This paper extends Iwasawa's theorem on class groups from $ ext{Z}_p$-extensions to more complex $ ext{Z}_p^d$-extensions, broadening the understanding of class group behavior in number theory.
Contribution
It introduces a generalization of Iwasawa's theorem applicable to all $ ext{Z}_p^d$-extensions, expanding the scope of class group analysis.
Findings
01
Generalization of Iwasawa's theorem to $ ext{Z}_p^d$-extensions
02
New insights into capitulation kernels in higher-dimensional extensions
03
Enhanced understanding of class group structure in complex extensions
Abstract
We generalize Iwasawa's theorem on class group over Zp-extensions to all Zpd-extensions.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric and Algebraic Topology · Advanced Algebra and Geometry
Full text
capitulation kernels of class groups over Zpd-extensions
We generalize Iwasawa’s theorems on class groups over Zp-extensions to all Zpd-extensions.
Key words and phrases:
Zpd extensions of number fields, class groups,
capitulation map, Iwasawa theory, algebraic functional equations.
2010 Mathematics Subject Classification:
11R29 (primary), 11R23, 12G05, 11R65 (secondary)
† The author was supported in part by the Ministry of
Science and Technology of Taiwan, MOST 103-2115-M-002 -008 -MY2,
MOST 105-2115-M-002 -009 -MY2.
It is our pleasure to thank NCTS/TPE for supporting a number of meetings of the authors in National Taiwan University.
1. Introduction
Let k be a number field and let K/k be an abelian extension.
If K/k is a finite extension,
we define a map from the group of ideals of k to that of K
by taking a to the fractional ideal aOK.
This map induces the homomorphism cK/k:Ck⟶CK of class groups which is called the
capitulation map. Previous studies indicate that there is certain connection between the order of ker(cK/k) and the ramification of K/k.
A famous theorem of Suzuki [Suz91] that generalizes both Hilbert’s theorem 94 and
the principal ideal theorem of class field theory
says if K/k is unramified, then the
order of ker(cK/k) is
divisible by the degree [K:k].
More results in this aspect can be found in [Gon07].
In contrast,
Iwasawa [Iwa73] proves that if K/k is a Zp-extension, so it is almost totally ramified at some place dividing p,
then the the capitulation kernel X˙K (see below)
is pseudo-null.
In this note, we generalize this result of Iwasawa to every Zpd-extension of k, and then use it to establish a pseudo-isomorphism
of Iwasawa modules that generalizes [Iwa73, Theorem 11].
From now on let K/k be a Zpd-extension.
For a number field E⊂K, let AE denote the Sylow p-subgroup
of the class group CE of E and let AE′ be the quotient of AE modulo the subgroup generated by ideals above p.
Define
[TABLE]
where E′/E runs through finite sub-extensions in K/E and
[TABLE]
is induced from cE/E′. Note that A˙E is a subgroup of AE. Put
[TABLE]
with the limit taken over the norm maps.
Let ∼ denote pseudo-isomorphism.
Theorem 1**.**
We have X˙K∼0 and X˙K′∼0.
Theorem 1
is proved in §2.3 by following the path of Iwasawa [Iwa73, Theorem 10].
For d≥2, the Iwasawa algebra is much more
complicated than that of a Zp-extension, this might be the reason why the content of the theorem has seldom discussed since
the publication of Iwasawa’s paper. The breakthrough in our proof is the duality in Lemma 2.1.3(c). Having it, we use
Monsky’s theorem [Mon81] (see §2.1) to show Proposition 2.2.3, which then leads to the proof of the theorem.
Write Γ for the Galois group of K/k and ΛΓ for the
Iwasawa algebra Zp[[Γ]].
Define the ΛΓ-modules
[TABLE]
with the limits taken over the homomorphisms dual to the capitulation maps,
and
[TABLE]
The following theorem might be known to experts, for the convenience of the readers, we include
a proof of it in §3.3.
Theorem 2**.**
The ΛΓ-modules WK and WK′ are finitely generated.
Let
[TABLE]
be the involution of Zp-algebra induced from Γ⟶Γ, γ↦γ−1.
For a ΛΓ-module D, let
D♯ denote the module with D as the underlying Zp-module while Γ acts on D
via ♯:Γ⟶Γ.
Theorem 3**.**
We have
[TABLE]
Suppose k0 is a subfield of k such that K/k0 is an abelian extension having Galois group
Gal(K/k0)=Γ×Θ with Θ finite of order prime to p. Then every pro-pGal(K/k0)-module
D can be written as
[TABLE]
where χ runs through all Qˉp∗-valued characters of Θ and
[TABLE]
It is clear that (D♯)χ=(Dχ−1)♯, and we have (see the last paragraph of §3.3)
[TABLE]
These pseudo-isomorphisms imply the equalities of characteristic ideals
[TABLE]
Theorem 3, proved in §3.3, is not entirely new, when d=1, it is by Iwasawa
[Iwa73, Theorem 11]. Suppose K/k has ramification locus S.
Nekovář defines in [Nek06, §9.5] a morphism from XK to the Iwasawa adjoint E1(WK♯)
of WK♯ (they are pseudo-isomorphic) and shows the cokernel is pseudo-null under the condition:
(Dec 2): For every v∈S, the decomposition group Γv≃Zpr(v), with r(v)≥2.
A similar result for XK′ and WK′♯ is given in [Nek06, §9.4], while Vauclair, in collaboration with Nekovář,
proves in [Vau09, Theorem 7.6]
that there is a natural morphism α:WK′⟶E1(XK′♯) with pseudo-isomorphic kernel and cokernel, and if (Dec 2) holds then α is a pseudo-isomorphism. This also implies unconditionally the equality of characteristic ideals as in (2) (see [Vau09, Corollary 6.8])
[TABLE]
Our proof of Theorem 3 uses the theory of Γ-system established in [LLTT18]. It turns out Theorem 1 is
the key ingredient for having the theory work in our situation. Indeed, a sensible homomorphism XK⟶WK♯
should be compatible with the duality between AE and Hom(AE,Qp/Zp) at all finite layers, and hence must factors through
the quotient of XK modulo X˙K. Therefore, the pseudo isomorphism between XK and WK♯ cannot established
without having X˙K∼0, the same with XK′ and WK′♯.
It is worthwhile to mention that over global function fields of characteristic p
our theorems hold trivially, because the global unit group UK has trivial p-primary part (cf. §3.1 and §3.3, especially the exact sequence
(17), Lemma 3.1.1 and the proof of Theorem 3).
We thank I. Longhi for helping us with the proof of Lemma 2.1.4,
we also thank D. Vauclair for a communication on our and his work.
1.1. Notation and preliminary remarks
Let E stand for a finite intermediate extension of K/k and denote ΓE=Gal(K/E).
Put Γ(n):=Γpn, kn:=KΓ(n), the nth layer of K/k.
Let IΓ denote
the augmentation ideal of ΛΓ and put In:=ker(ΛΓ⟶Zp[Gal(kn/k)]) so that I0=IΓ.
For a set T of places of k, let
TE denote the set of places of E sitting over T. Let P be the set of places of k above p.
For a Galois group G of number field extension, let Gw and Gw0 denote respectively the decomposition subgroup and
the inertia subgroup at a place w.
For a topological group B, let B∨ denote Hom(B,Qp/Zp), where Qp/Zp is endowed with the discrete topology.
In most cases, B
is pro-p or p-primary and discrete so that B∨ is its Pontryagin dual. We also identify B∨ with H1(B,Qp/Zp)
when B is a compact group.
For a functor ◊ on the category of finite intermediate extension of K/k, we abbreviate ◊kn=◊n.
If Q⟶D is a morphism of two such functotors
with QE, DE compact and
QE⟶DE
surjective for every E, then QK⟶DK is also surjective. To see this, we apply the dual
\textstyle{\mathfrak{D}_{E}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{Q}_{E}^{\vee}}
, which implies the injectivity of
\textstyle{\mathfrak{D}_{K}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{Q}_{K}^{\vee}}
, and then obtain the desired surjectivity via the duality [Kap50].
Since XK is torsion, it is annihilated by some non-zero f∈ΛΓ.
Then WK is annihilated by f♯, whence torsion.
To see this, we may assume that for every intermediate extension E of K/k, the extension
K/E contains no nontrivial unramified intermediate extension.
Then the restriction map Hom(AE,Qp/Zp)⟶Hom(AE′,Qp/Zp) is injective for E⊂E′⊂K, and hence
XK⟶AE is surjective.
This implies f⋅AE=0, so
f♯⋅Hom(AE,Qp/Zp)=0 and f♯⋅WK=0.
2. Towers of Γ-modules
2.1. Monsky’s Theorem
Endow \mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}} with the discrete topology. Let Γ^ denote all continuous characters from Γ to \mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}}.
Every χ∈Γ^ is of finite order, it factors through Γ⟶Γ/Γ(n), for some n,
and hence extends uniquely to a continuous Zp-algebra homomorphism χ:ΛΓ⟶O, where O
is the ring of integers of Qˉp.
Definition 2.1.1**.**
(1)
Define the zero set of an element θ∈ΛΓ to be
[TABLE]
2. (2)
A Zp-flat Z of codimension m is a subset of Γ^ consisting of solutions χ to
the system of equations
[TABLE]
*where ξ1,...,ξm are elements of Γ, extendable to a Zp-basis, \zeta_{1},...,\zeta_{m}\in\mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}}. *
3. (3)
We say a set τ1,....τc of Zp-generators of Γ is tight, if each τi∈/Γ(1) and the topological closure of <τi> and <τj> are distinct for i=j.
The zero set Δf of a non-zero f∈ΛΓ is a proper subset of Γ^. There are Zp-flats Z1,...,Zl such that
[TABLE]
Let us introduce more notation. For n≥0, σ∈Γ, σ=id,
denote
[TABLE]
which is regarded as an element of ΛΓ. Put ωσ,−1=1. For m≥n≥−1, write
[TABLE]
which is also an element of ΛΓ.
Because νσ,r,r+1 is the pr+1th cyclotomic polynomial in σ, it is irreducible in Zp[σ].
Hence in ΛΓ, if r=r′, then νσ,r′,r′+1 is relatively prime to
νσ,r,r+1. Also, if σ and σ′ are linearly independent over Zp, then νσ,n,m
and νσ′,n′,m′ are relatively prime in ΛΓ.
Due to technical reason, we will need to deal with ideals other than In,
especially when d≥2. We introduce the ideals
[TABLE]
where τ1,....τc is a chosen set of tightZp-generators of Γ. It is clear that for m≥n, the inclusion Jm⊂Jn holds.
Since ωτ,0⋅ντ,0,n=ωτ,n, the ideal
In=(ωτ1,n,...,ωτc,n) is inside Jn.
For a chosen Zp-basis σ1,...,σd of Γ, we also need to consider ideals,
[TABLE]
with n=(n1,...,nd), r=(r1,...,rd)∈Zd such that ri≥−1 and ni>ri, for every i,
the latter condition will be abbreviated as n>r.
For an ideal I=(θ1,...,θl)⊂ΛΓ, denote
[TABLE]
Write Δr,n for
ΔIr,n.
If in (3) each ζj is of order prj, then
[TABLE]
Lemma 2.1.3**.**
The following statements hold true:
(a)
If the order of χ∈Γ^ does not exceed pn or pni, for every i=1,...,d, then for all
x∈Ir,n+Jn the value χ(x) is divisible by the minimum of
pn and pn1−r1,...,pnd−rd.
2. (b)
For a fixed r, the intersection n>r,n>0⋂(Ir,n+Jn)=0.
3. (c)
Let x∈ΛΓ. Then x∈Ir,n if and only if χ(x)=0 for all
χ∈Δr,n.
Proof.
Let pα denote the order of χ(σ). For m≥α,
we have χ(νσ,r,m)=0, if α>r, while χ(νσ,r,m)=pm−r, if α≤r.
The assertion (a) follows.
To prove (b), let x be an element of the intersection in question. In view of Theorem 2.1.2, we have to show
χ(x)=0 for all χ∈Γ^.
Let α denote the maximum of r1,...,rd . For an integer β>α, put n=(β,...,β), n=β.
Then by (a), the value χ(x) is divisible by pβ−α. Since this holds for all β, we must have χ(x)=0.
We prove (c) by induction on d. The statement trivially holds for d=0. Suppose d>0, let Γ′⊂Γ be the
subgroup topologically generated by elements of the basis σ1,...,σd other than σ1.
If d=1, then both Γ′ and Γ′^ are the trivial group and ΛΓ′=Zp. In this case,
put Ir′,n′=(0) and Δr′,n′=Γ′^.
If d≥2, let r′=(r2,...,rd) and n′=(n2,...,nd). Put T:=σ1−1∈ΛΓ.
Then νσ1,r1,n1 is a distinguished polynomial in T of degree δ:=pn1−[pr1], where [⋅] is the
Gauss symbol.
The natural map
Γ^⟶Γ′^, χ↦χˉ, induces a surjection
ρ:Δr,n⟶Δr′,n′ such that the fibre ρ−1(χˉ)
for each χˉ∈Δr′,n′ is of cardinality δ and a character χ∈ρ−1(χˉ) is determined
by the value χ(σ1), or equivalently, by χ(T). By the Weierstrass division theorem [Bou72, VII, §3, Proposition 5], we may assume that
[TABLE]
Thus, for χ∈ρ−1(χˉ),
[TABLE]
This means the polynomial ∑i=0δ−1χˉ(yi)⋅Xi in X has δ distinct roots, hence must be trivial.
Therefore, each yi is annihilated by characters in Δr′,n′.
The induction hypothesis implies yi∈Ir′,n′ for every i, so x∈Ir,n.
∎
Lemma 2.1.4**.**
Suppose Y is a finitely generated ΛΓ-module. Then
⋂nJnY=0 so that the natural maps Y⟶Y/JnY induce the isomorphism
[TABLE]
Proof.
The homomorphism Y⟶Yˉ has
dense image, and it is surjective because both Y and Yˉ are compact. It is sufficient to show
⋂nJnY=0. Lemma 2.1.3 (b) says the assertion holds for Y=ΛΓ.
In general, we have a surjective homomorphism ϕ:⨁i=1lΛΓ⟶Y,
with JnY=ϕ(⨁i=1lJn). Suppose x∈⋂nJnY.
Denote X=ϕ−1(x) which is a compact subset of ⨁i=1lΛΓ.
Since X∩⨁i=1lJn=∅, for every n, and ⋂n⨁i=1lJn=0,
we must have 0∈X. Therefore, x=ϕ(0)=0.
∎
2.2. The norm maps
Fix a basis σ1,...,σd of Γ and set
[TABLE]
Since In is gernerated by ωσi,n and νσi,n,m⋅ωσi,n=ωσi,m, we have
νn,m⋅In⊂Im.
For a ΛΓ-module Y, the map sending y∈Y to νn,m⋅y
induces an endomorphism
\textstyle{{\mathfrak{Y}}/\mathscr{I}_{m}{\mathfrak{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{n,m}}$$\textstyle{{\mathfrak{Y}}/\mathscr{I}_{m}{\mathfrak{Y}}}
,
which turns out to be the norm map of Γ(n)/Γ(m) acting on Y/ImY, hence independent of the choice of the basis.
In general, if for each n, there associates a
ΛΓ submodule
In⊂Y containing InY such that νn,m⋅In⊂Im for m≥n, we shall
also denote the ΛΓ-map induced from νn,m as
[TABLE]
Let Jn be defined by choosing a tight set τ1,...,τc of generators of Γ.
Lemma 2.2.1**.**
We have νn,m⋅Jn⊂Jm.
Proof.
We have to show that νn,m⋅ντi,0,n∈Jm. Since modulo Im, the image
νn,m⋅ντi,0,n is
independent of the choice of the basis, we may assume that τi=σ1. Then
νn,m(ντi,0,n)=νσ1,0,m⋅νσ2,n,m⋅⋯⋅νσd,n,m∈Jm.
∎
Definition 2.2.2**.**
For a ΛΓ-module Y, define
[TABLE]
[TABLE]
and
[TABLE]
Proposition 2.2.3**.**
Suppose Y is a finitely generated ΛΓ-module. Then Y¨∼0.
Proof.
Consider an exact sequence
[TABLE]
with N, N′ pseudo-null and Z a direct sum ⊕i=1rΛΓ/(fi) where fi is either [math] or a power of an irreducible element of ΛΓ. Since N⊕N′ is pseudo-null,
there are relatively prime g1,g2∈ΛΓ annihilating both N and N′. These would lead to an exact seuqence
[TABLE]
with both a and b annihilated by g12 and g22, hence pseudo-null. Thus, it remains
to show that Y¨=0 for
Y=ΛΓ/(f).
Let xˉ∈Y¨⊂Y (Lemma 2.1.4) and let x∈ΛΓ be a lifting of xˉ.
For each n, there is m>n such that
[TABLE]
for some a∈ΛΓ. Put ω:=ωτ1,0⋅⋯⋅ωτc,0, y:=ωx. Since
ω⋅Jm⊂Im,
[TABLE]
Suppose f=0. For every χ∈ΔIn, we have χ(νn,m)=0, while χ(Im)=0, so χ(y)=0. Then it follows that y∈In (Lemma 2.1.3(c)). Since this holds for all n, Lemma 2.1.4 says y=0,
whence x=0 as desired.
Suppose f=0 and let Z1,…,Zl be the Zp-flats in Theorem 2.1.2. For each i,
let χ(ξi)=ζi be one of the defining equations (3) of Zi, so that (4) says Zi⊂Δϵi,
if ζi is of order pri and ϵi:=νξi,ri−1,ri. Then we put ϵ:=ϵ1⋅⋯⋅ϵl
to have
[TABLE]
Denote νn,m(i):=νσi+1,n,m⋅⋯⋅νσd,n,m, −1=(−1,...,−1),
and
m(i)=(m1(i),...,md(d)), with
mj(i)=n for j≤i; mj(i)=m, for j>i. We claim that if
[TABLE]
holds for ti,bi∈ΛΓ, then there is some bi+1∈ΛΓ such that
[TABLE]
Then, beginning with (6), for which i=0, by repeatedly applying the above implication, we can deduce
[TABLE]
which means ωϵd−1x∈In+(f). As this happens for every n, Lemma 2.1.4 says
ωϵd−1x∈(f). The proposition is proved, if f is relatively prime to ωϵ, and
it remains to treat the case
where f is a power of νσ,α−1,α, α≥0, σ extendable to a Zp-basis of Γ,
because every irreducible factor of
ωϵ is of such type.
We may also assume that σ=σ1.
Because τ1,...,τc form a tight generating
set, we can write
[TABLE]
with δ=0,1, and ω′ relatively prime to f. Note that if δ=1, then α=0 and σ1=τj, for some j, we may assume that the basis σ1,...,σd
are taken from {τ1,...,τc}. Put z=ω′x, a=(a1,...,ad) with a1=α, aj=−1 for j=1. By (5),
[TABLE]
Again, we claim that for n>α, if
[TABLE]
holds for si,ci∈ΛΓ, then there is some ci+1∈ΛΓ such that
[TABLE]
Then we can deduce that ω′x∈Ia,n(0)+(f), for all n>α,
and hence
ω′x∈(f). Therefore, xˉ=0, since ω′ is relatively prime to f.
The proofs of the claims rely on Lemma 2.1.3(c). Denote b(i):=(u1,...,ud), with ui+1=n and uj=−1 for j=i+1.
Let χ∈Δb(i),m(i). The inclusion
[TABLE]
leads to χ(νσi+1,n,m)=0=χ(I−1,m(i)), so (8) yields χ(bi⋅f)=0.
Now, if χ(f)=0, then (7) implies χ(ϵ)=0; otherwise, we have χ(bi)=0. Therefore, χ(ϵbi)=0
always holds. Lemma 2.1.3(c) says ϵbi∈Ib(i),m(i).
Then we can write
[TABLE]
with bi+1′∈(νσ1,−1,n,...,νσi,−1,n,νσi+2,−1,m,...,νσd,−1,m)⊂I−1,m(i+1). By (8) again,
[TABLE]
In view of Lemma 2.1.3, this proves the first claim, because Δνσi+1,n,m∩Δ−1,m(i+1)=∅,
if χ∈Δ−1,m(i+1), then χ(νσi+1,n,m)=0 and consequently
χ(νn,m(i+1)⋅ti−bi+1⋅f)=0.
The proof of the second claims is similar to the previous one, the basic difference lies in
that all characters applied will be in Δνσ,α,β, for some β>α,
so that χ(f) is never zero. Let a(i)=(a1(i),...,ad(i)) be such that aj(i)=aj, for j=i+1, ai+1(i)=n. Let
χ∈Δa(i),m(i). Since χ(νσi+1,n,m)=0=χ(Ia,m(i)), the congruence (10) yields χ(ci⋅f)=0,
hence χ(ci)=0. Lemma 2.1.3(c) says ci∈Ia(i),m(i),
and we can write
[TABLE]
with ci+1′∈(νσ1,a1,n,...,νσi,ai,n,νσi+2,ad+2,m,...,νσd,ad,m)⊂Ia,m(i+1). Thus, by (10),
[TABLE]
But χ(νσi+1,n,m)=0, for all χ∈Δa,m(i+1),
it follows that νn,m(i+1)⋅si−ci+1⋅f is contained in Ia,m(i+1).
∎
Remark 2.2.4**.**
If we replace JnY, JmY in Definition 2.2.2 by smaller
In,
Im satisfying νn,m(In)⊂Im, for m>n, and let Y˙˙(n), Y˙˙ be the resulting
counterpart of Y¨(n), Y¨, then Y˙˙⊂Y¨⊂Y. Therefore, Y˙˙∼0 as well.
See the proof of Theorem 1.
2.3. The module X˙K
Now we prove Theorem 1.
By replacing k by kn if necessary, we may assume that
for each v in the ramification locus S, there is some integer e such that
[TABLE]
and
[TABLE]
Let Kn be the maximal unramified abelian p-extension over kn so that by Class Field Theory,
An:=Akn=Gal(Kn/kn). For m>n, the norm map Am⟶An is compatible with the
restriction of Galois action Gal(Km/km)⟶Gal(Kn/kn), we have XK=Gal(L/K), for
L=⋃nKn. Denote G:=Gal(L/k). Then G/XK=Γ and since L/K is unramified,
at every place v of k, we have Gv0≃Γv0, a commutative group.
[TABLE]
Suppose S={v1,...,vs}. For each j, choose a place uj of L sitting
over vj and then choose a Zp-basis ξ~1(j),...,ξ~dj(j) of Guj0. By (11) and (12),
we can choose these bases to have the union of their images under G⟶Γ form a tight set
g:={τ1,...,τc} of generators of Γ. Then among g, we choose a Zp-basis
{σ1,...,σd} of Γ. We lift each σi to some ξ~l(j) and denote it by σ~i.
Every g∈G can be uniquely written as
[TABLE]
Let Jram denote the Zp-submodule of XK generated by
[TABLE]
The commutator σ~i⋅σ~j⋅σ~i−1⋅σ~j−1∈Jram, because
g=σ~i⋅σ~j⋅σ~i−1 in (13)
yields
[TABLE]
The canonical action of Γ on XK coincides with the conjugation in G, namely, if γ∈Γ,
x∈XK, and γ~∈G is a lifting of γ, then
[TABLE]
Hence \tensor[γ−1]x is the same as the commutator γ~⋅x⋅γ~−1⋅x−1 in G.
Set
[TABLE]
Note that J is a Γ-module, since Γ acts trivially on XK/I0XK.
Let J~⊂G denote the closed subgroup generated by J and {σ~1,...,σ~d}.
Then for every place u of L, the inertia subgroup Gu0⊂J~, because u=\tensor[γ]uj for some j,
and hence J~ contains the Zp-basis γ~⋅ξ~i(j)⋅γ~−1, i=1,...,dj, of Gu0.
Thus, J~ is the closed subgroup of G topologically generated by all commutators and all inertia subgroups of G.
Therefore, the fixed field LJ~ is the maximal unramified abelian p-extension K0 of k, with Gal(K0/k)=A0.
By the uniqueness of the expression (13), we have
[TABLE]
Also, (12) implies K∩K0=k, hence Gal(KK0/K)≃Gal(K0/k). By (14),
[TABLE]
To apply the above argument to kn, let G(n) denote the pre-image of Γ(n) under G⟶Γ and set
σ~i,n:=σ~ipn. Every element g∈G(n) can be uniquely expressed as
[TABLE]
Write ξ~1,n(j),...,ξ~dj,n(j) for (ξ~1(j))pn,...,(ξ~dj(j))pn. They form a Zp-basis of
the inertia subgroup of Guj(n).
Let Jram(n) denote the Zp-submodule of XK generated by
[TABLE]
Set
[TABLE]
Let J~n⊂G denote the closed subgroup generated by Jn and {σ~1,n,...,σ~d,n}.
Lemma 2.3.1**.**
The fixed field of J~n⊂G(n) is Kn.
We have
[TABLE]
Furthermore, Jn is a Γ-module, hence a ΛΓ-module.
Proof.
It remains to show that Jn is invariant under the action of Γ, or equivalently Kn/k is a Galois extension.
But this follows from the fact that kn/k is Galois and Kn is the maximal unramified abelian p-extension
over kn.
∎
If m≥n, then νn,m(Jn)⊂Jm and that induces the commutative diagram
[TABLE]
Proof.
Since (12) says K/k contains no non-trivial unramified subextension, the restriction of Galois action XK⟶An is surjective for every n. Let xn∈An and let
x∈XK such that x∣kn=xn. Denote x∣km:=xm. Let []m denote the Artin map and let L be
an ideal in Okm with [L]m=xm. If l=Nkm/kn(L), then xn=[l]n, hence
[TABLE]
In particular, if x∈Jn, then the left-hand side is trivial, hence νn,m(x)∈Jm.
∎
By Lemma 2.3.2, we can write for the capitulation kernel
[TABLE]
This makes it possible to apply the technique developed in the previous sections. Because J⊂XK is of finite index,
for convenience, we replace XK by J. To be more precise, since In⋅XK⊂Jn, we have In⋅J⊂Jn, put \dot{J}_{(n,m)}:=\ker(\,\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 15.63199pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-15.63199pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{J/J_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 19.391pt\raise 5.32639pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.6875pt\hbox{\scriptstyle{\nu_{n,m}}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 39.63199pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 39.63199pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{J/J_{m}}}}}}}}}\ignorespaces}}}}\ignorespaces\,),
J˙(n):=⋃m≥nJ˙(n,m), and J˙:=limnJ˙(n).
Then it is sufficient to show J˙∼0.
Take Y:=J, In:=Jn∩Jn⋅Y and let Y˙˙(n), Y˙˙ be as in Remark 2.2.4. The homomorphisms Y/In⟶Y/Jn⋅Y, for all n,
yield a homomorphism Y˙˙⟶Y¨ fitting into the commutative diagram
[TABLE]
Since the down-arrows are injective, Proposition 2.2.3 implies Y˙˙∼0.
To see the difference between In and Jn, we first check that for x∈XK,
the element ωσi,n⋅x=νσi,0,n⋅ωσi,0⋅x∈Jn⋅J, because σi∈g and
ωσi,0⋅x∈I0⋅XK⊂J. This shows In⋅XK⊂Jn⋅J∩Jn=In. Denote ξ~i(j)=:g, ξ~i,n(j)=:gn and let τ∈g
be the image of g under G⟶Γ. Let x=xg∈Jram so that g=gˉ⋅x with gˉ=σ~1a1⋅⋯⋅σ~dad. Then
[TABLE]
Similarly, if g′=γ~⋅g⋅γ~−1=gˉ⋅x′, with x′=xg′∈Jram, and
gn′=γ~⋅gn⋅γ~−1, then
[TABLE]
Let ρ~:=(σ~1,na1⋅⋯⋅σ~d,nad)−1gˉpn and write XK additively. Then,
since xgn,xgn′∈Jram(n),
[TABLE]
Therefore, we have shown that the Zp-module Jn/In is generated by the classes of xgn, with g=ξ~i(j),
j=1,...,s, i=1,...dj. Then we observe that xgn is trivial if the above τ=σi. This shows the p-rank of
Jn/In is at most c−d, which equals [math] if d=1. The exact sequence
[TABLE]
gives rise to the exact sequence
[TABLE]
with p-ranks of both an and bn bounded by c−d. Consequently, the cokernel of the induced map
Y˙˙⟶J˙ is of p-ranks bounded by c−d. Hence J˙ is pseudo-null and the theorem is proved for
X˙K.
The corresponding assertion for X˙K′ can be proved by similar approach, of which we only give a sketch.
Write
[TABLE]
where P1 consists of place v such that
Γw0=Γw.
For each v∈P, choose a place
u of L above v and denote w=u∣K, if v∈S, let u be as before.
If v∈P1, then by (11), we have non-canonically
Γw=Zp×Γw0, whence Γw≃Gu and w splits completely over L.
In this case, choose an ηu∈Gu representing a topological generator of Gu/Gu0 such that the map G⟶Γ
sends ηu into g. Write ηu,n for
ηupn and let Jun(n) denote the
Zp-module generated by
[TABLE]
If v∈P2, then we have canonically Gu≃Gal(Lu/Kw)×Γw with Gal(Lu/Kw)⊂XK, Gu0≃Γw. Let [XK] be the Γ-submodule of XK generated by Gal(Lu/Kw), for all v∈P2.
Set
[TABLE]
let {Jn} denote the closed subgroup of G generated by {Jn} and {σ~1,n,...,σ~d,n},
and let Kn′ be the fixed field of {Jn}. Then Kn′ is the maximal unramified p-extension of kn with all places
above p splitting completely, so Gal(Kn′/kn)=An′. Let [A]n be the image of {Jn} under XK⟶An=XK/Jn.
Then the exact sequences
[TABLE]
and the fact that
[TABLE]
yield XK′=XK/[XK]. Let Jn′ be the image of {Jn} under XK⟶XK′.
Then (XK′,Jn′,Kn′)-version of Lemma 2.3.1 holds, namely,
[TABLE]
Again by the calss field theory, the (XK′,Jn′,An′)-version of Lemma 2.3.2 holds.
Then take Y′:=J0′, In′:=Jn′∩Jn⋅Y′ and proceed as above.
∎
3. Cohomology groups of global units
The proof of Theorem 2 involves cohomology of unit groups.
Denote the group of global units of E by UE:=OE∗ and put UK:=⋃EUE.
The cohomology groups Hi(Γ,UK), i=1,2, has been studied in
[Iwa83, Yam84] for d=1 case. We are going to show that for general d, they are co-finitely generated Zp-modules.
Put
[TABLE]
and
[TABLE]
where Eˉab,p is the maximal pro-p abelian extension of E, unramified outside p.
3.1. An exact sequence
Let DivE and PE denote the groups of divisors and principal divisors of E.
In [Iwa83, Proposition 1], Iwasawa deduces the seven-term exact sequence:
[TABLE]
with G:=Gal(E/k) and
bE:=ker(H2(G,E∗)⟶H2(G,DivE)).
The exact sequence
identifies H1(G,UE) with the kernel of αE/k. The restriction of αE/k to Ck⊂DivEG/Pk is
the capitulation homomorphism cE/k. Therefore, we have the exact sequence
[TABLE]
If v∈S and w is a place of E sitting over v, then the sum
[TABLE]
is fixed by G. Set gE/k,v:=∣Gv0∣. In DivE, we have gE/k,v⋅ϵv=v. The sequence
[TABLE]
with lE/k the projection onto the SE-component, is exact. Combining (17) and (18), we obtain the exact sequence
[TABLE]
By taking the injective limit of the above sequence, one can show that H1(Γ,UK) is co-finitely generated over Zp.
As for H2(Γ,UK), we have the following. Denote
[TABLE]
Lemma 3.1.1**.**
The following statements are equivalent:
(a)
The ΛΓ-module WK is finitely generated.
2. (b)
The abelian group AKΓ has finite p-rank.
3. (c)
The abelian group H2(Γ,UK) has finite p-rank.
Proof.
Since AKΓ[p] is the Pontryagin dual of WK/(pΛΓ+IΓ)WK,
the equivalence between (a) and (b) follows from Nakayama’s Lemma.
Denote QE:=Im(αE/k)∩AEΓ. Then (16) induces the exact sequence
[TABLE]
Now, the p-rank of QK is ≤ the p-rank of Ak plus the cardinality of S. Also,
if v∈S, then the composition of
\textstyle{\operatorname{H}^{2}(\Gamma,U_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\operatorname{H}^{2}(k,K^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{H}^{2}(k_{v},K_{v}^{*})}
is the trivial homomorphism. Hence Im(β) is embedded into
∏v∈SQp/Zp by the local invariant maps and is of p-rank bounded by the cardinality of S. Therefore, (b) and (c) are
equivalent. ∎
3.2. The structure of UK∨
Recall that UK denotes the group of global units of K and
UK∨=Hom(Qp/Zp⊗UK,Qp/Zp).
Let VE denote the group of PE-units of E.
Kummer’s theory yields the commutative diagram
[TABLE]
Since E∗/VE and VE/UE are torsion free, arrows in the upper row of the diagram are injective.
Let n goes to ∞. We obtain the commutative diagram
[TABLE]
Lemma 3.2.1**.**
The restriction map
\textstyle{\operatorname{H}^{1}(\mathfrak{M}_{E},\mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{E}}$$\textstyle{\operatorname{H}^{1}(\mathfrak{M}_{K},\mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}})^{\Gamma_{E}}}
has finite kernel and cokernel.
*The cokernel of
\textstyle{{\mathcal{V}}_{E}\otimes{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{H}^{1}(\mathfrak{M}_{E},\mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{\infty}})}
is finite.*
Proof.
We show that as
n→∞,
the corkernel of
[TABLE]
remains bounded. By Kummer’s theory, an element of \operatorname{H}^{1}(\mathfrak{M}_{E},\mbox{\raisebox{-2.54025pt}{l}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{\color[rgb]{1,1,1}.}}\raisebox{3.7889pt}{\color[rgb]{1,1,1}.}\hskip 4.60007pt}{}_{p^{n}}) is represented by an f∈E∗ modulo
(E∗)pn such that pn∣ordv(f) for every v∈/PE.
Let w1,...,wr be a set of generators of AE and let pm be the exponent of AE
so that each wipm=(fi) for some fi∈E∗. If n>m, then modulo (E∗)pn,
f can be expressed as a product of powers of f1pn−m,...,frpn−m together with elements of VE. Hence the
cokernel in question has order bounded by pmr.
∎
Corollary 3.2.3**.**
The natural map UE⟶UKΓE has finite kernel, its cokernel is cofinitely generated over Zp of corank bounded by ∣SE∣.
Proof.
The exact sequence
[TABLE]
where ℧~E is defined by taking valuations at all v∈PE induces the commutative diagram of complexes
[TABLE]
Write QE:=ker(℧E), RE:=ker(℧K∘βE). Then QE/UE is finite.
In view of the diagram (20) and Lemma 3.2.1, 3.2.2, we need to show that the cokernel of
\textstyle{Q_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{E}}
has corank bounded by ∣SE∣. If w is a place of K sitting over v of E with
v∈SE, then w is unramified under K/k, and hence the map
[TABLE]
is injective. Therefore, we have the exact sequence
[TABLE]
∎
Lemma 3.2.4**.**
Let QK be a finitely generated ΛΓ-module.
If QK is of rank r over ΛΓ, then QK/InQK has Zp-rank
[TABLE]
Proof.
There exists an exact sequence
[TABLE]
Since T is torsion, [Tan14, Lemma 1.13] says T/InT has Zp-rank =O(p(d−1)n). Hence
[TABLE]
To obtain an inequality in the other direction, we use an exact sequence
[TABLE]
with T′ torsion.
∎
Let r1 and r2 denote the number of
real and complex places of K.
Proposition 3.2.5**.**
The ΛΓ-module UK∨ is finitely generated, of rank r1+r2.
Proof.
Since ∣Skn∣=O(p(d−1)n), Corollary 3.2.3 says
UK∨/InUK∨, which is the Pontryagin dual of UKΓ(n),
has Zp-rank equals (r1+r2)⋅pdn+O(p(d−1)n). In particular, UK∨/(pΛΓ+IΓ)UK∨ is finite. Hence UK∨ is finitely generated. By Lemma 3.2.4, the rank of UK∨
equals r1+r2.
∎
3.3. The module WK
Let K′/k be an intermediate Zpd−1-extension of K/k with Galois group Γ′.
We shall fix a Zp-extension k(∞)/k linearly disjoint from K′/k so that K=K′k(∞) and K′∩k(∞)=k.
Let k(n) denote the nth layer of k(∞)/k. From now on, the symbol F will denote a finite intermediate extension of K′/k, while E denotes a finite intermediate extension of K/k such that
E∩K′=F and E=Fk(n) for some n. Put F(∞):=Fk(∞).
Denote Ψ=Gal(K/K′) and let ψ be a topological generator of Ψ.
[TABLE]
For each F and for m≥n, we have the commutative diagram
[TABLE]
where the left down-arrow is induced by the natural map Z[pn1]⟶Z[pm1] and the right down-arrow
is the inflation map. This yields
Since WK′⊂WK, it is sufficient to treat WK.
Lemma 3.1.1 says
the theorem is equivalent to the assertion that H2(Γ,UK) has finite p-rank.
For d=1, the value of this p-rank is shown in
[Iwa83, Yam84], the equality (21) also implies it
is bounded by the rank of Uk. We prove the theorem by induction on d.
where C˙E/F is generated by DivEΨ. Denote CK:=limCE. Let E goes to K and take the direct limit to obtain
the exact sequence
[TABLE]
Since the p-part of CKΓ is Pontryagin dual to WK/IΓWK,
it remains to show both C˙K/K′Γ′ and H2(Ψ,UK)Γ′ are of finite p-ranks.
The surjection (21) says H2(Ψ,UK) is a quotient of UK′. Hence its Pontryagin dual, denoted JK′,
is a ΛΓ′ submodule
of UK′∨. Proposition 3.2.5 implies JK′ is ΛΓ′ finitely generated.
Then H2(Ψ,UK)Γ′, being the Pontryagin dual of JK′/IΓ′JK′,
must have finite p-rank.
Let T⊂S be the subset consisting of places v with Γv0 of rank d. We choose K′ such that K/K′ is unramified outside
TK′, which is a finite set. Apply (18) to the case where k=F and then let E goes to K to obtain the exact sequence
[TABLE]
where CˉK′ is the image of the capitulation homomorphism CK′⟶CK and V is a quotient of ∏w∈TK′Qp/Zp⋅w. Obviously, the p-rank of V is finite. Now CˉK′∨ is a ΛΓ′ submodule of WK′, whence finitely generated by the induction hypothesis. Consequently,
CˉK′Γ′, the Pontryagin dual of CˉK′∨/IΓ′CˉK′∨,
must have finite p-rank. The above exact sequence implies the p-rank of C˙K/K′Γ′ is finite.
∎
Write aE:=AE, bE:=Hom(AE,Qp/Zp). For E⊂E′⊂K, let rEE′:bE⟶bE′
and kEE′:bE′⟶bE denote respectively
the restriction and the corestriction. Extend them to homomorphisms
[TABLE]
and
[TABLE]
such that the restrictions to the first factors are respectively the capitulation and the norm. Then
[TABLE]
the norm map of Gal(E′/E)-action.
Let ⟨,⟩E:aE×bE⟶Qp/Zp be the natural pairing.
For (a,b)∈aE×bE, (a′,b′)∈aE′×bE′, we have
[TABLE]
Thus, in the terminology of [LLTT18, §3.1.1], the collection
[TABLE]
form a complete Γ-system. Indeed, it is a
T-system [op. cit.], because for every intermediate Zpe-extension M/k of K/k, d≥e≥1, if
[TABLE]
then aM×bM is finitely generated torsion over the Iwasawa algebra of M/k.
In view of [LLTT18, Theorem 1], for proving WK∼XK♯,
we need to show the system A is pseudo-control, in the sense that
[TABLE]
is pseudo-null. That a˙K∼0 is by Theorem 1. To show b˙K∼0,
we may assume that K/k has no nontrivial unramified extension. In this case
\textstyle{\mathfrak{b}_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{r}^{E^{\prime}}_{E}}$$\textstyle{\mathfrak{b}_{E^{\prime}}}
is injective, whence b˙K=0.
The assertion WK′∼XK′♯ is proved similarly. We first form the Γ-system A′ with aE′=AE′,
bE′=Hom(aE′,Qp/ZP). Since aM′ is quotient of aM, bM′⊂bM,
A′ is a T-system. Theorem 1 says a˙K′∼0, and since
b˙K′⊂b˙K∼0, A′ is pseudo-controlled. Then apply [LLTT18, Theorem 1].
∎
As for (1), form the Γ-system Aχ={aχ,E,bχ,E,rEE′,kEE′} with aχ,E=(AE)χ−1,
bE,χ=Hom(aχ,E,Qp/ZP), it is a pseudo-controlled T-system, and hence the first pseudo-isomorphism follows.
The second is proved similarly.
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