Equivariant Iwasawa Theory: An Example

Jürgen Ritter and Alfred Weiss^{1}

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 ℤ_{l}-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

\frac{ver(λ_{K_{ab}/k})}{λ_{K/k′}} ≡ 1 mod T′

1We acknowledge financial support provided by NSERC and the University of Augsburg.

Documenta Mathematica 13 (2008) 117–129

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 = ℤ_{l}[[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)

K_{1}(Λ_{∧}G)  ⟶^{𝕃}  T (Q_{∧}G)
Det ↓  \begin{matrix}Tr \\ ≃ \end{matrix} ↓
HOM(R_{l}G, (Λ_{∧}^{c}Γ_{k})^{×}) \underrightarrow{L} Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k})

is the logarithmic diagram defining the logarithmic pseudomeasure

t_{K/k} ∈ T (Q_{∧}G) by Tr(t_{K/k}) = L(L_{K/k})

where L_{K/k} = L_{K/k,S} ∈ HOM(R_{l}G, (Λ_{∧}^{c}Γ_{k})^{×}) is the Iwasawa L-function.

Theorem. With K/k and S as at the beginning and G = G(K/k) a pro-l group, t_{K/k} is integral (i.e., t_{K/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], L_{K/k} ∈ Det K_{1}(Λ_{∧}G) , which implies the conjecture (see [3, Theorem A]), up to its uniqueness assertion. However, SK_{1}(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 l^{2} 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

Res_{G}^{G′} : T (Q_{∧}G) → T (Q_{∧}G′)

and

Res_{G}^{G′} : Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k}) → Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′})

2R_{l}G is the ring of all (virtual) ℚ_{l}^{c}-characters of G with open kernel; Γ_{k} = G(k_{∞}/k) ; Λ_{∧}^{c}Γ_{k} = ℤ_{l}^{c} ⊗_{ℤ_{l}} Λ_{∧}Γ_{k} with ℤ_{l}^{c} the ring of integers in a fixed algebraic closure ℚ_{l}^{c} of ℚ_{l}

Documenta Mathematica 13 (2008) 117–129

making the diagram

K_{1}(Λ_{∧}G) ⟶^{𝕃}  T (Q_{∧}G)  ⟶^{Tr} Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k})
res _{G}^{G′} ↓  Res_{G}^{G′} ↓  Res_{G}^{G′} ↓
K_{1}(Λ_{∧}G′) ⟶^{𝕃′}  T (Q_{∧}G′)  ⟶^{Tr′} Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′})

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 Res_{G}^{G′} by the “natural” restriction map,

(res _{G}^{G′}f)(χ′) = f(ind _{G′}^{G}χ′) , f ∈ Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k}) , χ′ ∈ R_{l}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 : Λ_{∧}^{c}Γ_{k} → Λ_{∧}^{c}Γ_{k} is the map induced by Ψ(γ) = γ^{l} for γ ∈ Γ_{k} (compare [5,§1]) and that ψ_{l} is the l^{th} Adams operation on R_{l}(−).

Definition. Res_{G}^{G′} : Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k}) → Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′}) sends f to

Res_{G}^{G′}f = [ χ′ ↦ f(ind _{G′}^{G}χ′) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} (f(ψ_{l}^{r−1}χ)) ] ,

where χ = ψ_{l}(ind _{G′}^{G}χ′) − ind _{G′}^{G}(ψ_{l}χ′) .

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

m(g) = min{r ≥ 0 : g^{l^{r}} ∈ G′} for g ∈ G .

Then

ind _{G′}^{G}χ′(g) = ∑_{t} χ̇′(g^{t}) = ∑_{{t:m(g^{t})=0}} χ′(g^{t}) ,

if, as usual, χ̇′ coincides with χ′ on G′ and vanishes on G \ G′. Hence,

\begin{matrix}χ(g) = (ind _{G′}^{G}χ′)(g^{l}) − ind _{G′}^{G}(ψ_{l}χ′)(g) \\ = ∑_{m(g^{lt})=0} χ′(g^{lt}) − ∑_{m(g^{t})=0} χ′(g^{lt}) = ∑_{m(g^{t})=1} χ′(g^{lt}) . \end{matrix}

If r_{0} is such that G^{l^{r_{0}}} ⊂ G′, then ψ_{l}^{r_{0}−1}χ = 0 and ∑_{r≥1} = ∑_{r=1}^{r_{0}−2} , because the sum ∑_{m(g^{t})=1} is empty when g ∈ G^{l^{r_{0}−1}} .

3For finite G, this Res_{G}^{G′}, making the left square commute, appears already in [1, p.14].

For us the properties (HD),(TD) of Res_{G}^{G′} shown in §1 are equally necessary for the proof of the Theorem above.

Documenta Mathematica 13 (2008) 117–129

It remains to show that Res_{G}^{G′}f ∈ Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′}) , i.e., Res_{G}^{G′}f is a Galois stable homomorphism, compatible with W-twists (see [5,§1]), and taking values in Q_{∧}^{c}Γ_{k′}. 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 Γ_{k′}, and write ρ′ = res _{G}^{G′}ρ with ρ inflated from Γ_{k} to G. Then

f(ind _{G′}^{G}(ρ′χ′)) = f(ρ · ind _{G′}^{G}χ′) = ρ^{♯}(f(ind _{G′}^{G}χ′)) = (ρ′)^{♯}(f(ind _{G′}^{G}χ′))

as f(ind _{G′}^{G}χ′) ∈ Q_{∧}^{c}Γ_{k′}. Moreover, since ψ_{l} is multiplicative,

ψ_{l}(ind _{G′}^{G}(ρ′χ′)) − ind _{G′}^{G}(ψ_{l}(ρ′χ′)) = ψ_{l}(ρ · ind _{G′}^{G}χ′) − ind _{G′}^{G}((ρ′)^{l} · ψ_{l}χ′) = ρ^{l} · χ

and thus

\begin{matrix}\frac{Ψ^{r}}{l^{r}} (f(ψ_{l}^{r−1}(ρ^{l} · χ)))) = \frac{Ψ^{r}}{l^{r}} f(ρ^{l^{r}} · ψ_{l}^{r−1}χ) = \\ \frac{Ψ^{r}}{l^{r}} ((ρ^{l^{r}})^{♯}(f(ψ_{l}^{r−1}χ))) = ρ^{♯}( \frac{Ψ^{r}}{l^{r}} f(ψ_{l}^{r−1}χ)) = (ρ′)^{♯}( \frac{Ψ^{r}}{l^{r}} f(ψ_{l}^{r−1}χ)) . \end{matrix}

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) \begin{matrix}HOM(R_{l}G, (Λ_{∧}^{c}Γ_{k})^{×}) & \underrightarrow{L} & Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k}) \\ res _{G}^{G′} ↓ & & Res_{G}^{G′} ↓ \\ HOM(R_{l}G′, (Λ_{∧}^{c}Γ_{k′})^{×}) & \underrightarrow{L′} & Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′}) . \end{matrix}

Indeed, for f ∈ HOM(R_{l}G, (Λ_{∧}^{c}Γ_{k})^{×}) we get

(Res_{G}^{G′}Lf)(χ′) = (Lf)(ind _{G′}^{G}χ′) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} [(Lf)(ψ_{l}^{r−1}χ)]

˙=(Lf)(ind _{G′}^{G}χ′) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} [log(f(ψ_{l}^{r−1}χ)) − \frac{Ψ}{l} log(f(ψ_{l}^{r}χ))]

= (Lf)(ind _{G′}^{G}χ′) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} log(f(ψ_{l}^{r−1}χ)) − ∑_{r≥2} \frac{Ψ^{r}}{l^{r}} log(f(ψ_{l}^{r−1}χ))

= (Lf)(ind _{G′}^{G}χ′) + \frac{Ψ}{l} log(f(χ)) = \frac{1}{l} log \frac{f(ind χ′)^{l}}{Ψ(f(ψ_{l}ind χ′))} + \frac{Ψ}{l} log \frac{f(ψ_{l}ind χ′)}{f(ind ψ_{l}χ′)}

= \frac{1}{l} log \frac{f(ind χ′)^{l}·Ψf(ψ_{l}ind χ′)}{Ψf(ψ_{l}ind χ′)·Ψf(ind ψ_{l}χ′)} = \frac{1}{l} log \frac{f(ind χ′)^{l}}{Ψf(ind ψ_{l}χ′)}

= (L′res _{G}^{G′}f)(χ′) .

The dotted equality sign, ˙=, is due to the congruence \frac{f(χ)^{l}}{Ψf(ψ_{l}χ)} ≡ 1 mod lΛ_{∧}^{c}Γ_{k} (see [5,§1]) and to χ(1) = 0, so (ψ_{l}^{r−1}χ)(1) = 0 for every r. In fact, with χ̃ =^{def} ψ_{l}^{r−1}χ, we have

f(χ̃)^{l} ≡ Ψf(ψ_{l}χ̃) mod lΛ_{∧}^{c}Γ_{k} ⟹
f(χ̃)^{l^{s}} ≡ Ψ^{s}f(ψ_{l}^{s}χ̃) ≡ Ψ^{s}f(χ̃(1)1) = 1 mod lΛ_{∧}^{c}Γ_{k}

for big enough s. Thus (Lf)(χ̃) = log(f(χ̃)) − \frac{Ψ}{l} log(f(ψ_{l}χ̃)) as ‘log’ converges on an element a power of which is ≡ 1 mod lΛ_{∧}^{c}Γ_{k}.

The proof of Lemma 1 is complete.

By means of the trace isomorphism Tr : T (−) → Hom^{∗}(−) we next transport Res_{G}^{G′} to Res_{G}^{G′} : T (Q_{∧}G) → T (Q_{∧}G′), i.e., the diagram

(TD)

T (Q_{∧}G)  ⟶^{Tr} Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k})
Res_{G}^{G′} ↓  Res_{G}^{G′} ↓
T (Q_{∧}G′)  ⟶^{Tr′} Hom^{∗}(R_{l}G′, Q_{∧}^{c}Γ_{k′})

commutes.

Lemma 2.

K_{1}(Λ_{∧}G) →^{𝕃}  T (Q_{∧}G) res _{G}^{G′} ↓  Res_{G}^{G′} ↓ K_{1}(Λ_{∧}G′) →^{𝕃′}  T (Q_{∧}G′)  commutes and Res_{G}^{G′}t_{K/k} = t_{K/k′} .

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′}L_{K/k} = L_{K/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′ = ⟨a⟩ 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

Res_{G}^{G′}(τg) = \{ \begin{matrix}∑_{i=0}^{l−1} g^{a^{i}} & if g ∈ G′ \\ g^{l} & if g ∉ G′ . \end{matrix}

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 m_{G}^{H}(g) = min{r ≥ 0 : g^{l^{r}} ∈ H} ^{5}, and let t run through a set of left representatives of H in G, i.e., G = ˙⋃tH . Then

1. Res_{G}^{H}τ_{G}(g) = ∑_{t} τ_{H}((t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}})/l^{m_{G}^{H}(t^{−1}gt)} ,

2. Res is transitive ,

3. Res_{G}^{H} is integral, i.e., Res_{G}^{H}(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)\overline{g} with \overline{g} denoting the image of g ∈ G′ in Γ_{k′} (see [5,§1]) :

4whence τ′ : Λ_{∧}G′ → T (Λ_{∧}G′) is the identity map

5so m_{G}^{H}(g) is the m(g) defined earlier with H = G′

Documenta Mathematica 13 (2008) 117–129

1. (Tr′Res_{G}^{G′}(τg))(χ′) = Res_{G}^{G′}(Tr(τg))(χ′) = Tr(τg)(ind _{G′}^{G}χ′) + \frac{Ψ}{l} Tr(τg)(χ) since G^{l} ⊂ G′. Now, if g ∈ G′, Tr(τg)(ind _{G′}^{G}χ′) = ∑_{i=0}^{l−1} χ′(g^{a^{i}})\overline{g} and χ(g) = 0. On the other hand, if g ∉ G′, Tr(τg)(ind _{G′}^{G}χ′) = 0 and \frac{Ψ}{l} Tr(τg)(χ) = \frac{1}{l} ind _{G′}^{G}χ′(g^{l})\overline{g}^{l} = χ′(g^{l})\overline{g}^{l} since we may choose a = g mod G′.

2. Tr′(∑_{i=0}^{l−1} g^{a^{i}})(χ′) = ∑_{i=0}^{l−1} χ′(g^{a^{i}})\overline{g}, since g^{a^{i}} and g have the same image in Γ_{k} and so in Γ_{k′}. On the other hand, Tr′(g^{l})(χ′) = χ′(g^{l})\overline{g^{l}}.

The lemma is established. We note that if Γ (≃ ℤ_{l}) 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 as ∑_{g} β_{g}τ(g) with β_{g} ∈ 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

Res_{G}^{G′}(β_{g}τ(g)) = \{ \begin{matrix}∑_{i=0}^{l−1} β_{g}g^{a^{i}} & if g ∈ G′ \\ Ψ(β_{g})g^{l} & if g ∉ G′ . \end{matrix}

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′ = ⟨a⟩, and we set Â = 1 + a + ⋯ + a^{l−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 defl_{G}^{G^{ab}}x = defl_{G}^{G^{ab}}t_{K/k} and Res_{G}^{G′}x = Res_{G}^{G′}t_{K/k}. Then t_{K/k} ∈ T (Λ_{∧}G).

Denoting by K_{ab} the fixed field of the finite group [G, G], we first observe, because of [5, Lemma 2.1] and Lemma 2, that defl_{G}^{G^{ab}}t_{K/k} = t_{K_{ab}/k} and Res_{G}^{G′}t_{K/k} = t_{K/k′} are integral : indeed, a logarithmic pseudomeasure is integral whenever the group is abelian. From [4, Proposition 9] we obtain a power l^{n} of l such that l^{n}t_{K/k} ∈ T (Λ_{∧}G). Consider the element x̃ = l^{n}(x − t_{K/k}) ∈ T (Λ_{∧}G). It satisfies defl_{G}^{G^{ab}} x̃ = 0 = Res_{G}^{G′} x̃. We are going to prove x̃ = 0 which implies x = t_{K/k} because Hom^{∗}(R_{l}G, Q_{∧}^{c}Γ_{k}), and so T (Q_{∧}G), is torsionfree; whence the the lemma will be verified. The proof of x̃ = 0 employs the commutative diagram shown in the proof of [5, Proposition 2.2] :

1 + 𝔞_{∧} ↣ (Λ_{∧}G)^{×} ↠^{defl_{G}^{G^{ab}}} (Λ_{∧}G^{ab})^{×}
↓ ˇ  𝕃 ↓  𝕃^{ab} ↓
τ(𝔞_{∧}) ↣ T (Λ_{∧}G) ↠^{defl_{G}^{G^{ab}}} Λ_{∧}G^{ab} ,

in which 𝕃 is extended to (Λ_{∧}G)^{×} by means of the canonical surjection (Λ_{∧}G)^{×} ↠ K_{1}(Λ_{∧}G) and 𝔞_{∧} = ker(Λ_{∧}G → Λ_{∧}G^{ab}). The diagram yields a v ∈ (Λ_{∧}G)^{×} with 𝕃(v) = x̃, simply because defl_{G}^{G^{ab}} x̃ = 0. Combining diagrams (HD), (LD) and (TD), we arrive at

L′(res_{G}^{G′}(Det v)) = Res_{G}^{G′}(L(Det v)) = Res_{G}^{G′}(Tr 𝕃(v)) = Tr′(Res_{G}^{G′} x̃) = 0

and, with res_{G}^{G′} replaced by defl_{G}^{G^{ab}}, at

L^{ab}(defl_{G}^{G^{ab}}(Det v)) = defl_{G}^{G^{ab}}(L(Det v)) =
= defl_{G}^{G^{ab}}(Tr 𝕃(v)) = Tr^{ab}(defl_{G}^{G^{ab}} x̃) = 0 ,

since L and Tr commute with deflation. The first displayed formula in [3, p.46] now implies that res_{G}^{G′}(Det v) and defl_{G}^{G^{ab}}(Det v) are torsion elements in HOM(R_{l}G′, (Λ_{∧}^{c}Γ_{k′})^{×}) and HOM(R_{l}(G^{ab}), (Λ_{∧}^{c}Γ_{k})^{×}), 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(R_{l}G, (Λ_{∧}^{c}Γ_{k})^{×}). Consequently, for some natural number m, (Det v)^{l^{m}} = 1, so l^{m}L(Det v) = 0 = l^{m}Tr(𝕃v) = Tr(l^{m}x̃) , and x̃ = 0 follows, as has been claimed.

We now introduce the commutative diagram

τ(𝔞_{∧})  ↣  T (Λ_{∧}G)  ↠  Λ_{∧}G^{ab} = T (Λ_{∧}G^{ab})
Res ↓  Res_{G}^{G′} ↓  Res ↓
𝔟′_{∧} = τ′(𝔟′_{∧}) ↣Λ_{∧}G′ = T (Λ_{∧}G′) ↠ 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_{∧})  ↣  T (Λ_{∧}G)  ↠  Λ_{∧}G^{ab}
Res ↓  Res_{G}^{G′} ↓  Res ↓
𝔟′_{∧}^{A}  ↣  (Λ_{∧}G′)^{A}  → (Λ_{∧}(G′/[G, G]))^{A}
↓ ˇ  ↓ ˇ  ↓ ˇ
Ĥ^{0}(A, 𝔟′_{∧}) → Ĥ^{0}(A, Λ_{∧}G′) → Ĥ^{0}(A, Λ_{∧}(G′/[G, G]))

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 Res_{G}^{G′}τ(g(c−1)), using Lemma 3 :

1. if g ∈ G′, Res_{G}^{G′}τ(g(c − 1)) = ∑_{i=0}^{l−1}((gc)^{a^{i}} − g^{a^{i}}) = ∑_{i=0}^{l−1} (g(c − 1))^{a^{i}} ∈

tr_{A}𝔟′_{∧},

2. if g ∉ G′, Res_{G}^{G′}τ(g(c−1)) = Res_{G}^{G′}(τ(gc)−τ(g)) = (gc)^{l}−g^{l} = g^{l}cÂ−g^{l} = 0 , since

(⋆)  [G, G]Â = 1

by [G, G] ˙=[G, G′] and [G, G′]Â = ((G′)^{a−1})Â = 1 as (a − 1)Â = 0. Here, the dotted equality sign, ˙=, results from the equation

\begin{matrix}[bg′_{1}, b^{i}g′_{2}] = (g′_{1})^{−1}b^{−1}(g′_{2})^{−1}b^{−i}bg′_{1}b^{i}g′_{2} = (g′_{1})^{−1}(g′_{2}^{−1})^{b}(g′_{1})^{b^{i}}g′_{2} = \\ ((g′_{1})^{−1}(g′_{1})^{b^{i}})((g′_{2}^{−1})^{b}g′_{2}) ∈ [G′, G] · [G, G′] ≤ [G, G′] \end{matrix}

for g′_{1}, g′_{2} ∈ G′ and b ∈ G \ G′, because G′ is abelian and normal in G.

Thus, Res_{G}^{G′}τ(𝔞_{∧}) = tr_{A}𝔟′_{∧}, which proves the first claim of the lemma. The second claim follows from Ĥ^{−1}(A, Λ_{∧}(G′/[G, G])) = 0 and this in turn from the trivial action of A on G′/[G, G] and the torsion freeness of Λ_{∧}(G′/[G, G]). Lemma 5 is established.

As seen in diagram (D), there is an element x_{1} ∈ T (Λ_{∧}G) with defl_{G}^{G^{ab}}x_{1} = t_{K_{ab}/k}. We define x′_{1} ∈ Λ_{∧}G′ by Res_{G}^{G′}x_{1} = t_{K/k′} + x′_{1}. Because of [5, Lemma 3.1], x′_{1} is fixed by A. We want to change x_{1} 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} ∈ Res_{G}^{G′}(τ(𝔞_{∧})) and so, because of Lemma 5, if x′_{1} is in T′ =^{def} tr_{A}(Λ_{∧}G′) , the A-trace ideal of the A-action on Λ_{∧}G′.

Proposition. There exists an element x_{1} of T (Λ_{∧}G) with defl_{G}^{G^{ab}}x_{1} = t_{K_{ab}/k} and x′_{1} ∈ T′.

This is seen as follows. From [5,§1] we recall the existence of pseudomeasures λ_{K_{ab}/k}, λ_{K/k′} in K_{1}(Λ_{∧}G^{ab}) and K_{1}(Λ_{∧}G′), respectively, satisfying Det λ_{K_{ab}/k} = L_{K_{ab}/k}, Det λ_{K/k′} = L_{K/k′} (so 𝕃^{ab}(λ_{K_{ab}/k}) = t_{K_{ab}/k}, , 𝕃′(λ_{K/k′}) = t_{K/k′}). From [5, 2. of Proposition 3.2] and [6, Theorem] we know that

\frac{ver(λ_{K_{ab}/k})}{λ_{K/k′}} ≡ 1 mod T′

where ‘ver’ is the map induced from the transfer homomorphism G^{ab} → G′.

Let y ∈ (Λ_{∧}G)^{×} have defl_{G}^{G^{ab}}y = λ_{K_{ab}/k} and set res_{G}^{G′}y = λ_{K/k′} · y′. Then

y′ = \frac{res_{G}^{G′}y}{λ_{K/k′}} ≡ \frac{ver(λ_{K_{ab}/k})}{λ_{K/k′}} ≡ 1 mod T′

(see the proof of [5, Proposition 3.2]). Moreover, y′ ∈ 1 + 𝔟′_{∧}. Now, x_{1} =^{def} 𝕃(y) has Res_{G}^{G′}x_{1} = Res_{G}^{G′}𝕃(y) = t_{K/k′} + x′_{1} with x′_{1} =^{def} 𝕃′(y′), and x′_{1} ∈ 𝔟′_{∧} because of the commutativity of

1 + 𝔟′_{∧} ↣ (Λ_{∧}G′)^{×} ↠ (Λ_{∧}(G′/[G, G]))^{×}
𝕃′ ↓ 𝕃′ ↓ \overline{𝕃′} ↓
𝔟′_{∧} ↣ Λ_{∧}G′ ↠ Λ_{∧}(G′/[G, G]) .

Hence, the proposition (and therefore the theorem) will be proved, if

x′_{1} = 𝕃′(y′) ∈ T′ .

However, Lemma 5 gives

y′ ∈ (1 + 𝔟′_{∧}^{A}) ∩ (1 + T′) = 1 + (𝔟′_{∧}^{A} ∩ T′) = 1 + tr_{A}𝔟′_{∧}

and as 𝕃′(y′) = \frac{1}{l} log \frac{y′^{l}}{Ψ(y′)} (compare [3, p.39]), we see that

(1)  𝕃′(y′) ∈ T′ if \frac{y′^{l}}{Ψ(y′)} ≡ 1 mod lT′ .

So it suffices to show this last congruence. Write y′ = 1 + tr_{A}β′ with β′ ∈ 𝔟′_{∧}. Since (1 + tr_{A}β′)^{l} ≡ 1 + (tr_{A}β′)^{l} mod lT′, the congruence in (1) is equivalent to

(2)  (tr_{A}β′)^{l} ≡ Ψ(tr_{A}β′) mod lT′.

On picking a central subgroup Γ (≃ ℤ_{l}) of G and writing β′ = ∑_{g′,c} β_{g′,c} g′(c− 1) with elements β_{g′,c} ∈ Λ_{∧}Γ, g′ ∈ G′, c ∈ [G, G], we obtain

(a)

(tr_{A}β′)^{l} = ( ∑_{g′,c} β_{g′,c}tr_{A}(g′(c − 1)))^{l}

≡ ∑_{g′,c}(β_{g′,c})^{l}(tr_{A}(g′(c − 1)))^{l}

≡ ∑_{g′,c} Ψ(β_{g′,c})((tr_{A}(g′c))^{l} − (tr_{A}g′)^{l}) mod lT′

and

(b)  Ψ(tr_{A}β′) = ∑_{g′,c} Ψ(β_{g′,c})(tr_{A}((g′c)^{l}) − tr_{A}(g′^{l}))

as Ψ and tr_{A} commute. Thus congruence (2) will result from Lemma 6 below, since then subtracting (b) from (a) yields the sum

∑_{g′,c} Ψ(β_{g′,c})((tr_{A}(g′c))^{l} − tr_{A}((g′c)^{l}) − (tr_{A}g′)^{l} + tr_{A}(g′^{l})) ≡

∑_{g′,c} Ψ(β_{g′,c})( − l(g′c)Â + lg′Â) ≡

∑_{g′,c}(−l)Ψ(β_{g′,c})g′Â(cÂ − 1) ≡ 0 mod lT′ ,

by (⋆) of the proof of Lemma 5.

Lemma 6. (tr_{A}g′)^{l} − tr_{A}(g′^{l}) ≡ −lg′Â mod lT′ for g′ ∈ G′.

Proof. Set Ã = ℤ/l × A and make M = Maps(ℤ/l, A) into an Ã-set by defining m^{(z,a^{i})}(x) = m(x − z) · a^{i}. Then

(tr_{A}g′)^{l} = (∑_{i=0}^{l−1} g′^{a^{i}})^{l} = ∑_{m∈M} ∏_{z∈ℤ/l} g′^{m(z)} = ∑_{m∈M} g′^{∑_{z∈ℤ/l} m(z)}

with ∑_{z} m(z) read in ℤ[A].

We compute the subsums of ∑_{m} in which m is constrained to an Ã-orbit.

If m ∈ M has stabilizer {(0, 1)} in Ã, then the Ã-orbit sum is

\begin{matrix}∑_{(z,a^{i})∈Ã} g′^{∑_{v∈ℤ/l} m^{(z,a^{i})}(v)} = ∑_{(z,a^{i})∈Ã} g′^{∑_{v∈ℤ/l} m(v−z)a^{i}} = \\ ∑_{(z,a^{i})} g′^{∑_{v} m(v)a^{i}} = l ∑_{i}(g′^{∑_{v} m(v)})^{a^{i}} = l · tr_{A}(g′^{∑_{v} m(v)}) ∈ lT′ . \end{matrix}

Note that no m ∈ M is stabilized by (0, a^{i}) with a^{i} ≠ 1 : for m(z) = m^{(0,a^{i})}(z) = m(z)a^{i} implies a^{i} = 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, a^{i}) : 0 ≤ i ≤ l − 1} and therefore = ⟨(1, a^{j})⟩ for a unique j mod l.

One now checks that for each j there is exactly one Ã-orbit with stabilizer ⟨(1, a^{j})⟩ and that it is represented by m_{j} , m_{j}(z) = a^{jz}. Moreover, {(0, a^{i}) : 0 ≤ i ≤ l − 1} is a transversal of the stabilizer of m_{j} in Ã.

For each j, the sum of g′^{∑_{z} m(z)} over the Ã-orbit of m_{j} is ∑_{i} g′^{∑_{z} m_{j}^{(0,a^{i})}(z)} = ∑_{i} g′^{∑_{z} a^{jz}a^{i}} . If j = 0, this is ∑_{i} g′^{la^{i}} = tr_{A}(g′^{l}) , accounting for that term

in the claim. If j ≠ 0, it is ∑_{i} g′^{Â·a^{i}} = lg′Â, and summing over j ≠ 0 gives

(l − 1)l · g′Â ≡ −l · g′Â mod lT′ because l^{2} · g′Â = l · tr_{A}(g′Â) ∈ lT′.

This finishes the proof of Lemma 6.

3 . Appendix

Proof of Proposition A :

We start from the formula χ(g) = ∑_{m_{G}^{H}(g^{t})=1} χ′(g^{lt}) which appeared in the

justification of the definition in §1 (now with H = G′). It implies χ(g^{l^{r−1}}) = ∑_{m_{G}^{H}(t^{−1}gt)=r} χ′((t^{−1}gt)^{l^{r}}) for r ≥ 1. Exploiting [3, Lemma 6] for the equality

Equivariant Iwasawa Theory: An Example 127

˙= and [3, Proposition 3] for ¨= we obtain (Res_{G}^{H}Tr_{G}τ_{G}(g))(χ′)

= (Tr_{G}τ_{G}(g))(ind _{H}^{G}χ′) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} [(Tr_{G}τ_{G}(g))(ψ_{l}^{r−1}χ)]
˙=trace(g | 𝔙_{ind _{H}^{G}χ′}) + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} [trace(g | 𝔙_{ψ_{l}^{r−1}χ})]
¨=(ind _{H}^{G}χ′)(g)\overline{g} + ∑_{r≥1} \frac{Ψ^{r}}{l^{r}} [(ψ_{l}^{r−1}χ)(g)\overline{g}]
= (ind _{H}^{G}χ′)(g)\overline{g} + ∑_{r≥1} χ(g^{l^{r−1}}) \frac{\overline{g}^{l^{r}}}{l^{r}}
= ∑_{m_{G}^{H}(t^{−1}gt)=0} χ′(t^{−1}gt)\overline{g} + ∑_{r≥1} ∑_{m_{G}^{H}(t^{−1}gt)=r} χ′((t^{−1}gt)^{l^{r}}) \frac{\overline{g}^{l^{r}}}{l^{r}}
= ∑_{t} χ′((t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}) \frac{\overline{g}^{l^{m^{H}_{G} (t^{−1}gt)}}}{l^{m^{H}_{G} (t^{−1}gt)}} .

On the other hand, if h ∈ H, the same calculation on the H-level gives (Tr_{H}τ_{H}(h))(χ′) = trace(h | 𝔙_{χ′}) = χ′(h)\overline{h} ,

and therefore, with τ_{H}(∑_{t} \frac{(t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}}{l^{m^{H}_{G} (t^{−1}gt)}} ) ∈ T (Q(H)) , we get

(Tr_{H}τ_{H} ∑_{t} \frac{(t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}}{l^{m^{H}_{G} (t^{−1}gt)}} )(χ′)
= ∑_{t} \frac{1}{l^{m^{H}_{G} (t^{−1}gt)}} χ′((t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}})\overline{(t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}}
= ∑_{t} χ′((t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}) \frac{\overline{g}^{l^{m^{H}_{G} (t^{−1}gt)}}}{l^{m^{H}_{G} (t^{−1}gt)}}

via Q^{c}(Γ_{k′}) ↪ Q^{c}(Γ_{k}) (where k′ is the fixed field of H). Since these formulae agree for all characters χ′ of H (with open kernel) it follows that Res_{G}^{H}τ_{G}(g) = τ_{H}(∑_{t} \frac{(t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}}}{l^{m^{H}_{G} (t^{−1}gt)}} ) by the uniqueness of this element, proving 1. of Proposition A. For 2. we first consider the situation H ≤ G′ ≤ G with [G : G′] = l and show Res_{G}^{H} = Res_{G′}^{H} ∘ Res_{G}^{G′} : Write G = ˙⋃_{x}xG′ , G′ = ˙⋃_{y}yH, hence G = ˙⋃_{x,y}xyH, and recall^{6} from 1. that

Res_{G}^{G′}g = \{ \begin{matrix}τ_{G′}(∑_{x} x^{−1}gx) & , g ∈ G′ \\ τ_{G′}(g^{l}) & , g ∉ G′ \end{matrix} .

6if g ∉ G′, then we may use {t} = {g^{i} : 0 ≤ i ≤ l − 1}

Documenta Mathematica 13 (2008) 117–129

Then, accordingly,

Res_{G′}^{H}Res_{G}^{G′}τ_{G}(g)
= \{ \begin{matrix}∑_{x} Res_{G′}^{H}τ_{G′}(x^{−1}gx) \\ Res_{G′}^{H}τ_{G′}(g^{l}) \end{matrix}

= \{ \begin{matrix}∑_{x} ∑_{y} τ_{H}((y^{−1}x^{−1}gxy)^{l^{m{H \atop G′} (y^{−1}x^{−1}gxy)}})/l^{m_{G′}^{H} (y^{−1}x^{−1}gxy)} \\ ∑_{y} τ_{H}((y^{−1}g^{l}y)^{l^{m{H \atop G′} (y^{−1}g^{l}y)}})/l^{m_{G′}^{H} (y^{−1}g^{l}y)} \end{matrix}

= \{ \begin{matrix}∑_{t} τ_{H}((t^{−1}gt)^{l^{m^{H}_{G} (t^{−1}gt)}})/l^{m_{G}^{H}(t^{−1}gt)} \\ ∑_{t} τ_{H}((t^{−1}g^{l}t)^{l^{m{H \atop G′} (t^{−1}g^{l}t)}})/l · l^{m_{G′}^{H} (t^{−1}g^{l}t)} \end{matrix}
= ∑_{t} τ_{H}((t^{−1}gt)^{l^{m^{H}_{G}} (t^{−1}gt)}/l^{m_{G}^{H}(t^{−1}gt)} = Res_{G}^{H}τ_{G}(g) ,

using G′ ⊲ G and m_{G′}^{H}(x^{l}) + 1 = m_{G}^{H}(x) for x ∉ G′.

Induction on [G : H] now proves 2., Res_{G}^{H} = Res_{G′′}^{H} ∘ Res_{G}^{G′′} for H ≤ G′′ ≤ G. Indeed, if G′′ ≠ G, find G′ ≥ G′′ with [G : G′] = l and use

Res_{G′′}^{H}Res_{G}^{G′′} = Res_{G′′}^{H}(Res_{G′}^{G′′}Res_{G}^{G′}) =
= (Res_{G′′}^{H}Res_{G′}^{G′′})Res_{G}^{G′} = Res_{G′}^{H}Res_{G}^{G′} = Res_{G}^{H} .

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] 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

[2-6] 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), 155-171

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)

[7] Wang, S., On the commutator group of a simple algebra. Amer. J. Math. 72 (1950), 323-334 Jürgen Ritter Institut für Mathematik Universität Augsburg 86135 Augsburg Germany Alfred Weiss Department of Mathematics University of Alberta Edmonton, AB Canada T6G 2G1 130

Documenta Mathematica 13 (2008)