ON THE WORK OF RITTER AND WEISS IN COMPARISON WITH KAKDE’S APPROACH

OTMAR VENJAKOB

1. Introduction

Almost simultaneously Ritter and Weiss [21] on the one hand and Kakde [8] on the other hand gave a proof of the non-commutative Iwasawa main conjecture over totally real fields for the Tate motive under the assumption that a certain μ-invariant vanishes as has been conjectured also by Iwasawa. Actually the article of Ritter and Weiss appeared slightly earlier on the arXive and deals with the case of onedimensional p-adic Lie-extensions, which can be combined with Burns’ well-known insight in [2, thm. 2.1] (this crucial idea has been presented by Burns already during a seminar at the University of Kyoto in early 2006) based on Fukaya and Kato’s result [4, prop. 1.5.1] to obtain the conjecture for general p-adic (admissible) Lie-extensions. Kakde’s paper, which got its final version practically during this workshop, contains - using the same result of Fukaya and Kato - directly a full proof in the general situation. Various special cases have been known by the work of (in alphabetical order) Hara [5], Kakde [6], Kato [9] as well as Ritter and Weiss [18].

Since in both approaches the reduction steps from the general case of one-dimensional extensions to the l-elementary extensions (in the language of Ritter and Weiss) or to essential pro-p-extensions (in the language of Kakde) are based on the same principles (even though at different places: for Ritter and Weiss with respect to the Hom-description while in Kakde’s case for the K-theory) and since again the generalisation from the pure pro-p case to this slightly more general case follows certain standard techniques, see [24], we restrict in this survey from the very beginning to the one-dimensional pro-p-case. In these notes we shall use the same notation as in [22], but for the convenience of the reader we have collected the crucial notation from [21] in a glossary below comparing it with Kakde’s and our notation, respectively.

Since during the workshop and hence in this volume Kakde’s approach has been discussed in great detail and we may hence assume greater familiarity with his techniques, many comments in this comparison will be made from the perspective of Kade’s point of view. So, we want to stress that this certainly does not reflect the historical development as Kakde has probably been influenced by a couple of ideas from Ritter and Weiss. E.g. the analytic techniques for the actual proof that the abelian p-adic L-functions satisfy the required conditions in order to induce the “non-commutative” p-adic L-function has been applied first by Ritter and Weiss as well as by Kato (in an unpublished preprint about Heisenberg type p-adic Lie group extensions). For that reason we won’t discuss this analytic part at all in this comparison because again the methods are essentially the same, see [21, §3] and [8, §6], respectively.

One evident difference between the two works consists of the way of presentation: While Kakde, who also had partial results in previous publications, delivers an almost self-contained account of a full proof of the main conjecture (even dealing with arbitrary admissible p-adic Lie extensions) in one ingenious paper, the impressive work of Ritter and Weiss is spread over at least 10 articles [12, 13, 14, 16, 15, 19, 20, 17, 18, 21], viz naturally in the way as their theory has been developed over the recent decade. The last article, which contains the general result, is rather an instruction how to modify and extend the proofs of earlier results (in less general cases) in previous publications combined with an extensive discussion of the new Möbius-Wall-congruence in order to complete the proof in the general case than a self-contained proof. A general outline of the overall strategy of the proof is missing and the reader is forced even to collect the notation from all the other articles. This is somewhat unfortunate as otherwise the strategy of their nice proof could have been much more easily accessible for the reader.^{1}

Another obvious difference is the fact that while Kakde describes K_{1}(Λ(G)) modulo SK_{1}(Λ(G)) in terms of certain relations (some call them congruences) among elements in the units of Iwasawa algebras for certain abelian subquotients of G Ritter and Weiss are only interested whether the particular system of abelian pseudomeassures is in the image of K_{1}. But in the last section of this survey we try to clarify what can be proved by the methods of Ritter and Weiss towards describing the image of K_{1} in the product of the K_{1}’s of appropriate subquotients of G in the manner of Kakde.

I am very grateful to Cornelius Greither and Mahesh Kakde for reading an earlier version and suggesting a couple of improvements.

2. Glossary

In the following the notation in the different columns usually denote the same object, but sometimes it only indicates that they are just closely related.

1In order to remedy this Ritter and Weiss are apparently thinking about writing a lecture note volume about their approach.

Kakde  Ritter/Weiss

p ≠ 2  l ≠ 2
F_{∞}/F  K/k
G = G(F_{∞}/F)  G = G(K/k)
F^{cyc}  k_{∞}
Γ  Γ_{k}
H = G(F_{∞}/F^{cyc})  H
A := A(G) := Λ(G)_{S} = Λ(G)_{(p)}  Λ_{•}(G)
B := B(G) = Λ(G)_{S}^{}Λ_{∧}(G)
Λ(G)_{S^{∗}} = A(G)[\frac{1}{p}]  QG
O/ℤ_{p} unramified, (Λ = Λ_{O})  𝔒
Σ ⊇ Σ_{ram}(F_{∞}/F) finite set of ; primes of F  S ⊇ S_{ram}(K/k) ∪ S_{∞}  finite set of ; primes of k
K_{0}(Λ(G), Λ(G)_{S}) = K_{0}(H_{S}) = K_{0}(C_{S})  K_{0}T Λ(G)
[RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨}]  ℧ = ℧_{S} (“mhO” or “agemO”)
ζ_{F_{∞}/F}  Θ, λ_{F_{∞}/F} (pseudomeasure)
if F_{∞}/F is abelian
O[[Cong (G)]] = Λ_{O}(G)^{ab}  T Λ_{𝔒}(G)

μ = 0 ⇔ There exists an open pro-p subgroup H′ ⊂ H such that X_{F} = G(𝔉_{Σ}/𝔉)^{ab}(p) is finitely generated over ℤ_{p} for 𝔉 = F_{∞}^{H′} and 𝔉_{Σ} the maximal outside Σ unramified extension of 𝔉.

μ_{Γ′}(X_{F_{∞}}) = 0

for some open subgroup Γ′ ⊂ Γ

Kakde - Schneider/Venjakob or this article  Ritter/Weiss
π_{V}^{U}, pr_{V}^{U}  U ↠ V  defl_{U}^{V}
N_{V}^{U}  V ⊆ U  res_{U}^{V}
T̃r_{V}^{U}  Res_{U}^{V}
L, L_{B}  𝕃
L  L, ∏_{U} 𝕃_{U}

3. 2-extensions versus perfect complexes

The statement of the Main Conjecture (MC) affirms in both approaches the existence of an element Θ, respectively ζ ∈ K_{1}(Λ(G)_{S}) satisfying firstly a certain interpolation property we shall discuss later and secondly is mapped under the connecting homomorphism of the localisation sequence of K-theory

K_{1}(Λ(G)_{S}) → K_{0}(H_{S}) ≅ K_{0}(C_{S})

to the class ℧_{S}, respectively [RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨}]. To be more precise, one uses in general the localisation with respect to the Ore set S^{∗} for the statement of the MC, but assuming the vanishing of μ, it is easily reduced in both approaches to the above statement. Thus the first question which naturally arises is:

Why do the classes of RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨} and ℧ = ℧_{Σ} in the relative K_{0} agree?

Let Ω denote the maximal outside Σ unramified p-extension of F_{∞}^{Γ}; here we fixed a splitting Γ ⊆ G of the projection G ↠ Γ. We denote the Galois group G(Ω/F) by G′_{Σ} and note that this group is well-known to be (topologically) finitely generated. Hence, choosing d generators we obtain the following commutative diagram

[figure: 1; 1; N; N; 1; R; F_{d}; G; 1; 1; H′_{Σ}; G′_{Σ}; G; 1,; 1; 1]

in which F_{d} denotes the free pro-finite group on d elements and all the other groups are defined by exactness of the diagram. By [10, prop. 5.6.6] there is a canonical complex associated to the above diagram

\xymatrix{0 \ar[r] & N^{ab}(p) \ar[r] & ℤ_{p}[[G′_{Σ}]]^{d} \ar[r] & ℤ_{p}[[G′_{Σ}]] \ar[r] & 0}

which forms a resolution of ℤ_{p} by projective ℤ_{p}[[G′_{Σ}]]-modules, whence flat ℤ_{p}[[H′_{Σ}]]- modules. Therefore, the complex

\xymatrix{0 \ar[r] & N^{ab}(p)_{H′_{Σ}} \ar[r] & ℤ_{p}[[G]]^{d} \ar[r] & ℤ_{p}[[G]] \ar[r] & 0,}

which arises by taking H′_{Σ}-coinvariants, represents

RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨} ≅ RΓ(H′_{Σ}, ℚ_{p}/ℤ_{p})^{∨}

by Pontryagin duality and using a theorem of Neumann.

By [10, prop. 5.6.7] and using the weak Leopoldt statement H_{2}(H′_{Σ}, ℤ_{p}) = 0 there is a quasi-isomorphism

[figure: 0; N^{ab}(p)_{H′_{Σ}}; ℤ_{p}[[G]]^{d}; ℤ_{p}[[G]]; 0; 0; 0; Y; d_{Y}; ℤ_{p}[[G]]; 0,]

where Y := I_{G′_{Σ}}/I_{H′_{Σ}}I_{G′_{Σ}} and the map d_{Y} : Y → ℤ_{p}[[G]] factorises over the augmentation ideal as

Y ↠ I_{G} ↪ ℤ_{p}[[G]].

Now consider the exact sequence of (vertical) complexes

[figure: 0; ℤ_{p}[[G]]; ψ; Y; d_{Y}; M; 0; 0; ℤ_{p}[[G]]; ψ′; ℤ_{p}[[G]]; N; 0,]

5

where ψ is a suitably chosen homomorphism such that ψ′ := d_{Y} ∘ ψ, which has by definition image in I_{G}, is injective. Then, by definition

℧ = [M] − [N] = [M → N] ∈ K_{0}(H_{S}) ≅ K_{0}(Ch^{b}(H_{S})),

where H_{S} denotes the exact category of S-torsion Λ(G)-modules with finite projective dimension while Ch^{b}(H_{S}) denotes the category of bounded complexes in H_{S} and the above identification is shown in [25, II. thm. 9.2.2]. We also shall write C_{S} := Ch_{perf,S}(Λ(G)) for the category of perfect complexes of Λ(G)-modules which become acyclic after tensoring with Λ(G)_{S}. Under the identification K_{0}(Ch^{b}(H_{S})) = K_{0}(Ch_{perf,S}(Λ(G))) (cp. [25, II exerc. 9.2, V exerc. 3.14]) we then have

[M → N] = [Y → ℤ_{p}[[G]] ] − [ℤ_{p}[[G]] → ℤ_{p}[[G]] ]
= [Y → ℤ_{p}[[G]] ]
= [RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨}]

because the complex ℤ_{p}[[G]] → ℤ_{p}[[G]] is acyclic and quasi-isomorphisms induce identities in K_{0}. Thus we have shown the following

Proposition 3.1.

℧ = [RΓ(G(F_{Σ}/F_{∞}), ℚ_{p}/ℤ_{p})^{∨}] in K_{0}(H_{S}) ≅ K_{0}(Ch_{perf,S}(Λ(G))).

4. Hom-description

We shall write Irr(G) for the set of \overline{ℚ}_{p}-valued irreducible representations of G with finite image. Considering elements of K_{1}(Λ(G)) as maps on (irreducible) representations we obtain a natural homomorphism

Det : K_{1}(Λ(G)) → Maps(Irr(G), \overline{ℤ_{p}}^{×})

into maps from Irr(G) to the units of the ring of integers \overline{ℤ_{p}} of \overline{ℚ}_{p}, which allows for example requiring interpolation properties for elements of the K-group. Of course, the target is “much bigger” than the image and one crucial question for the proof of the main conjecture consists of finding the most appropriate intermediate target in which to work. Ritter and Weiss follow the philosophy of Fröhlich’s so called Hom-description:

Hom_{G_{ℚp}}(R(G), \overline{ℤ_{p}}^{×}) ⊆ Hom(R(G), \overline{ℤ_{p}}^{×}) = Maps(Irr(G), \overline{ℤ_{p}}^{×}),

R(G) being the group of virtual characters, i.e., the free abelian group generated by Irr(G), and they refine it slightly by an Iwasawa-theoretic variant

\xymatrix{K_{1}(Λ(G) \ar[r]^{Det} & Hom_{G_{ℚp}, R(Γ)}(R(G), Λ\overline{ℤ_{p}}(Γ)^{×})} ⊆ Hom_{G_{ℚp}}(R(G), \overline{ℤ_{p}}^{×})

whose definition will be recalled in the case of the localised Iwasawa algebra Λ(G)_{S^{∗}} at the end of this section.

In contrast, Kakde uses a version which is closer to the side of K-theory

\xymatrix{K_{1}(Λ(G)) \ar[r]^{θ} & (∏_{U∈S(G,Z)} K_{1}(Λ(U^{ab})))^{G} = (∏_{U∈S(G,Z)} Λ(U^{ab})^{×})^{G}}

where G acts by inner conjugation on S(G, Z), the set of all subgroups Z ⊆ U ⊆ G for a (fixed) central subgroup Z ⊆ G, and among the Iwasawa algebras: (x_{U})_{U} ↦ (g^{−1}x_{gUg^{−1}}g)_{U}. Recall from [22, §4] that the component θ_{U} of θ is the composite

θ_{U} : K_{1}(Λ(G)) ⟶^{N_{U}^{G}} K_{1}(Λ(U)) ⟶ K_{1}(Λ(U^{ab})) = Λ(U^{ab})^{×}

of the norm map and the homomorphism induced by the canonical surjection U ↠ U^{ab}.

By explicit Brauer induction [1, 23] we obtain a splitting a_{G} of the canonical map

(⊕_{U∈S(G,Z)}R(U^{ab}/Z))_{G} → R(G/Z),

which is induced by sending a tuple (n_{U}χ_{U}) of one-dimensional characters χ_{U} of U/Z and integers n_{U} to ∑ n_{U}Ind_{U/Z}^{G/Z}(χ_{U}); by abuse of notation we shall also write χ_{U} for the associated character infl_{U/Z}^{U}(χ) of U by inflation. Again, G is acting by inner conjugation: g(n_{U}χ_{U})_{U∈S(G,Z)} := (n_{gUg^{−1}}χ_{gUg^{−1}}(g · g^{−1}))_{U∈S(G,Z)}, where · denotes the missing argument. Hence, using also the fact [8, lem. 92] that we have a surjection

(1)  Irr(Γ) × Irr(G/Z) ↠ Irr(G), (χ, ρ) ↦ infl_{Γ}^{G}(χ) · infl_{G/Z}^{G}(ρ)

and hence a surjection

R(Γ) ⊗_{ℤ} R(G/Z) ↠ R(G),

we may define a homomorphism

BrInd : ( ∏_{U∈S(G,Z)} Λ(U^{ab})^{×})^{G} → Hom_{G_{ℚp}, R(Γ)}(R(G), Λ\overline{ℤ_{p}}(Γ)^{×})

such that

BrInd((x_{U})_{U})(ρ) := ∏_{U} Det_{U^{ab}}(x_{U})(χ_{U})^{n_{U}} ,

if a_{G}(ρ) is represented by (n_{U}χ_{U})_{U∈S(G,Z)} (and ρ is induced from G/Z via inflation). Here Det_{U^{ab}} denotes the natural map

Det_{U^{ab}} : K_{1}(Λ(U^{ab})) → Hom_{G_{ℚp}, R(Γ_{U})}(R(U^{ab}), Λ\overline{ℤ_{p}}(Γ_{U})^{×})

with Γ_{U} the image of U under the fixed projection G ↠ Γ and we use the embedding Λ\overline{ℤ_{p}}(Γ_{U}) ⊆ Λ\overline{ℤ_{p}}(Γ) in order to consider the values of Det_{U^{ab}} in Λ\overline{ℤ_{p}}(Γ)^{×}.

Lemma 4.1. The map BrInd is well-defined and BrInd((x_{U})_{U}) is G_{ℚ_{p}}- and R(Γ)- invariant, where these action are recalled at the end of this section.

Proof. Assume first that ρ arises by inflation from ρ̃ ∈ Irr(G/Z). First we check that the defining term ∏_{U} Det_{U^{ab}}(x_{U})(χ_{U})^{n_{U}} of BrInd((x_{U})_{U})(ρ) is independent of the choice of representatives (n_{U}χ_{U})_{U∈S(G,Z)} for a_{G}(ρ) = a_{G}(ρ̃) (using our lax notational convention): For any conjugate g(n_{U}χ_{U})_{U∈S(G,Z)} = (n_{gUg^{−1}}χ_{gUg^{−1}}(g · g^{−1}))_{U∈S(G,Z)} we have by the G-invariance of (x_{U})_{U}

∏_{U} Det_{U^{ab}}(x_{U})(χ_{gUg^{−1}}(g · g^{−1}))^{n_{gUg^{−1}}}

= ∏_{U} Det_{U^{ab}}(g^{−1}x_{gUg^{−1}}g)(χ_{gUg^{−1}}(g · g^{−1}))^{n_{gUg^{−1}}}

= ∏_{U} χ_{gUg^{−1}}(g(g^{−1}x_{gUg^{−1}}g)g^{−1})^{n_{gUg^{−1}}}

= ∏_{gUg^{−1}} χ_{gUg^{−1}}(x_{gUg^{−1}})^{n_{gUg^{−1}}}

= ∏_{U} Det_{U^{ab}}(x_{U})(χ_{U})^{n_{U}} .

Next we show that our (partial) definition sofar is invariant under characters of the kind χ : G ↠ Γ/Z → \overline{ℤ_{p}}^{×}. Indeed,

χ · ∏_{U} Det_{U^{ab}}(x_{U})(χ_{U})^{n_{U}} = ∏_{U} Res_{U}^{G}(χ) · Det_{U^{ab}}(x_{U})(χ_{U})^{n_{U}}
= ∏_{U} Det_{U^{ab}}(x_{U})(Res_{U}^{G}(χ)χ_{U})^{n_{U}}

by the R(Γ_{U})-invariance of Det_{U^{ab}}(x_{U}). But by Lemma 4.2 below (n_{U}Res_{U}^{G}(χ)χ_{U}) represents a_{G}(χρ), whence we have shown that

χ · BrInd((x_{U})_{U})(ρ) = BrInd((x_{U})_{U})(χρ)

as claimed.

If ρ ∈ Irr(G) is arbitrary, using (1) we choose some χ′ : G ↠ Γ → \overline{ℤ_{p}}^{×} such that χ′ ⊗ ρ is inflated from G/Z. We then define

BrInd((x_{U})_{U})(ρ) := (χ′)^{−1} · BrInd((x_{U})_{U})(χ′ ⊗ ρ).

If χ′′ is a second such character, we conclude that χ := (χ′)^{−1}χ′′ comes from a character of G/Z. Using the invariance of BrInd((x_{U})_{U}) with respect to such characters as shown above we conclude that BrInd((x_{U})_{U}) is well-defined on arbitrary irreducible representations of G and hence extends to a homomorphism on R(G) being R(Γ)-invariant by construction. Finally the G_{ℚ_{p}}-invariance follows from the G_{ℚ_{p}}-invariance of all the Det_{U^{ab}}(x_{U}) and clearly BrInd is a homomorphism. □

For a finite group G let R_{+}(G) denote the free abelian group on the G-conjugacy classes of characters ϕ : U → \overline{ℤ_{p}}^{×}, where U is any subgroup of G. We shall write (U, ϕ)^{G} for the G-conjugacy class of (U, ϕ). We clearly have a natural isomorphism

R_{+}(G) ≅ (⊕_{U}R(U^{ab}))_{G},

where U runs through all subgroups of G and usually explicit Brauer induction is defined in terms of a section a_{G} of

R_{+}(G) ↠ R(G),

i.e.,

a_{G}(ρ) = ∑_{(U,ϕ)^{G}} α_{(U,ϕ)^{G}}(ρ)(U, ϕ)^{G}

for some integers α_{(U,ϕ)^{G}}(ρ). Note that there is a natural action of any character χ : G → \overline{ℤ_{p}}^{×} on R_{+}(G), sending (U, ϕ)^{G} to χ · (U, ϕ)^{G} := (U, Res_{U}^{G}(χ)ϕ)^{G}.

Lemma 4.2. For every ρ ∈ R(G) we have α_{(U,Res_{U}^{G}(χ)ϕ)^{G}}(χρ) = α_{(U,ϕ)^{G}}(ρ), i.e. a_{G}(χρ) = χ · a_{G}(ρ).

Proof. According to [23, thm. 2.3.15] the explicit formula for a_{G} is given by

α_{(U,ϕ)^{G}}(ρ) = \frac{|U|}{|G|} ∑_{\begin{matrix}(U′,ϕ′)∈(U,ϕ)^{G}, \\ (U′,ϕ′)⊆(U′′,ϕ′′) \end{matrix}} μ_{(U′,ϕ′),(U′′,ϕ′′)} < ϕ′′, Res_{U′′}^{G}(ρ) >,

where μ denotes the Möbius function on the partially ordered set M_{G} consisting of the characters on subgroups (U, ϕ) of G with the (U, ϕ) ⊆ (U′, ϕ′) if and only if U ⊆ U′ and Res_{U}^{U′}(ϕ′) = ϕ while < , > denotes the usual Schur inner product (loc. cit., def. 1.2.7). Since sending (U, ϕ) to (U, Res_{U}^{G}(χ)ϕ) induces an G-equivariant isomorphism χ : M_{G} → M_{G}, of partially ordered sets, we obtain that

μ_{χ(U′,ϕ′),χ(U′′,ϕ′′)} = μ_{(U′,ϕ′),(U′′,ϕ′′)}.

Taking into account that < Res_{U}^{G}(χ)ϕ′′, Res_{U′′}^{G}(χρ) >=< ϕ′′, Res_{U′′}^{G}(ρ) > it follows that

α_{(U,Res_{U}^{G}(χ)ϕ)^{G}}(ρ) = \frac{|U|}{|G|} ∑_{\begin{matrix}(U′,ϕ′)∈(U,ϕ)^{G}, \\ (U′,ϕ′)⊆(U′′,ϕ′′) \end{matrix}} μ_{χ(U′,ϕ′),χ(U′′,ϕ′′)} < Res_{U}^{G}(χ)ϕ′′, Res_{U′′}^{G}(χρ) >,

equals α_{(U,ϕ)^{G}}(ρ). □

By construction, the definition of θ and the functorial behaviour of Det with respect to norm maps and induction ([14, Lem. 9]):

Det_{G}(λ)(Ind_{U}^{G}χ) = Det_{U}(N_{U}^{G}(λ))(χ) = Det_{U^{ab}}(θ_{U}(λ))(χ)

we have thus a commutative diagram

\xymatrix{K_{1}(Λ(G) \ar[r]^{^{Det^{G}}} \ar[d]_{θ} & Hom_{G_{ℚp}, R(Γ)}(R(G), Λ\overline{ℤ_{p}}(Γ)^{×}) \\ _{(∏_{U∈S(G,Z)} Λ(U^{ab})^{×})_{G}} \ar[r]_{_{∏ Det_{U^{ab}}}} \ar@{-->}[ur]^{BrInd} & (∏_{U∈S(G,Z)} Hom_{G_{ℚp}, R(Γ_{U})}(R(U^{ab}), Λ\overline{ℤ_{p}}(Γ_{U})^{×}))_{G}, \ar[u]_{brInd}}

where brInd is defined in an analogous way as BrInd.

While all of Kakde’s congruences are among the tuples (x_{U})_{U} ∈ ∏_{U∈S(G,Z)} Λ(U^{ab})^{×} in the work of Ritter and Weiss some are in a similar product (as speculated below, but Ritter and Weiss do not formalise this) and others are expressed within their Hom-description (over all with the aim to construct an integral logarithm).

Now we are going to describe the setting of Ritter and Weiss’ approach in more detail. First we set Q\overline{ℚ_{p}}(Γ) := Quot(Λ\overline{ℤ_{p}}[[Γ]]) and note that the homomorphism

Det( )(χ) : K_{1}(Λ(G)_{S^{∗}}) ⟶ Q_{L}(Γ)^{×} ⊆ Q\overline{ℚ_{p}}(Γ)

in [RW2, §3] defined using the reduced norm K_{1}(Λ(G)_{S^{∗}}) ⟶^{nr} Z(Λ(G)_{S^{∗}})^{×} coincides with Φ_{χ} of [3] by [2, lem. 3.1] and defines a homomorphism

K_{1}(Λ(G)_{S^{∗}}) ⟶^{Det} Hom_{G_{ℚp},R(Γ)}(R(G), Q\overline{ℚ_{p}}(Γ)^{×})

into the group of G_{ℚ_{p}}- and R(Γ)-invariant homomorphism, where:

a) σ ∈ G_{ℚ_{p}} acts coefficientwise on Q\overline{ℚ_{p}}(Γ) and R(G),

b) φ ∈ R(Γ) acts on R(G) via the tensor product χ ⟼ infl_{Γ}^{G}(φ)χ and on Q\overline{ℚ_{p}}(Γ) via the homomorphism induced by twisting

\begin{matrix}T w_{φ} : Λ\overline{ℤ_{p}}(Γ) & ⟶ Λ\overline{ℤ_{p}}(Γ) \\ γ & ⟼ φ(γ)γ. \end{matrix}

In particular, from RW’s construction (see Rem. E in [RW2, §4]):

kerDet = ker(nr_{Λ(G)_{S^{∗}}}) = SK_{1}(Λ(G)_{S^{∗}}).

By [14, Lem. 9] Det behaves functorially with respect to open subgroups U ⊆ G and the map included by Induction (Ind_{G}^{U})^{∗}, as well as to factor groups G ↠ \overline{G} and the map induced by inflation (infl\frac{G}{G})^{∗} on the Hom-description.

5. The interpolation property versus L_{F_{∞}/F}

By

L_{F_{∞}/F} (χ) := L_{χ,Σ} = \frac{G_{χ,Σ}(γ − 1)}{H_{χ}(γ − 1)} ∈ Q\overline{ℚ_{p}}(Γ)^{×}

we denote the (Σ-truncated) p-adic Artin L-function for F^{cyc}/F attached to χ ∈ R(G), for γ a fixed topological generator of Γ [RW2, §4]. Then L_{F_{∞}/F} belongs to Hom_{G_{Qp},R(Γ)}(R(G), Q\overline{ℚ_{p}}(Γ)^{×}) and is independent of choice of γ (loc. cit. prop. 11).

Applying the extended localised augmentation map φ′ : Q\overline{ℚ_{p}}(Γ)^{×} ⟶ \overline{ℚ_{p}} ∪ \{∞\}, which is induced by sending any γ ∈ Γ to 1, (see before [24, thm. 2.4]) one sees that giving L_{F_{∞}/F} is morally the same as requiring the usual interpolation property:

Det(ζ) = L_{F_{∞}/F}

for some ζ ∈ K_{1}(Λ(G)_{S^{∗}}) corresponds to

ζ(χ) = φ′ (Det(ζ)(χ)) = φ′(L_{F_{∞}/F} (χ)) = L_{Σ}(χ, 1)

Extending this interpolation property also to the cyclotomic character κ, it even determines L_{F_{∞}/F} uniquely. In fact, Ritter and Weiss show that

L_{F_{∞}/F} ∈ Hom_{G_{Op},R(Γ)}^{(1)}(R(G), B\overline{ℤ_{p}}(Γ)^{×}),

where ^{(1)} indicates that f satisfies the congruence

(2)  \frac{f(χ)^{p}}{Ψf(ψ_{p}χ)} ≡ 1 mod p

with the ring endomorphism Ψ : B\overline{ℤ_{p}}(Γ) → B\overline{ℤ_{p}}(Γ) being induced by sending γ to γ^{p}, while ψ_{p} : R(G) → R(G) denotes the p^{th} Adams operator, i.e.,

(ψ_{p}χ)(g) = χ(g^{p})

for any character χ. This result uses twice explicit Brauer induction: Firstly to generalise a theorem of Snaith [23, thm. 4.1.6] saying that the image of Det lies in Hom_{G_{Op},R(Γ)}^{(1)}(R(G), B\overline{ℤ_{p}}(Γ)^{×}). Due to the existence of Serre’s pseudomeasures which can be interpreted as elements in K_{1}(Λ(G)_{S}) for G abelian, it follows that L_{F_{∞}/F} satisfies the generalised Snaith congruences. But secondly by explicit Brauer induction the values of L_{F_{∞}/F} in the general case can be expressed by the values of suitable abelian L’s, hence implying the congruences. In both cases it is crucial that the Brauer induction can be arranged in a compatible way with respect to the p^{th} Adams operator, which in general does not behave well under induction.

Theorem 5.1 ([RW3] Thm. B_{∧} in §6). For G pro-p we have

Det K_{1}(B(G)) ∩ Hom_{G_{ℚp}, R(Γ)}(R(G), Λ\overline{ℤ_{p}}(Γ)^{×}) ⊂ DetK_{1}(Λ(G)).

in Hom_{G_{ℚp}, R(Γ)}(R(G)), B\overline{ℤ_{p}}(Γ)^{×}).

This should be compared to

Φ_{B} ∩ ∏_{U∈S(G,Z)} Λ(U^{ab})^{×} = Φ

in Kakde’s work and to Burns’ result [2, thm. 6.1, rem. 6.2] - the latter approach avoids the analysis of integral logarithms. Thus, by the usual argument (compare ????reference into some article of the same volume to be inserted later) the Main Conjecture is equivalent to showing that

(3)  L_{F_{∞}/F} ∈ Det K_{1}(B(G)).

In order to verify the latter condition, Ritter and Weiss introduce a new integral group logarithm, which makes it possible to translate this multiplicative statement in a additive statement plus a statement about the kernel of the integral group logarithm which is referred to as the “torsion” part (since for a finite p-group G this kernel is actually a torsion group) even though in this setting it may contain a torsionfree part!

Set T B := B/[B, B] for any ring B, where [B, B] denotes the set of additive commutators ab − ba, a, b ∈ B and consider the diagram

\xymatrix{K_{1}(B(G)) \ar[r]^{𝕃} \ar[d]_{Det} & T B(G) ⊂ T B(G)[\frac{1}{p}] \ar[d]^{Tr}_{≅} \\ Hom_{G_{ℚp},R(Γ)}^{(1)}(R(G), B\overline{ℤ_{p}}(Γ)^{×}) \ar[r]^{L} & Hom_{G_{ℚp},R(Γ)}(R(G), B\overline{ℤ_{p}}(Γ)[\frac{1}{p}])}

f ↦ L(f)(χ) = \frac{1}{p} log(\frac{f(χ)^{p}}{Ψ(f(ψ_{p}χ))})

in which Lf is well-defined due to the congruence (2) (note that due the Galois invariance, for any f in Hom_{G_{ℚp},R(Γ)}(R(G), B\overline{ℤ_{p}}(Γ)^{×}) the value f(ρ) belongs to some B_{O_{L}}(Γ)^{×} for some finite extension L of ℚ_{p}) and which defines the integral logarithm 𝕃 = 𝕃_{G} in the upper row following the approach of Snaith in [23]. The trace map Tr is the analogue of Det in the additive setting, see [16].

Question: Does 𝕃 coincide with Kakde’s integral logarithm L_{B} in [22, §5]?

As mentioned above now Ritter and Weiss divide the condition (3) into the question whether the “additive” element L(L_{F_{∞/F}}) lies in Tr(im(𝕃)) and under which conditions a (torsion) element in ker(L), viz the defect of L_{F_{∞/F}} not being determined by L(L_{F_{∞/F}}), is in the image of Det? To this aim they introduce

t_{F_{∞/F}} := T r^{−1}L(L_{F_{∞/F}}) ∈ T B(G)[\frac{1}{p}],

and call it the logarithmic pseudomeasure.

Theorem 5.2. L_{F_{∞}/F} ∈ DetK_{1}(B(G)) if and only if

(i ) t_{F_{∞}/F} ∈ T B(G) (integrality) and (ii) ver_{U}^{V} ζ_{F_{∞}^{[V,V ]}/F_{∞}^{V}} ≡ ζ_{F_{∞}/F_{∞}^{U}} mod im(σ_{U}^{V}) (torsion congruence)

for all U ⊆ V ⊆ G with U abelian, [V : U] = p where ver_{U}^{V} is induced from ver_{U}^{V} : V^{ab} → U^{ab} by linear extension.

Remark 5.3. Ritter and Weiss’ strategy of decomposing the problem into an additive/logarithmic and torsion part by various diagram chases should be compared with the use of the 5-lemma in Kakde’s approach with respect to the following diagram in [22, between (8) and lem. 4.7]

(4) \xymatrix{1 \ar[r] & μ(O) × G^{ab} \ar[d]_{=} \ar[r]^{ι} & K′_{1}(Λ(G)) \ar[d]_{θ} \ar[r]^{L} & O[[Conj(G)]] \ar[d]_{β}^{≅} \ar[r]^{ω} & G^{ab} \ar[d]_{=} \ar[r] & 1 \\ 1 \ar[r] & μ(O) × G^{ab} \ar[r]^{θ∘ι} & Φ \ar[r]^{L} & Ψ \ar[r]^{ω∘β^{−1}} & G^{ab} \ar[r] & 1.}

combined with diagram [22, (11)] for the B-situation.

Idea of Proof: (for the converse direction). If t_{F_{∞}/F} ∈ T B(G), then there exists y ∈ K_{1}(B(G)) such that

(5)  t_{F_{∞}/F} = 𝕃(y)

and

(6)  y ↦ ζ_{F_{∞}^{[G,G]}/F}

under the canonical map pr_{G^{ab}}^{G} : K_{1}(B(G)) → K_{1}(B(G^{ab})). This follows immediately from the abelian case 𝕃_{G^{ab}}(ζ_{F_{∞}^{[G,G]}/F}) = t_{F_{∞}^{[G,G]}/F} and the following commutative diagram [21, lem. 7.1 ii)] with exact rows

\xymatrix{1 \ar[r] & 1 + 𝔞 \ar[r] \ar@{->>}[d] & B(G)^{×} \ar@{->>}[r] \ar[d]_{𝕃_{G}} & B(G^{ab})^{×} \ar[r] \ar[d]_{𝕃_{G^{ab}}} & 0 \\ 0 \ar[r] & τ(𝔞) \ar[r] & T B(G) \ar[r] & T B(G^{ab}) \ar[r] & 0 \\  &  & t_{F_{∞}/F} \ar@{|->}[r] & t_{F_{∞}^{[G,G]}/F}}

where 𝔞 := ker(B(G) → B(G^{ab})) and τ(𝔞) is the image of 𝔞 with respect to the canonical map τ : B(G) ↠ T B(G). Setting ω := Det(y)^{−1} · L_{F_{∞}/F} we have by (6)

that

(7)  ω_{|R(G^{ab})} ≡ 1

and by the definition of t_{F^{∞}/F}

L(ω) = 0,

whence

\frac{ω(χ)^{p}}{Ψω(ψ_{p}χ)} = 1,

as log_{|1+p B_{O_{L}}(Γ)} is injective. Therefore ω(χ)^{p^{n}} = Ψ^{n}ω(ψ_{p}^{n}χ) = Ψ^{n}(ω(𝟙))^{χ(𝟙)} = 1 for n sufficiently big such that ψ_{p}^{n}R(G/Z) = {𝟙}.

That means that ω is a torsion element. In [RW5, prop. 2.4] even uniqueness of ω is shown. We want to show that ω = 1. Assume first that G contains an abelian subgroup G′ of index p. Since then any irreducible representation χ of G is either inflated from an abelian character α of G^{ab}

χ = infl_{G^{ab}}^{G}α

or induced from an abelian character β of G′

χ = ind_{G′}^{G}(β′),

it suffices by (7) to verify that

(8)  ω|Ind_{G′}^{G}R(G′) ≡ 1 or (Ind_{G′}^{G})^{∗}ω ≡ 1

But by the functoriality properties (Ind_{G′}^{G})^{∗}L_{F_{∞}/F} = L_{F_{∞}/F_{∞}^{G′}} and Det N_{G′}^{G}(y) = (Ind_{G′}^{G})^{∗}Det(y) we obtain

(Ind_{G′}^{G})^{∗} ω = \frac{(Ind_{G′}^{G})^{∗} L_{F_{∞}/F}}{Det (N_{G′}^{G}(y))} = Det (\frac{ζ_{F_{∞/F{G′ \atop ∞}}}}{N_{G′}^{G}(y)} )

where N_{G′}^{G} : K_{1}(B(G)) → K_{1}(B(G′)) denotes the norm map. Since Det is injective on K_{1}(B(G′)), we see that

\frac{ζ_{F_{∞}/F_{∞}^{G′}}}{N_{G′}^{G}(y)}

is a torsion element. By the Wall-congruence ([RW3, proof of Lemma 12])

N_{G′}^{G}(y) ≡ ver_{G′}^{G^{ab}} (pr_{G^{ab}}^{G}(y)) mod im (σ_{G′}^{G})

(which corresponds in this special case to (M3) in Kakde’s work, see [22]: N_{G′}^{G}(θ_{G′}(x)) ≡ ver_{G′}^{G^{ab}} (θ_{G}(x)) mod im (σ_{G′}^{G})) it follows that by our choice of y with pr_{G^{ab}}^{G}(y) = ζ_{F_{∞}^{[G,G]}/F_{∞}}

e := \frac{ζ_{F∞/F∞^{G′}}}{N_{G′}^{G}(y)} = ^{\frac{ver_{G′}^{G^{ab}}(pr_{G^{ab}}^{G}(y))}{N_{G′}^{G}(y)}} \frac{ζ_{F∞/F∞^{G′}}}{ver_{G′}^{G^{ab}}(ζ_{_{F∞}_{[G,G]}/F∞})}

≡ \frac{ζ_{F∞/F{G′ \atop ∞}}}{ver_{G′}^{G^{ab}}(ζ_{F{[G,G] \atop ∞}/F∞})} ≡^{(ii)} 1 mod im(σ_{G′}^{G})

by our assumption. The Theorem follows from the

Claim: If e ∈ B(G′)^{×} is torsion and satisfies e ≡ 1 mod im(σ_{G′}^{G}), then e = 1, cp. with [22, proof of lem. 4.9].

By Higman’s theorem we have: e = ζh ≡ 1 mod im(σ_{G′}^{G}) with h ∈ H′ (for the group ring B(G′) = B(Γ′)[H′], if G′ decomposes as Γ′ × H′). The augmentation map ϵ : B(G′) → B(Γ′) induces ζ ≡ 1 mod pB(Γ′), whence ζ = 1 and h′ − 1 ∈ im(σ_{G}^{G′}). Thus h′ = 1, e = 1 and (8) holds.

By an inductive argument this argument can be extended to arbitrary G, see proof of Theorem in [RW5, §3]. □

The hard part is now to inductively show that

(9)  t_{F_{∞}/F} ∈ T B(G)

holds, which requires the new Möbius-Wall congruence

∑_{A⊆U⊆G} μ_{G/Z}(U/Z)ver_{A}^{U^{ab}}(ζ_{F_{∞}^{[U,U]}/F_{∞}^{U}}) ∈ im(σ_{A}^{G})

for any abelian normal open subgroup A ⊴ G (actually for each such one-dimensional subextension) introduced in [21], more precisely in (loc. cit.) only a similar relation for units in B(G) is called Möbius-Wall congruence. Recall that for a finite p-group G, the Möbius-function is defined inductively as follows

\begin{aligned}μ_{G}(1) & = 1 \\ μ_{G}(U) & = − ∑_{V ⊊U} μ_{G}(V) for 1 ≠ U ⊆ G. \end{aligned}

How this condition enters the proof will be explained in the next section, in which we try first to abstract and formalise what the methods of Ritter and Weiss actually prove.

6. The abstract setting - a reinterpretation of Ritter and Weiss’ approach

Fix a one dimensional pro-p-group G with projection onto Γ ≅ ℤ_{p}, let H denote its kernel and define the following index sets

S := S_{G} := {U one-dimensional subquotient of G}
= {U|U ⊆ G/C open for some C ⊴ G with C ⊆ H}

and

S^{ab} := S_{G}^{ab} = {U ∈ S_{G}|U abelian}.

Note that for any U ∈ S the quotient U^{ab} also belongs to S and in particular to S^{ab}. Define Φ̃_{B} to be the subgroup of ∏_{U∈S_{G}^{ab}} B(U)^{×} consisting of those (λ_{U})_{U∈S_{G}^{ab}} satisfying the following conditions:

(RW1) For every surjection U ↠ V in S^{ab} we have pr_{V}^{U} λ_{U} = λ_{V} , where pr_{V}^{U} : K_{1}(B(U)) → K_{1}(B(V)) is the natural map induced by the projection, and for every inclusion V ⊆ U in S^{ab} we have N_{V}^{U}λ_{U} = λ_{V} , where N_{V}^{U} denotes the norm map.

(RW2) For all U ∈ S_{G} the (sub-)tuple (λ_{V^{ab}})_{V ⊆U} is U-invariant.

(RW3) (Möbius-Wall congruence) For all U ∈ S_{G} and all abelian normal open subgroups A ⊴ U we have ∑_{A⊆V ⊆U} μ_{U/A}(V/A)ver_{A}^{V}(λ_{V^{ab}}) ∈ σ_{A}^{U}(B(A)).

In particular the torsion congruence

(RW3a) (Torsion congruence) For all U ∈ S_{G} and all abelian normal open sub- groups A ⊴ U of index p we have ver_{A}^{U}(λ_{U^{ab}}) − λ_{A} ∈ σ_{A}^{U}(B(A)).

Actually Ritter and Weiss show that (RW3) holds for every tuple which arises from an element ϑ ∈ K_{1}(B(G)), see [21, thm. 2]. Strictly speaking, they only call the relation for such ϑ Möbius-Wall congruence, but we extend this notation to tuples in Φ̃_{B}. The proof generalises Wall’s proof of (RW3a) by analysing the Leibniz-formula for determinants; for combinatorial reasons the Möbius function shows up. Is it by chance that this or a similar Möbius function also shows up in the explicit formula of Brauer induction? While the proof of (RW3) is rather tedious it is straightforward to check (RW1) and (RW2).

Using explicit Brauer induction (as at the beginning of section 4), for any U ∈ S_{G} one can assign to a tuple (λ_{V}) ∈ Φ̃_{B}, or rather to its sub-tuple (λ_{V})_{V ⊆U}, elements

Ξ_{U} ∈ Hom_{G_{ℚp},R(Γ_{U})}^{(1)}(R(U), B_{O_{L}}(Γ_{U})^{×})

and

t_{U} := T r^{−1}(L_{U}(Ξ_{U})) ∈ T B(U)[\frac{1}{p}],

such that t_{U} = 𝕃_{U}(λ_{U}) ∈ T B(U) for all U ∈ S_{G}^{ab}. Indeed, for every one-dimensional character ρ of U (which is trivial on some central subgroup Z_{U} of U) a_{U}(ρ) is represented by ρ itsself under explicit Brauer induction by [1]. Finally we require (RW4) For any U ∈ S_{G}, the definition of Ξ_{U} does not depend on the above chosen way by explicit Brauer induction, i.e. Ξ_{U}(ρ) = ∏_{V} Det_{V^{ab}}(λ_{V^{ab}})(χ_{V})^{n_{V}} , whenever ρ = ∑ n_{V} Ind_{U}^{V} χ_{V} in R(U) for certain subgroups V ⊆ U, one-dimensional representations χ_{V} of V and (finitely many nonzero) integers n_{V} .

This conditions looks a little weird, but whenever one is interested in p-adic Lfunctions, it is completely harmless, as it is always satisfied by the usual behaviour of L-functions under induction.

Questions: Do conditions (RW1-3) imply already (RW4)? Is there a way of proving the next Lemma 6.1 without requiring (RW4)?

All what one needs to extend Ritter and Weiss proof of property (9) to Theorem 6.3 below are the following functoriality properties.

Lemma 6.1. pr_{V}^{U} (Ξ_{U}) = Ξ_{V} and N_{V}^{U}(Ξ_{U}) = Ξ_{V} as well as similarly pr_{V}^{U} (t_{U}) = t_{V} and T̃r_{V}^{U} (t_{U}) = t_{V} for all possible U, V ∈ S_{G}, where the modified trace T̃r_{V}^{U} is introduced in (10) below.

Proof. While it is well-known that explicit Brauer induction (given by a_{G} as above) behaves well under inflation, it does not behave well under induction. Therefore we need at present condition (RW4) here to prove the correct behaviour under the norm: For V ⊆ U, let

ρ = ∑_{W ⊆V} n_{W} Ind_{V}^{W} χ_{W}

in R(V). Then, by the transitivity of induction we obtain

Ind(ρ) = ∑_{W ⊆V} n_{W} Ind_{U}^{W} χ_{W}

in R(U). By the definition of the norm on the Hom-description we thus have

(N_{V}^{U}Ξ_{U})(ρ) = Ξ_{U}(Ind(ρ))
= ∏_{W ⊆V} Det_{W^{ab}}(λ_{W^{ab}})(χ_{W})^{n_{W}}
= Ξ_{V} (ρ).

For pr : U ↠ V we obtain

infl_{V}^{U} (ρ) = ∑_{W ⊆V} n_{W} Ind_{U}^{W′}(infl_{W}^{W′}χ_{W}),

where W′ := pr^{−1}(W) is the full preimage of W under pr. Hence

pr_{V}^{U} (Ξ_{U})(ρ) = Ξ_{U}(infl_{V}^{U} (ρ))
= ∏_{W ⊆V} Det_{W^{′ab}}(λ_{W^{′ab}})(infl_{W}^{W′}χ_{W})^{n_{W}}
= ∏_{W ⊆V} Det_{W^{ab}}(pr_{W^{ab}}^{W′^{ab}}(λ_{W′}_{ab}))(χ_{W})^{n_{W}}
= ∏_{W ⊆V} Det_{W^{ab}}(λ_{W^{ab}})(χ_{W})^{n_{W}}
= Ξ_{V} (ρ).

(the last property does not require (RW4)!). The corresponding statements for t_{U} follow from the functorial properties of T r^{−1} ∘ L_{U}, see [21, lem. 7.2]. □

Theorem 6.2. For every (λ_{U})_{U} ∈ Φ̃_{B} there exist λ_{G} ∈ K_{1}(B(G)) such that

Det_{G}(λ_{G}) = Λ_{G}.

Question: How big is the kernel of Det_{G}?

In order to obtain (3) from this Theorem we just have to observe that for the tuple (λ_{U})_{U∈S^{ab}} consisting of the abelian pseudomeasures λ_{U} = λ_{F_{∞}^{C}/F_{∞}^{V}} if U = V/C ∈ S^{ab} for some subgroup V ⊆ G = G(F_{∞}/F) and some normal subgroup C ⊴ G contained in H the associated element Ξ_{G} equals L, supposed of course that (λ_{U})_{U∈S^{ab}} belongs to Φ̃_{B}. While (RW1), (RW2) and (RW4) are well-known properties, (RW3) forms a completely new property which is proved by Ritter and Weiss using the q-expansion principle of Deligne-Ribet.

Now we are going to prove the Theorem: using (RW3a) and the analogue of Theorem 5.2, which also can be proved in this general setting, it suffices to prove

Theorem 6.3. In the situation of the above Theorem we have

t_{G} ∈ T B(G).

Idea of proof: Actually we shall show that t_{U} ∈ T B(U) for all U ∈ S_{G} by contraposition (alternatively one could formulate the argument using induction): If this statement is false we firstly may consider among the counterexamples those U for which the order of [U, U] is minimal (this order must be different from one as the claim of the proposition holds for all abelian U). Among those U we may assume that the order of [U : Z(U)] is minimal, where Z(U) denotes the centre of U. Without loss of generality we may and do assume that these minimality conditions already hold for G itself. Then we choose a central element c ∈ [G, G] of order p and set C :=< c > . Furthermore we choose a maximal (with respect to inclusion) abelian normal subgroup A of G, which then automatically contains C. Also we fix a central subgroup Z ≅ ℤ_{p} of G contained in A.

In order to arrive at a contradiction we will use the following lemma of Ritter and Weiss, in which T̃r_{V}^{U} for V ⊆ U denotes the modified trace map defined by Ritter and Weiss (extending the definition in [11] to their Iwasawa theoretic Hom-description) to make the following diagram, in which N_{V}^{U} denotes the norm map, commutative

(10)  \xymatrix{K_{1}(B(U)) \ar[d]_{N_{V}^{U}} \ar[r]^{𝕃_{U}} & T B(U) \ar[d]^{_{T̃r{U \atop V}}} \\ K_{1}(B(V)) \ar[r]^{𝕃_{V}} & T B(V).}

Note that here we encounter another significant difference in comparison to Kakde’s approach, who uses in a similar situation the original trace map Tr_{U}^{V} , but then has to take a modified integral logarithm map 𝕃̃_{V} in order to obtain a similar commutative diagram (which then induces the diagram in [22, prop. 4.4 and (11)] involving Φ and Ψ).

Lemma 6.4. Defining β̃ = (β̃_{U}) by β̃_{U} = pr_{U^{ab}}^{U}T̃r_{U}^{G} for U in S(G, A) := {U ∈ S_{G}|A ⊆ U} we obtain a commutative diagram of injective homomorphisms

\xymatrix{T B(G) \ar[d] \ar[r]^{β̃} & ∏_{U∈S(G,A)} B(U^{ab}) \ar[d] \\ T B(G)[\frac{1}{p}] \ar[r]^{β̃[ \frac{1}{p}]} & ∏_{U∈S(G,A)} B(U^{ab})[\frac{1}{p}].}

For the desired contradiction it thus suffices to show that there exist a t ∈ T B(G) such that β̃(t) = (t_{U^{ab}})_{U∈S(G,A)}, because then t = t_{G} as t_{G} is mapped to the same tuple under β̃ by Lemma 6.1. We first will search for a t such that

(11)  t ↦ t_{A}, t_{G^{ab}}.

Setting Ḡ := G/C and Ā:= A/C we define the ideals 𝔞 and 𝔟 by exactness of rows in the following commutative diagram

\xymatrix{0 \ar[r] & 𝔞 \ar[r] & B(G) \ar[r] & B(Ḡ) \ar[r] & 0 \\ 0 \ar[r] & 𝔟 \ar[r] \ar[u] & B(A) \ar[r] \ar[u] & B(Ā) \ar[r] \ar[u] & 0.}

Ritter and Weiss show that one has a commutative diagram

\xymatrix{0 \ar[r] & τ𝔞 \ar[r] \ar@{->>}[d] & T B(G) \ar[dd]_{_{T̃r{G \atop A}}} \ar[r] & T B(Ḡ) \ar[dd]_{_{T̃rḠ ; Ā}} \ar[r] & 0 \\  & σ_{A}^{G}(𝔟) \ar@{^{(}->}[d] \\ 0 \ar[r] & 𝔟 \ar[r] & B(A) \ar[r] & B(Ā) \ar[r] & 0,}

By the minimality of [G, G] we know that tḠ belongs to T B(Ḡ). We choose any lift t = 𝕃_{G}(ϑ) ∈ T B(G) of tḠ in the image of 𝕃_{G} (which is possible by the same reasoning as in the proof of Theorem 5.2 with the additional property that prA ; ĀN_{A}^{G}(ϑ) = λĀ). Then the condition (RW3) for the tuple (λ_{U})_{U} reads

∑_{A⊆V ⊆G} μ_{G/A}(V/A)ver_{A}^{V}(λ_{V^{ab}}) ∈ σ_{A}^{G}(B(A))

while one easily shows - using the fact that N_{A}^{G}(ϑ)λ_{A}^{−1} lies in the kernel of B(A)^{×} → B(Ā)^{×} - that for the tuple (pr_{U^{ab}}^{U}N_{U}^{G}(ϑ))_{U} induced by ϑ it reads

N_{A}^{G}(ϑ) + ∑_{A⫋V ⊆G} μ_{G/A}(V/A)ver_{A}^{V}(λ_{V^{ab}}) ∈ σ_{A}^{G}(B(A))

since

ver_{A}^{V}(pr_{V^{ab}}^{V}N_{V}^{G}(ϑ)) = ver_{A}^{V}(λ_{V^{ab}}) for V ≠ A.

This implies that

N_{A}^{G}(ϑ)λ_{A}^{−1} ∈ 1 + σ_{A}^{G}(B(A)),

from which Ritter and Weiss derive that

x := T̃r_{A}^{G}(t) − t_{A} = 𝕃_{G}(N_{A}^{G}(ϑ)λ_{A}^{−1}) ∈ 𝔟^{G} ∩ σ_{A}^{G}(B(A)) = σ_{A}^{G}(𝔟)

lies in the image of τ(𝔞) under T̃r_{A}^{G}. Upon modifying t accordingly, we may and do thus assume that

T̃r_{A}^{G}(t) = t_{A}.

as claimed above.

In fact, it follows easily that then t is also mapped to t_{U^{ab}} under β̃_{U} for all U such that c is contained in [U, U]. Otherwise we have C ∩ [U, U] = 1, i.e., the order of [U, U] is strictly smaller than that of [G, G], whence

t_{U} ∈ T B(U)

by assumption. We set

x_{U} := x_{U}(t) := T̃r_{U}^{G}(t) − t_{U} ∈ T B(U).

By our contraposition the support, i.e., the set

supp(t) := {U ∈ S(G, A)|C ∩ [U, U] = 1, x_{U}(t) ≠ 0},

is non-empty for all choices of t satisfying (11). Let t be such a choice with the order of supp(t) being minimal and let U ∈ supp(t) have minimal [U : A]. Explicit calculations with T̃r using [21, lem. 5.1], the minimality of [G : Z(G)] and the uniqueness of expressing elements in T B(U) as linear combinations over B(Z) similarly as for the proof of Ψ → Ψ_{cyc} being injective in [22, lem. 3.5] show firstly that U/A must be cyclic and secondly that in this cyclic case one can modify t to an element t′ (using [21, §6, claim 6.A]) with strictly smaller support than t, a contradiction. □

At the end we want to investigate whether for Λ(G) itself instead of B(G) the analogous group Φ̃ in terms of Λ(−) satisfying again (RW1-4) describes, i.e., equals the image of K_{1}(Λ(G)). From Kakde’s result we know that his condition (M4) (together with (M1-M3) at least) imply the Möbius-Wall congruence (RW3), even though there does not seem to be a direct link between them. For example (M4) involves only cyclic subgroups while (RW3) ranges over all subgroups among A and G. In fact we will now prove the converse, viz that (RW1-4) also implies (M1-4).

Theorem 6.5.

K′_{1}(Λ(G)) := K_{1}(Λ(G))/SK_{1}(Λ(G)) ≅ Φ̃.

Proof. Consider the commutative diagram with exact rows

\xymatrix{0 \ar[r] & μ × G^{ab} \ar[d]_{ι} \ar[r] & K′_{1}(Λ(G)) \ar[d]_{θ̃} \ar[r]^{𝕃} & im(𝕃) \ar[d]^{β̃|im(𝕃)} \ar[r] & 0 \\ 0 \ar[r] & Φ̃ ∩ ∏(μ × U^{ab}) \ar[r] & Φ̃ \ar[r]^{∏ 𝕃_{U}} & im(Φ̃) \ar[r] & 0}

analogous to (4). The map ι is an isomorphism by (RW3a) (cp. [22, lem. 4.9]) while β̃ is an isomorphism by Theorem 6.3 combined with (5) and Lemma 6.4 above. □

Note that in contrast to Kakde’s approach an explicit description of the image of β̃ is avoided in the above proof. As Kakde pointed out to me such description would be very messy, because the description of the image of θ̃ is - among others - in terms of the additive congruences (RW3) which would translate into terms of the logarithm applied to them for the image of β̃.

Recently, also the kernel of 𝕃_{B(G)} has been determined by Kakde [7] using the result for abelian G by Ritter and Weiss in [20]:

ker(𝕃_{B(G)}) = SK_{1}(B(G)) × μ × G^{ab},

where

SK_{1}(B(G)) := ker (K_{1}(B(G)) → K_{1}(B(G)[\frac{1}{p}])) .

Hence the same proof as above also shows the following

Theorem 6.6.

K′_{1}(B(G)) := K_{1}(B(G))/SK_{1}(B(G)) ≅ Φ̃_{B}.

Question: Does an analogous statement hold for A(G)? Finally we want to remark that - fixing a central subgroup Z ≅ ℤ_{p} of G contained in Γ - we may replace the infinite index sets S_{G}, S_{G}^{ab} by the finite subsets

S_{G,Z} := {U|Z ⊆ U ⊆ G/C for some C ⊴ G with C ⊆ H}

(note that Z can be considered canonically as subgroup of G/C, because Z ∩C = 1) and

S_{G,Z}^{ab} := {U ∈ S_{G,Z}|U abelian}.

For the corresponding Φ̃ we obtain the same statements as in Theorems 6.5 and 6.6.

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Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany, http://www.mathi.uni-heidelberg.de/˜ venjakob/

E-mail address: venjakob@mathi.uni-heidelberg.de