Directory UMM :Journals:Journal_of_mathematics:OTHER:
117
Documenta Math.
Equivariant Iwasawa Theory: An Example
J¨
urgen Ritter and Alfred Weiss1
Received: November 6, 2007
Revised: April 22, 2008
Communicated by Peter Schneider
Abstract. The equivariant main conjecture of Iwasawa theory is
shown to hold for a Galois extension K/k of totally real number fields
with Galois group an l-adic pro-l Lie group of dimension 1 containing
an abelian subgroup of index l, provided that Iwasawa’s µ-invariant
µ(K/k) vanishes.
2000 Mathematics Subject Classification: 11R23, 11R32, 11R42
Keywords and Phrases: Iwasawa theory, Iwasawa L-functions
This note justifies a remark made in the introduction of [6] according to which
the “main conjecture” of equivariant Iwasawa theory, as formulated in [2,
p.564], holds when G = G(K/k) is a pro-l group with an abelian subgroup
G′ of index l.
We quickly repeat the general set-up and, in doing so, refer the reader to [5,§1]
for facts and notation that is taken from our earlier papers on Iwasawa theory.
Namely, l is a fixed odd prime number and K/k a Galois extension of totally
real number fields, with k/Q and K/k∞ finite, where k∞ is the cyclotomic
Zl -extension of k. Throughout it will be assumed that Iwasawa’s µ-invariant
µ(K/k) vanishes. We also fix a finite set S of primes of k containing all primes
above ∞ and all those whose ramification index in K/k is divisible by l.
In this situation it is shown in [5] that the “main conjecture” of equivariant
Iwasawa theory would follow from two kinds of hypothetical congruences between values of Iwasawa L-functions. One of these kinds, the so-called torsion
congruences (see [5, Proposition 3.2]), stated as
ver(λKab /k )
≡1
λK/k′
1 We
mod T ′
acknowledge financial support provided by NSERC and the University of Augsburg.
Documenta Mathematica 13 (2008) 117–129
118
J¨
urgen Ritter and Alfred Weiss
in the proof of the proposition in §2, has meanwhile been verified in [6].
The purpose of the present paper is to show that the torsion congruences already suffice to obtain the whole conjecture in the special case when G is a
pro-l group with an abelian subgroup G′ of index l. Before stating the precise
theorem we need to recall some notation (compare [5,§1]).
Λ∧ G is the l-completion of the localization Λ• G which is obtained from
the Iwasawa algebra ΛG = Zl [[G]] by inverting all central elements which
are regular in ΛG/lΛG; Q∧ G is the total ring of fractions of Λ∧ G;
T (Q∧ G) = Q∧ G/[Q∧ G, Q∧ G] is the quotient of Q∧ G by Lie commutators;
if G is a pro-l group, then (see [5,§2] 2 )
(LD)
K1 (Λ∧ G)
Det ↓
−→
L
T (Q∧ G)
Tr ↓
≃
HOM(Rl G, (Λc∧ Γk )× )
L
−→
Hom∗ (Rl G, Qc∧ Γk )
is the logarithmic diagram defining the logarithmic pseudomeasure
tK/k ∈ T (Q∧ G)
by
Tr(tK/k ) = L(LK/k )
where LK/k = LK/k,S ∈ HOM(Rl G, (Λc∧ Γk )× ) is the Iwasawa L-function.
Theorem. With K/k and S as at the beginning and G = G(K/k) a pro-l
group, tK/k is integral (i.e., tK/k ∈ T (Λ∧ G)) whenever G has an abelian
subgroup G′ of index l.
As a corollary, by [5, Proposition 3.2] and [6, Theorem], LK/k ∈ Det K1 (Λ∧ G) ,
which implies the conjecture (see [3, Theorem A]), up to its uniqueness assertion. However, SK1 (QG) = 1 because each simple component, after tensoring
up with a suitable extension field of its centre, becomes isomorphic to a matrix ring of dimension a divisor of l2 by the proof of [2, Proposition 6], as the
character degrees χ(1) all divide l. Now apply [7, p.334, Corollary].
The proof of the theorem is carried out in §2; before, in a short §1, we introduce
restriction maps
′
′
ResG
G : T (Q∧ G) → T (Q∧ G )
and
′
∗
∗
c
′
c
ResG
G : Hom (Rl G, Q∧ Γk ) → Hom (Rl G , Q∧ Γk′ )
2 R G is the ring of all (virtual)
l
G(k∞ /k) ; Λc∧ Γk = Zlc ⊗Zl Λ∧ Γk with
ure Qlc of Ql
Qlc -characters of G with open kernel; Γk =
Zlc the ring of integers in a fixed algebraic clos-
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
119
making the diagram
L
T (Q∧ G)
′
ResG
G ↓
−→
Tr
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
L′
T (Q∧ G′ )
−→
Tr′
Hom∗ (Rl G′ , Qc∧ Γk′ )
K1 (Λ∧ G)
′
res G
G ↓
−→
K1 (Λ∧ G′ )
−→
commute 3 for any pair of pro-l groups G = G(K/k) and G′ = G(K/k ′ ) ≤ G
′
such that [G : G′ ] is finite. We remark that replacing ResG
G by the “natural”
restriction map,
′
G ′
∗
′
′
′
c
(res G
G f )(χ ) = f (ind G′ χ ) , f ∈ Hom (Rl G, Q∧ Γk ) , χ ∈ Rl G ,
does not work, because induction and Adams operations do not commute.
1 . Res
Let G = G(K/k) be a pro-l group and G′ = G(K/k ′ ) ≤ G an open subgroup.
Recall that Ψ : Λc∧ Γk → Λc∧ Γk is the map induced by Ψ(γ) = γ l for γ ∈ Γk
(compare [5,§1]) and that ψl is the lth Adams operation on Rl (−).
Definition.
′
∗
∗
c
′
c
ResG
G : Hom (Rl G, Q∧ Γk ) → Hom (Rl G , Q∧ Γk′ ) sends f to
X Ψr
′
′
G ′
ResG
f
=
χ
→
7
f
(ind
χ
)
+
(f (ψlr−1 χ)) ,
′
G
G
r
l
r≥1
G
′
′
where χ = ψl (ind G
G′ χ ) − ind G′ (ψl χ ) .
P
To justify the definition we must show that the sum r≥1 is actually a finite
sum. For this, let {t} be a set of coset representatives of G′ in G, so G = ∪˙ t tG′ ,
and define
r
m(g) = min{r ≥ 0 : g l ∈ G′ } for g ∈ G .
Then
′
ind G
G′ χ (g) =
X
t
χ˙ ′ (g t ) =
X
χ′ (g t ) ,
{t:m(gt )=0}
if, as usual, χ˙ ′ coincides with χ′ on G′ and vanishes on G \ G′ . Hence,
G ′
′
l
) −Pind G
χ(g)
G′ (ψl χ )(g)
P
P = (ind G′ χ′ )(g
′ lt
lt
= m(glt )=0 χ (g ) − m(gt )=0 χ (g ) = m(gt )=1 χ′ (g lt ) .
r0
If r0 is such that Gl ⊂ G′ , then ψlr0 −1 χ = 0 and
P
r0 −1
.
the sum m(gt )=1 is empty when g ∈ Gl
P
r≥1
=
′
Pr0 −2
r=1
, because
finite G, this ResG
G , making the left square commute, appears already in [1, p.14].
′
For us the properties (HD),(TD) of ResG
G shown in §1 are equally necessary for the proof of
the Theorem above.
3 For
Documenta Mathematica 13 (2008) 117–129
120
J¨
urgen Ritter and Alfred Weiss
′
′
G
∗
′
c
It remains to show that ResG
G f ∈ Hom (Rl G , Q∧ Γk′ ) , i.e., ResG f is a Galois
stable homomorphism, compatible with W-twists (see [5,§1]), and taking values
in Qc∧ Γk′ . The first property is easily checked and the third follows from the
second as in [2, proof of Lemma 9]. We turn to twisting.
Let ρ′ be a type-W character of G′ , so ρ′ is inflated from Γk′ , and write ρ′ =
′
res G
G ρ with ρ inflated from Γk to G. Then
G ′
G ′
G ′
′ ′
♯
′ ♯
f (ind G
G′ (ρ χ )) = f (ρ · ind G′ χ ) = ρ (f (ind G′ χ )) = (ρ ) (f (ind G′ χ ))
′
c
as f (ind G
G′ χ ) ∈ Q∧ Γk′ . Moreover, since ψl is multiplicative,
G
G ′
G
′ ′
′ ′
′ l
′
l
ψl (ind G
G′ (ρ χ )) − ind G′ (ψl (ρ χ )) = ψl (ρ · ind G′ χ ) − ind G′ ((ρ ) · ψl χ ) = ρ · χ
and thus
Ψr
lr
Ψr
lr
r
r
(f (ψlr−1 (ρl · χ)))) = Ψlr f (ρl · ψlr−1 χ) =
r
r
r
((ρl )♯ (f (ψlr−1 χ))) = ρ♯ ( Ψlr f (ψlr−1 χ)) = (ρ′ )♯ ( Ψlr f (ψlr−1 χ)) .
Lemma 1. The diagram below commutes. In it, L and L′ are the lower horizontal maps of the logarithmic diagram (LD) for G and G′ , respectively.
(HD)
HOM(Rl G, (Λc∧ Γk )× )
′
res G
G ↓
HOM(Rl G′ , (Λc∧ Γk′ )× )
L
−→
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
L′
−→
Hom∗ (Rl G′ , Qc∧ Γk′ ) .
Indeed, for f ∈ HOM(Rl G, (Λc∧ Γk )× ) we get
P
′
r−1
Ψr
′
′
χ)]
(ResG
)(ind G
G Lf )(χ ) = (Lf
G′ χ ) +
r≥1 lr [(Lf )(ψl
P
r
G ′
r−1
Ψ
Ψ
=(Lf
˙
)(ind G′ χ ) + r≥1 lr [log(f (ψl χ)) − l log(f (ψlr χ))]
P
P
r
r−1
Ψr
′
χ)) − r≥2 Ψlr log(f (ψlr−1 χ))
= (Lf )(ind G
G′ χ ) +
r≥1 lr log(f (ψl
′
= (Lf )(ind G
G′ χ ) +
=
=
Ψ
l
log(f (χ)) =
f (ind χ′ )l ·Ψf (ψl ind χ′ )
Ψf (ψl ind χ′ )·Ψf (ind ψl χ′ )
′
′
(L′ res G
G f )(χ ) .
1
l
log
=
1
l
1
l
log
log
f (ind χ′ )l
Ψ(f (ψl ind χ′ ))
+
Ψ
l
log
f (ψl ind χ′ )
f (ind ψl χ′ )
f (ind χ′ )l
Ψf (ind ψl χ′ )
The dotted equality sign, =,
˙ is due to the congruence
f (χ)l
Ψf (ψl χ)
≡ 1 mod lΛc∧ Γk
(see [5,§1]) and to χ(1) = 0, so (ψlr−1 χ)(1) = 0 for every r. In fact, with
def
χ
˜ = ψlr−1 χ, we have
f (χ)
˜ l ≡ Ψf (ψl χ)
˜ mod lΛc∧ Γk =⇒
s
f (χ)
˜ l ≡ Ψs f (ψls χ)
˜ ≡ Ψs f (χ(1)1)
˜
= 1 mod lΛc∧ Γk
˜ as ‘log’ converges
for big enough s. Thus (Lf )(χ)
˜ = log(f (χ))
˜ − Ψl log(f (ψl χ))
on an element a power of which is ≡ 1 mod lΛc∧ Γk .
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
121
The proof of Lemma 1 is complete.
By means of the trace isomorphism Tr : T (−) → Hom∗ (−) we next transport
′
G′
′
ResG
G to ResG : T (Q∧ G) → T (Q∧ G ), i.e., the diagram
(TD)
T (Q∧ G)
′
ResG
G ↓
−→
Tr
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
T (Q∧ G′ )
−→
Tr′
Hom∗ (Rl G′ , Qc∧ Γk′ )
commutes.
Lemma 2.
K1 (Λ∧ G)
′
res G
G ↓
→
K1 (Λ∧ G′ )
→
L
L
′
T (Q∧ G)
′
ResG
G ↓
′
commutes and ResG
G tK/k = tK/k′ .
T (Q∧ G′ )
The first claim follows from gluing together the diagrams (LD), (HD), (TD)
′
and applying [2, Lemma 9]; the second claim follows from res G
G LK/k = LK/k′
[2, Proposition 12].
The next lemma already concentrates on the case when G′ is abelian and
[G : G′ ] = l. We set A = G/G′ = hai and observe that a acts on G′ by
conjugation.
Lemma 3. Let τ : Λ∧ G → T (Λ∧ G) denote the canonical map and g ∈ G. If G′
is abelian 4 and of index l in G, then
Pl−1 i
a
if g ∈ G′
G′
i=0 g
ResG (τ g) =
l
g
if g ∈
/ G′ .
The lemma is just a special case of
Proposition A. Let H be an open subgroup of G = G(K/k). For g ∈ G set
lr
mH
∈ H} 5 , and let t run through a set of left
G (g) = min{r ≥ 0 : g
S
representatives of H in G, i.e., G = ˙ tH . Then
P
mH (t−1 gt)
H −1
−1
1. ResH
gt)l G
)/lmG (t gt) ,
G τG (g) =
t τH ((t
2. Res is transitive ,
H
3. ResH
G is integral, i.e., ResG (T (Λ∧ G)) ⊂ T (Λ∧ H) for H ≤ G of finite
index.
Proposition A will be shown in the Appendix.
For the purpose of this paper it is enough to know Lemma 3 which we quickly prove directly on applying Tr′ to both sides and employing the formula
Tr′ (τ ′ g)(χ′ ) = χ′ (g)g with g denoting the image of g ∈ G′ in Γk′ (see [5,§1]) :
4 whence
5 so
τ ′ : Λ∧ G′ → T (Λ∧ G′ ) is the identity map
(g)
is the m(g) defined earlier with H = G′
mH
G
Documenta Mathematica 13 (2008) 117–129
122
J¨
urgen Ritter and Alfred Weiss
′
′
′
′
′
1. (Tr′ ResG
= ResG
= Tr(τ g)(ind G
G (τ g))(χ )
G (Tr(τ g))(χ )
G′ χ ) +
G ′
Ψ
l
′
′
⊂ G . Now, if g ∈ G , Tr(τ g)(ind G′ χ ) =
l Tr(τ g)(χ) since G
P
l−1 ′ ai
/ G′ ,
i=0 χ (g )g and χ(g) = 0. On the other hand, if g ∈
G ′ l l
G ′
1
Ψ
Tr(τ g)(ind G′ χ ) = 0 and l Tr(τ g)(χ) = l ind G′ χ (g )g = χ′ (g l )g l
since we may choose a = g mod G′ .
Pl−1
Pl−1 i
i
i
2. Tr′ ( i=0 g a )(χ′ ) = i=0 χ′ (g a )g, since g a and g have the same image
in Γk and so in Γk′ . On the other hand, Tr′ (g l )(χ′ ) = χ′ (g l )g l .
The lemma is established.
We note that if Γ (≃ Zl ) is a central subgroup of G contained in the abelian
subgoup
G′ of index l, then the elements of T (Λ∧ G) can uniquely be written
P
as
g βg τ (g) with βg ∈ Λ∧ Γ and g running through a set of preimages of
conjugacy classes of G/Γ (see [3, Lemma 5]). For each summand we have
Pl−1
ai
if g ∈ G′
G′
i=0 βg g
ResG (βg τ (g)) =
l
Ψ(βg )g
if g ∈
/ G′ .
2 . Proof of the theorem
In this section G = G(K/k) is a pro-l group and G′ = G(K/k ′ ) an abelian
subgroup of index l (K/k is as in the introduction). As before, A = G/G′ = hai ,
and we set Aˆ = 1 + a + · · · + al−1 .
If G itself is abelian, the theorem holds by [4,§5, Example 1], whence we assume
that G is non-abelian.
Lemma 4. Assume that there exists an element x ∈ T (Λ∧ G) such that
ab
Gab
G′
G′
deflG
G x = deflG tK/k and ResG x = ResG tK/k . Then tK/k ∈ T (Λ∧ G).
Denoting by Kab the fixed field of the finite group [G, G], we first obserab
ve, because of [5, Lemma 2.1] and Lemma 2, that deflG
G tK/k = tKab /k and
′
ResG
G tK/k = tK/k′ are integral : indeed, a logarithmic pseudomeasure is integral
whenever the group is abelian.
From [4, Proposition 9] we obtain a power ln of l such that ln tK/k ∈ T (Λ∧ G).
ab
Consider the element x
˜ = ln (x − tK/k ) ∈ T (Λ∧ G). It satisfies deflG
˜ =
G x
′
G
0 = ResG x
˜ . We are going to prove x˜ = 0 which implies x = tK/k because
Hom∗ (Rl G, Qc∧ Γk ), and so T (Q∧ G), is torsionfree; whence the the lemma will
be verified.
The proof of x
˜ = 0 employs the commutative diagram shown in the proof of
[5, Proposition 2.2] :
ab
1 + a∧
↓
ˇ
τ (a∧ )
(Λ∧ G)×
L↓
deflG
G
։
(Λ∧ Gab )×
Lab ↓
ab
T (Λ∧ G)
deflG
G
։
Λ∧ Gab
Documenta Mathematica 13 (2008) 117–129
,
Equivariant Iwasawa Theory: An Example
123
in which L is extended to (Λ∧ G)× by means of the canonical surjection
(Λ∧ G)× ։ K1 (Λ∧ G) and a∧ = ker(Λ∧ G → Λ∧ Gab ). The diagram yields a
Gab
v ∈ (Λ∧ G)× with L(v) = x
˜, simply because deflG
x
˜ = 0. Combining diagrams
(HD), (LD) and (TD), we arrive at
′
′
′
′
G
G
′
G
L′ (resG
˜) = 0
G (Det v)) = ResG (L(Det v)) = ResG (Tr L(v)) = Tr (ResG x
ab
G
and, with resG
G replaced by deflG , at
′
ab
ab
G
Lab (deflG
G (Det v)) = deflG (L(Det v)) =
ab
ab
G
ab
= deflG
˜) = 0 ,
G (Tr L(v)) = Tr (deflG x
since L and Tr commute with deflation.
′
The first displayed formula in [3, p.46] now implies that resG
G (Det v)
ab
×
′
c
and deflG
G (Det v) are torsion elements in HOM(Rl G , (Λ∧ Γk′ ) ) and
ab
c
×
HOM(Rl (G ), (Λ∧ Γk ) ), respectively. Moreover, the first paragraph of
the proof of [5, Proposition 3.2] therefore shows that Det v itself is a torsion
element in HOM(Rl G, (Λc∧ Γk )× ). Consequently, for some natural number m,
m
(Det v)l = 1, so lm L(Det v) = 0 = lm Tr(Lv) = Tr(lm x˜) , and x
˜ = 0 follows,
as has been claimed.
We now introduce the commutative diagram
τ (a∧ )
Res ↓
b′∧ = τ ′ (b′∧ )
T (Λ∧ G)
′
ResG
G ↓
Λ∧ G′ = T (Λ∧ G′ )
Λ∧ Gab = T (Λ∧ Gab )
Res ↓
։ Λ∧ (G′ /[G, G]) = T (Λ∧ (G′ /[G, G]))
։
with exact rows (of which the upper one has already appeared in the diagram
shown in the proof of the preceding lemma). The images of all vertical maps
are fixed elementwise by A because of Lemma 3. Thus we can turn the diagram
into
(D)
τ (a∧ )
Res ↓
A
b′∧
↓
ˇ
ˆ 0 (A, b′ ) →
H
∧
T (Λ∧ G)
′
ResG
G ↓
(Λ∧ G′ )A
↓
ˇ
ˆ 0 (A, Λ∧ G′ )
H
Λ∧ Gab
Res ↓
→
(Λ∧ (G′ /[G, G]))A
↓
ˇ
ˆ 0 (A, Λ∧ (G′ /[G, G]))
→ H
։
with exact rows and canonical lower vertical maps.
Lemma 5. In (D), the left vertical column is exact and the left bottom horizontal map is injective.
Proof. The ideal a∧ is (additively) generated by the elements g(c − 1) with
′
g ∈ G, c ∈ [G, G] ; those with g ∈ G′ generate b′∧ . We compute ResG
G τ (g(c−1)),
using Lemma 3 :
Documenta Mathematica 13 (2008) 117–129
124
J¨
urgen Ritter and Alfred Weiss
′
1. if g ∈ G′ , ResG
G τ (g(c − 1)) =
trA b′∧ ,
ai
i=0 ((gc)
Pl−1
′
i
− ga ) =
Pl−1
i=0
ai
g(c − 1)
∈
ˆ
′
G
l
l
l A
l
2. if g ∈
/ G′ , ResG
G τ (g(c−1)) = ResG (τ (gc)−τ (g)) = (gc) −g = g c −g =
0 , since
ˆ
(⋆)
[G, G]A = 1
ˆ
ˆ
by [G, G]=[G,
˙
G′ ] and [G, G′ ]A = ((G′ )a−1 )A = 1 as (a − 1)Aˆ = 0. Here,
the dotted equality sign, =,
˙ results from the equation
i
−1
′ −1 −1 ′ −1 −i ′ i ′
[bg1′ , bi g2′ ] = (g
b (g2 ) b bg1 b g2 = (g1′ )−1 (g2′ )b (g1′ )b g2′ =
1)
′ −1 b ′
′ −1 ′ bi
(g2 ) g2 ∈ [G′ , G] · [G, G′ ] ≤ [G, G′ ]
(g1 ) (g1 )
for g1′ , g2′ ∈ G′ and b ∈ G \ G′ , because G′ is abelian and normal in G.
′
′
Thus, ResG
G τ (a∧ ) = trA b∧ , which proves the first claim of the lemma.
ˆ −1 (A, Λ∧ (G′ /[G, G])) = 0 and this in turn from
The second claim follows from H
′
the trivial action of A on G /[G, G] and the torsion freeness of Λ∧ (G′ /[G, G]).
Lemma 5 is established.
ab
As seen in diagram (D), there is an element x1 ∈ T (Λ∧ G) with deflG
G x1 =
G′
′
′
′
tKab /k . We define x1 ∈ Λ∧ G by ResG x1 = tK/k′ + x1 . Because of [5, Lemma
3.1], x′1 is fixed by A. We want to change x1 modulo τ (a∧ ) so that the new x′1
becomes zero: then we have arrived at an x ∈ T (Λ∧ G) as assumed in Lemma
4 and the theorem will have been confirmed.
′
The above change is possible if, and only if, x′1 ∈ ResG
G (τ (a∧ )) and so, because
def
of Lemma 5, if x′1 is in T ′ = trA (Λ∧ G′ ) , the A-trace ideal of the A-action on
Λ∧ G′ .
ab
Proposition. There exists an element x1 of T (Λ∧ G) with deflG
G x1 = tKab /k
and x′1 ∈ T ′ .
This is seen as follows. From [5,§1] we recall the existence of pseudomeasures λKab /k , λK/k′ in K1 (Λ∧ Gab ) and K1 (Λ∧ G′ ), respectively, satisfying Det λKab /k = LKab /k , Det λK/k′ = LK/k′ (so Lab (λKab /k ) =
tKab /k , , L′ (λK/k′ ) = tK/k′ ). From [5, 2. of Proposition 3.2] and [6, Theorem]
we know that
ver(λKab /k )
≡ 1 mod T ′
λK/k′
where ‘ver’ is the map induced from the transfer homomorphism Gab → G′ .
ab
G
′
Let y ∈ (Λ∧ G)× have deflG
G y = λKab /k and set resG y = λK/k′ · y . Then
′
′
y′ =
ver(λKab /k )
resG
G y
≡
≡1
λK/k′
λK/k′
mod T ′
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
125
def
(see the proof of [5, Proposition 3.2]). Moreover, y ′ ∈ 1 + b′∧ . Now, x1 = L(y)
def
′
′
G
′
′
′ ′
′
′
has ResG
G x1 = ResG L(y) = tK/k′ + x1 with x1 = L (y ), and x1 ∈ b∧ because
of the commutativity of
1 + b′∧
L′ ↓
b′∧
(Λ∧ G′ )×
L′ ↓
Λ∧ G′
։ (Λ∧ (G′ /[G, G]))×
L′ ↓
։
Λ∧ (G′ /[G, G])
.
Hence, the proposition (and therefore the theorem) will be proved, if
x′1 = L′ (y ′ ) ∈ T ′ .
However, Lemma 5 gives
A
A
y ′ ∈ (1 + b′∧ ) ∩ (1 + T ′ ) = 1 + (b′∧ ∩ T ′ ) = 1 + trA b′∧
and as L′ (y ′ ) =
(1)
1
l
log
y ′l
Ψ(y ′ )
(compare [3, p.39]), we see that
L′ (y ′ ) ∈ T ′
if
y ′l
Ψ(y ′ )
≡ 1 mod lT ′ .
So it suffices to show this last congruence.
Write y ′ = 1 + trA β ′ with β ′ ∈ b′∧ . Since (1 + trA β ′ )l ≡ 1 + (trA β ′ )l mod lT ′ ,
the congruence in (1) is equivalent to
(2)
(trA β ′ )l ≡ Ψ(trA β ′ ) mod lT ′ .
P
On picking a central subgroup Γ (≃ Zl ) of G and writing β ′ = g′ ,c βg′ ,c g ′ (c−
1) with elements βg′ ,c ∈ Λ∧ Γ, g ′ ∈ G′ , c ∈ [G, G], we obtain
l
P
′
(trA β ′ )l =
g′ ,c βg′ ,c trA (g (c − 1))
l
P
(a)
≡ g′ ,c (βg′ ,c )l trA (g ′ (c − 1))
P
≡ g′ ,c Ψ(βg′ ,c ) (trA (g ′ c))l − (trA g ′ )l mod lT ′
and
(b)
Ψ(trA β ′ ) =
P
g′ ,c
l
Ψ(βg′ ,c ) trA ((g ′ c)l ) − trA (g ′ )
as Ψ and trA commute. Thus congruence (2) will result from Lemma 6 below,
since then subtracting (b) from (a) yields the sum
P
′
l
′ l
′ l
′l
g′ ,c Ψ(βg′ ,c ) (trA (g c)) − trA ((g c) ) − (trA g ) + trA (g ) ≡
P
ˆ
ˆ
′ A
′A
′ ,c )
≡
Ψ(β
−
l(g
c)
+
lg
′
g
g ,c
P
ˆ
ˆ
A
′
A
′
g′ ,c (−l)Ψ(βg′ ,c )g (c − 1) ≡ 0 mod lT ,
by (⋆) of the proof of Lemma 5.
Documenta Mathematica 13 (2008) 117–129
126
J¨
urgen Ritter and Alfred Weiss
l
(trA g ′ )l − trA (g ′ ) ≡ −lg ′
Lemma 6.
ˆ
A
mod lT ′ for g ′ ∈ G′ .
˜
Proof. Set A˜ = Z/l × A and make M = Maps(Z/l, A) into an A-set
by
(z,ai )
i
defining m
(x) = m(x − z) · a . Then
l−1
X P
X Y m(z)
X
m(z)
ai
g′
=
g ′ z∈Z/l
g ′ )l =
(trA g ′ )l = (
i=0
with
P
z
m∈M
m∈M z∈Z/l
m(z) read in Z[A].
We compute the subsums of
P
m
˜
in which m is constrained to an A-orbit.
˜ then the A-orbit
˜
If m ∈ M has stabilizer {(0, 1)} in A,
sum is
P
P
˜
(z,ai )∈A
(z,ai )
g
′
g′
P
P
v
i
v∈Z/l
m(v)ai
m(z,a ) (v)
=l
P
=
i (g
′
P
P
v
˜
(z,ai )∈A
m(v) ai
)
g′
P
v∈Z/l
m(v−z)ai
= l · trA (g
′
P
v
m(v)
=
) ∈ lT ′ .
i
Note that no m ∈ M is stabilized by (0, ai ) with ai 6= 1 : for m(z) = m(0,a ) (z) =
m(z)ai implies ai = 1. It follows that the stabilizers of the elements with
stabilizer different from {(0, 1)} must be cyclic of order l and different from
{(0, ai ) : 0 ≤ i ≤ l − 1} and therefore = h(1, aj )i for a unique j mod l.
˜
One now checks that for each j there is exactly one A-orbit
with stabilizer
j
jz
h(1, a )i and that it is represented by mj , mj (z) = a . Moreover, {(0, ai ) :
˜
0 ≤ i ≤ l − 1} is a transversal of the stabilizer of mj in A.
P
P P m(0,ai ) (z)
m(z)
˜
=
over the A-orbit
of mj is i g ′ z j
For each j, the sum of g ′ z
P ′ lai
P ′ Pz ajz ai
′l
= trA (g ) , accounting for that term
. If j = 0, this is i g
ig
P ′ A·a
ˆ
ˆ i
= lg ′ A , and summing over j 6= 0 gives
in the claim. If j 6= 0, it is i g
ˆ
ˆ
ˆ
ˆ
(l − 1)l · g ′ A ≡ −l · g ′ A mod lT ′ because l2 · g ′ A = l · trA (g ′ A ) ∈ lT ′ .
This finishes the proof of Lemma 6.
3 . Appendix
Proof of Proposition A :
We start from the formula χ(g) =
P
t
mH
G (g )=1
χ′ (g lt ) which appeared in the
r−1
justification of the definition in §1 (now with H = G′ ). It implies χ(g l ) =
P
r
′
−1
gt)l ) for r ≥ 1. Exploiting [3, Lemma 6] for the equality
mH (t−1 gt)=r χ ((t
G
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
127
=
˙ and [3, Proposition 3] for =
¨ we obtain
′
(ResH
G TrG τG (g))(χ )
P
r
G ′
= (TrG τG (g))(ind H χ ) + r≥1 Ψlr [(TrG τG (g))(ψlr−1 χ)]
P
r
Ψ
=trace(g
˙
| Vind G
′) +
r≥1 lr [trace(g | Vψlr−1 χ )]
Hχ
P
r
G ′
r−1
Ψ
χ)(g)g]
=(ind
¨
H χ )(g)g +
r≥1 lr [(ψl
r
P
r−1
G ′
gl
l
= (ind H χ )(g)g + r≥1 χ(g
) lr
lr
P
P
P
r
= mH (t−1 gt)=0 χ′ (t−1 gt)g + r≥1 mH (t−1 gt)=r χ′ ((t−1 gt)l ) glr
G
G
mH (t−1 gt)
−1 gt)
P ′ −1 lmH
gl G
G (t
) mH (t−1 gt) .
= t χ ((t gt)
l
G
On the other hand, if h ∈ H, the same calculation on the H-level gives
(TrH τH (h))(χ′ ) = trace(h | Vχ′ ) = χ′ (h)h ,
P
and therefore, with τH ( t
(t−1 gt)l
H
−1 gt)
mH
G (t
−1 gt)
lmG (t
H
) ∈ T (Q(H)) , we get
−1
P −1 lmG (t gt)
(TrH τH t (t gt)
)(χ′ )
H −1
lmG (t gt)
P
mH (t−1 gt)
mH (t−1 gt)
= t mH (t1−1 gt) χ′ ((t−1 gt)l G
)(t−1 gt)l G
l G
mH (t−1 gt)
l G
P ′ −1 lmH
(t−1 gt)
) g mH (t−1 gt)
= t χ ((t gt) G
l
G
via Qc (Γk′ ) ֒→ Qc (Γk ) (where k ′ is the fixed field of H).
Since these formulae agree for all characters χ′ of H (with open kernel) it
H −1
P (t−1 gt)lmG (t gt)
) by the uniqueness of this
follows that ResH
τ
(g)
=
τ
(
H
H −1
G G
t
lmG (t gt)
element, proving 1. of Proposition A.
For 2. we first consider the situation H ≤ G′ ≤ G with [G : G′ ] = l and show
H
G′
ResH
G = ResG′ ◦ ResG :
S
S
S
Write G = ˙ x xG′ , G′ = ˙ y yH , hence G = ˙ x,y xyH , and recall 6 from 1. that
′
ResG
G g
6 if
=
P
τG′ ( x x−1 gx) , g ∈ G′
.
, g∈
/ G′
τG′ (g l )
g∈
/ G′ , then we may use {t} = {g i : 0 ≤ i ≤ l − 1}
Documenta Mathematica 13 (2008) 117–129
128
J¨
urgen Ritter and Alfred Weiss
Then, accordingly,
′
G
ResH
G′ Res
GPτG (g) H
−1
gx)
x ResG′ τG′ (x
=
H
ResG′ τG′ (g l )
H
−1 −1
P P τ ((y −1 x−1 gxy)lmG′ (y x gxy) )/lmH
(y −1 x−1 gxy)
G′
x
y H
=
H
−1 l
P τ ((y −1 g l y)lmG′ (y g y) )/lmH
(y −1 gl y)
G′
H
y
( P
H
−1
m (t
gt)
H −1
−1
gt)l G
)/lmG (t gt)
t τH ((t
=
H
−1
l
m
(t
g t)
P
H
−1 l
−1 l l G′
g t)
)/l · lmG′ (t g t)
t τH ((t
H
P
m
−1
H −1
−1
=
gt)l G (t gt) /lmG (t gt) = ResH
G τG (g) ,
t τH ((t
l
H
using G′ ✁ G and mH
/ G′ .
G′ (x ) + 1 = mG (x) for x ∈
′′
H
G
′′
Induction on [G : H] now proves 2., ResH
G = ResG′′ ◦ ResG for H ≤ G ≤ G.
′′
′
′′
′
Indeed, if G 6= G, find G ≥ G with [G : G ] = l and use
′′
′′
′
G
H
G
G
ResH
G′′ ResG = ResG′′ (ResG′ ResG ) =
′′
′
′
G
G
H
G
H
= (ResH
G′′ ResG′ )ResG = ResG′ ResG = ResG .
Finally, for 3., proceed by induction on [G : H] and use 1. if H has index l in
G and 2. when the index is bigger, in which case there is a subgroup G′ of G
with H ≤ G , [G : G′ ] = l.
The proof of Proposition A is complete.
References
[1]
[2-6]
[7]
Oliver, R. and Taylor, L.R., Logarithmic descriptions of Whitehead groups and class groups for p-groups. Memoirs of the AMS
(1988), vol. 76, no. 392
Ritter, J. and Weiss, A. ,
2. Towards equivariant Iwasawa theory, II. Indag. Mathemat. 15
(2004), 549-572
3. · · · , III. Math. Ann. 336 (2006), 27-49
4. · · · , IV. Homology, Homotopy and Applications 7 (2005), 155171
5. Non-abelian pseudomeasures and congruences between abelian
Iwasawa L-functions. Pure and Applied Math. Quarterly 4 (2008),
1085-1106
6. Congruences between abelian pseudomeasures. To appear in
Math.Res. Lett. 15 (2008)
Wang, S., On the commutator group of a simple algebra.
Amer. J. Math. 72 (1950), 323-334
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
J¨
urgen Ritter
Institut f¨
ur Mathematik
Universit¨
at Augsburg
86135 Augsburg
Germany
Alfred Weiss
Department of Mathematics
University of Alberta
Edmonton, AB
Canada T6G 2G1
Documenta Mathematica 13 (2008) 117–129
129
130
Documenta Mathematica 13 (2008)
Documenta Math.
Equivariant Iwasawa Theory: An Example
J¨
urgen Ritter and Alfred Weiss1
Received: November 6, 2007
Revised: April 22, 2008
Communicated by Peter Schneider
Abstract. The equivariant main conjecture of Iwasawa theory is
shown to hold for a Galois extension K/k of totally real number fields
with Galois group an l-adic pro-l Lie group of dimension 1 containing
an abelian subgroup of index l, provided that Iwasawa’s µ-invariant
µ(K/k) vanishes.
2000 Mathematics Subject Classification: 11R23, 11R32, 11R42
Keywords and Phrases: Iwasawa theory, Iwasawa L-functions
This note justifies a remark made in the introduction of [6] according to which
the “main conjecture” of equivariant Iwasawa theory, as formulated in [2,
p.564], holds when G = G(K/k) is a pro-l group with an abelian subgroup
G′ of index l.
We quickly repeat the general set-up and, in doing so, refer the reader to [5,§1]
for facts and notation that is taken from our earlier papers on Iwasawa theory.
Namely, l is a fixed odd prime number and K/k a Galois extension of totally
real number fields, with k/Q and K/k∞ finite, where k∞ is the cyclotomic
Zl -extension of k. Throughout it will be assumed that Iwasawa’s µ-invariant
µ(K/k) vanishes. We also fix a finite set S of primes of k containing all primes
above ∞ and all those whose ramification index in K/k is divisible by l.
In this situation it is shown in [5] that the “main conjecture” of equivariant
Iwasawa theory would follow from two kinds of hypothetical congruences between values of Iwasawa L-functions. One of these kinds, the so-called torsion
congruences (see [5, Proposition 3.2]), stated as
ver(λKab /k )
≡1
λK/k′
1 We
mod T ′
acknowledge financial support provided by NSERC and the University of Augsburg.
Documenta Mathematica 13 (2008) 117–129
118
J¨
urgen Ritter and Alfred Weiss
in the proof of the proposition in §2, has meanwhile been verified in [6].
The purpose of the present paper is to show that the torsion congruences already suffice to obtain the whole conjecture in the special case when G is a
pro-l group with an abelian subgroup G′ of index l. Before stating the precise
theorem we need to recall some notation (compare [5,§1]).
Λ∧ G is the l-completion of the localization Λ• G which is obtained from
the Iwasawa algebra ΛG = Zl [[G]] by inverting all central elements which
are regular in ΛG/lΛG; Q∧ G is the total ring of fractions of Λ∧ G;
T (Q∧ G) = Q∧ G/[Q∧ G, Q∧ G] is the quotient of Q∧ G by Lie commutators;
if G is a pro-l group, then (see [5,§2] 2 )
(LD)
K1 (Λ∧ G)
Det ↓
−→
L
T (Q∧ G)
Tr ↓
≃
HOM(Rl G, (Λc∧ Γk )× )
L
−→
Hom∗ (Rl G, Qc∧ Γk )
is the logarithmic diagram defining the logarithmic pseudomeasure
tK/k ∈ T (Q∧ G)
by
Tr(tK/k ) = L(LK/k )
where LK/k = LK/k,S ∈ HOM(Rl G, (Λc∧ Γk )× ) is the Iwasawa L-function.
Theorem. With K/k and S as at the beginning and G = G(K/k) a pro-l
group, tK/k is integral (i.e., tK/k ∈ T (Λ∧ G)) whenever G has an abelian
subgroup G′ of index l.
As a corollary, by [5, Proposition 3.2] and [6, Theorem], LK/k ∈ Det K1 (Λ∧ G) ,
which implies the conjecture (see [3, Theorem A]), up to its uniqueness assertion. However, SK1 (QG) = 1 because each simple component, after tensoring
up with a suitable extension field of its centre, becomes isomorphic to a matrix ring of dimension a divisor of l2 by the proof of [2, Proposition 6], as the
character degrees χ(1) all divide l. Now apply [7, p.334, Corollary].
The proof of the theorem is carried out in §2; before, in a short §1, we introduce
restriction maps
′
′
ResG
G : T (Q∧ G) → T (Q∧ G )
and
′
∗
∗
c
′
c
ResG
G : Hom (Rl G, Q∧ Γk ) → Hom (Rl G , Q∧ Γk′ )
2 R G is the ring of all (virtual)
l
G(k∞ /k) ; Λc∧ Γk = Zlc ⊗Zl Λ∧ Γk with
ure Qlc of Ql
Qlc -characters of G with open kernel; Γk =
Zlc the ring of integers in a fixed algebraic clos-
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
119
making the diagram
L
T (Q∧ G)
′
ResG
G ↓
−→
Tr
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
L′
T (Q∧ G′ )
−→
Tr′
Hom∗ (Rl G′ , Qc∧ Γk′ )
K1 (Λ∧ G)
′
res G
G ↓
−→
K1 (Λ∧ G′ )
−→
commute 3 for any pair of pro-l groups G = G(K/k) and G′ = G(K/k ′ ) ≤ G
′
such that [G : G′ ] is finite. We remark that replacing ResG
G by the “natural”
restriction map,
′
G ′
∗
′
′
′
c
(res G
G f )(χ ) = f (ind G′ χ ) , f ∈ Hom (Rl G, Q∧ Γk ) , χ ∈ Rl G ,
does not work, because induction and Adams operations do not commute.
1 . Res
Let G = G(K/k) be a pro-l group and G′ = G(K/k ′ ) ≤ G an open subgroup.
Recall that Ψ : Λc∧ Γk → Λc∧ Γk is the map induced by Ψ(γ) = γ l for γ ∈ Γk
(compare [5,§1]) and that ψl is the lth Adams operation on Rl (−).
Definition.
′
∗
∗
c
′
c
ResG
G : Hom (Rl G, Q∧ Γk ) → Hom (Rl G , Q∧ Γk′ ) sends f to
X Ψr
′
′
G ′
ResG
f
=
χ
→
7
f
(ind
χ
)
+
(f (ψlr−1 χ)) ,
′
G
G
r
l
r≥1
G
′
′
where χ = ψl (ind G
G′ χ ) − ind G′ (ψl χ ) .
P
To justify the definition we must show that the sum r≥1 is actually a finite
sum. For this, let {t} be a set of coset representatives of G′ in G, so G = ∪˙ t tG′ ,
and define
r
m(g) = min{r ≥ 0 : g l ∈ G′ } for g ∈ G .
Then
′
ind G
G′ χ (g) =
X
t
χ˙ ′ (g t ) =
X
χ′ (g t ) ,
{t:m(gt )=0}
if, as usual, χ˙ ′ coincides with χ′ on G′ and vanishes on G \ G′ . Hence,
G ′
′
l
) −Pind G
χ(g)
G′ (ψl χ )(g)
P
P = (ind G′ χ′ )(g
′ lt
lt
= m(glt )=0 χ (g ) − m(gt )=0 χ (g ) = m(gt )=1 χ′ (g lt ) .
r0
If r0 is such that Gl ⊂ G′ , then ψlr0 −1 χ = 0 and
P
r0 −1
.
the sum m(gt )=1 is empty when g ∈ Gl
P
r≥1
=
′
Pr0 −2
r=1
, because
finite G, this ResG
G , making the left square commute, appears already in [1, p.14].
′
For us the properties (HD),(TD) of ResG
G shown in §1 are equally necessary for the proof of
the Theorem above.
3 For
Documenta Mathematica 13 (2008) 117–129
120
J¨
urgen Ritter and Alfred Weiss
′
′
G
∗
′
c
It remains to show that ResG
G f ∈ Hom (Rl G , Q∧ Γk′ ) , i.e., ResG f is a Galois
stable homomorphism, compatible with W-twists (see [5,§1]), and taking values
in Qc∧ Γk′ . The first property is easily checked and the third follows from the
second as in [2, proof of Lemma 9]. We turn to twisting.
Let ρ′ be a type-W character of G′ , so ρ′ is inflated from Γk′ , and write ρ′ =
′
res G
G ρ with ρ inflated from Γk to G. Then
G ′
G ′
G ′
′ ′
♯
′ ♯
f (ind G
G′ (ρ χ )) = f (ρ · ind G′ χ ) = ρ (f (ind G′ χ )) = (ρ ) (f (ind G′ χ ))
′
c
as f (ind G
G′ χ ) ∈ Q∧ Γk′ . Moreover, since ψl is multiplicative,
G
G ′
G
′ ′
′ ′
′ l
′
l
ψl (ind G
G′ (ρ χ )) − ind G′ (ψl (ρ χ )) = ψl (ρ · ind G′ χ ) − ind G′ ((ρ ) · ψl χ ) = ρ · χ
and thus
Ψr
lr
Ψr
lr
r
r
(f (ψlr−1 (ρl · χ)))) = Ψlr f (ρl · ψlr−1 χ) =
r
r
r
((ρl )♯ (f (ψlr−1 χ))) = ρ♯ ( Ψlr f (ψlr−1 χ)) = (ρ′ )♯ ( Ψlr f (ψlr−1 χ)) .
Lemma 1. The diagram below commutes. In it, L and L′ are the lower horizontal maps of the logarithmic diagram (LD) for G and G′ , respectively.
(HD)
HOM(Rl G, (Λc∧ Γk )× )
′
res G
G ↓
HOM(Rl G′ , (Λc∧ Γk′ )× )
L
−→
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
L′
−→
Hom∗ (Rl G′ , Qc∧ Γk′ ) .
Indeed, for f ∈ HOM(Rl G, (Λc∧ Γk )× ) we get
P
′
r−1
Ψr
′
′
χ)]
(ResG
)(ind G
G Lf )(χ ) = (Lf
G′ χ ) +
r≥1 lr [(Lf )(ψl
P
r
G ′
r−1
Ψ
Ψ
=(Lf
˙
)(ind G′ χ ) + r≥1 lr [log(f (ψl χ)) − l log(f (ψlr χ))]
P
P
r
r−1
Ψr
′
χ)) − r≥2 Ψlr log(f (ψlr−1 χ))
= (Lf )(ind G
G′ χ ) +
r≥1 lr log(f (ψl
′
= (Lf )(ind G
G′ χ ) +
=
=
Ψ
l
log(f (χ)) =
f (ind χ′ )l ·Ψf (ψl ind χ′ )
Ψf (ψl ind χ′ )·Ψf (ind ψl χ′ )
′
′
(L′ res G
G f )(χ ) .
1
l
log
=
1
l
1
l
log
log
f (ind χ′ )l
Ψ(f (ψl ind χ′ ))
+
Ψ
l
log
f (ψl ind χ′ )
f (ind ψl χ′ )
f (ind χ′ )l
Ψf (ind ψl χ′ )
The dotted equality sign, =,
˙ is due to the congruence
f (χ)l
Ψf (ψl χ)
≡ 1 mod lΛc∧ Γk
(see [5,§1]) and to χ(1) = 0, so (ψlr−1 χ)(1) = 0 for every r. In fact, with
def
χ
˜ = ψlr−1 χ, we have
f (χ)
˜ l ≡ Ψf (ψl χ)
˜ mod lΛc∧ Γk =⇒
s
f (χ)
˜ l ≡ Ψs f (ψls χ)
˜ ≡ Ψs f (χ(1)1)
˜
= 1 mod lΛc∧ Γk
˜ as ‘log’ converges
for big enough s. Thus (Lf )(χ)
˜ = log(f (χ))
˜ − Ψl log(f (ψl χ))
on an element a power of which is ≡ 1 mod lΛc∧ Γk .
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
121
The proof of Lemma 1 is complete.
By means of the trace isomorphism Tr : T (−) → Hom∗ (−) we next transport
′
G′
′
ResG
G to ResG : T (Q∧ G) → T (Q∧ G ), i.e., the diagram
(TD)
T (Q∧ G)
′
ResG
G ↓
−→
Tr
Hom∗ (Rl G, Qc∧ Γk )
′
ResG
G ↓
T (Q∧ G′ )
−→
Tr′
Hom∗ (Rl G′ , Qc∧ Γk′ )
commutes.
Lemma 2.
K1 (Λ∧ G)
′
res G
G ↓
→
K1 (Λ∧ G′ )
→
L
L
′
T (Q∧ G)
′
ResG
G ↓
′
commutes and ResG
G tK/k = tK/k′ .
T (Q∧ G′ )
The first claim follows from gluing together the diagrams (LD), (HD), (TD)
′
and applying [2, Lemma 9]; the second claim follows from res G
G LK/k = LK/k′
[2, Proposition 12].
The next lemma already concentrates on the case when G′ is abelian and
[G : G′ ] = l. We set A = G/G′ = hai and observe that a acts on G′ by
conjugation.
Lemma 3. Let τ : Λ∧ G → T (Λ∧ G) denote the canonical map and g ∈ G. If G′
is abelian 4 and of index l in G, then
Pl−1 i
a
if g ∈ G′
G′
i=0 g
ResG (τ g) =
l
g
if g ∈
/ G′ .
The lemma is just a special case of
Proposition A. Let H be an open subgroup of G = G(K/k). For g ∈ G set
lr
mH
∈ H} 5 , and let t run through a set of left
G (g) = min{r ≥ 0 : g
S
representatives of H in G, i.e., G = ˙ tH . Then
P
mH (t−1 gt)
H −1
−1
1. ResH
gt)l G
)/lmG (t gt) ,
G τG (g) =
t τH ((t
2. Res is transitive ,
H
3. ResH
G is integral, i.e., ResG (T (Λ∧ G)) ⊂ T (Λ∧ H) for H ≤ G of finite
index.
Proposition A will be shown in the Appendix.
For the purpose of this paper it is enough to know Lemma 3 which we quickly prove directly on applying Tr′ to both sides and employing the formula
Tr′ (τ ′ g)(χ′ ) = χ′ (g)g with g denoting the image of g ∈ G′ in Γk′ (see [5,§1]) :
4 whence
5 so
τ ′ : Λ∧ G′ → T (Λ∧ G′ ) is the identity map
(g)
is the m(g) defined earlier with H = G′
mH
G
Documenta Mathematica 13 (2008) 117–129
122
J¨
urgen Ritter and Alfred Weiss
′
′
′
′
′
1. (Tr′ ResG
= ResG
= Tr(τ g)(ind G
G (τ g))(χ )
G (Tr(τ g))(χ )
G′ χ ) +
G ′
Ψ
l
′
′
⊂ G . Now, if g ∈ G , Tr(τ g)(ind G′ χ ) =
l Tr(τ g)(χ) since G
P
l−1 ′ ai
/ G′ ,
i=0 χ (g )g and χ(g) = 0. On the other hand, if g ∈
G ′ l l
G ′
1
Ψ
Tr(τ g)(ind G′ χ ) = 0 and l Tr(τ g)(χ) = l ind G′ χ (g )g = χ′ (g l )g l
since we may choose a = g mod G′ .
Pl−1
Pl−1 i
i
i
2. Tr′ ( i=0 g a )(χ′ ) = i=0 χ′ (g a )g, since g a and g have the same image
in Γk and so in Γk′ . On the other hand, Tr′ (g l )(χ′ ) = χ′ (g l )g l .
The lemma is established.
We note that if Γ (≃ Zl ) is a central subgroup of G contained in the abelian
subgoup
G′ of index l, then the elements of T (Λ∧ G) can uniquely be written
P
as
g βg τ (g) with βg ∈ Λ∧ Γ and g running through a set of preimages of
conjugacy classes of G/Γ (see [3, Lemma 5]). For each summand we have
Pl−1
ai
if g ∈ G′
G′
i=0 βg g
ResG (βg τ (g)) =
l
Ψ(βg )g
if g ∈
/ G′ .
2 . Proof of the theorem
In this section G = G(K/k) is a pro-l group and G′ = G(K/k ′ ) an abelian
subgroup of index l (K/k is as in the introduction). As before, A = G/G′ = hai ,
and we set Aˆ = 1 + a + · · · + al−1 .
If G itself is abelian, the theorem holds by [4,§5, Example 1], whence we assume
that G is non-abelian.
Lemma 4. Assume that there exists an element x ∈ T (Λ∧ G) such that
ab
Gab
G′
G′
deflG
G x = deflG tK/k and ResG x = ResG tK/k . Then tK/k ∈ T (Λ∧ G).
Denoting by Kab the fixed field of the finite group [G, G], we first obserab
ve, because of [5, Lemma 2.1] and Lemma 2, that deflG
G tK/k = tKab /k and
′
ResG
G tK/k = tK/k′ are integral : indeed, a logarithmic pseudomeasure is integral
whenever the group is abelian.
From [4, Proposition 9] we obtain a power ln of l such that ln tK/k ∈ T (Λ∧ G).
ab
Consider the element x
˜ = ln (x − tK/k ) ∈ T (Λ∧ G). It satisfies deflG
˜ =
G x
′
G
0 = ResG x
˜ . We are going to prove x˜ = 0 which implies x = tK/k because
Hom∗ (Rl G, Qc∧ Γk ), and so T (Q∧ G), is torsionfree; whence the the lemma will
be verified.
The proof of x
˜ = 0 employs the commutative diagram shown in the proof of
[5, Proposition 2.2] :
ab
1 + a∧
↓
ˇ
τ (a∧ )
(Λ∧ G)×
L↓
deflG
G
։
(Λ∧ Gab )×
Lab ↓
ab
T (Λ∧ G)
deflG
G
։
Λ∧ Gab
Documenta Mathematica 13 (2008) 117–129
,
Equivariant Iwasawa Theory: An Example
123
in which L is extended to (Λ∧ G)× by means of the canonical surjection
(Λ∧ G)× ։ K1 (Λ∧ G) and a∧ = ker(Λ∧ G → Λ∧ Gab ). The diagram yields a
Gab
v ∈ (Λ∧ G)× with L(v) = x
˜, simply because deflG
x
˜ = 0. Combining diagrams
(HD), (LD) and (TD), we arrive at
′
′
′
′
G
G
′
G
L′ (resG
˜) = 0
G (Det v)) = ResG (L(Det v)) = ResG (Tr L(v)) = Tr (ResG x
ab
G
and, with resG
G replaced by deflG , at
′
ab
ab
G
Lab (deflG
G (Det v)) = deflG (L(Det v)) =
ab
ab
G
ab
= deflG
˜) = 0 ,
G (Tr L(v)) = Tr (deflG x
since L and Tr commute with deflation.
′
The first displayed formula in [3, p.46] now implies that resG
G (Det v)
ab
×
′
c
and deflG
G (Det v) are torsion elements in HOM(Rl G , (Λ∧ Γk′ ) ) and
ab
c
×
HOM(Rl (G ), (Λ∧ Γk ) ), respectively. Moreover, the first paragraph of
the proof of [5, Proposition 3.2] therefore shows that Det v itself is a torsion
element in HOM(Rl G, (Λc∧ Γk )× ). Consequently, for some natural number m,
m
(Det v)l = 1, so lm L(Det v) = 0 = lm Tr(Lv) = Tr(lm x˜) , and x
˜ = 0 follows,
as has been claimed.
We now introduce the commutative diagram
τ (a∧ )
Res ↓
b′∧ = τ ′ (b′∧ )
T (Λ∧ G)
′
ResG
G ↓
Λ∧ G′ = T (Λ∧ G′ )
Λ∧ Gab = T (Λ∧ Gab )
Res ↓
։ Λ∧ (G′ /[G, G]) = T (Λ∧ (G′ /[G, G]))
։
with exact rows (of which the upper one has already appeared in the diagram
shown in the proof of the preceding lemma). The images of all vertical maps
are fixed elementwise by A because of Lemma 3. Thus we can turn the diagram
into
(D)
τ (a∧ )
Res ↓
A
b′∧
↓
ˇ
ˆ 0 (A, b′ ) →
H
∧
T (Λ∧ G)
′
ResG
G ↓
(Λ∧ G′ )A
↓
ˇ
ˆ 0 (A, Λ∧ G′ )
H
Λ∧ Gab
Res ↓
→
(Λ∧ (G′ /[G, G]))A
↓
ˇ
ˆ 0 (A, Λ∧ (G′ /[G, G]))
→ H
։
with exact rows and canonical lower vertical maps.
Lemma 5. In (D), the left vertical column is exact and the left bottom horizontal map is injective.
Proof. The ideal a∧ is (additively) generated by the elements g(c − 1) with
′
g ∈ G, c ∈ [G, G] ; those with g ∈ G′ generate b′∧ . We compute ResG
G τ (g(c−1)),
using Lemma 3 :
Documenta Mathematica 13 (2008) 117–129
124
J¨
urgen Ritter and Alfred Weiss
′
1. if g ∈ G′ , ResG
G τ (g(c − 1)) =
trA b′∧ ,
ai
i=0 ((gc)
Pl−1
′
i
− ga ) =
Pl−1
i=0
ai
g(c − 1)
∈
ˆ
′
G
l
l
l A
l
2. if g ∈
/ G′ , ResG
G τ (g(c−1)) = ResG (τ (gc)−τ (g)) = (gc) −g = g c −g =
0 , since
ˆ
(⋆)
[G, G]A = 1
ˆ
ˆ
by [G, G]=[G,
˙
G′ ] and [G, G′ ]A = ((G′ )a−1 )A = 1 as (a − 1)Aˆ = 0. Here,
the dotted equality sign, =,
˙ results from the equation
i
−1
′ −1 −1 ′ −1 −i ′ i ′
[bg1′ , bi g2′ ] = (g
b (g2 ) b bg1 b g2 = (g1′ )−1 (g2′ )b (g1′ )b g2′ =
1)
′ −1 b ′
′ −1 ′ bi
(g2 ) g2 ∈ [G′ , G] · [G, G′ ] ≤ [G, G′ ]
(g1 ) (g1 )
for g1′ , g2′ ∈ G′ and b ∈ G \ G′ , because G′ is abelian and normal in G.
′
′
Thus, ResG
G τ (a∧ ) = trA b∧ , which proves the first claim of the lemma.
ˆ −1 (A, Λ∧ (G′ /[G, G])) = 0 and this in turn from
The second claim follows from H
′
the trivial action of A on G /[G, G] and the torsion freeness of Λ∧ (G′ /[G, G]).
Lemma 5 is established.
ab
As seen in diagram (D), there is an element x1 ∈ T (Λ∧ G) with deflG
G x1 =
G′
′
′
′
tKab /k . We define x1 ∈ Λ∧ G by ResG x1 = tK/k′ + x1 . Because of [5, Lemma
3.1], x′1 is fixed by A. We want to change x1 modulo τ (a∧ ) so that the new x′1
becomes zero: then we have arrived at an x ∈ T (Λ∧ G) as assumed in Lemma
4 and the theorem will have been confirmed.
′
The above change is possible if, and only if, x′1 ∈ ResG
G (τ (a∧ )) and so, because
def
of Lemma 5, if x′1 is in T ′ = trA (Λ∧ G′ ) , the A-trace ideal of the A-action on
Λ∧ G′ .
ab
Proposition. There exists an element x1 of T (Λ∧ G) with deflG
G x1 = tKab /k
and x′1 ∈ T ′ .
This is seen as follows. From [5,§1] we recall the existence of pseudomeasures λKab /k , λK/k′ in K1 (Λ∧ Gab ) and K1 (Λ∧ G′ ), respectively, satisfying Det λKab /k = LKab /k , Det λK/k′ = LK/k′ (so Lab (λKab /k ) =
tKab /k , , L′ (λK/k′ ) = tK/k′ ). From [5, 2. of Proposition 3.2] and [6, Theorem]
we know that
ver(λKab /k )
≡ 1 mod T ′
λK/k′
where ‘ver’ is the map induced from the transfer homomorphism Gab → G′ .
ab
G
′
Let y ∈ (Λ∧ G)× have deflG
G y = λKab /k and set resG y = λK/k′ · y . Then
′
′
y′ =
ver(λKab /k )
resG
G y
≡
≡1
λK/k′
λK/k′
mod T ′
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
125
def
(see the proof of [5, Proposition 3.2]). Moreover, y ′ ∈ 1 + b′∧ . Now, x1 = L(y)
def
′
′
G
′
′
′ ′
′
′
has ResG
G x1 = ResG L(y) = tK/k′ + x1 with x1 = L (y ), and x1 ∈ b∧ because
of the commutativity of
1 + b′∧
L′ ↓
b′∧
(Λ∧ G′ )×
L′ ↓
Λ∧ G′
։ (Λ∧ (G′ /[G, G]))×
L′ ↓
։
Λ∧ (G′ /[G, G])
.
Hence, the proposition (and therefore the theorem) will be proved, if
x′1 = L′ (y ′ ) ∈ T ′ .
However, Lemma 5 gives
A
A
y ′ ∈ (1 + b′∧ ) ∩ (1 + T ′ ) = 1 + (b′∧ ∩ T ′ ) = 1 + trA b′∧
and as L′ (y ′ ) =
(1)
1
l
log
y ′l
Ψ(y ′ )
(compare [3, p.39]), we see that
L′ (y ′ ) ∈ T ′
if
y ′l
Ψ(y ′ )
≡ 1 mod lT ′ .
So it suffices to show this last congruence.
Write y ′ = 1 + trA β ′ with β ′ ∈ b′∧ . Since (1 + trA β ′ )l ≡ 1 + (trA β ′ )l mod lT ′ ,
the congruence in (1) is equivalent to
(2)
(trA β ′ )l ≡ Ψ(trA β ′ ) mod lT ′ .
P
On picking a central subgroup Γ (≃ Zl ) of G and writing β ′ = g′ ,c βg′ ,c g ′ (c−
1) with elements βg′ ,c ∈ Λ∧ Γ, g ′ ∈ G′ , c ∈ [G, G], we obtain
l
P
′
(trA β ′ )l =
g′ ,c βg′ ,c trA (g (c − 1))
l
P
(a)
≡ g′ ,c (βg′ ,c )l trA (g ′ (c − 1))
P
≡ g′ ,c Ψ(βg′ ,c ) (trA (g ′ c))l − (trA g ′ )l mod lT ′
and
(b)
Ψ(trA β ′ ) =
P
g′ ,c
l
Ψ(βg′ ,c ) trA ((g ′ c)l ) − trA (g ′ )
as Ψ and trA commute. Thus congruence (2) will result from Lemma 6 below,
since then subtracting (b) from (a) yields the sum
P
′
l
′ l
′ l
′l
g′ ,c Ψ(βg′ ,c ) (trA (g c)) − trA ((g c) ) − (trA g ) + trA (g ) ≡
P
ˆ
ˆ
′ A
′A
′ ,c )
≡
Ψ(β
−
l(g
c)
+
lg
′
g
g ,c
P
ˆ
ˆ
A
′
A
′
g′ ,c (−l)Ψ(βg′ ,c )g (c − 1) ≡ 0 mod lT ,
by (⋆) of the proof of Lemma 5.
Documenta Mathematica 13 (2008) 117–129
126
J¨
urgen Ritter and Alfred Weiss
l
(trA g ′ )l − trA (g ′ ) ≡ −lg ′
Lemma 6.
ˆ
A
mod lT ′ for g ′ ∈ G′ .
˜
Proof. Set A˜ = Z/l × A and make M = Maps(Z/l, A) into an A-set
by
(z,ai )
i
defining m
(x) = m(x − z) · a . Then
l−1
X P
X Y m(z)
X
m(z)
ai
g′
=
g ′ z∈Z/l
g ′ )l =
(trA g ′ )l = (
i=0
with
P
z
m∈M
m∈M z∈Z/l
m(z) read in Z[A].
We compute the subsums of
P
m
˜
in which m is constrained to an A-orbit.
˜ then the A-orbit
˜
If m ∈ M has stabilizer {(0, 1)} in A,
sum is
P
P
˜
(z,ai )∈A
(z,ai )
g
′
g′
P
P
v
i
v∈Z/l
m(v)ai
m(z,a ) (v)
=l
P
=
i (g
′
P
P
v
˜
(z,ai )∈A
m(v) ai
)
g′
P
v∈Z/l
m(v−z)ai
= l · trA (g
′
P
v
m(v)
=
) ∈ lT ′ .
i
Note that no m ∈ M is stabilized by (0, ai ) with ai 6= 1 : for m(z) = m(0,a ) (z) =
m(z)ai implies ai = 1. It follows that the stabilizers of the elements with
stabilizer different from {(0, 1)} must be cyclic of order l and different from
{(0, ai ) : 0 ≤ i ≤ l − 1} and therefore = h(1, aj )i for a unique j mod l.
˜
One now checks that for each j there is exactly one A-orbit
with stabilizer
j
jz
h(1, a )i and that it is represented by mj , mj (z) = a . Moreover, {(0, ai ) :
˜
0 ≤ i ≤ l − 1} is a transversal of the stabilizer of mj in A.
P
P P m(0,ai ) (z)
m(z)
˜
=
over the A-orbit
of mj is i g ′ z j
For each j, the sum of g ′ z
P ′ lai
P ′ Pz ajz ai
′l
= trA (g ) , accounting for that term
. If j = 0, this is i g
ig
P ′ A·a
ˆ
ˆ i
= lg ′ A , and summing over j 6= 0 gives
in the claim. If j 6= 0, it is i g
ˆ
ˆ
ˆ
ˆ
(l − 1)l · g ′ A ≡ −l · g ′ A mod lT ′ because l2 · g ′ A = l · trA (g ′ A ) ∈ lT ′ .
This finishes the proof of Lemma 6.
3 . Appendix
Proof of Proposition A :
We start from the formula χ(g) =
P
t
mH
G (g )=1
χ′ (g lt ) which appeared in the
r−1
justification of the definition in §1 (now with H = G′ ). It implies χ(g l ) =
P
r
′
−1
gt)l ) for r ≥ 1. Exploiting [3, Lemma 6] for the equality
mH (t−1 gt)=r χ ((t
G
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
127
=
˙ and [3, Proposition 3] for =
¨ we obtain
′
(ResH
G TrG τG (g))(χ )
P
r
G ′
= (TrG τG (g))(ind H χ ) + r≥1 Ψlr [(TrG τG (g))(ψlr−1 χ)]
P
r
Ψ
=trace(g
˙
| Vind G
′) +
r≥1 lr [trace(g | Vψlr−1 χ )]
Hχ
P
r
G ′
r−1
Ψ
χ)(g)g]
=(ind
¨
H χ )(g)g +
r≥1 lr [(ψl
r
P
r−1
G ′
gl
l
= (ind H χ )(g)g + r≥1 χ(g
) lr
lr
P
P
P
r
= mH (t−1 gt)=0 χ′ (t−1 gt)g + r≥1 mH (t−1 gt)=r χ′ ((t−1 gt)l ) glr
G
G
mH (t−1 gt)
−1 gt)
P ′ −1 lmH
gl G
G (t
) mH (t−1 gt) .
= t χ ((t gt)
l
G
On the other hand, if h ∈ H, the same calculation on the H-level gives
(TrH τH (h))(χ′ ) = trace(h | Vχ′ ) = χ′ (h)h ,
P
and therefore, with τH ( t
(t−1 gt)l
H
−1 gt)
mH
G (t
−1 gt)
lmG (t
H
) ∈ T (Q(H)) , we get
−1
P −1 lmG (t gt)
(TrH τH t (t gt)
)(χ′ )
H −1
lmG (t gt)
P
mH (t−1 gt)
mH (t−1 gt)
= t mH (t1−1 gt) χ′ ((t−1 gt)l G
)(t−1 gt)l G
l G
mH (t−1 gt)
l G
P ′ −1 lmH
(t−1 gt)
) g mH (t−1 gt)
= t χ ((t gt) G
l
G
via Qc (Γk′ ) ֒→ Qc (Γk ) (where k ′ is the fixed field of H).
Since these formulae agree for all characters χ′ of H (with open kernel) it
H −1
P (t−1 gt)lmG (t gt)
) by the uniqueness of this
follows that ResH
τ
(g)
=
τ
(
H
H −1
G G
t
lmG (t gt)
element, proving 1. of Proposition A.
For 2. we first consider the situation H ≤ G′ ≤ G with [G : G′ ] = l and show
H
G′
ResH
G = ResG′ ◦ ResG :
S
S
S
Write G = ˙ x xG′ , G′ = ˙ y yH , hence G = ˙ x,y xyH , and recall 6 from 1. that
′
ResG
G g
6 if
=
P
τG′ ( x x−1 gx) , g ∈ G′
.
, g∈
/ G′
τG′ (g l )
g∈
/ G′ , then we may use {t} = {g i : 0 ≤ i ≤ l − 1}
Documenta Mathematica 13 (2008) 117–129
128
J¨
urgen Ritter and Alfred Weiss
Then, accordingly,
′
G
ResH
G′ Res
GPτG (g) H
−1
gx)
x ResG′ τG′ (x
=
H
ResG′ τG′ (g l )
H
−1 −1
P P τ ((y −1 x−1 gxy)lmG′ (y x gxy) )/lmH
(y −1 x−1 gxy)
G′
x
y H
=
H
−1 l
P τ ((y −1 g l y)lmG′ (y g y) )/lmH
(y −1 gl y)
G′
H
y
( P
H
−1
m (t
gt)
H −1
−1
gt)l G
)/lmG (t gt)
t τH ((t
=
H
−1
l
m
(t
g t)
P
H
−1 l
−1 l l G′
g t)
)/l · lmG′ (t g t)
t τH ((t
H
P
m
−1
H −1
−1
=
gt)l G (t gt) /lmG (t gt) = ResH
G τG (g) ,
t τH ((t
l
H
using G′ ✁ G and mH
/ G′ .
G′ (x ) + 1 = mG (x) for x ∈
′′
H
G
′′
Induction on [G : H] now proves 2., ResH
G = ResG′′ ◦ ResG for H ≤ G ≤ G.
′′
′
′′
′
Indeed, if G 6= G, find G ≥ G with [G : G ] = l and use
′′
′′
′
G
H
G
G
ResH
G′′ ResG = ResG′′ (ResG′ ResG ) =
′′
′
′
G
G
H
G
H
= (ResH
G′′ ResG′ )ResG = ResG′ ResG = ResG .
Finally, for 3., proceed by induction on [G : H] and use 1. if H has index l in
G and 2. when the index is bigger, in which case there is a subgroup G′ of G
with H ≤ G , [G : G′ ] = l.
The proof of Proposition A is complete.
References
[1]
[2-6]
[7]
Oliver, R. and Taylor, L.R., Logarithmic descriptions of Whitehead groups and class groups for p-groups. Memoirs of the AMS
(1988), vol. 76, no. 392
Ritter, J. and Weiss, A. ,
2. Towards equivariant Iwasawa theory, II. Indag. Mathemat. 15
(2004), 549-572
3. · · · , III. Math. Ann. 336 (2006), 27-49
4. · · · , IV. Homology, Homotopy and Applications 7 (2005), 155171
5. Non-abelian pseudomeasures and congruences between abelian
Iwasawa L-functions. Pure and Applied Math. Quarterly 4 (2008),
1085-1106
6. Congruences between abelian pseudomeasures. To appear in
Math.Res. Lett. 15 (2008)
Wang, S., On the commutator group of a simple algebra.
Amer. J. Math. 72 (1950), 323-334
Documenta Mathematica 13 (2008) 117–129
Equivariant Iwasawa Theory: An Example
J¨
urgen Ritter
Institut f¨
ur Mathematik
Universit¨
at Augsburg
86135 Augsburg
Germany
Alfred Weiss
Department of Mathematics
University of Alberta
Edmonton, AB
Canada T6G 2G1
Documenta Mathematica 13 (2008) 117–129
129
130
Documenta Mathematica 13 (2008)