On capitulation cokernels in Iwasawa theory
M. Le Floc’h
A. Movahhedi
T. Nguyen Quang Do
March 1, 2004
Abstract
For a number field F and an odd prime p, we study the “capitulation cokernels” coker(A0n → A0 Γ∞n ) associated with the (p)-class
groups of the cyclotomic Zp -extension of F . We prove that these
cokernels stabilize and we characterize their direct limit in Iwasawa
theoretic terms, thus generalizing previous partial results obtained
by H. Ichimura. This problem is intimately related to Greenberg’s
Conjecture.
Introduction
Let F be a number field and p an odd prime. Let F∞ be the cyclotomic Zp extension of F , with finite layers Fn for all integers n, and let us write as usual
Γn := Gal(F∞ /Fn ). In this introduction (and only therein, since this is not
standard vocabulary), the natural maps An → AΓ∞n and A0n → A0 Γ∞n between
the p-primary part of the class group (resp. (p)-class group) of Fn and the Γn fixed points of A∞ := lim An (resp. A0∞ := lim A0n ) will be called capitulation
−→
−→
maps. Their study is an interesting problem in Iwasawa theory, especially in
connection with Greenberg’s Conjecture, which predicts the triviality of A∞
and A0∞ in the totally real case. The capitulation kernels have been studied
intensively ([G1], [GJ], [Iw1], [Ku], etc.) but – strangely enough – the same
is not true for the cokernels. To the best of our knowledge, the capitulation
cokernels have been touched upon by Iwasawa in [Iw2], p. 198, but almost all
the substantial results obtained so far are contained in papers by H. Ichimura
([I1, I2, I3]; see also [S]), who only deals with totally real abelian fields of
degree prime to p, or totally real fields satisfying Leopoldt’s Conjecture in
which p is totally split.
In this article, we shall only consider the capitulation maps jn : A0n →
0 Γn
A ∞ , assuming the finiteness of A0 Γ∞n (the Gross Conjecture). One good
1
reason for this is that, except in the totally real case (where A∞ = A0∞ modulo Leopoldt’s Conjecture), the groups AΓ∞n are in general not finite. We
completely solve the problem of describing the asymptotical behaviour of
the capitulation cokernels in the general case (and of course we retrieve all
previously known results). More precisely, in Section 1, we show that the capitulation cokernels stabilize and their direct limit measures the asymptotical
0
deviation between the class groups A0n and the co-descent modules (X∞
) Γn ,
0
0
where X∞ = lim An (Theorem 1.4). In section 2, we show that the stabi←−
lization starts from level n0 , where n0 is the smallest integer n such that the
0
natural map (X∞
)Γn → A0n becomes surjective; supposing (for simplification)
that F contains a primitive p-th root of unity, we characterize lim coker jn as
−→
the Kummer dual of a certain Galois module related to embedding problems
in Iwasawa theory (Theorem 2.4). In Section 3, we take another look at this
module, using Gross kernels (Theorem 3.2). Finally, the abelian semi-simple
case (i.e., when F is abelian of degree prime to p over Q) is studied in detail:
explicit formulæ and numerical examples are given in Section 4.
First, we set up some notations.
| |p
γ
Λ = Zp [[Γ]] ∼
= Zp [[T ]]
Σ, S
A0n
A0∞ = lim A0n
−→
Mn , M∞
L∞
L0n , L0∞
Un0
0
U∞
= lim Un0
−→
0
N∞
BP
FnBP , F∞
GS (Fn )
p-adic valuation;
topological generator of Γ;
the Iwasawa algebra,
the isomorphism above being obtained
by mapping γ to 1 + T ;
places of F over p (resp. over p and ∞);
p-primary part of the (p)-class group of Fn ;
p-primary part of the (p)-class group of F∞ ;
maximal p-ramified abelian pro-p-extension of Fn
(resp. of F∞ );
maximal unramified abelian pro-p-extension of F∞ ;
maximal unramified abelian pro-p-extension of Fn
(resp. of F∞ ) in which every prime over p splits completely;
group of (p)-units in Fn ;
group of (p)-units in F∞ ;
field generated over F∞ by the pn -th roots
0
of all ε ∈ U∞
for all n ≥ 0;
field of Bertrandias-Payan over Fn (resp. F∞ ), i.e.,
the compositum of all p-extensions of Fn (resp. F∞ )
which are infinitely embeddable in cyclic p-extensions;
Galois group of the maximum S-ramified
extension of Fn ;
2
Galois group of the maximum S-ramified abelian
pro-p-extension of Fn ;
X Fn
X∞ = Gal(M∞ /F∞ );
X∞ = Gal(L∞ /F∞ );
0
X∞
= Gal(L0∞ /F∞ );
00
0
BP
N∞
= N∞
∩ F∞
;
0
0
Z∞ = Gal(N∞ /F∞ );
BP
BP∞ = Gal(F∞
/F∞ );
BP
W∞ = Gal(M∞ /F∞
);
T∞
fixed field of torΛ X∞ ;
The fields above fit into the following diagram:
0
N∞
M∞
00
N∞
BP
F∞
T∞
F∞
1
Asymptotical results
In this section, we shall use the theory of adjoints (and co-adjoints) to give an
asymptotical description of the cokernels coker jn . For further details about
adjoints, see [Iw1] or [W]. Throughout this paper, we will only consider
finitely generated Λ-modules. Let Z be a (finitely generated) torsion Λmodule. For each prime ideal p of height one in Λ, let Zp := Z ⊗Λ Λp ,
where Λp is the localization of Λ at p. The kernel of the canonical map
Z → ⊕p Zp is the maximal finite Λ-submodule of Z, which we shall write
Z 0 , and the cokernel is by definition the co-adjoint of Z, denoted β(Z). Let
α(Z) := HomZp (β(Z), Qp /Zp ) which we make into a Λ-module by defining
(σ.f )(y) := f (σ −1 .y) for σ ∈ Λ, y ∈ β(Z) and f ∈ α(Z). We then see that
α(Z) is a finitely generated Λ-torsion module, called the adjoint of Z.
We recall two important facts about adjoints that we shall subsequently
use: an adjoint has no non-trivial finite submodules and α(Z) is pseudoisomorphic to Z (−1) , where (−1) means that the Galois action has been in-
3
n
verted. If we denote as usual γ p − 1 by ωn , we have natural homomorphisms
Z/ωn Z → Z/ωm Z, x mod ωn Z 7→ (ωm /ωn )x mod ωm Z,
for all m ≥ n ≥ 0. If we also assume that the principal divisors ωn are
disjoint from the divisor of Z, then β(Z) is obtained as the direct limit of the
Z/ωn Z’s with respect to the above homomorphisms. Throughout the paper,
the co-invariants Z/ωn Z will be denoted ZΓn . We can now state an algebraic
lemma which is interesting in itself:
Lemma 1.1. Let Z be a torsion Λ-module such that the principal divisors
ωn are disjoint from the divisor of Z ( i.e., ZΓn is finite) for all sufficiently
large n. Then, for n large enough, the equality |β(Z)Γn | = |ZΓn |/|Z Γn | holds,
in
(lim ZΓn )Γn are surjective.
and the natural homomorphisms ZΓn −→
−→
Proof. Under our hypotheses, the co-adjoint β(Z) is isomorphic to lim ZΓn .
−→
Since Z and α(Z)] have the same characteristic polynomial as Λn := Zp [[Γn ]]modules, say fn (T ), these two Λn -modules have the same Herbrand quotient
with respect to Γn ; besides, the characteristic polynomials of α(Z) and α(Z)]
have the same constant term. Putting this together, we get
Γn
Γn
|fn (0)|−1
p = |ZΓn |/|Z | = |α(Z)Γn |/|α(Z) | = |α(Z)Γn |,
because α(Z)Γn is finite, hence trivial. By duality, we obtain |ZΓn |/|Z Γn | =
|β(Z)Γn |. The surjectivity of the map in : ZΓn → β(Z)Γn is then equivalent
to the equality of orders |Z Γn | = | ker in |. But it is known (see [Ku] where it
is not assumed that Z Γn is finite) that for n large enough, the kernels ker in
stabilize and are canonically isomorphic to the maximal finite submodule Z 0
of Z.
We need some finiteness assumption in order to apply Lemma 1.1 to class
groups. This will be:
The Gross Conjecture (see [FGS], [J], or “hypothesis 3” of [Ku])
A number field satisfies the (generalized) Gross Conjecture at p if the following properties, which are equivalent, hold:
(i) X 0 Γ∞ is finite;
0
(ii) (X∞
)Γ is finite;
(iii) A0 Γ∞ is finite;
4
(iv) (A0∞ )Γ is trivial.
The equivalence of (iii) and (iv) is discussed below after Lemma 1.2. The
Gross Conjecture is known to be true for abelian fields and in the totally
real case, it is implied by Leopoldt’s Conjecture. In the sequel, we will often
assume the Gross Conjecture for all fields Fn , n 0, in the cyclotomic Zp tower of F . In short, “the Gross Conjecture for all n large enough”. Unless
F is totally real, the analogous properties for X∞ are known to be false in
general.
Let us now make a short digression and discuss a twisted version of the
Gross Conjecture, in relation with étale wild kernels:
ét
by definition, the higher étale wild kernel W K2i
(Fn ) of Fn , i ≥ 1, is the
kernel of the map
2
Hét
(OFS , Zp (i + 1)) → ⊕v∈S H 2 (Fv , Zp (i + 1)).
The étale wild kernels play a similar role in étale cohomology, étale K-theory
and Iwasawa theory as the p-primary parts A0n of the (p)-class groups of Fn .
Assume that F contains a primitive p-th root of unity. It is then well known
ét
0
([Sc], §6, Lemma 1) that W K2i
(Fn ) ∼
(i)Γn in a canonical way (here, for
= X∞
a Λ-module M and an integer i, M (i) means M twisted i-times à la Tate).
0
ét
In particular, the groups X∞
(i)Γn are finite. Let us define W̃ K2i
(F∞ ) :=
ét
lim W K2i (Fn ). An immediate application of Lemma 1.1 shows that the
−→
canonical maps
ét
ét
W K2i
(Fn ) → W̃ K2i
(F∞ )Γn
are surjective. Note that the same property holds for the étale K-groups
2
(OFS , Zp (i + 1)) (see [Ka] and [KM]). This will be in sharp contrast with
Hét
the class groups (see Theorems 1.4, 2.4 and 3.2 below).
Going back to class groups, a by-product of Lemma 1.1 is the following
asymptotical formula:
Lemma 1.2. Assume the Gross Conjecture for all n large enough. Then,
Γ
Γ
0
0
|A0 ∞n | = |(X∞
)Γn |/|(X∞
) n | = |fn (0)|−1
p ,
(1)
0
where fn (T ) is the characteristic series of X∞
as a Zp [[Γn ]]-module.
Proof. Recall that L0n0 is the maximal p-decomposed extension contained in
the p-class field of Fn0 , and following Iwasawa, let Y∞0 := Gal(L0∞ /F∞ L0n0 ),
0
and νn,n0 = ωn /ωn0 . Then we have A0n ∼
/νn,n0 Y∞0 for all n ≥ n0 (cf.
= X∞
0
0
) and β(X∞
)∼
[Iw1], Theorem 7). From this, we deduce that A0∞ ∼
= β(Y∞
0
A∞ . Whence the lemma.
5
This proof also indicates why the equivalence between properties (iii) and
0 Γ
(iv) holds: (A0∞ )Γ is isomorphic to the Pontrjagin dual of α(Y∞
) , hence is
trivial if it is finite.
In view of Lemma 1.2, we must obviously compute the deviation between
0
0
(X∞
)Γn and A0n . For n ≥ 0, let us call Ψn the kernel of the map (X∞
) Γn →
0
An . In the introduction, we defined n0 to be the smallest integer n such that
this map becomes surjective; n0 is easily seen to be the smallest integer n for
which all p-adic primes are totally ramify in F∞ /Fn .
Lemma 1.3. Assuming the Gross Conjecture for n large, the kernels Ψn are
finite and stabilize, i.e., Ψn ∼
Ψ =: Ψ∞ for n 0.
= lim
−→ k
Proof. By class field theory (see [Iw1], §4.3), we have a commutative diagram
0
0
(X∞
) Γm
Ψm
ωm /ωn
0
A0m
ωm /ωn
0
(X∞
) Γn
Ψn
0
nat.
A0n
0
for m ≥ n ≥ n0 . If m n 0, we have
0 0
0
0
ker(Ψn → Ψm ) ⊆ ker((X∞
)Γn → (X∞
) Γm ) ∼
),
= (X∞
0 0
0
where as before (X∞
) stands for the maximal finite submodule of X∞
and
the isomorphism is as explained in the proof of Lemma 1.1. Now, notice that
0 0
0
the intersection (X∞
) ∩ Gal(L0∞ /F∞ L0n ) is trivial for n 0. Indeed, let F∞
0 0
0
0
0
0
be the fixed field of (X∞ ) . Then, since L∞ /F∞ is finite and L∞ = ∪Ln ,
0
0
the sequence [L0n .F∞ .F∞
: F∞
] is increasing and bounded, hence constant
0
0
0
0 0
for n 0, so that L∞ = Ln .F∞ .F∞
for n large. By identifying (X∞
) with
0
0 0
Gal(L0n F∞ /F∞
∩L0n F∞ ), we see that (X∞
) injects into Gal(L0n F∞ /F∞ ) ∼
= A0n .
This implies that Ψn → Ψm becomes injective for all m ≥ n 0.
0
Secondly, since A0∞ ∼ β(X∞
), Iwasawa’s asymptotical formulæ for the
0
0
two sequences (X∞ )Γn and An read
0
0 n +ν
0
∀n 0, |(X∞
)Γn | = pλ n+µ p
1
0
0 n +ν
and |A0n | = pλ n+µ p
2
,
so that the order of Ψn is bounded independently of n 0. We conclude
'
that Ψ∞ = lim Ψn is finite and that Ψn → Ψm for m ≥ n 0.
−→
Note that we get the extra result that
|Ψ∞ | = pν1 −ν2 .
6
As a consequence, we find that
0 0
ker(A0n → A0m ) ∼
)
= (X∞
for m n 0. This result is implicitly contained in [Iw1] and explicitly in
[Ku]; our argument above is merely a clarification of Kuz’min’s proof ([Ku],
Theorem 3.1). See also [GJ].
Taking the inductive limit of the sequences
0
0 → Ψn → (X∞
)Γn → A0n → 0,
we get an exact sequence
0
0 → Ψ∞ → β(X∞
) → A0∞ → 0.
Taking Γn -invariants and co-invariants for n large enough, we obtain the top
exact sequence in the following commutative diagram:
0
ΨΓ∞n = Ψ∞
A0 Γ∞n
0 Γn
β(X∞
)
(Ψ∞ )Γn = Ψ∞
k
0
o
0
Ψn
0
β(X∞
) Γn
0
(X∞
) Γn
A0n
0
0 Γn
0
) , hence trivial). An easy diagram
(note that β(X∞
)Γn is dual to α(X∞
chase gives
Γ
coker(A0n → A0 ∞n ) ∼
= Ψ∞ ,
whence the following:
Theorem 1.4. Let F be a number field and F∞ /F the cyclotomic Zp -extension
for an odd prime p. Assume the Gross Conjecture for all n 0; then, for all
n large enough, the cokernels coker(A0n → A0 Γ∞n ) stabilize and are isomorphic
0
to Ψ∞ := lim Ψk , where Ψk = ker((X∞
)Γk → A0k ).
−→
Remark 1.5. Without the Gross Conjecture, we can no longer apply Lemma
1.1, so that Theorem 1.4 is no longer available; but see Remark 2.7.
'
0
Corollary 1.6. Suppose that (X∞
)ΓN −→ A0N for a certain N ≥ n0 (this
happens e.g. if there is only one prime over p in F∞ ). Then, for all n ≥ N ,
the natural map A0n −→ A0 Γ∞n is onto.
7
0
0
Proof. Recall that A0n ∼
/νn,n0 Y∞
for all n ≥ n0 , where Y∞0 = Gal(L0∞ /F∞ L0n0 ),
= X∞
0
and νn,n0 = ωn /ωn0 . The assumption of the corollary means that νN,n0 Y∞
=
0
ωN X∞ . From this, we derive
0
0
0
0
0 ∼
/ωn X∞
= X∞
/νN,n ωN X∞
(X∞
) Γn = X ∞
= A0n
for n ≥ N . Thus, ∀n ≥ N, Ψn = 0, leading to Ψ∞ = 0.
Remark 1.7. (i) this corollary is Theorem 3 in [I3], but here we do not
need the assumption of [I3] that p splits completely nor that F is real;
0
(ii) of course, the condition (X∞
) Γn ∼
= A0n0 is not in general necessary for
0
the map A0n −→ A0 Γ∞n to be surjective
(but see the
√
√
√ subsection 4.1) as
shown by the examples of Q( 67), Q( 103), Q( 106), etc. in Table
2. For these fields, Greenberg’s Conjecture is true (i.e., A0∞ = 0), so
0
the cokernel is trivial, while n0 = 0 and (X∞
)Γ is not isomorphic to A00
(indeed, we have an isomorphism if and only if Ψ0 = 0). We shall see
in the next section that unlike the Ψk ’s, the cokernels coker jn stabilize
starting from n0 .
Moreover, we retrieve Theorem 1 in [S] (all the results in Section 1 remain
valid for any Zp -extension modulo the finiteness assumptions in the Gross
Conjecture):
Corollary 1.8. Assume the Gross Conjecture for all n 0. Then,
jn
Γ
A0∞ = 0 ⇔ ∃n 0, ker(A0n −→ A0 ∞n ) = A0n and Ψn = 0.
2
Kummer duality and the Bertrandias-Payan
module
In this section, we aim to give a non-asymptotical version of Theorem 1.4
by using Kummer theory and techniques of embedding problems in Iwasawa
theory. For this, we suppose that F contains the group µp of p-th roots
of unity. This assumption is not restrictive since one can always take the
invariants under ∆ := Gal(F (µp )/F ) to descend to the field F (p 6= 2). We
0
0
first state a lemma that generalizes Lemma 7 of [I1]. Let E∞
= U∞
⊗ Qp /Zp
0
1
∞
be the Kummer dual of Gal(N∞ /F∞ ) and V∞ = H (GS (F∞ ), µp ) be that of
X∞ , where GS (F∞ ) is the Galois group over F∞ of the maximum S-ramified
extension of F∞ . Finally, let Vn := H 1 (GS (Fn ), µp∞ ) and En0 := Un0 ⊗ Qp /Zp .
8
Lemma 2.1.
and
jn
Γ
Γ
∀n ≥ 0, ker(A0n −→ A0 ∞n ) ∼
= E 0 ∞n /En0 ,
n
n
jn
Γ
0
0
coker(A0n −→ A0 ∞n ) ∼
∩ (γ p − 1)V∞ )/(γ p − 1)E∞
.
= (E∞
Proof. Consider the following commutative diagram from Kummer theory:
0
0 → E∞
→ V∞ → A0∞ → 0
↑
↑
↑
0
0 → En → Vn → A0n → 0
Γn
It is well known that Vn ∼
; this stems from Tate’s lemma (i.e.,
= V∞
1
2
H (Γn , µp∞ ) = H (Γn , µp∞ ) = 0) and from the inflation-restriction sequence.
By the snake lemma, we therefore obtain a commutative exact diagram
0
E 0 Γ∞n
Γn
V∞
0
En0
Vn
o
A0 Γ∞n
0
(E∞
) Γn
fn
(V∞ )Γn
jn
A0n
0
Now, an easy diagram chase gives
Γ
Γ
ker jn ∼
= coker(En0 → E 0 ∞n ) = E 0 ∞n /En0 ,
and
n
n
0
0
coker jn ∼
∩ (γ p − 1)V∞ )/(γ p − 1)E∞
.
= ker fn = (E∞
The point now is to give a more tractable description of the Kummer
radical in Lemma 2.1. In the special cases studied in [I1, I2], this is done in
a rather technical and lengthy way. Here we solve the general problem by
studying the so-called “field of Bertrandias-Payan” (in reference to [BP]).
Definition 2.2. For a global field K, the field of Bertrandias-Payan over K
is the compositum of all the p-extensions of K which are embeddable in cyclic
extensions of degree pm for all m ≥ 1. This is the same as the compositum of
all the extensions of K which are locally Zp -embeddable for any finite place
(see [N1], Lemme 4.1).
BP
Let us denote by FnBP (resp. by F∞
) the field of Bertandias-Payan over
Fn (resp. over F∞ ) and by BPn (resp. BP∞ ) the Galois group of FnBP over Fn
BP
BP
(resp. of F∞
over F∞ ). One readily sees that T∞ ⊆ F∞
⊆ M∞ , where T∞
BP
is the fixed field of torΛ X∞ , so that torΛ BP∞ = Gal(F∞ /T∞ ). As before,
n0 is the smallest integer n such that the extension F∞ /Fn is totally ramified
over the p-places. A crucial step towards the proof of the main theorem in
this section is the following proposition:
9
00
Proposition 2.3. Assume the Gross Conjecture for all n ≥ n0 . Let N∞
be
0
BP
00
the intersection of N∞ and F∞ . Then Gal(N∞ /T∞ ) is finite and
00 ∼
0
BP
Gal(M∞ /N∞
) = Gal(M∞ /N∞
) × Gal(M∞ /F∞
).
Proof. Let Xn := Gal(Mn /Fn ) and Wn := Gal(Mn /FnBP ). In [N2], §1, it has
been shown by techniques of embedding problems that there is a canonical
exact sequence
0
µF n
⊕v|p µFn,v
Xn
BPn
0.
Wn
For v|p, we denote by Γv its decomposition group (with respect to F∞ /F )
and we set Λv := Zp [[Γv ]]. Then, the induced module of a Λv -module M is
defined by Indv M := M ⊗Λv Λ. By taking inverse limits, we obtain an exact
sequence of Λ-modules
0
Zp (1)
⊕v|p Indv Zp (1)
X∞
BP∞
0,
W∞
from which we deduce an exact sequence of Galois modules
BP
0 → W∞ → torΛ X∞ = Gal(M∞ /T∞ ) → Gal(F∞
/T∞ ) → 0.
(2)
But Gal(T∞ /F∞ ) = frΛ X∞ is the maximal Λ-torsion free quotient of X∞ ,
BP
hence also of BP∞ and Gal(F∞
/T∞ ) = torΛ BP∞ .
0
Now, let us first show that the intersection I := W∞ ∩ Gal(M∞ /N∞
) is
Γn
trivial. By the explicit description of W∞ , we have W∞ (−1) = W∞ (−1)
for n ≥ n0 (recall that for a Λ-module M and an integer i, the notation M (i)
means M twisted i-times à la Tate). Besides, Kummer’s exact sequence in
Lemma 2.1 shows that
0
Gal(M∞ /N∞
) = Hom(A0∞ , µp∞ ) = Hom(A0∞ , Qp /Zp )(1),
0
and by Iwasawa’s theory of adjoints, we know that A0∞ ∼ β(X∞
), hence
0
0
0
Γn
Gal(M∞ /N∞ )(−1) ∼ α(X∞ ), and Gal(M∞ /N∞ )(−1) is finite by the Gross
Conjecture. This means that I(−1) = I(−1)Γn is finite, hence trivial because
X∞ has no non-trivial finite submodules. The triviality of the intersection I
shows the second assertion of the proposition.
0
As for the first statement, it suffices to recall that Gal(N∞
/T∞ ) is known
s∞ −1
to be isomorphic as a Zp -module to Zp
, where s∞ is the number of places
over p in F∞ (see [Iw1], [Ku]).
10
Actually, the proof of Proposition 2.3 yields extra information. It gives
0
the precise Λ-module structure of Gal(N∞
/T∞ ) (only the associated elementary module was previously known; see [Iw1], [Ku]). See Corollary 2.5 below.
0
As a corollary of Proposition 2.3, we also find that torΛ BP∞ ∼ α(X∞
)(−1),
Γn
Γn
so that torΛ BP∞ (−1) is finite; because both W∞ (−1) = W∞ (−1) and
torΛ X∞ (−1)Γn are Zp -free, it follows by (2) that
torΛ X∞ (−1)Γn ∼
= W∞ (−1)
(3)
for all n ≥ n0 . We can now state the main result of this section. We recall
that T∞ is the fixed field of torΛ X∞ inside M∞ (or of torΛ BP∞ inside M∞ )
00
BP
0
and N∞
is the intersection of F∞
and N∞
.
Theorem 2.4. Assume the Gross Conjecture for all n ≥ n0 . Then, for all
n ≥ n0 ,
Γ
00
coker(A0n → A0 ∞n ) ∼
/T∞ ), µp∞ )
= Hom(Gal(N∞
0
0
∼
/L0∞ ∩ T∞ ), µp∞ ).
= Hom(Gal(L∞ ∩ N∞
Proof. Consider the exact sequence
0
)Γn → (V∞ )Γn → (A0∞ )Γn = 0.
0 → ker fn → (E∞
Taking Kummer duals, which we denote by ( )∨ , we obtain:
∨
∨
0
0 → (V∞ )Γn → (E∞
)Γn → (ker fn )∨ → 0.
We have
(V∞ )Γn
∨
∼
= X∞ (−1)Γn = (torΛ X∞ )(−1)Γn ,
(the last equality holds because the Γn -fixed points of a Λ-module are in its
Λ-torsion submodule) and likewise,
∨
0
0
0
(E∞
) Γn ∼
(−1)Γn = (torΛ Z∞
)(−1)Γn ,
= Z∞
0
0
where Z∞
= Gal(N∞
/F∞ ). It is known (see [Iw1], Theorem 15) that
t
0
(torΛ Z∞
)(−1) ,→ ⊕ Λ/ξni Λ
i=1
P
with finite cokernel, where ξni = ωni /ωni −1 and ti=1 ni = s∞ − 1 (s∞ is the
0
number of primes over p in F∞ ); from this it follows that torΛ Z∞
(−1) ∼
=
s∞ −1
0
Zp
as abstract groups, and that torΛ Z∞ (−1) is invariant under Γn for
0
n 0. But here we need to be more precise and prove that torΛ Z∞
(−1) is
11
invariant under Γn for all n ≥ n0 . This can be obtained by scrutinizing the
results of [Ku]. The details are as follows:
0
0
0
0
Let E ∞ := lim U n ; then E ∞ / torΛ E ∞ is a free Λ-module of rank r1 + r2
←−
0
0
(Theorem 7.2 of [Ku]). By taking co-invariants, (E ∞ )Γn injects into U n
0
e 0 (∼
(Theorem 7.3 of [Ku]). If we denote by U
n = (E ∞ )Γn ) its image, we get
what we will call the Kuz’min exact sequence
0
e 0 → U → Vn → 0,
0→U
n
n
where Vn is tautologically defined (Vn is denoted FZl (U (kn ) ⊗ Zl )/ im κn in
[Ku]).
0
e 0 = µ(Fn ),
Now, suppose that µp ⊂ F ; then torΛ E ∞ = Zp (1) and torZp U
n
0
0
r2
e
e
therefore Un / torZp Un is isomorphic to Zp [Γ/Γn ] . In his proof of Proposition
8.2, Kuz’min shows that the Vn ’s stabilize, starting from n0 and he also proves
Γ
that torZp Vn ∼
= torZp X 0 ∞n (written TZl (Tl (k))Γn in the text of Kuz’min).
Tensoring by Qp /Zp over Zp gives
Γ
en0 ⊗ Qp /Zp → Un0 ⊗ Qp /Zp → Vn ⊗ Qp /Zp → 0.
0 → torZp Vn = torZp X 0 ∞n → U
e 0 ⊗ Qp /Zp ) in U 0 ⊗ Qp /Zp is the
We deduce from this that the image of lim(U
n
∞
−
→
0
0
Kummer dual of frΛ Z∞
= frΛ X∞ , hence lim Vn ⊗ Qp /Zp is that of torΛ Z∞
.
−
→
0
It follows in particular that torΛ Z∞ (−1) is fixed under Γn for n ≥ n0 , as
required.
Notice that we did not need the Gross Conjecture to establish that
0
torΛ Z∞
(−1) is fixed under Γn , n ≥ n0 (cp. Theorem 2.7 below).
Going back to the proof of Theorem 2.4, we finally get
0
(coker jn )∨ ∼
)(−1)Γn )
= coker(W∞ (−1)Γn → (torΛ Z∞
= (ker fn )∨ ∼
0
∼
)(−1))
= coker(W∞ (−1) → (torΛ Z∞
00
∼
= Gal(N∞ /T∞ )(−1)
for n ≥ n0 . The first isomorphism stems from (3) and the second from the
digression above. This proves the first assertion of the theorem.
As for the second assertion, note that the canonical homomorphism θ∞ :
0
BP
00
torΛ BP∞ → X∞
maps torΛ BP∞ onto Gal(L0∞ /L0∞ ∩T∞ ) and Gal(F∞
/N∞
)
0
0
0
0
0
0
0
onto Gal(L∞ /(L∞ T∞ ∩ N∞ ∩ L∞ )) = Gal(L∞ /N∞ ∩ L∞ ). Hence,
'
BP
00
0
torΛ BP∞ / Gal(F∞
/N∞
) −→ Gal(N∞
∩ L0∞ /L0∞ ∩ T∞ )
which proves the second isomorphism in the theorem. The situation is illustrated in the diagram of fields:
12
0
N∞
M∞
W∞
00
N∞
BP
F∞
ker θ∞
T∞
L0∞ T∞
L0∞ ∩ T∞
L0∞
im θ∞
F∞
The proof of the first isomorphism in Theorem 2.4 actually shows that
there is a natural exact sequence of Galois modules
0
0 → W∞ → torΛ Z∞
→ Hom(coker jn , µp∞ ) → 0 (∀n ≥ n0 ).
0
0
This leads to the following description of Z∞
= Gal(N∞
/F∞ ):
Corollary 2.5. There is an exact sequence
0
0 → W∞ → torΛ Z∞
→ Hom(Ψ∞ , µp∞ ) → 0.
Note that unlike the results of the previous section, we have obtained now
a non-asymptotical formulation of Theorem 1.4.
Corollary 2.6. In the situation of Theorem 2.4, the cokernels coker jn stabilize for n ≥ n0 and are isomorphic to Ψ∞ = lim Ψk .
−→
Proof. Obvious from Theorem 1.4 and Theorem 2.4.
N.B.: this does not mean that the kernels Ψn stabilize starting from n0
(for a counter-example, see Remark 1.7.)
Remark 2.7. One may wonder what can be said without the Gross Conjec00
ture; it turns out that some relevant modules (e.g. Gal(N∞
/T∞ )) are no
longer finite and have Zp -rank equal to δ∞ , the asymptotical “defect” of the
Gross Conjecture. As for Theorem 2.4, we note that the key-point in its proof
0
\
is the determination of ker
fn as coker(W∞ (−1)Γn → (torΛ Z∞
)(−1)Γn ). But
it is known in full generality (without the Gross Conjecture) that
torΛ X∞ (−1)Γn ∼
= W∞ (−1)
for n ≥ n0 ([Ku], 7-1; [Wi], 7-13), so that Theorem 2.4 remains valid without
the Gross Conjecture.
13
Now, assume that F is a CM-field; then, for a Zp [Gal(F/Q)]-module M ,
the “plus”-part is defined by M + := (1 + j)/2 M and the “minus”-part by
M − := (1 − j)/2 M , where j denotes complex conjugation.
Corollary 2.8. Suppose that F is a CM-field. Then, for n ≥ n0 , (coker jn )±
0
is the Kummer dual of the maximal finite submodule of Gal(L0∞ ∩ N∞
/F∞ )∓ .
N.B.: the statement about (coker jn )+ generalizes the main result of Ichimura
([I1], Theorem 2; [I2], Theorem 2).
0
Proof. Let us first look at (coker jn )− : its Kummer dual is Gal(L0∞ ∩N∞
/L0∞ ∩
0
T∞ )+ , which is thus a finite submodule of Gal(L0∞ ∩N∞
/F∞ )+ . But the weak
Leopoldt Conjecture (which is valid for any cyclotomic Zp -extension) implies
that Gal(T∞ /F∞ ) is reduced to its “minus” part, hence Gal(L0∞ ∩ T∞ /F∞ ) =
0
Gal(L0∞ ∩ T∞ /F∞ )− , and Gal(L0∞ ∩ N∞
/L0∞ ∩ T∞ )+ is the maximal finite
0
0
+
submodule of Gal(L∞ ∩ N∞ /F∞ ) .
Things are more complicated for (coker jn )+ : its Kummer dual is Gal(L0∞ ∩
0
N∞
/L0∞ ∩ T∞ )− , and we have to show that Gal(L0∞ ∩ T∞ /F∞ ) has no nontrivial finite submodule. This is a result of [B], which we reprove for the
convenience of the reader.
We adopt some of the notations in [N2]. Set D∞ := Gal(M∞ /L0∞ ); as
BP
before, W∞ = Gal(M∞ /F∞
) and θ∞ is the canonical homomorphism from
0
torΛ BP∞ to X∞ . These Galois groups fit into the commutative diagram
0
W∞
torΛ X∞
torΛ BP∞
X∞
0
X∞
0
θ∞
0
D∞
0
which, by the snake lemma, gives the exact sequence
0 → ker θ∞ → D∞ /W∞ → frΛ X∞ → coker θ∞ → 0.
(4)
It is shown in [N2] that torΛ BP∞ has no non-trivial finite submodules. It
follows from (4) that D∞ /W∞ shares this property. Next, let us record some
0
useful properties of the modules E ∞ := lim Un0 ⊗ Zp and H∞ := ⊕v|p Xv,∞ ,
←−
where Xv,∞ is the maximal abelian pro-p-extension of the completed field
Fv,∞ . The weak Leopoldt Conjecture (which holds for any cyclotomic Zp 0
extension) states that frΛ X∞ ∼ Λr2 and Kuz’min has shown that frΛ E ∞ ∼
=
2r2
([Ku],
Proposition
7.1).
Λ
Λr2 ([Ku], Theorem 7.2) and frΛ H∞ ∼
=
Let us now stick to the case of a CM-field F . Invoking the weak Leopoldt
+
Conjecture again, frΛ X+
∞ = 0, hence also (coker θ∞ ) = 0. Besides, it has
14
been shown in [N2] that ker θ∞ is pseudo-dual to coker θ∞ (duality with values
in Zp (1)), hence the “minus” part of the exact sequence (4) reads:
0 → (D∞ /W∞ )− → frΛ X∞ → coker θ∞ → 0.
(5)
0
−
Because the U 0 −
= 0 and
n ’s stabilize modulo roots of unity, frΛ (E ∞ )
−
−
− ∼
frΛ (H∞ ) = (D∞ /W∞ ) . It follows in particular that (D∞ /W∞ ) is a direct
factor of Λ2r2 , hence is Λ-free, of rank r2 because of (5). Besides, the structure
theorem for Λ-modules gives a monomorphism frΛ X∞ ,→ Λr2 with finite
cokernel.
Putting all this together, we obtain a commutative diagram
0
(D∞ /W∞ )−
frΛ X∞
coker θ∞
0
Λr2
M
0
o
f
Λr2
where the map f is such that the left square commutes and the module M
is induced by the right part of the diagram. This shows that the projective
dimension of M over Λ is at most 1, hence M (and consequently coker θ∞ )
has no non-trivial finite submodules. This proves the corollary.
3
Kummer duality and Gross kernels
We saw in Section 2 that Theorems 1.4 and 2.4 together give the Kummer
00
radical of Gal(N∞
/T∞ ). In this section, we shall give a direct proof which is
interesting for itself.
×
Let us denote by (.) the p-adic completion and by Fbn,v
the universal norms
in the cyclotomic Zp -extension of Fn,v , the completion of Fn at the place v.
Then, the so-called Sinnott exact sequence for Fn reads (see [FGS], [Ko], [J],
etc.):
0
gn
×
×
0 → ker gn → U n −→ ⊕v|p F n,v /Fbn,v
→ Gal(L̃Fn /Fn ) → A0n ,
for all n ≥ 0. Here, L̃Fn stands for the maximal abelian extension of Fn
contained in L0∞ . The ker gn ’s are called Gross kernels in [Ko]. By the
b×
e v|p F ×
product formula, im gn is contained in ⊕
n,v /Fn,v , the elements with trivial
sum of components. Now, the modified sequence reads
0
ker gn
0
Un
b×
e v|p F ×
⊕
n,v /Fn,v
0
(X∞
) Γn
A0n
Ψn
for all n ≥ n0 . Following Kolster, let us write ker g∞ := lim ker gn .
−→
15
0,
Theorem 3.1 ([Ko, Ku]). Assume the Gross Conjecture for all n 0.
Then:
Gal(T∞ /F∞ ) = frΛ X∞ ∼
= Hom(ker g∞ ⊗ Qp /Zp , µp∞ ).
In other words, the Kummer radical of frΛ X∞ is ker g∞ ⊗ Qp /Zp . We
00
use this to determine the Kummer dual of Gal(N∞
/T∞ ), which is a priori a
subgroup of
0
0
(U∞
⊗ Qp /Zp )/(ker g∞ ⊗ Qp /Zp ) = lim(U n ⊗ Qp /Zp ) lim(ker gn ⊗ Qp /Zp )
−→
−→
(inductive limits commute with tensor products). Consider an element a ⊗
0
1/pk ∈ U∞
⊗ Qp /Zp . We may assume that a to belong to Un0 for n ≥ k.
k
The Kummer extension Fn (a1/p )/Fn with k ≤ n is infinitely embeddable
1/pk
in cyclic p-extensions if and only if Fn,v (av )/Fn,v is Zp -embeddable for all
v|p and unramified for v - p, where av is the image of a in Fn,v . This, in
k
× p b×
turn, is equivalent to av ∈ Fn,v
Fn,v for all v|p (see [BP], exemple p. 526).
00
This shows that the Kummer dual of Gal(N∞
/T∞ ) is lim Φn , where Φn is the
−→ ×
0
×
kernel of the natural map (U n / ker gn ) ⊗ Qp /Zp → ⊕v|p (F n,v /Fbn,v
) ⊗ Qp /Zp .
After tensoring Sinnott’s modified exact sequence by Qp /Zp , we obtain the
following exact sequence
0
b×
e v|p F ×
0 → torZp Ψn = Ψn → (U n / ker gn ) ⊗ Qp /Zp −→ (⊕
n,v /Fn,v ) ⊗ Qp /Zp → 0, (6)
where exactness on the right (resp. left) is due to the finiteness of Ψn (resp.
b×
e v|p F ×
to the Zp -freeness of (⊕
n,v /Fn,v )). It follows that Φn = Ψn , hence:
Theorem 3.2. Under the assumptions of Theorem 3.1:
00
• the Kummer dual of Gal(N∞
/T∞ ) is Ψ∞ := lim Ψk ;
−→
×
0
00
e F /Fb × ) ⊗ Qp /Zp .
• the Kummer dual of Gal(N∞
/N∞
) is lim(⊕
−→ v|p n,v n,v
We see that Proposition 2.3, Corollary 2.5 and Theorem 3.2 together provide a complete description of the module Gal(M∞ /T∞ ) = torΛ X∞ .
For the sake of completeness, let us compare the exact sequences of
Kuz’min and Sinnott. Let gn be, as before, the map defined in Sinnott’s
exact sequence. Then there is a commutative diagram
0
Vn
0
0
im gn
0
0
e0
U
n
Un
0
ker gn
Un
16
Γ
en0 ∼
But ker gn /U
= X 0 ∞n by Proposition 7.5 of [Ku], so that
Γ
0 → X 0 ∞n → Vn → im gn → 0.
With the Gross Conjecture, we then have Vn ⊗ Qp /Zp = im gn ⊗ Qp /Zp and
0
we retrieve the fact that the latter module is the Kummer dual of torΛ Z∞
,
as was shown in this section.
Remark 3.3. Without the Gross Conjecture, only a part of the kummerian
description of coker jn using Gross kernels remains available. Moreover, in
the exact sequence (6), we no longer have torZp Ψn = Ψn . By taking direct
limits and applying Theorem 2.4, one can show that the Kummer dual of
00
lim(torZp Ψn ) is the maximal finite submodule of Gal(N∞
/T∞ ).
−→
4
Explicit formulæ in the abelian semi-simple
case
In this section we study the abelian semi-simple case in detail, i.e., the case
when G = Gal(F/Q) is abelian of order prime to P
p. For any one-dimensional
∗
p-adic character χ ∈ Hom(G, Qp ), let eχ = 1/|G| σ∈G χ−1 (σ)σ be the usual
idempotent, which lives in Oχ [G], where Oχ denotes the ring generated over
Zp by the values of χ. The “χ-part” of a Zp [G]-module M is the Oχ [G]module defined by M (χ) := eχ (M ⊗Zp Oχ ).
In the semi-simple case, the invariant n0 of the preceding sections is obvie0 , we aim to give explicit formulæ
ously equal to zero. Denoting j0 (A00 ) by A
0
0Γ
0
e
for the orders of the χ-parts (A ∞ /A0 )(χ). If F contains µp , Corollary 2.8
e0 )(χ) is the Kummer dual of the maximal finite subtells us that (A0 Γ∞ /A
0
0
module of Gal(L0∞ ∩ N∞
/F∞ )(χ−1 ω), where ω is the Teichmüller character
(cp. [I2], Theorem 2). Note right away that for the trivial character χtriv , we
clearly have A0∞ (χtriv ) = (0) = A00 (χtriv ), since the idempotent eχtriv is just
the norm up to a p-adic unit. In the general case, the starting point is a
straightforward “χ-version” of Lemma 1.2:
Γ
e0 )(χ)|
|(A0 ∞ /A
0
0
|X∞
(χ)Γ | | ker j0 (χ)|
·
.
=
0
|A0 (χ)| |X 0 ∞ (χ)Γ |
(7)
Adopting the additional notations |A00 (χ)| = |A0F (χ)| ∼p h0χ , and
0
ε0F (χ) := | ker j0 (χ)|/|(X 0 ∞ )(χ)Γ |,
we get
Γ
e0 )(χ)| = |Ψ0 (χ)| · ε0 (χ) ∼p
|(A0 ∞ /A
0
F
17
0
|X∞
(χ)Γ | 0
· εF (χ).
0
hF (χ)
(8)
Our primary goal is now to compute the order of Ψ0 (χ) for a non-trivial
character χ. We consider the χ-part of Sinnott’s modified sequence:
0
b 0 (χ)
U
F
0
U F (χ)
gχ
×
(⊕v|p F v /Fbv× )(χ)
Artin
0
X∞
(χ)Γ
A0F (χ)
0.
Ψ0 (χ)
(9)
0
b
Here UF (χ) denotes the kernel of gχ and we note that we have a direct sum
as well as surjectivity on the right, since χ is non-trivial. We shall need a
×
more precise description of the middle term (⊕v|p F v /Fbv× )(χ). Let Σ be the
set of p-places of F and Oχ [Σ] be the free Oχ -module on Σ. The third term
of the exact sequence (9) is clearly isomorphic to Oχ [Σ](χ); but the choice
'
of a given place v ∈ Σ yields an isomorphism Oχ [Σ] −→ Oχ [G/D] induced
by w ∈ Σ 7→ τw ∈ G/D, where D is the decomposition subgroup of v and
τw is the unique element of G/D sending v to w. Summarizing, we have an
isomorphism
×
(⊕ F v /Fbv× )(χ) ∼
(10)
= Oχ [G/D](χ).
v|p
We can now give another interpretation of the exact sequence (9), closer
to that of Sinnott in [FGS], Proposition 6.5: for w|p, let us denote by Nw =
NFw /Qp the local norm map. Composing the map gF with logp ◦ Nw , we get
0
b 0 → NF := ⊕w∈Σ logp (Nw (F × ))w. But F is linearly disjoint
a map λ : U F /U
F
w
from Q∞ , and the same is true for the completions at all w|p because [F : Q]
is prime to p. Local class field theory then gives a chain of isomorphisms
×
(F w /Fbw× )
Nw
'
−→
×
b×
Q p /Q
p
logp
'
∼
= 1 + pZp −→ pZp ,
hence, the exact sequence (9) becomes, for any χ 6= χtriv ,
0
0
b 0 (χ)
U F (χ)/U
F
λχ
0
X∞
(χ)Γ
NF (χ) = (⊕w∈Σ pZp )(χ)
A0F (χ)
Ψ0 (χ)
(11)
We can now show Proposition 2 of [I1], namely
e0 )(χ) and
Proposition 4.1. For any character χ such that χ(p) 6= 1, (A0 Γ∞ /A
0
Ψ0 (χ) are trivial.
18
0.
Proof. By Corollary 1.6, it is sufficient to show the triviality of Ψ0 (χ), i.e.,
the surjectivity of the map gχ in (9). We see with isomorphisme (10) that
χ(p) = 1 if and only if χ factors through G/D, hence Oχ [Σ](χ) = (0) if
χ(p) 6= 1. This implies Proposition 4.1.
Remark 4.2. The proof actually shows that if χ(p) = 1, then Oχ [Σ](χ) ∼
=
Oχ [G/D](χ) ∼
O
·
e
(viewing
χ
as
a
character
of
G/D).
It
follows
from
= χ χ
0
b 0 )(χ) has Oχ -rank 1.
(9) that for χ 6= χtriv and χ(p) = 1, (U F /U
F
We now tackle the “split” case, for which we will give two formulæ, depending on the parity of χ. Still, one can formulate a unified theorem as
follows (see Theorems 4.5 and 4.6):
Theorem 4.3. Let χ be a non-trivial character such that χ(p) = 1. Choose
0
0
b 0 (χ). Fix a p-place
u0 ∈ U F such that eχ ⊗ u0 gives an Oχ -basis of U F (χ)/U
F
0
0
v of F and set mχ := ordp (logp (Nv (u ))) − 1 and dχ := [Oχ : Zp ]. Then
Γ
e0 )(χ)| = pm0χ dχ · ε0 (χ).
|(A0 ∞ /A
0
F
Let us now investigate the “minus” and “plus” parts separately.
4.1
The “minus” part in the “split” case
If χ is an odd character, the capitulation kernel ker j0 (χ) is trivial (see [FG]
e0 )(χ)| by (7). In particular, the
p. 101 for instance), hence |Ψ0 (χ)| = |(A0 Γ∞ /A
0
0
map A00 (χ) → A0 Γ∞ (χ) is surjective if and only if the map X∞
(χ)Γ → A0F (χ)
is injective (compare with Corollary 1.6). Moreover, using Remark 4.2, we
0
can determine U F (χ):
0
Lemma 4.4. For χ odd and χ(p) = 1, U F (χ) has Oχ -rank 1 and its torsion
b 0 (χ).
is U
F
Proof. Since F is Galois over Q, it is known (see e.g. [BN], §1.1) that there
is an exact sequence
0
val
nat
0 → U F → U F −→ Zp [Σ] → AF −→ A0F → 0,
(12)
where the map “val” is induced by the p-adic valuations. For an odd character
0
χ, U F (χ) is finite, hence U F (χ) and Oχ [Σ](χ) have the same Oχ -rank. We
conclude thanks to the remark following Proposition 4.1.
We can now compute |Ψ0 (χ)|:
19
Theorem 4.5. Let χ be an odd character such that χ(p) = 1. Choose u 0 ∈
0
0
U F such that eχ ⊗ u0 gives an Oχ -basis of U F (χ) modulo torsion. Fix a pplace v of F and let L0p (.) be the derivative of the Kubota-Leopoldt p-adic L
function, m0χ = ordp (logp (Nv (u0 ))) − 1, and dχ = [Oχ : Zp ]. Then
Γ
e0 )(χ)| = pm0χ dχ ∼p h0 −1 (L0 (χ−1 ω, 0)/w(F (µp)))dχ .
|Ψ0 (χ)| = |(A0 ∞ /A
0
χ
p
(Here w(k) denotes, as usual, the number of roots of unity contained in a
field k.)
0
b 0 (χ) → NF (χ) in
Proof. We aim to compute the cokernel of λχ : U F (χ)/U
F
the exact sequence (11). As in the proof of Proposition 4.1, the choice of the
p-place v allows one to identify Oχ [Σ] with Oχ [G/D], and χ can be viewed
as a character of G/D since χ(p) = 1. We have coker λχ ∼
= Oχ eχ / im λχ , the
“denominator” being Oχ -free of rank 1 (since Ψ0 (χ) is finite), generated by
X
logp (Nw (u0 ))χ(τw )−1 τw = |G|(logp (Nv (u0 )))eχ
w|p
(recall that τw is characterized by τw (v) = w). This gives the first formula of
Theorem 4.5.
The second formula is a direct consequence of the Main Conjecture: let
0
fχ (T ) and gχ (T ) resp. be the characteristic series of X∞
(χ) and X∞ (χ) over
Oχ [[T ]]. Under our hypothesis that χ(p) = 1 and χ is odd, it is known (cp.
[G2]) that gχ (T ) has a zero of order 1 at T = 0, hence gχ (T ) = T fχ (T ). Still
0
because χ is odd, |X∞
(χ)Γ | ∼p fχ (0)dχ and fχ (0) is equal to
Gχ−1 ω (T )
gχ (T )
∼p lim
,
T →0
T →0
T
T
lim
where the series Gχ−1 ω (T ) satisfies the following interpolation property (by
our assumption, χ−1 ω is not trivial)
Lp (χ−1 ω, s) = Gχ−1 ω (us − 1) for s ∈ Zp ,
where u = κ(γ), κ is the cyclotomic character and γ is a topological generator
of Γ = Gal(F∞ /F ). Then, one finds
Gχ−1 ω (T )
= L0p (χ−1 ω, 0)/ logp u.
T →0
T
lim
Since logp u ∼p wF (µp ) , we finally get
fχ (0) ∼p L0p (χ−1 ω, 0)/wF (µp ) ,
whence the theorem. Notice that wF (µp ) ∼p p since [F : Q] is prime to p.
20
One may remark that we have essentially computed the “χ-part” of the
Gross regulator (cp. [FGS], 3.8). An analytic expression of L0p (χ−1 ω, 0)
in terms of p-adic Γ-functions and Bernoulli numbers is available in [FG],
Proposition 1. Also, for χ odd quadratic, the formula for L0p (χ−1 ω, 0) (or
|A0 Γ∞ (χ)|) given on p. 100 of [FG] can be considered as a particular case of
Theorem 4.5.
4.2
The “plus” part in the “split” case
0
b 0 (χ) has
Let χ be an even non-trivial character. By Remark 4.2, U F (χ)/U
F
Oχ -rank 1 if χ(p) = 1. Then the arguments in the first part of Theorem 4.5
remain entirely valid and we get
Theorem 4.6. Let χ be an even non-trivial character such that χ(p) = 1.
0
0
b 0 (χ). Then
Choose u0 ∈ U F such that eχ ⊗ u0 gives an Oχ -basis of U F (χ)/U
F
Γ
e0 )(χ)| = pm0χ dχ · ε0 (χ),
|(A0 ∞ /A
0
F
with m0χ , dχ as in Theorem 4.5.
This result is not entirely satisfactory, for many reasons:
(i) For an even character, the capitulation kernels are not trivial in general,
hence the fuss factor ε0F (χ), which is not entirely under control (but
see Proposition 4.13 below).
b 0 of locally universal norms. However
(ii) Neither do we control the group U
F
0
we can show (see Lemma 4.8 below) that U F (χ) has rank 2, so that,
in practice, it should not be difficult to exhibit the special (p)-unit u0
in Theorem 4.6.
(iii) The expression in terms of p-adic L-functions is a priori not available,
because the Main Conjecture does not apply to X∞ (χ) for χ even.
Fortunately, this difficulty can be overcome thanks to a lemma by Ozaki
& Taya:
Lemma 4.7 ([T], Lemma 2). Let k be a totally real field satisfying Leopoldt’s
Conjecture. If p is totally split in k, we have a canonical isomorphism
torZp Xk ∼
= (X∞ )Γ
(see the notations after the introduction).
21
Moreover, torZp Xk is a finite group, the order of which is given by Coates’s
p-adic formula in terms of regulators ([C], appendix), or by the Main Conjecture in terms of p-adic L-functions.
For even characters χ, we have A∞ (χ) ∼
= A0∞ (χ) since Leopoldt’s Conjece0 (χ) ∼
ture holds true all along the cyclotomic tower, and therefore also A
=
0
0
e
A0 (χ), since the map A0 (χ) → A0 is onto. As a result, we can rewrite formula
(8) as:
e0 )(χ)| ∼p |X∞ (χ)Γ | · εF (χ),
|(AΓ∞ /A
hχ
0 Γ
where εF = | ker(A0 → AΓ∞ )|/|(X∞
) |. Moreover, if χ(p) = 1, Lemma 4.7
allows us to replace X∞ (χ)Γ by torZp XF (χ), which sits in the following exact
sequence of class field theory relative to inertia (for χ 6= χtriv ):
0
Artin
0 → U F (χ) → UF (χ) := (⊕v|p Uv1 )(χ) −→ torZp XF (χ) → AF (χ) → 0. (13)
Since torZp Xk is finite, (13) implies that U F (χ) and UF (χ) have the same
Oχ -rank; but it is well known (see e.g. [K], p. 3) that U (χ) has rank 1.
0
Therefore U F (χ) has rank 2 by (12). Whence the
Lemma 4.8. For any non-trivial even character χ, U F (χ) has Oχ -rank 1
0
(hence U F (χ) has Oχ -rank 2).
We can now give the actual “real” analogue of Theorem 4.5:
Theorem 4.9. Let χ be an even non-trivial character such that χ(p) = 1.
Choose u ∈ U F such that eχ ⊗ u gives an Oχ -basis of U F (χ) modulo torsion.
Fix a p-place v of F and define mχ := ordp (logp (Nv (u))) − 1. Then
Γ
e0 )(χ)| = |(AΓ /A
e0 )(χ)| = pmχ dχ · εF (χ) ∼p h−1 Lp (χ, 1)dχ · εF (χ).
|(A0 ∞ /A
χ
0
∞
bF (χ) = (1) inside
Proof. Since χ(p) = 1, it is easy to see that U F (χ) ∩ U
0
U F (χ) (for details, see [BN], Lemma 1.1). Applying the map λχ in the exact
sequence (11), we then get an isomorphism UF (χ)/U F (χ) ∼
= NF (χ)/λχ (u)Oχ ,
and we proceed exactly as in the proof of the first part of Theorem 4.5 to get
the first formula of Theorem 4.9. The second formula follows directly from
the Main Conjecture applied to the characteristic series of X∞ (χ) for χ even,
non-trivial.
Remark 4.10. At this point, it is interesting in view of the examples given in
section 5.4 to compare the formulæ given in Theorem 4.6 and Theorem 4.9.
Since the fuss factors εF (χ) and ε0F (χ) are not easily accessible, in practice,
we shall only compute Ψ0 (χ) or its analog X∞ (χ)Γ /A0 (χ). Obviously, the
22
triviality of the latter implies that of Ψ0 (χ), but the converse is not true.
Hence, for the applications, it is more efficient to test the triviality of Ψ0 (χ)
and therefore apply Theorem 4.6.
Actually, the proof of Theorem 4.9, together with Theorem 4.6, also yields
Corollary 4.11 (cp. [BN], Theorem 3.7).
0
bF0 (χ)] = εF (χ)/ε0F (χ).
[U F (χ) : U F (χ) ⊕ U
One may notice again that we have essentially computed the “χ-part” of
the Leopoldt regulator. In the statements of Theorem 4.9, we can go a bit
further and give another expression of the fuss factor
0 Γ
εF (χ) = | ker(A0 → AΓ∞ )(χ)|/|(X∞
) (χ)|
j0
featuring the capitulation kernel ker(A0 → AΓ∞ ) (or ker(A00 → A0 Γ∞ )) alone.
In the quotient εF (χ), the denominator is usually hard to compute, because
it is an asymptotical invariant. We can try and remove it by using decomposition subgroups. Let us define DF , the decomposition at p, by the exact
sequence 0 → DF → AF → A0F → 0. By taking projective limits along the
0
cyclotomic tower, we find the exact sequence 0 → D∞ → X∞ → X∞
→ 0.
By definition, D∞ is invariant under Γ, and D∞ (χ) is known to be finite when
χ is even (see e.g. [G1]). Note that the “χ-part” of Greenberg’s Conjecture
is equivalent to |D∞ (χ)| = |X∞ (χ)Γ |. The following weaker statement is
available:
Lemma 4.12 (see [B]). For an even character χ such that χ(p) = 1,
D∞ (χ) ∼
= X Γ (χ).
∞
for even characters, another easy application of
Since A∞ (χ) ∼
=
the snake lemma yields the exact sequence
A0∞ (χ)
0 → DF (χ) → ker(A0 → AΓ∞ )(χ) → (ker j0 )(χ) → 0.
We finally get the following expression for εF (χ):
Proposition 4.13. For an even character χ such that χ(p) = 1,
εF (χ) ∼p | ker j0 (χ)| · hχ /h0χ · |D∞ (χ)|−1 .
Remark 4.14. Greenberg’s Conjecture is equivalent to the triviality of A0 ∞ (χ)Γ
e00 )(χ) for
for all even χ. A weaker conjecture would be the triviality of (A0 Γ∞ /A
all even χ; numerically, it requires the computation of the fuss factor εF (χ).
In the formula given in Proposition 4.13, only the first factor |(ker j0 )(χ)| is
not easily accessible (the same difficulty arises when checking Greenberg’s
Conjecture).
23
4.3
Quadratic examples
e0 in the most
In this subsection, we study the vanishing of coker j0 = A0 Γ∞ /A
0
simple case: we take for F a quadratic field in which p splits (when there is
only one prime√over p, we already know that the map j0 is surjective). We
write F = Q( d), where d is a squarefree integer. Besides, χ denotes the
non-trivial character of Gal(F/Q).
Let us suppose first that F is an imaginary quadratic field. Here, as
explained in Section 5.2 and since A0∞ (χtriv ) vanishes, the surjectivity of the
map A00 → A0 Γ∞ is equivalent to the triviality of Ψ0 (χ). By Theorem 4.5, we
0
have |Ψ0 (χ)| = pmχ dχ and here dχ = 1. We just need to exhibit an element
0
0
u0 ∈ U F which generates U F (χ) modulo torsion.
Let p1 and p2 be the two primes above p and let n be the order of p in
the class group Cl(F ); then pn is principal, and we may pick a generator α.
0
Then, one easily sees that U F /µ(F ) = hα, pi. Indeed, a (p)-unit generates a
m0
m0 m−m0
principal ideal of the form pm
, hence m − m0 is divisible by n,
1 p2 = p p1
0
0
and pm−m
is a power of (α). Therefore, U F (χ) modulo torsion is generated
1
by α and we may take u0 := α.
The element u0 , when embedded in Fp ∼
= Qp , can be written as u0 = pn u1
×
with u1 ∈ Zp . Then, it follows from the definition of m0χ that
|Ψ0 | = |Ψ0 (χ)| = pvp (logp (u1 ))−1 .
For the examples below, we fix the prime p = 3 and provide a table com0
puted with the program PARI that gives the value of |Ψ0 (χ)| = 3v3 (log3 (u ))−1
(hence of coker j0 ), for various values of d (see Table 1).
Let us now turn to the case when F is a real quadratic field in which p
splits; we denote by p1 and p2 the two primes over p. The completed (p)-units
0
U F have rank 3 and, with the same kind of arguments as above, we can write
0
U F = h0 , α, pi, where 0 is the fundamental unit and α is a generator of the
principal ideal pn1 (n is the order of p1 in Cl(F )).
0
Theorem 4.6 tells us that |Ψ0 (χ)| = pmχ , where m0χ = vp (logp (u0 )) − 1
0
and u0 ∈ U F is such that λχ (u0 ) generates im λχ . The elements 0 and α,
when embedded in Fp ∼
= Qp , can be written respectively as 0 = ζ1 u1 and
(1)
n
α = p ζ2 u2 , with ζ1 , ζ2 ∈ µp−1 and u1 , u2 ∈ Up , the principal units of Zp .
b × = µp−1 × pZ , we have λχ (α) = u2 − 1 and λχ (0 ) = u1 − 1, so that
Since Q
p
im λχ is generated by u2 − 1 or u1 − 1, depending on whether vp (u2 − 1) ≤
vp (u1 − 1) or not. Accordingly, we take u0 := α or u0 := 0 . Note that
vp (u0 − 1) = vp (logp (u0 )), so that
0
|Ψ0 | = pvp (u −1)−1 .
24
d
h0
-2
1
-5
1
-11 1
-14 1
-17 1
-23 1
-26 2
-29 1
-35 1
-38 1
-41 1
-47 1
-53 1
-59 1
-62 1
-65 2
-71 1
-74 2
-77 2
-83 1
-86 1
-89 1
-95 1
-101 1
-107 1
-110 2
-113 1
-119 1
-122 2
-131 1
-134 1
|Ψ0 |
1
1
1
3
1
1
1
1
3
1
27
9
1
1
1
3
1
9
1
1
3
1
1
9
9
1
3
1
1
1
1
d
h0
-137 1
-143 2
-146 2
-149 1
-155 1
-158 1
-161 2
-167 1
-170 2
-173 1
-179 1
-182 2
-185 2
-191 1
-194 4
-197 1
-203 1
-206 1
-209 2
-215 1
-218 2
-221 2
-227 1
-230 2
-233 1
-239 3
··· ···
-461 3
··· ···
-629 6
|Ψ0 |
1
1
1
9
1
3
1
1
1
3
1
1
1
1
1
1
1
1
1
1
3
1
3
9
1
1
···
1
···
1
Table 1: imaginary quadratic fields
25
In Table 2, we provide for p = 3 and various values of d the 3-adic
valuation of u1 − 1 (denoted k1 ) and the 3-adic valuation of u2 − 1 denoted
k2 ). The third column gives the order of |Ψ0 |.
26
d
7
10
13
19
22
31
34
37
43
46
55
58
61
67
70
73
79
82
85
91
94
97
103
106
109
115
118
127
130
133
139
142
k1
1
1
1
1
1
1
1
1
2
1
1
2
1
3
1
1
2
2
2
1
1
1
2
2
2
1
1
1
1
1
2
1
k2
1
1
1
1
1
1
2
1
1
2
1
1
2
2
1
2
1
1
1
2
2
1
2
3
1
1
2
1
1
2
2
3
|Ψ0 |
1
1
1
1
1
1
1
1
1
1
1
1
1
3
1
1
1
1
1
1
1
1
3
3
1
1
1
1
1
1
3
1
d
145
151
154
157
163
166
178
181
187
190
193
202
205
211
214
217
223
229
235
238
241
247
253
259
262
265
271
274
277
283
286
295
298
k1
1
2
1
1
1
1
1
2
1
1
1
2
1
1
1
1
1
1
1
3
1
4
2
1
1
1
3
1
1
1
1
2
1
k2
1
1
1
2
1
1
2
1
2
2
1
1
2
1
1
1
1
1
3
2
1
1
2
1
1
1
1
1
1
2
1
3
1
|Ψ0 |
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
1
1
3
1
1
1
1
1
1
1
1
3
1
Table 2: real quadratic fields
27
An infinite family of real quadratic fields
In this subsection, for p = 3, we give an infinite family of real quadratic
fields for which the cokernel coker j0 is trivial. Let
√ a be a non-negative integer
and let F be the real √
quadratic field F = Q( 4a − 1). Then we can easily
a
check that a := 2 + 4a − 1 is a fundamental unit for the field F . In the
following, we shall compute the 3-adic valuation of 2a − 1 (in doing so, we
compute what we called k1 in Table 2, as raising to the square gets rid of the
roots of unity).
Without necessarily assuming that p = 3, we first show that if any, there
are infinitely many a for which 4a − 1 is a square in Zp , where p is a fixed
odd prime. Suppose there is an a0 such that 4a0 − 1 is a square in Zp . Write
t
4a0 − 1 = p2r v 2 , v ∈ Z×
p . Since 4 is prime to p, there is a t satisfying 4 ≡ 1
mod p2r+1 . We claim that 4a0 +kt − 1 is a square in Zp for all integers k ≥ 0.
Indeed, by Hensel’s lemma, P (X) := X 2 − (4a0 +kt − 1) has a root in Zp if
there exists y ∈ Zp such that vp (P (y)/P 0(y)2 ) > 0.
Let us take y := pr v; then P 0 (y) = 2pr v and P (y) = 4a0 (1 − 4kt ). But
vp (P (y)) ≥ vp (4t − 1) > 2r = 2vp (P 0 (y)).
Now, we look at the example p = 3. Then 3 is a suitable choice for a0
and we get r = 1. The order of 4 in (Z/33 Z)× is t = 9. We deduce that all
43+9k − 1, k ≥ 0 are squares
in Z3 . We want to compute the 3-adic valuation
√
3+9k
2
3+9k
− 1)2 − 1. An easy calculation shows that the
+ 4
of k − 1 = (2
beginning of the 3-adic expansion of 2k − 1 is 2k − 1 = 2(−1)3k+1 3 + · · · , so
that v3 (2k − 1) = 1. Hence, as explained in the quadratic examples above,
for each k, the corresponding k is a suitable choice for the element u0 in
0
Theorem 4.6, i.e., it generates im λχ , and |Ψ0 | = 3v3 (u −1)−1 = 1.
Therefore, for all k ≥ √
0 and p = 3, the cokernel coker(A00 → A0 Γ∞ ) associated to the field Fk = Q( 43+9k − 1) is trivial.
References
[B] R. Badino, Sur les égalités du miroir en théorie d’Iwasawa, thèse, Besançon, 2003.
[BN] J.-R. Belliard et T. Nguyen Quang Do, On modified circular units and
annihilation of real classes, to appear in Nagoya Math. J., 2002.
[BP] F. Bertrandias et J.-J. Payan, Γ-extensions et invariants cyclotomiques,
Ann. Sci. Éc. Norm. Sup., 5:517-543, 1972.
28
[C] J. Coates, p-adic L-functions and Iwasawa’s theory, in Algebraic Number
Fields, edited by A. Fröhlich and Press, 1977.
[FG] B. Ferrero and R. Greenberg, On the behavior of p-adic L-functions at
s = 0, Invent. Math. 50:91-102, 1978.
[FGS] L. Federer and B. H. Gross (with an appendix by W. Sinnott), Regulators and Iwasawa modules, Invent. Math. 62:443-457, 1981.
[G1] R. Greenberg, On the Iwasawa invariants of totally real number fields,
Amer. J. Math., 98:263-284, 1976.
[G2] R. Greenberg, On a certain `-adic representation, Invent. Math. 21:117124, 1973.
[GJ] M. Grandet et J.-F. Jaulent, Sur la capitulation dans une Zl -extension,
J. Reine Angew. Math., 362:213-217, 1985.
[I1] H. Ichimura, On a quotient of the unramified Iwasawa module over an
abelian number field, J. of Num. Th., 88:175-190, 2001.
[I2] H. Ichimura, On a quotient of the unramified Iwasawa module over an
abelian number field, II, Pac. J. of Math., 206:129-137, 2002.
[I3] H. Ichimura, A note on the ideal classgroup of the cyclotomic Zp extension of a totally real number field, Acta Arithm., 105:29-34, 2002.
[Iw1] K. Iwasawa, On Zl -extensions of algebraic number fields, Ann. of
Math., 98:246-326, 1973.
[Iw2] K. Iwasawa, On cohomology groups of units for Zp -extensions, Amer.
J. Math., 105:189-200, 1982.
[J] J.-F. Jaulent, Sur les conjectures de Leopoldt et de Gross, J. Arithm. de
Besançon (1985), Astérisque, 147-148:107-120, 1987.
[K] H. Kisilevsky, On Iwasawa’s λ-invariant for certain Zl -extensions, Acta
Arithm., XL:1-8, 1981.
[KM] M. Kolster and A. Movahhedi, Galois co-descent for étale wild kernels
and capitulation, Ann. Inst. Fourier, 50:35-65, 2000.
[Ka] B. Kahn, Descente galoisienne et K2 des corps de nombres, K-Theory,
7:55-100, 1993.
29
[Ko] M. Kolster, An idelic approach to the wild kernel, Invent. Math. 103:924, 1991.
[Ku] L. V. Kuz’min, The Tate module for algebraic number fields, Math.
USSR Izv., 6-2:263-321, 1972.
[N1] T. Nguyen Quang Do, Sur la Zp -torsion de certains modules galoisiens,
Ann. Inst. Fourier, 36-2:27-46, 1986.
[N2] T. Nguyen Quang Do, Sur la torsion de certains modules galoisiens II,
Sémin. Théorie des Nombres de Paris, 271-297, 1986-87.
[S] H. Sumida, On capitulation of S-ideals in Zp -extensions, J. Number Th.,
86:163-174, 2001.
[Sc] P. Schneider, Über gewisse Galoiskohomologiegruppen, Math. Z.,
168:181-205, 1979.
[T] H. Taya, On p-adic L functions and Zp -extensions of certain totally real
fields, J. Number Th., 75:170-184, 1999.
[W] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag,
2nd edition, New-York, 1997.
[Wi] K. Wingberg, Duality theorems for Γ-extensions of algebraic number
fields, Compos. Math., 55:333-381, 1985.
M. Le Floc’h & A. Movahhedi
Université de Limoges
LACO
CNRS UMR 6090
123 av. Albert-Thomas
87060 LIMOGES CEDEX
FRANCE
T. Nguyen Quang Do
Université de Franche-Comté
Laboratoire de Mathématiques
CNRS UMR 6623
16 route de Gray
25030 BESANÇON CEDEX
FRANCE
30