/F has a Galois group isomorphicto the symmetric group Sn of n letters. Then K/F has no sub-extensions otherthan F and itself.

Proof. This is equivalent to the assertion that the stabilizer subgroup Sn−1 of theletter 1 ∈ {1, · · · , n} is a maximal subgroup of Sn. It suffices to show that anysubgroup H properly containing Sn−1 acts transitively on the letters {1, · · · , n}.If n = 1, 2, this is clear. Assume that n ≥ 3. Let τ be an element of H , τ(1) = i,i = 1. For any j ∈ {1, · · · , n}, different from 1 and i, the permutation σ := (ij)τin H sends 1 to j. !Proposition 7.3. (Ekedahl) Let K be a number field, and OK its ring of integers.Let S be a dense open sub-scheme of Spec(OK). Let X, Y be two schemes of finitetype over S, and let g : Y → X be a finite etale surjective S-morphism. Suppose

ON HYPER-SYMMETRIC ABELIAN VARIETIES 23

that YK := Y ×S Spec(K) is geometrically irreducible and XK := X ×S Spec(K)satisfies the property of weak approximation. Then the set of K-rational points x ofX such that g−1(x) is connected satisfies also the property of weak approximation.

Remark 7.4. Let X be a scheme of finite type over a number field K. Recallthat a subset E of X(K) is said to satisfy the property of weak approximation, iffor 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 thatof Kv. In particular, the K-scheme X is said to satisfy the property of weakapproximation, if X(K) does.

Proposition 7.5. 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 ℓ ∈ Σ let K ′

ℓ bea finite etale algebra over Kℓ of rank n. Then there is a totally real extensionK ′/K of degree n, such that its normal hull has a Galois group isomorphic to thesymmetric group Sn of n letters, and K ′ ⊗K Kℓ ≃ K ′

ℓ, for all ℓ ∈ Σ.

Proof. We consider the following situation. Let S = Spec(OK),X ′ = S[a1, · · · , an],an S-affine space with coordinates a1, · · · , an. Let Y ′ be the hyper-surface in X ′[t]defined by the equation

f = tn + a1tn−1 + · · ·+ an.

Let R be the resultant of f and its derivative f ′. We denote by X the complementof {R = 0} in X ′ and by Y := Y ′ ×X′ X ; Y is an etale cover of X of rank n.The scheme XK , being a non-empty open sub-scheme of an affine space, clearlysatisfies the property of weak approximation. The geometric fibre YK := YK ⊗K Kis affine of ring Γ(OYK

) = (K[a1, · · · , an, t]/(f))R. We will prove in the next lemmathat Γ(OYK

) is an integral domain. Now it is ready to apply Ekedahl’s Hilbertirreducibility theorem (7.3) according to which, the subset M of the K-rationalpoints x where Yx is connected, i.e. Yx is the spectrum of a field extension K ′

of K of degree n, satisfies the property of weak approximation. Requiring theKl-algebras K ′ ⊗K Kl to be isomorphic to some given etale algebras at finitelymany places l of K imposes a weak approximation question on the parametersa1, · · · , an ∈ K. The condition on the Galois group of the normal hull is a weakapproximation property, cf. [5]. The proposition follows by modifying a little thecontent but not the proof of Ekedahl’s theorem [4]. !Lemma 7.6. Let K be a factorial domain, A = K[a1, · · · , an] a polynomial algebraover K. The “generic” polynomial f = tn+a1tn−1+ · · ·+an is irreducible in A[t].

Proof. Let B = K[b1, · · · , bn], where bi = ai/an, for 1 ≤ i ≤ n − 1, and bn = an.As A is a subring of B, it suffices to prove that f is irreducible in B[t]. This is sobecause f is an Eisenstein polynomial in B[t] with respect to the prime an. !

24 YING ZONG

Now consider an (S)-restricted partitioned isocrystal x = h.sΓ∨y. For proving

the “if” part, it suffices to show that each component of y is realizable by a Γ-isotypic hyper-symmetric abelian variety. From now on, we assume that y =(M,P ) is a simply partitioned isocrystal. By the supersingular restriction (S),there is a CM extension B/F such that P is equivalent to PB/F |TΓ. Let B0 bethe maximal totally real subfield of B. We also let N be the common number ofisotypic components of y at all places v of F above p.

These reductions and hypothesis are in force for the rest. Let us now finish theproof of the main theorem (5.1). First, assume that F is a CM field. Let F0 bethe maximal totally real subfield of F .

Proposition 7.7. Assume that F is a CM field. Suppose that y = (M,P ) is an(S)-restricted simply partitioned isocrystal. Then there exists a Γ-isotypic hyper-symmetric abelian variety Y such that M is Γ-isomorphic to H1(Y ).

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

Tv|F0 =

⎧⎪⎨

⎪⎩

(F0)Nv|F0, if v = v

(F0)v|F0 × F (N−1)/2v , if v = v,N odd

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

It follows from Proposition (7.5) that there is a totally real extension E/F0 ofrelative 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 ℓ ∈ TΓ, E ⊗F0 (F0)ℓ|F0 ≃ B0 ⊗F0 (F0)ℓ|F0.

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

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

(ii) for each ℓ ∈ TΓ, K ⊗F Fℓ ≃ B ⊗F Fℓ,

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

The property (iii) allows us to index the slopes of y at v as {λw;w|v}, where wruns over the places of K above v. One can even arrange that λw + λw = 1, sincethe underlying isocrystal M of y is polarized, cf. (2.7). We apply (7.1) to get aninteger a ≥ 1 and a pa-Weil number π ∈ K, so that

ordw(π)/ordw(pa) = λw, for all w|p

Note that the field F (π) must be equal to K. Indeed, if N = 1, this is clearbecause F = F (π) = K. If N > 1, π is not an element of F , because, otherwise,we would have ordw1(π) = ordw2(π), for any two places w1, w2 above v. This isabsurd in view of the choice of π. By (7.2) and (i), we have F (π) = K.

ON HYPER-SYMMETRIC ABELIAN VARIETIES 25

According to the theorem of Honda-Tate (2.12), up to isogeny there is a uniqueΓ-simple abelian variety Y ′

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

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

is

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

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

′) is totally splitat every place v|p of F , the abelian variety Y ′ is therefore Γ-hyper-symmetric, cf.(3.4).

The pair y′ = (H1(Y ′), PK/F |TΓ) is a simply partitioned isocrystal satisfying thesupersingular restriction (S) by (6.1). By construction, y′ and y have the sameslopes at every place v of F above p.

Now we prove that the multiplicity n(y′) divides n(y). In fact, n(y′) is the orderof [End0

Γ(Y )] in the Brauer group of K, cf. (6.1). Since y satisfies the condition(SPI2) (4.11), one has n(y).invℓ(Γ)[Kℓ′ : Fℓ] = 0 in Q/Z. Look at the local Brauerinvariants of C := End0

Γ(Y )

invν(C) =

{−[Fv : Qp]λν − invv(Γ), if ν | v−[Kν : Fℓ].invℓ(Γ), if ν ! p.

By Kottwitz 11.5 [8], n(y).invw(C) = 0 in Q/Z, for all w above p. These twoequations together show that n(y′) divides n(y). Let e be the integer such thatn(y) = e.n(y′).

It remains to prove that the underlying isocrystals of y and e.y′ are isomorphic aspolarized Γ-linear isocrystals. Indeed, we can modify the polarization on Y := Y ′e

so that e.y′ with this modified polarization is isomorphic to y. For a proof, letS be the Q-vector space of the symmetric elements in Hom0

Γ(Y, Y∗), where Y ∗

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

is isomorphic to the symmetric elements of HomΓ(H1(Y ∗), H1(Y )). The space Sbeing dense in S⊗QQp, our claim is clearly justified and the proof in the case thatF is a CM field is now complete. !Proposition 7.8. Assume that F is a totally real number field. And suppose thaty = (M,P ) is an (S)-restricted simply partitioned isocrystal. Then there exists aΓ-isotypic hyper-symmetric abelian variety Y such that M ≃ H1(Y ).

Proof. As y is Γ-linearly polarized and contains no slope 1/2 part by (S2), N is aneven integer. By Proposition (7.5), there is a totally real extension E/F of degreeN/2 such that

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

(2) for each ℓ ∈ TΓ, there is an Fℓ-isomorphism fℓ : E ⊗F Fℓ ≃ B0 ⊗F Fℓ,

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

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

26 YING ZONG

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

(ii) for each ℓ ∈ TΓ, there is an isomorphism gℓ : K⊗F Fℓ ≃ B⊗F Fℓ compatiblewith fℓ,

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

The properties (1) and (i) show that K/F is totally split everywhere above v.Thus we can index the slopes of y at v as {λw;w|v} with w running over the placesof K above v. Moreover, as y is Γ-linearly polarized, one can even arrange thatλw+λw = 1, cf. (2.7). Similarly as in the preceding proposition, there is a pa-Weilnumber π, for a suitable integer a ≥ 1, such that F (π) = K, and

ordw(π)/ordw(pa) = λw,

for all places w of K above p.

We assume that a is sufficiently divisible. The unique Γ-simple abelian varietyY ′Fpa

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

[8]. Let Y ′ := Y ′Fpa

⊗FpaFp, which is by construction Γ-hyper-symmetric. We then

modify, if necessary, the polarization on Y ′ so that a copy Y := Y ′e realizes y. Theargument is the same as that in (7.7). We have proved the Proposition (7.8). !

References

[1] C. L. Chai, F. Oort. Hypersymmetric Abelian Varieties. Pure and Applied Mathematics Quar-terly, 2: 1-27, 2006.

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

[3] C. L. Chai. Methods for p-adic Monodromy. To appear in J.Inst.Math.Jessieu.[4] T. Ekedahl. An effective version of Hilbert’s irreduciblity theorem. Seminaire de Theorie des

Nombres, Paris 1988-89. Progress in Math. 91, Birkhauser: 241-249, 1990.[5] J. Ellenberg, A. Venkatesh. The number of extensions of a number field with fixed degree

and bounded discriminant. Ann.of.Math., 163: 723-741, 2006.[6] N. Katz. Slope filtration of F -crystals. Journ.Geom.Alg.Rennes., Asterisque, 63: 113-164,

1979.[7] N. Katz, W. Messing. Some consequences of the Riemann Hypothesis for Varieties over Finite

Fields. Inv.Math., 23: 73-77, 1974.[8] R. E. Kottwitz. Points on some Shimura varieties over finite fields. J.Amer.Math.Soc, 2:

373-444, 1992.[9] D. Mumford. Geometric Invariant Theory. Springer Verlag.[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: 399-408, 1973.[11] J. Tate. Classes d’isogenie de varietes abeliennes sur un corps fini (d’apres T.Honda). Sem.

Bourbaki, 21 1968/69, Exp. 352, LNM 179: 95-110, 1971.