On hypersymmetric abelian varieties. - Penn Math

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ON HYPER-SYMMETRIC ABELIAN VARIETIES
YING ZONG
Abstract. MotivatedbyOort’sHecke-orbitconjecture,Chaiintroducedhyper-
symmetric points in the study of fine structures of modular varieties in positive
characteristics. Weproveanecessaryandsufficientconditiontodeterminewhich
Newton 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 initiated
by Chai-Oort [1]. The notion is motivated by the Hecke-orbit conjecture.
For the reduction of a PEL-type Shimura variety, the conjecture claims that
every orbit under the Hecke correspondences is Zariski dense in the leaf containing
it. In positive characterisitic p, the decomposition of a Shimura variety into leaves
is a refinement of the decomposition into disjoint union of Newton polygon strata.
A leaf is a smooth quasi-affine scheme over Fp. Its completion at a closed point is
a successive fibration whose fibres are torsors under certain Barsotti-Tate groups.
The resulting canonical coordinates, a terminology of Chai, provides the basic tool
for understanding its structure.
Fix an integer g 1 and a prime number p. Consider the Siegel modular variety
≥
incharacteristicp. Denoteby (x)theleafpassingthroughaclosedpointx. By
g
A C
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 (x) is big. Later, in their
C
solution of the Hecke-orbit conjecture for , Chai and Oort used the technique
g
A
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 1.1. A Γ-linear polarized abelian variety (Y,λ) over an algebraically
closed field k of characteristic p is Γ-hyper-symmetric, if the natural map
End0Γ(Y)⊗Q Qp → EndΓ(H1(Y))
is a bijection.
1
2 YINGZONG
For simplicity we denote by H1(Y) the isocrystal H1 (Y/W(k)) Q. The goal
crys ⊗Z
of this paper is to answer the following question:
Question. Doesevery Newtonpolygonstratumcontainahyper-symmetric point?
The answer to the question in general is no; a Newton polygon stratum must
satisfy certain conditions to contain a Γ-hyper-symmetric point. See (5.3) for an
example when Γ is a real quadratic field split at p, and (5.7) whenΓis a division
algebra 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) for
Γ-hyper-symmetric abelian varieties Y as the underlying isocrystals of partitioned
isocrystals with supersingular restriction (S).
Consider a typical situation. Let Y = Y′⊗FpaFp be a Γ-simple hyper-symmetric
abelian variety over Fp, where Y′ is a Γ-simple abelian variety over a finite field
Fpa. By the theory of Honda-Tate, up to isogeny, Y′ is completely characterized
by its Frobenius endomorphism πY . Let F be the center of Γ. Assume that Fpa
′
is sufficiently large. We show in (3.4) that Y is Γ-hyper-symmetric if and only if
the extension F(π )/F is totally split everywhere above p, that is,
Y
′
F(π ) F F F ,
Y F v v v
′ ⊗ ≃ ×···×
for every prime v of F above p. Thus Y is Γ-hyper-symmetric if and only if it is
F-hyper-symmetric.
Denote by T the set of finite prime-to-p places ℓ of F where Γ is ramified. To
Γ
Y, one can associate its isocrystal H1(Y) as well as a family of partitions P = (P )
ℓ
of the integer N = [F(π ) : F] indexed by ℓ T . For each ℓ T , P is given by
Y Γ Γ ℓ
′ ∈ ∈
P (ℓ) = [F(π ) : F ]
ℓ ′ Y ℓ ℓ
′ ′
with ℓ ranging over the places of F(π ) above ℓ. The pair (H1(Y),P) is the
′ Y
′
partitioned isocrystal attached to Y. In particular, we denote by s the pair
Γ
attached to the unique Γ-simple supersingular abelian variety up to isogeny over
Fp, 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 End (H1(Y))
F
→
for the ring homomorphism defining the Γ-action induced by functoriality on its
isocrystal H1(Y). In essence, the definition (4.11) of partitioned isocrystals is a
purely combinatorial formulation of the conditions that Y is F-hyper-symmetric
and ρ factors through the endomorphism algebra End0(Y) of the F-linear abelian
F
variety Y.
The introduction of supersingular restriction (S) (4.20) 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)
ON HYPER-SYMMETRIC ABELIAN VARIETIES 3
is isomorphic to M. In the proof of the main theorem (5.1), 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 s component, the construction of a Γ-hyper-symmetric abelian variety
Γ
Y realizing y goes as follows. Let N be the integer such that P = (P ) is
ℓ ℓ∈TΓ
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
(P ) given by
K/F, ℓ ℓ∈TΓ
P (ℓ) := [K : F ], ℓ ℓ
K/F, ℓ ′ ℓ ℓ ′
′ ∀ |
concide with (P ). Then a simple formula (7.1) gives directly a pa-Weil number
ℓ
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 λ = ord (π)/ord (pa), for w v. Let Y be the
w w w ′
|
unique abelian variety up to Γ-isogeny corresponding to π. For some integer e,
(Y′)e⊗Fpa Fp equipped with a suitable polarization is a desired Γ-hyper-symmetric
abelian variety.
The organization of this paper is as follows. In section 2 we set up the notations
and review the fundamentals of isocrystals with extra structures, Dieudonn´e’s
theorem on the classification of isocrystals and the Honda-Tate theory. In section
3, we show that every Γ-hyper-symmetric abelianvariety is isogenous toan abelian
variety defined over Fp (3.2). Then we prove a criterion of hyper-symmetry in
terms of endomorphism algebras (3.4). In the next section, we define partitions
and partitioned isocrystals. The main theorem (5.1) is stated in section 5. Several
examples are provided to illustrate how to determine which data of slopes are
realizable by hyper-symmetric abelian varieties. The proof of (5.1) is divided into
two parts. The “only-if” part, in section 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 ) has rational
Fpa
coefficients. In section 7 we prove the inverse, the “if” part.
Acknowledgement. This thesis is under the supervision of my advisor Ching-Li
Chai, 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 rational
numbers. We fix a positive involution on Γ. Let F be the center of Γ; F is either
∗
a totally real number field or a CM field. Let v , ,v be the places of F above
1 t
···
p. We have
Γ⊗Q Qp =Γ v1 ×···× Γvt.
4 YINGZONG
Let T denote the following set
Γ
TΓ = ℓ Spec( F) ℓ ! p,ℓ = (0),invℓ(Γ) = 0 .
{ ∈ O | ̸ ̸ }
2.2. Recall the computation of Brauer invariants. Let K be a finite extension of
Qp. Let A be a central simple K-algebra of dimension d2. By Hasse, A contains
a d-dimensional unramified extension L/K such that for an element u A, the
∈
vectors 1,u, ,ud 1 form an L-basis of A, and
−
···
ua = σ(a)u, a L
∀ ∈
ud = α L
! ∈
where σ Gal(L/K) is the Frobenius automorphism of L/K. Then we define the
∈
Brauer invariant invK(A) Br(K) Q/Z as
∈ ≃
inv (A) = ord (α)/d,
K L
−
where ord is the normalized valuation of L, i.e. ord (π) = 1, for a uniformizer
L L
π .
L
∈O
2.3. If k is a perfect field of characteristic p, we denote by W(k) the ring of Witt
vectors of k. Let K(k) be the fraction field of W(k). The Frobenius automorphism
of k induces by functoriality an automorphism σ of W(k), namely,
σ(a ,a , ) = (ap,ap, )
0 1 ··· 0 1 ···
for all a ,a , k.
0 1
···∈
2.4. An isocrystal over k is a finite dimensional K(k)-vector space M equipped
with a σ-linear automorphism Φ. A morphism f : (M,Φ) (M ,Φ) is a K(k)-
′ ′
→
linear map f : M M such that fΦ=Φ f. Isocrystals over k form an abelian
′ ′
→
category.
2.5. 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)
−
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 = M(r)
"r∈Q
where M(r) is the largest sub-isocrystal of slope r, i.e.
M(r)⊗K(k) K(k) ≃ Emr r
ON HYPER-SYMMETRIC ABELIAN VARIETIES 5
for an integer m .
r
The rational numbers occurred in the decomposition M = M(r) are called
r Q
the 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 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. 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 , theΓ -linear isocrystals over k.
v v
C
(2) For each place v of F above p, the simple objects of are parametrized
v
C
by 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 = M M is the decomposition defined
v1×···× vt
in (1), we call the slopes of M the slopes of M at v and define the multiplicity of
v
a slope r at v by
multMv(r) = dimK(k)Mv(r)/([Fv : Qp][Γ : F]1/2)
2.8. AΓ -linear polarized isocrystal is a quadruple (M,Φ,i,ψ), where (M,Φ) is an
isocrystal, i :Γ End(M,Φ) is a ring homomorphism, and ψ is a polarization on
→
M such that
ψ(γx,y) = ψ(x,γ y)
∗
for all γ Γ,x,y M. If F is a totally real number field, the slopes of M at each
∈ ∈
place v of F above p, arranged in increasing order, are symmetric about 1/2. If
F is a CM field, the slopes at v and v collected together, arranged in increasing
order, are symmetric with respect to 1/2.
2.9. Recall that a morphism of abelian varieties f : X X is an isogeny if it
′
→
is surjective with a finite kernel. Let X be an abelian variety over a finite field
k = Fpa. The relative Frobenius morphism
F : X X(p)
X/k
→
is an isogeny. We call π = Fa the Frobenius endomorphism of X. If X is a
X X/k
simple abelian variety, the Frobenius endomorphism π is a pa-Weil number, that
X
is, an algebraic integer π such that for every complex imbedding ι : Q(π) * C,
→
one has
ι(π) = pa/2.
| |
6 YINGZONG
Here is a basic result, due to Honda-Tate [11]:
(1) The map X π defines a bijection from the isogeny classes of simple
X
+→
abelian varieties over k to the conjugacy classes of pa-Weil numbers.
(2) The endomorphism algebra End0(X ) of a simple abelian variety X cor-
π π
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 D , the quaternion division algebra
p,
∞
over Q, ramified exactly at p and the infinity.
(b) If a Z 2Z, and π = pa/2, then Xπ kk′ is isogenous to the product
∈ − ⊗
of two supersingular elliptic curves, where k is the unique quadratic
′
extension of k.
(c) If π is totally imaginary, the division algebra D = End0(X ) is unram-
π
ified away from p. For a place w of Q(π) above p, the local invariant
of D at w is
inv (D) = ord (π)/ord (pa).
w w w
−
2.10. AΓ -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 require
→
that i is compatible with the involution and the Rosati involution on End0(Y)
∗
associated to the polarization λ. The category ofΓ-linear polarized abelian vari-
eties up to isogeny is semi-simple. In particular, any such abelian variety Y admits
aΓ- isotypic decomposition,
Y ∼Γ-isog Y1e1 ×···× Yrer
where each Y is Γ-simple and for i = j, Y and Y are not Γ-isogenous. For each
i i j
̸
i, there exist a simple abelian variety X and an integer e , such that Y Xei.
i i i ∼isog i
We say Y is of type X .
i i
2.11. Let Y be a Γ-simple abelian variety of type X, i.e. Y Xe, for an
isog
∼
integer e. Let Z , Z be the center of End0(X) and End0(Y), respectively. There
0 Γ
is the following relation [8],
e.[End0(X) : Z0]1/2[Z0 : Q] = [Γ : F]1/2[End0Γ(Y) : Z]1/2[Z : Q].
One deduces that the Q-dimension of any maximal ´etale sub-algebra of End0(Y)
is equal to [Γ : F]1/2 times the Q-dimension of any maximal ´etale sub-algebra of
End0(Y).
Γ
ON HYPER-SYMMETRIC ABELIAN VARIETIES 7
2.12. Let k = Fpa be a finite field. Kottwitz [8] proved a variant of the theorem
of Honda-Tate:
(1) The map Y π is a bijection from the set of isogeny classes of Γ-simple
Y
+→
abelian varieties over k to the F-conjugacy classes of pa-Weil numbers.
(2) The endomorphism algebra End0(Y ) of a Γ-simple abelian variety Y cor-
Γ π π
responding to π is a central division algebra over F(π). Let X be a simple
π
abelian variety up to isogeny corresponding to π as in (2.9); Y is of type
π
X . Let D = End0(X ), C = End0(Y ). Then one has the equality
π π Γ π
[C] = [D F(π)] [Γ F(π)]
⊗Q(π) − ⊗F
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 k
ofcharacteristic p, andletY ∼Γ-isog Y1e1×···×Yrer betheΓ-isotypicdecomposition
of Y, cf. (2.10). For the rest, H1(Y) stands for the first crystalline cohomology of
Y, H1 (Y/W(k)) Q.
crys ⊗Z
Lemma 3.1. The abelian variety Y is Γ-hyper-symmetric if and only if each Y is
i
Γ-hyper-symmetric and for any place v of F above p, for dffierent i,j, Y and Y
i j
have no common slopes at v.
Proof. This is clear. !
Proposition 3.2. If Y is Γ-hyper-symmetric, there exists a Γ-hyper-symmetric
abelian variety Y′ over Fp such that Y′ k is Γ-isogenous to Y.
⊗Fp
We first prove a weaker result.
Corollary 3.3. There is a Γ-hyper-symmetric abelian variety Y′ over Fp such that
the isocrystal H1(Y k) is isomorphic to H1(Y).
′
⊗Fp
Proof. There is a Γ-linear polarized abelian variety Y over a finitely generated
K
subfield K such that Y k is isomorphic to Y and End(Y ) = End(Y).
K K K
⊗
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
Y extends to an abelian scheme over S.
K
Y
By a theorem of Grothendieck-Katz [6], the function assigning any point x of S
the Newton polygon of the isocrystal H1( ) is constructible. Let S be the open
x ′
Y
subset 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 homomorphism
′
End( ) End(Y ) is an isomorphism. So there is a well defined specialization
S K
Y ′ →
map sp : End(Y ) End( ) for any point t S . By the rigidity lemma 6.1
K t ′
→ Y ∈
8 YINGZONG
[9], sp is injective. Let t be a closed point of S and = k(t). As Y is
′ Yt Yt ⊗k(t)
Γ-hyper-symmetric, End0Γ(YK)⊗QQp and EndΓ(H1(Yt)) have the same dimension.
Thus the composite map
End0Γ(YK)⊗Q Qp *→ End0Γ(Yt)⊗Q Qp *→ EndΓ(H1(Yt))
is bijective. It follows that is a desired Γ-hyper-symmetric abelian variety over
Yt
k(t) Fp. !
≃
Proof. of (3.2). 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 Xe for an integer e. Let Z , Z denote respectively the center
isog 0
∼
of End0(X) and End0(Y). The dimension r of any maximal ´etale sub-algebra of
Γ
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 isΓ-hyper-symmetric. Intheabove, E denotes thecenter ofEnd (H1(Y)).
Γ
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 End (H1(Y )).
′ Γ ′
By the choice of Y′, r and r′ are equal. As any abelian variety over Fp has
sufficiently many complex multiplication (2.9), we have r = r = 2.dim(Y ). This
′ ′
finishes the proof. !
In the following we prove a criterion of Γ-hyper-symmetry in terms of the center
Z of End0(Y).
Γ
Proposition 3.4. A Γ-linear polarized abelian variety Y over Fp is Γ-hyper-
symmetric if and only if the F -algebra Z F is completely decomposed, i.e.,
v F v
⊗
Z F F F , for every place v of F above p.
F v v v
⊗ ≃ ×···×
Proof. Let Y′ be a Γ-linear polarized abelian variety over a finite field Fpa, such
that Y′⊗Fpa Fp ≃ Y and End(Y′) = End(Y). The center Z can be identified with
F(π), the sub-algebra generated by the Frobenius endomorphism of Y . By Tate
′
[11], over Fpa, the map
End0(Y′)⊗Q Qp → End(H1(Y′))
ON HYPER-SYMMETRIC ABELIAN VARIETIES 9
is bijective.
Hence, the condition for Y to be Γ-hyper-symmetric is equivalent to
End (H1(Y )) = End (H1(Y)).
Γ ′ Γ
Let M := H1(Y ), and M = M be the decomposition defined in (2.7).
′ ′ ′ vp v′
|
The isocrystal M is Γ -linear and has a decomposition into isotypic components,
v′ v #
M = M (r).
v′ v′
"r∈Q
With these decompositions, the condition for Y to be Γ-hyper-symmetric is equiv-
alent to
EndΓv(Mv′(r)) = EndΓv(Mv′(r)⊗K(Fpa) K(Fp)),
for any v p, and r Q.
| ∈
On the left hand side, the center of End (M (r)) is F (π ), where π stands
Γv v′ v v,r v,r
for the endomorphism π M (r). On the right hand side, the center is isomorphic
| v′
to a direct product F F with the number of factors equal to the number
v v
×···×
of Γv-simple components of Mv′(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, anyΓ-linear endomorphism f of the isocrystal
(H1(Y),Φ) commutes with the operator π 1Φa, and thus stabilizes the invariant
−
sub-space of π 1Φa, i.e. H1(Y ). Hence f End (H1(Y )). This implies that Y
− ′ Γ ′
∈
is Γ-hyper-symmetric. !
4. Partitions and Partitioned Isocrystals
Definition 4.1. Let N be a positive integer. A partition of N with support in a
finite set I is a function P : I → Z>0, such that i I P(i) = N.
∈
Definition 4.2. Let f : X S be a surjective m$ap of sets such that for all s S,
→ ∈
f 1(s) is finite. An S-partition of N with support in the fibres of f is a function
−
P : X Z>0 such that for each s S, P f−1(s) is a partition of N with support
→ ∈ |
in f 1(s).
−
P
X Z>0
f
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 -partition
′ ∗ ′
→ ◦
of N, where p : X S X is the projection.
S ′
× →
10 YINGZONG
Definition 4.4. Let P be an S -partition of N, i = 1,2. We say that P is
i i 1
equivalent to P if there exist a bijection u : S S and a u-isomorphism
2 1 2
→
g : X X such that P = P g.
1 2 1 2
→ ◦
Definition 4.5. Consider S-partitions Pi of Ni, i = 1,2. Let fi : Xi Z>0 be
→
the structural maps. The sum P P is the following S-partition P of N +N ,
1 2 1 2
⊕
P
X1 X2 Z>0
f%
S
where P X = P , and f X = f , i = 1,2.
i i i i
| |
Example 4.6. 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 4.7. Let F be a number field, K/F a finite field extension of degree
N. Let S = Spec( ), I = Spec( ), and f : I S the structural morphism.
F K
O O →
Consider the function PK/F : I Z>0 defined as
→
[K : F ], if w is a finite prime
P (w) = w f(w)
K/F N, if w = (0)
!
This P defines an S-partition of N. The most interesting case is K = F(π ),
K/F Y
the field generated by the Frobenius endomorphism π of a Γ-simple non-super-
Y
singular abelian variety Y over a finite field k (2.12). We study this example in
more detail.
(a). F is totally real, K is a CM extension.
One has [K : F ] = [K : F ], and [K : F ] is an even integer if
w f(w) w f(w) w f(w)
w = w. Recall that T (2.1) denotes the set of finite prime-to-p places ℓ of F
Γ
where Γ is ramified. The restriction P T (4.3) is equivalent to a T -partition
K/F Γ Γ
|
Pℓ : [1,dℓ] Z>0 ℓ TΓ of N = [K : F], which satisfies the following property
{ → | ∈ }
P (2i 1) = P (2i), for i [1,c (ℓ)]
ℓ ℓ 1
− ∈
P (i) is even, for i [2c (ℓ)+1,d ]
ℓ 1 ℓ
! ∈
where d = Card(f 1(ℓ)), 2c (ℓ) = Card( w f 1(ℓ) w = w ).
ℓ − 1 −
{ ∈ | ̸ }
(b). F is a CM field, K is a CM extension.
One has [K : F ] = [K : F ]. The restriction P T is equivalent to
w f(w) w f(w) K/F Γ
|
Pℓ : [1,dℓ] Z>0 ℓ TΓ
{ → | ∈ }
which satisfies the property
P (2i 1) = P (2i), if ℓ = ℓ,i [1,c (ℓ)]
ℓ ℓ 1
− ∈
! Pℓ(i) = Pℓ(i), if ℓ ! ℓ