Contemporary Mathematics

The Equivariant Tamagawa Number Conjecture: A survey

Matthias Flach

(with an appendix by C. Greither)

Abstract. We give a survey of the equivariant Tamagawa number (a.k.a. Bloch-Kato) conjecture with particular emphasis on proven cases. The only new result is a proof of the 2-primary part of this conjecture for Tate-motives over abelian fields.

This article is an expanded version of a survey talk given at the conference on Stark’s conjecture, Johns Hopkins University, Baltimore, August 5-9, 2002. We have tried to retain the succinctness of the talk when covering generalities but have considerably expanded the section on examples.

Most of the following recapitulates well known material due to many people. Section 3 is joint work with D. Burns (for which [14], [15], [16], [17] are the main references). In section 5.1 we have given a detailed proof of the main result which also covers the prime l = 2 (unavailable in the literature so far).

Contents

Part 1. The Tamagawa Number Conjecture in the formulation of Fontaine and Perrin-Riou 2

1. The setup 2 2. Periods and Regulators 3 3. Galois cohomology 4

Part 2. The Equivariant Refinement 5 4. Commutative Coefficients 6 5. Proven cases 6

Part 3. Determinant Functors: Some Algebra 37 6. Noncommutative Coefficients 39 7. The Stark Conjectures 41

APPENDIX: On the vanishing of μ-invariants 42 References 45

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. 11G40, 11R23, 11R33, 11G18.

The author was supported by a grant from the National Science Foundation.

Part 1. The Tamagawa Number Conjecture in the formulation of Fontaine and Perrin-Riou

The Tamagawa number conjecture of Bloch and Kato [10] is a beautiful generalization of the analytic class number formula (this is a theorem!) on the one hand, and the conjecture of Birch and Swinnerton- Dyer on the other. It was inspired by the computation of Tamagawa numbers of algebraic groups with, roughly speaking, motivic cohomology groups playing the role of commutative algebraic groups. In [35] and [34] Fontaine and Perrin-Riou found an equivalent formulation of the conjecture which has two advantages over the original one: It applies to any integer argument of the L-function (rather than just those corresponding to motives of negative weight), and it generalizes to motives with coefficients in an algebra other than ℚ. Independently, Kato developed similar ideas in [48] and [49]. In this section we sketch this formulation.

1. The setup

Suppose given a smooth projective variety

X → Spec(ℚ)

and integers i, j ∈ ℤ. The ”motive” M = h^{i}(X)(j) is the key object to which both an L-function and all the data conjecturally describing the leading coefficient of this function are attached. For the purpose of discussing L-functions, one need not appeal to any more elaborate notion of motive than that which identifies M with this collection of data (the ”realisations” and the ”motivic cohomology” of M). One has

• M_{l} = H_{et}^{i}(Xℚ̄, ℚ_{l})(j) a continuous representation of the Galois group G_{ℚ}.

• The characteristic polynomial P_{p}(T) = det(1 − Fr_{p}^{−1} · T |M_{l}^{I_{p}}) ∈ ℚ_{l}[T] where Fr_{p} ∈ G_{ℚ} is a Frobenius element. It is conjectured, and known if X has good reduction at p, that P_{p}(T) lies in ℚ[T] and is independent of l.

• The L-function L(M, s) = ∏_{p} P_{p}(p^{−s})^{−1}, defined and analytic for ℜ(s) large enough.

• The Taylor expansion

L(M, s) = L^{∗}(M)s^{r(M)} + ⋯

at s = 0. That L(M, s) can be meromorphically continued to s = 0 is part of the conjectural framework. This continuation is known, for example, if X is of dimension 0, or X is an elliptic curve [13] or a Fermat curve x^{N} + y^{N} = z^{N} [76] and i and j are arbitrary.

Aim: Describe L^{∗}(M) ∈ ℝ^{×} and r(M) ∈ ℤ.

Examples. a) If M = h^{0}(Spec L)(0) for a number field L then L(M, s) coincides with the Dedekind Zeta function ζ_{L}(s). If we write L ⊗_{ℚ} ℝ ≅ ℝ^{r_{1}} × ℂ^{r_{2}} then r(M) = r_{1} + r_{2} − 1 and L^{∗}(M) = −hR/w (R the unit regulator of L, w the number of roots of unity in L and h the class number of O_{L}). This is the analytic class number formula.

b) If M = h^{1}(A)(1) for an abelian variety A over a number field (or in fact for any X as above), then L(M, s − 1) is the classical Hasse-Weil L-function of the dual abelian variety Ǎ (or the Picard variety Pic^{0}(X) of X). Since A and Ǎ are isogenous, L(M, s − 1) also coincides with the Hasse-Weil L-function of A.

The weight of M is the integer i − 2j.

2. Periods and Regulators

We have four ℚ-vector spaces attached to M.

• A finite dimensional space M_{B} = H^{i}(X(ℂ), ℚ)(j) which carries an action of complex conjugation and a Hodge structure (see [27] for more details).

• A finite dimensional filtered space M_{dR} = H_{dR}^{i}(X/ℚ)(j).

• Motivic cohomology spaces H_{f}^{0}(M) and H_{f}^{1}(M) which may be defined in terms of algebraic K-theory. For example, if X has a regular, proper flat model 𝔛 over Spec(ℤ) and i − 2j ≠ −1 then

H_{f}^{0}(M) = CH^{j}(X) ⊗ ℚ/hom. equiv. if M = h^{2j}(X)(j)
H_{f}^{1}(M) = im ((K_{2j−i−1}(𝔛) ⊗ ℚ)^{(j)} → (K_{2j−i−1}(X) ⊗ ℚ)^{(j)}) .

Using alterations this image space can also be defined without assuming the existence of a regular model 𝔛 [69].

The spaces H_{f}^{0}(M) and H_{f}^{1}(M) are conjectured to be finite dimensional but essentially the only examples where this is known are those mentioned above:

Examples continued. a) For M = h^{0}(Spec(L)) we have H_{f}^{0}(M) = ℚ and H_{f}^{1}(M) = 0 whereas for M = h^{0}(Spec(L))(1) we have H_{f}^{0}(M) = 0 and H_{f}^{1}(M) = O_{L}^{×} ⊗_{ℤ} ℚ. For M = h^{0}(Spec(L))(j) it is also known that H_{f}^{1}(M) = K_{2j−1}(L) ⊗ ℚ is finite dimensional [12].

b) For M = h^{1}(X)(1) we have H_{f}^{0}(M) = 0 and H_{f}^{1}(M) = Pic^{0}(X) ⊗_{ℤ} ℚ. This last space is finite dimensional by the Mordell-Weil theorem.

For a ℚ-vector space W and a ℚ-algebra R we put W_{R} = W ⊗_{ℚ} R. The period isomorphism M_{B,ℂ} ≅ M_{dR,ℂ} induces a map

α_{M} : M_{B,ℝ}^{+} → (M_{dR}/ Fil^{0} M_{dR})_{ℝ}.

For any motive M one has a dual motive M^{∗} with dual realizations. For example, if M = h^{i}(X)(j) where X is of dimension d then Poincare duality gives a perfect pairing

H^{i}(X)(j) × H^{2d−i}(X)(d − j) → H^{2d}(X)(d) ⟶^{tr} ℚ

which identifies M^{∗} with h^{2d−i}(X)(d − j).

Conjecture Mot_{∞}: There exists an exact sequence

0 → H_{f}^{0}(M)_{ℝ} ⟶^{c} ker(α_{M}) → H_{f}^{1}(M^{∗}(1))_{ℝ}^{∗} ⟶^{h}

H_{f}^{1}(M)_{ℝ} ⟶^{r} coker(α_{M}) → H_{f}^{0}(M^{∗}(1))_{ℝ}^{∗} → 0

Here c is a cycle class map, h a height pairing, and r the Beilinson regulator. Again, the exactness of this sequence is only known in a few cases, essentially those given by our standard examples a) and b).

Conjecture 1 (Vanishing Order):

r(M) = dim_{ℚ} H_{f}^{1}(M^{∗}(1)) − dim_{ℚ} H_{f}^{0}(M^{∗}(1))

Remark. The appearance of the dual motive M^{∗}(1) in the last two conjectures can be understood in two steps. First, one may conjecture that there are groups H_{c}^{i}(M) (”motivic cohomology with compact support and ℝ-coefficients”) which fit into a long exact sequence

⋯ → H_{c}^{i}(M) → H_{f}^{i}(M) → H_{D}^{i}(ℝ, M) → ⋯

and where H_{D}^{0}(ℝ, M) ≅ ker(α_{M}), H_{D}^{1}(ℝ, M) ≅ coker(α_{M}) (a definition of such groups, Arakelov Chow groups in his terminology, has been given by Goncharov in [40]). Secondly, one may conjecture a perfect duality of finite dimensional ℝ-vector spaces

H_{c}^{i}(M) × H_{f}^{2−i}(M^{∗}(1)) → H_{c}^{2}(ℚ(1)) ≅ ℝ,

an archimedean analogue of Poitou-Tate duality. Then Conjecture 1 says that r(M) is the Euler characteristic of motivic cohomology with compact support.

Define a ℚ-vector space of dimension 1

Ξ(M) := Det_{ℚ}(H_{f}^{0}(M)) ⊗ Det_{ℚ}^{−1}(H_{f}^{1}(M))
⊗ Det_{ℚ}(H_{f}^{1}(M^{∗}(1))^{∗}) ⊗ Det_{ℚ}^{−1}(H_{f}^{0}(M^{∗}(1))^{∗})
⊗ Det_{ℚ}^{−1}(M_{B}^{+}) ⊗ Det_{ℚ}(M_{dr}/ Fil^{0})

The exact sequence in Conjecture Mot_{∞} induces an isomorphism

ϑ_{∞} : ℝ ≅ Ξ(M) ⊗_{ℚ} ℝ

Conjecture 2 (Rationality):

ϑ_{∞}(L^{∗}(M)^{−1}) ∈ Ξ(M) ⊗ 1

This conjecture goes back to Deligne [27][Conj. 1.8] in the critical case (i.e. where α_{M} is an isomorphism) and Beilinson [5] in the general case.

3. Galois cohomology

Throughout this section we refer to [16] for unexplained notation and further details. Define for each prime p a complex RΓ_{f}(ℚ_{p}, M_{l})

= \begin{cases}M_{l}^{I_{p}} ⟶^{1−Fr_{p}} M_{l}^{I_{p}} & l ≠ p \\ D_{cris}(M_{l}) ⟶^{(1−Fr_{p},π)} D_{cris}(M_{l}) ⊕ D_{dR}(M_{l})/ Fil^{0} & l = p \end{cases}

One can construct a map of complexes RΓ_{f}(ℚ_{p}, M_{l}) → RΓ(ℚ_{p}, M_{l}) and one defines RΓ_{/f}(ℚ_{p}, M_{l}) as the mapping cone so that there is a distinguished triangle

RΓ_{f}(ℚ_{p}, M_{l}) → RΓ(ℚ_{p}, M_{l}) → RΓ_{/f}(ℚ_{p}, M_{l})

in the derived category of ℚ_{l} -vector spaces.

Let S be a finite set of primes containing l, ∞ and primes of bad reduction. There are distinguished triangles

RΓ_{c}(ℤ[\frac{1}{S}], M_{l}) → RΓ(ℤ[\frac{1}{S}], M_{l}) → ⨁_{p∈S} RΓ(ℚ_{p}, M_{l})

RΓ_{f}(ℚ, M_{l}) → RΓ(ℤ[\frac{1}{S}], M_{l}) → ⨁_{p∈S} RΓ_{/f}(ℚ_{p}, M_{l})

RΓ_{c}(ℤ[\frac{1}{S}], M_{l}) → RΓ_{f}(ℚ, M_{l}) → ⨁_{p∈S} RΓ_{f}(ℚ_{p}, M_{l}) (3.1)

Conjecture Mot_{l}: There are natural isomorphisms H_{f}^{0}(M)_{ℚ_{l}} ≅ H_{f}^{0}(ℚ, M_{l}) (cycle class map) and H_{f}^{1}(M)_{ℚ_{l}} ≅ H_{f}^{1}(ℚ, M_{l}) (Chern class map).

One can construct an isomorphism H_{f}^{i}(ℚ, M_{l}) ≅ H_{f}^{3−i}(ℚ, M_{l}^{∗}(1))^{∗} for all i. Hence Conjecture Mot_{l} computes the cohomology of RΓ_{f}(ℚ, M_{l}) in all degrees.

The exact triangle (3.1) induces an isomorphism

ϑ_{l} : Ξ(M) ⊗_{ℚ} ℚ_{l} ≅ Det_{ℚ_{l}} RΓ_{c}(ℤ[\frac{1}{S}], M_{l})

Let T_{l} ⊂ M_{l} be any G_{ℚ}-stable ℤ_{l}-lattice.

Conjecture 3 (Integrality):

ℤ_{l} · ϑ_{l}ϑ_{∞}(L^{∗}(M)^{−1}) = Det_{ℤ_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l})

This conjecture (for all l) determines L^{∗}(M) ∈ ℝ^{×} up to sign. It assumes Conjecture 2 and is independent of the choice of S and T_{l} [16][Lemma 5]. For M = h^{0}(Spec(L)) (resp. M = h^{1}(X)(1)) it is equivalent to the l-primary part of the analytic class number formula (resp. Birch and Swinnerton-Dyer conjecture). For a sketch of the argument giving this equivalence we refer to section 5.4 below.

Part 2. The Equivariant Refinement

In many situations one has ’extra symmetries’, more precisely there is a semisimple, finite dimensional ℚ-algebra A acting on M.

Examples:

• X an abelian variety, A = End(X) ⊗ ℚ

• X = X′ ×_{Spec(ℚ)} Spec(K), K/ℚ Galois with group G, A = ℚ[G]

• X a modular curve, A the Hecke algebra

4. Commutative Coefficients

If A is commutative (i.e. a product of number fields) one can construct L(_{A}M, s), Ξ(_{A}M), _{A}ϑ_{∞}, _{A}ϑ_{l} as before using determinants over A, A ⊗ ℝ, A ⊗ ℚ_{l}. L(_{A}M, s) is a meromorphic function with values in A ⊗ ℂ and

r(_{A}M) ∈H^{0}(Spec(A ⊗ ℝ), ℤ)
L^{∗}(_{A}M) ∈(A ⊗ ℝ)^{×}.

For a finitely generated A-module P we denote by dim_{A} P the function 𝔭 ↦ rank_{A_{𝔭}}(P_{𝔭}) on Spec(A).

One gets refinements of Conjectures 1 and 2 in a straightforward way.

Conjecture 1 (Equivariant Version):

r(_{A}M) = dim_{A} H_{f}^{1}(M^{∗}(1)) − dim_{A} H_{f}^{0}(M^{∗}(1))

Conjecture 2 (Equivariant Version):

Aϑ_{∞}(L^{∗}(_{A}M)^{−1}) ∈ Ξ(_{A}M) ⊗ 1

Somewhat more interesting is the generalization of Conjecture 3. There are many ℤ-orders 𝔄 ⊆ A unlike in the case A = ℚ. It turns out that in order to formulate a conjecture over 𝔄 one additional assumption is necessary.

Assume that there is a projective G_{ℚ}-stable 𝔄_{l} := 𝔄 ⊗ ℤ_{l} lattice T_{l} ⊂ M_{l}.

Then RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) is a perfect complex of 𝔄_{l}-modules and Det_{𝔄_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) is an invertible 𝔄_{l}-module. Since 𝔄_{l} is a product of local rings this means that Det_{𝔄_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) is in fact free of rank 1 over 𝔄_{l}. The existence of a projective lattice is guaranteed if 𝔄 is a maximal ℤ-order in A. If M = M_{0} ⊗ h^{0}(Spec(L)) arises by base change of a motive M_{0} to a finite Galois extension L/ℚ with group G then there is a projective lattice over the order 𝔄 = ℤ[G] in A = ℚ[G].

Conjecture 3 (Equivariant Version):

𝔄_{l} ·_{A}ϑ_{l} (_{A}ϑ_{∞}(L^{∗}(_{A}M)^{−1})) = Det_{𝔄_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l})

This conjecture (for all l) determines L^{∗}(_{A}M) ∈ (A ⊗ ℝ)^{×} up to 𝔄^{×}. Taking ”Norms form A to ℚ” one deduces the original Conjectures 1-3 from their equivariant refinements [16][Remark 11].

5. Proven cases

If 𝔄 is a maximal order in A, in particular if 𝔄 = ℤ, then Conjecture 3 has been considered traditionally in various cases, most notably in our Examples a) and b) above. In this section we review proven cases of Conjecture 3 with particular emphasis on non-maximal orders 𝔄. One should note here that Conjecture 3 over an order 𝔄 implies Conjecture 3 over any larger order 𝔄′ ⊇ 𝔄 but not vice versa.

5.1. Abelian extensions of ℚ and the main conjecture of Iwasawa Theory. The following theorem summarizes what is known about our Example a) when L/ℚ is abelian.

Theorem 5.1. (Burns-Greither) Let L/ℚ be a Galois extension with abelian group G, M = h^{0}(Spec(L))(j) with j ∈ ℤ, 𝔄 = ℤ[G] and l any prime number. Then Conjecture 3 holds.

Corollary 5.1. For any abelian field L and prime number l the cohomological Lichtenbaum conjecture (see [45][Thm. 1.4.1]) holds for the Dedekind Zeta-function at any j ∈ ℤ.

Proof. The cohomological Lichtenbaum conjecture is a rather immediate reformulation of Conjecture 3 for M = h^{0}(Spec(L))(j) and 𝔄 = ℤ (see the proof of [45][Thm. 1.4.1]). Hence it follows by general functoriality [16][Remark 11] from Theorem 5.1. □

We shall give the proof of Theorem 5.1 for j ≤ 0 in some detail in order to demonstrate how the formalism above unfolds in a concrete situation. One may also expect that the way in which we use the Iwasawa main conjecture and results on l-adic L-functions will be fairly typical for proofs of Conjecture 3 in a number of other situations. In essence we follow the proof [18] by Burns and Greither but our arguments cover the case l = 2 whereas those of [18] do not. We shall deduce Conjecture 3 for j ≤ 0 from an Iwasawa theoretic statement (Theorem 5.2 below). This descent argument is fairly direct except for difficulties arising from trivial zeros of the l-adic L-function for j = 0. These can be overcome by using the Theorem of Ferrero-Greenberg [33] for odd χ and results of Solomon [71] for even χ. The main difficulty for j < 0 (identification of the image of Beilinson’s elements in K_{1−2j}(L) under the étale Chern class map) has already been dealt with by Huber and Wildeshaus in [46]. Such a proof of Conjecture 3 by descent from Theorem 5.2 is also possible for j ≥ 1 provided one knows the non-vanishing of the l-adic L-function at j. Since this is currently the case only for j = 1 (as a consequence of Leopoldts conjecture for abelian fields [75][Cor. 5.30]) we do not give the details of this line of argument. Suffice it to say that Theorem 5.1 for j ≥ 1 is then proven somewhat indirectly by appealing to compatibility of Conjecture 3 with the functional equation of the L-function (see the recent preprint [6]). We do not address the issue of compatibility with the functional equation in this survey. We also do not go into the proof of the Iwasawa main conjecture because it is by now fairly well documented in the literature, even for l = 2 (see [41] for a proof via Euler systems).

Finally we remark that another proof of Theorem 5.1 (but only for 𝔄 a maximal order and l ≠ 2) has been given by Huber and Kings in [45]. This proof also appeals to compatibility with the functional equation. Theorem 5.1 for j < 0 and 𝔄 = ℤ was proven before by Kolster, Nguyen Quang-Do and Fleckinger [54] (with final corrections in [7]).

We need some notation. For an integer m ≥ 1 let ζ_{m} = e^{2πi/m}, L_{m} = ℚ(ζ_{m}), σ_{m} : L_{m} → ℂ the inclusion (which we also view as an archimedean place of L_{m}) and G_{m} = Gal(L_{m}/ℚ). By the Kronecker-Weber Theorem and general functoriality [16][Prop. 4.1b)] it suffices to prove Conjecture 3 for L_{m} in order to deduce it for all abelian L/ℚ. By the same argument we may, and occasionally will assume that m has at least two distinct prime factors. Let Ĝ_{m} be the set of complex characters of G_{m}, for η ∈ Ĝ_{m} let e_{η} ∈ ℂ[G_{m}] be the idempotent |G_{m}|^{−1} ∑_{g∈G_{m}} η(g)g^{−1} and denote by L(η, s) the Dirichlet L-function of η.

Explicit formulas for Dirichlet L-functions. In this section we fix 1 < m ≢ 2 mod 4 and write M = h^{0}(Spec(L_{m})), A = ℚ[G_{m}] and 𝔄 = ℤ[G_{m}]. One has

L(_{A}M, s) = (L(η, s))_{η∈Ĝ_{m}} ∈ ∏_{η∈Ĝ_{m}} ℂ = A ⊗ ℂ

and for M = h^{0}(Spec(L_{m})) the sequence in Conjecture Mot_{∞} is the ℝ-dual (with contragredient G_{m}-action) of the unit regulator sequence

0 → O_{L_{m}}^{×} ⊗_{ℤ} ℝ ⟶^{− log | |_{v}} ⨁_{v|∞} ℝ \underrightarrow{∑} ℝ → 0.

So one has

Ξ(_{A}M)^{#} = Det_{A}^{−1}(O_{L_{m}}^{×} ⊗_{ℤ} ℚ) ⊗ Det_{A}(X_{{v|∞}} ⊗_{ℤ} ℚ)

where for any set S of places of L_{m} we define Y_{S} = Y_{S}(L_{m}) := ⨁_{v∈S} ℤ and X_{S} = X_{S}(L_{m}) to be the kernel of the sum map Y_{S} → ℤ. Moreover, the superscript # indicates that the G_{m}-action has been twisted by the automorphism g ↦ g^{−1} of G_{m}.

It is well known that

L(η, 0) = − ∑_{a=1}^{f_{η}} (\frac{a}{f_{η}} − \frac{1}{2}) η(a)

\frac{d}{ds}L(η, s)|_{s=0} = − \frac{1}{2} ∑_{a=1}^{f_{η}} log |1 − e^{2πia/f_{η}}|η(a) η ≠ 1 even

where f_{η}|m is the conductor of η and η(a) = 0 for (a, f_{η}) > 1, and that L(η, 0) ≠ 0 if and only if η = 1 or η is odd [75][Ch. 4]. One deduces Conjecture 1 for M = h^{0}(Spec(L_{m})) since

dim_{ℂ} e_{η}(O_{L_{m}}^{×} ⊗_{ℤ} ℂ) = \begin{cases}1 & η ≠ 1 even \\ 0 & otherwise. \end{cases}

Before dealing with Conjecture 2 we introduce some further notation. We denote by Ĝ_{m}^{ℚ} the set of ℚ-rational characters , i.e. Aut(ℂ)-orbits on Ĝ_{m}. For each χ ∈ Ĝ_{m}^{ℚ} we put e_{χ} = ∑_{η∈χ} e_{η} ∈ A and we denote by ℚ(χ) the field generated by the values of η for any η ∈ χ. Then there is a ring isomorphism

A = ∏_{χ∈Ĝ_{m}^{ℚ}} ℚ(χ).

We put

L(χ, 0) := ∑_{η∈χ} L(η, 0)e_{η} ∈ Ae_{χ} ≅ ℚ(χ)

and note that L(χ, 0)^{#} := ∑_{η∈χ} L(η^{−1}, 0)e_{η}.

For f_{η} ≠ 1 the image of (1 − ζ_{f_{η}}) ∈ L_{m} under the regulator map is

(1 − ζ_{f_{η}}) ↦ − ∑_{a∈(ℤ/mℤ)^{×}/±1} log |1 − e^{2πia/f_{η}}|^{2}τ_{a}^{−1}(σ_{m})

where τ_{a} is the automorphism ζ_{m} ↦ ζ_{m}^{a} of L_{m}, and hence for any even character η ≠ 1 the image of e_{η}(1 − ζ_{f_{η}}) is L′(η^{−1}, 0) · 2 · [L_{m} : L_{f_{η}}] · e_{η}(σ_{m}). Note here that e_{η}τ_{a} = η(a)e_{η}, and that σ_{m} (resp. 1 − ζ_{f_{η}} if f_{η} is a prime power) lies in the larger 𝔄-module Y_{\{v|∞\}} ⊃ X_{\{v|∞\}} (resp. O_{L_{fη}} [\frac{1}{f_{η}}]^{×} ⊃ O_{L_{fη}}^{×}) but application of e_{η} (or equivalently − ⊗_{𝔄} ℚ(η)) turns this inclusion into an equality for η ≠ 1.

There is a canonical isomorphism

Ξ(_{A}M)^{#} ⟶^{∼} Det_{A}^{−1}(O_{L_{m}}^{×} ⊗_{ℤ} ℚ) ⊗ Det_{A}(X_{{v|∞}} ⊗_{ℤ} ℚ)

⟶^{∼} ∏_{\begin{matrix}χ≠1 \\ even \end{matrix}} Det_{ℚ(χ)}^{−1}(O_{L_{m}}^{×} ⊗_{𝔄} ℚ(χ)) ⊗ Det_{ℚ(χ)}(X_{\{v|∞\}} ⊗_{𝔄} ℚ(χ)) × ∏_{other χ} ℚ(χ)

⟶^{∼} ∏_{\begin{matrix}χ≠1 \\ even \end{matrix}}(O_{L_{m}}^{×} ⊗_{𝔄} ℚ(χ))^{−1} ⊗_{ℚ(χ)} (X_{\{v|∞\}} ⊗_{𝔄} ℚ(χ)) × ∏_{other χ} ℚ(χ)

and in this description _{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}) = (L^{∗}(_{A}M, 0)^{−1})^{#}_{A}ϑ_{∞}(1) has components

(5.1)  _{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1})_{χ} = \begin{cases}2 · [L_{m} : L_{f_{χ}}][1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m} & χ ≠ 1 even \\ (L(χ, 0)^{#})^{−1} & otherwise. \end{cases}

We now fix a prime number l and put A_{l} := A ⊗ ℚ_{l}. The isomorphism

Ξ(_{A}M)^{#} ⊗ ℚ_{l} ⟶^{_{A}ϑ_{l}} Det_{A_{l}} RΓ_{c}(ℤ[\frac{1}{ml}], M_{l})^{#}

is given by the composite

Det_{A_{l}}^{−1}(O_{L_{m}}^{×} ⊗_{ℤ} ℚ_{l}) ⊗ Det_{A_{l}}(X_{{v|∞}} ⊗_{ℤ} ℚ_{l})
(5.2)  ⟶^{∼} Det_{A_{l}}^{−1}(O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{ℤ} ℚ_{l}) ⊗ Det_{A_{l}}(X_{\{v|ml∞\}} ⊗_{ℤ} ℚ_{l})
(5.3)  ⟶^{∼} Det_{A_{l}}^{−1}(O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{ℤ} ℚ_{l}) ⊗ Det_{A_{l}}(X_{\{v|ml∞\}} ⊗_{ℤ} ℚ_{l})
(5.4)  ⟶^{∼} Det_{A_{l}} RΓ_{c}(ℤ[\frac{1}{ml}], M_{l})^{#}.

Here (5.2) is induced by the short exact sequences

0 → O_{L_{m}}^{×} ⊗_{ℤ} ℚ → O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{ℤ} ℚ ⟶^{val} Y_{\{v|ml\}} ⊗_{ℤ} ℚ → 0

0 → Y_{{v|ml}} ⊗_{ℤ} ℚ → X_{{v|ml∞}} ⊗_{ℤ} ℚ → X_{{v|∞}} ⊗_{ℤ} ℚ → 0

and the identity map on Y_{{v|ml}} ⊗_{ℤ} ℚ. The isomorphism (5.3) is multiplication with the (Euler-) factor (see [15][Lemma 2]) ∏_{p|ml} E_{p}^{#} ∈ A^{×} where E_{p} ∈ A^{×} is defined by

(5.5)  E_{p} = ∑_{η(D_{p})=1} |D_{p}/I_{p}|e_{η} + ∑_{η(D_{p})≠1}(1 − η(p))^{−1}e_{η}

with D_{p} (resp. I_{p}) denoting the decomposition subgroup (resp. inertia subgroup) of G_{m} at p.

Finally (5.4) arises as follows. Put

Δ(L_{m}) := R Hom_{ℤ_{l}}(RΓ_{c}(ℤ[\frac{1}{ml}], T_{l}), ℤ_{l})[−3].

Then Δ(L_{m}) is a perfect complex of 𝔄_{l}-modules and there is a natural isomorphism

Det_{𝔄_{l}} Δ(L_{m}) ≅ Det_{𝔄_{l}} RΓ_{c}(ℤ[\frac{1}{ml}], T_{l})^{#}.

On the other hand, the cohomology of Δ(L_{m}) can be computed by Tate-Poitou duality, the Kummer sequence and some additional arguments [15][Prop. 3.3]. One finds H^{i}(Δ(L_{m})) = 0 for i ≠ 1, 2, a canonical isomorphism

H^{1}(Δ(L_{m})) ≅ H^{1}(O_{L_{m}}[\frac{1}{ml}], ℤ_{l}(1)) ≅ O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{ℤ} ℤ_{l}

and a short exact sequence

0 → Pic(O_{L_{m}}[\frac{1}{ml}]) ⊗_{ℤ} ℤ_{l} → H^{2}(Δ(L_{m})) → X_{\{v|ml∞\}} ⊗_{ℤ} ℤ_{l} → 0.

After tensoring with ℚ_{l} this computation gives the isomorphism (5.4). Conjecture 3 then becomes the statement that the element ∏_{p|ml} E_{p}^{#} ·_{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}) described in (5.1) and (5.5) is not only a basis of Det_{A_{l}} Δ(L_{m}) ⊗ ℚ_{l} but in fact an 𝔄_{l}-basis of Det_{𝔄_{l}} Δ(L_{m}).

Iwasawa Theory. We fix l and m as in the last section and retain the notation introduced there. Put

Λ = \underleftarrow{lim}_{n} ℤ_{l}[G_{ml^{n}}] ≅ ℤ_{l}[G_{ℓm_{0}}][[T]]

where

m = m_{0}l^{ord_{l}(m)}; ℓ = \begin{cases}l & l ≠ 2 \\ 4 & l = 2. \end{cases}

The Iwasawa algebra Λ is a finite product of complete local 2-dimensional Cohen-Macaulay (even complete intersection) rings. However, Λ is regular if and only if l ∤ #G_{ℓm_{0}}. As usual, the element T = γ − 1 ∈ Λ depends on the choice of a topological generator γ of Gal(L_{ml^{∞}}/L_{ℓm_{0}}) ≅ ℤ_{l}.

Defining a perfect complex of Λ-modules

Δ^{∞} = \underleftarrow{lim}_{n} Δ(L_{m_{0}l^{n}})

we have H^{i}(Δ^{∞}) = 0 for i ≠ 1, 2, a canonical isomorphism

H^{1}(Δ^{∞}) ≅ U_{\{v|ml\}}^{∞} := \underleftarrow{lim}_{n} O_{L_{m_{0}l^{n}}} [\frac{1}{ml}]^{×} ⊗_{ℤ} ℤ_{l}

and a short exact sequence

0 → P_{{v|ml}}^{∞} → H^{2}(Δ^{∞}) → X_{{v|ml∞}}^{∞} → 0

where

P_{\{v|ml\}}^{∞} := \underleftarrow{lim}_{n} Pic(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}]) ⊗_{ℤ} ℤ_{l}, X_{\{v|ml∞\}}^{∞} := \underleftarrow{lim}_{n} X_{\{v|m_{0}l∞\}}(L_{m_{0}l^{n}}) ⊗_{ℤ} ℤ_{l}.

All limits are taken with respect to Norm maps (on Y_{S} this is the map sending a place to its restriction). For d | m_{0} put

η_{d} :=(1 − ζ_{ℓdl^{n}})_{n≥0} ∈ U_{{v|ml}}^{∞}
σ :=(σ_{ℓm_{0}l^{n}})_{n≥0} ∈ Y_{{v|ml∞}}^{∞}
θ_{d} :=(g_{ℓdl^{n}})_{n≥0} ∈ \frac{1}{[L_{m_{0}} : L_{d}]} · (γ − χ_{cyclo}(γ))^{−1}Λ

where

(5.6)  g_{k} = − ∑_{0<a<k,(a,k)=1} (\frac{a}{k} − \frac{1}{2}) τ_{a,k}^{−1} ∈ ℚ[G_{k}]

with τ_{a,k} ∈ G_{k} defined by τ_{a,k}(ζ_{k}) = ζ_{k}^{a}. Here we also view τ_{a,k} as an element of ℚ[G_{k′}] for k | k′ (which allows us to view θ_{d} as an element of the fraction field of Λ for d | m_{0}) by τ_{a,k} ↦ [G_{k′} : G_{k}]^{−1} ∑_{a′≡a mod k} τ_{a′,k′}. The relationship between θ_{d} and l-adic L-functions in the usual normalization is given by the interpolation formula

(5.7)  χχ_{cyclo}^{j}(θ_{d}) = (1 − χ^{−1}(l)l^{−j})L(χ^{−1}, j) =: L_{l}(χ^{−1}ω^{1−j}, j)

for all characters χ of conductor dl^{n} and j ≤ 0 (here ω denotes the Teichmueller character).

We fix an embedding ℚ̄_{l} → ℂ and identify Ĝ_{k} with the set of ℚ̄_{l}-valued characters. The total ring of fractions

(5.8)  Q(Λ) ≅ ∏_{ψ∈Ĝ_{ℓm_{0}}^{ℚ_{l}}} Q(ψ)

of Λ is a product of fields indexed by the ℚ_{l}-rational characters of G_{ℓm_{0}}. Since for any place w of ℚ the ℤ[G_{m_{0}l^{n}}]-module Y_{{v|w}}(L_{m_{0}l^{n}}) is induced from the trivial module ℤ on the decomposition group D_{w} ⊆ G_{m_{0}l^{n}}, and for w = ∞ (resp. nonarchimedean w) we have [G_{m_{0}l^{n}} : D_{w}] = [L_{m_{0}l^{n}} : ℚ]/2 (resp. the index [G_{m_{0}l^{n}} : D_{w}] is bounded as n → ∞) one computes easily

(5.9) dim_{Q(ψ)}(U_{\{v|ml\}}^{∞} ⊗_{Λ} Q(ψ)) = dim_{Q(ψ)}(Y_{\{v|ml∞\}}^{∞} ⊗_{Λ} Q(ψ)) = \begin{cases}1 & ψ even \\ 0 & ψ odd. \end{cases}

Note that the inclusion X_{{v|ml∞}}^{∞} ⊆ Y_{{v|ml∞}}^{∞} becomes an isomorphism after tensoring with Q(ψ) and that e_{ψ}(η_{m_{0}}^{−1} ⊗ σ) is a Q(ψ)-basis of

Det_{Q(ψ)}^{−1}(U_{{v|ml}}^{∞} ⊗_{Λ} Q(ψ)) ⊗ Det_{Q(ψ)}(X_{{v|ml∞}}^{∞} ⊗_{Λ} Q(ψ)
≅ Det_{Q(ψ)} (Δ^{∞} ⊗_{Λ} Q(ψ))

for even ψ. For odd ψ the complex Δ^{∞} ⊗_{Λ} Q(ψ) is acyclic and we can view e_{ψ}θ_{m_{0}} ∈ Q(ψ) as an element of

Det_{Q(ψ)} (Δ^{∞} ⊗_{Λ} Q(ψ)) ≅ Q(ψ).

Note also that e_{ψ}θ_{m_{0}} = 0 (resp. e_{ψ}(η_{m_{0}}^{−1} ⊗ σ) = 0) if ψ is even (resp. odd). Hence we obtain an element

L := θ_{m_{0}}^{−1} + 2 · η_{m_{0}}^{−1} ⊗ σ ∈ Det_{Q(Λ)} (Δ^{∞} ⊗_{Λ} Q(Λ)) .

Theorem 5.2. There is an equality of invertible Λ-submodules

Λ · L = Det_{Λ} Δ^{∞}

of Det_{Q(Λ)} (Δ^{∞} ⊗_{Λ} Q(Λ)).

This statement is an Iwasawa theoretic analogue of Conjecture 3 and, as we shall see below, it implies Conjecture 3 for M = h^{0}(Spec L_{d})(j), 𝔄 = ℤ[G_{d}] and any d | m_{0}l^{∞} and j ≤ 0. Our proof will also show that Theorem 5.2 is essentially equivalent to the main conjecture of Iwasawa theory, combined with the vanishing of μ-invariants of various Iwasawa modules. The idea to seek a conjecture that unifies the Tamagawa number conjecture on the one hand and the Iwasawa main conjecture on the other goes back to Kato [48], [49][3.3.8].

Proof of Theorem 5.2. The following Lemma allows a prime-by-prime analysis of the identity in Theorem 5.2.

Lemma 5.3. Let R be a Noetherian Cohen-Macaulay ring with total ring of fractions Q(R). Suppose R is a finite product of local rings. If I and J are invertible R-submodules of some invertible Q(R)-module M then I = J if and only if I_{𝔮} = J_{𝔮} (inside M_{𝔮}) for all height 1 prime ideals 𝔮 of R.

Proof. Since R is a product of local rings both I and J are free of rank 1 with bases b_{I}, b_{J} ∈ M, say. Since Q(R) is Artinian, hence a product of local rings, M is free with basis b_{M}, say. Writing b_{I} = \frac{x}{y}b_{M} we find that x cannot be a zero-divisor in R since I is R-free. Hence \frac{x}{y} is a unit in Q(R) and we may write

b_{J} = \frac{x′}{y′} b_{M} = \frac{x′y}{y′x}b_{I} =: \frac{a}{b}b_{I}.

Since R is Cohen-Macaulay and b is not a zero-divisor all prime divisors 𝔭_{1}, ..., 𝔭_{n} of the principal ideal bR have height 1 [59][Thm. 17.6]. By assumption bR_{𝔭_{i}} = aR_{𝔭_{i}} for i = 1, .., n. Hence a ∈ ϕ_{i}^{−1}(bR_{𝔭_{i}}) where ϕ_{i} : R → R_{𝔭_{i}} is the natural map, and the primary decomposition of the ideal bR [59][Th.6.8] gives

a ∈ ϕ_{1}^{−1}(bR_{𝔭_{1}}) ∩ ⋯ ∩ ϕ_{n}^{−1}(bR_{𝔭_{n}}) = bR.

So \frac{a}{b} ∈ R and I = Rb_{I} ⊇ Rb_{J} = J. By symmetry I = J. □

At this stage a fundamental distinction presents itself. We call a height 1 prime 𝔮 of Λ regular (resp. singular) if l ∉ 𝔮 (resp. l ∈ 𝔮). For a regular prime 𝔮 the ring Λ_{𝔮} is a discrete valuation ring with fraction field Q(ψ) for some ψ = ψ_{𝔮} ∈ Ĝ_{ℓm_{0}}^{ℚ_{l}}. The residue field ℚ_{l}(𝔮) of Λ_{𝔮} is a finite extension of the field of values ℚ_{l}(ψ) of ψ. For singular primes, on the other hand, the ring Λ_{𝔮} is regular if and only if Λ is.

Analysis of regular primes. The computation of the cohomology of Δ^{∞} given above shows that the identity

Λ_{𝔮} · L = Det_{Λ_{𝔮}} (Δ^{∞} ⊗_{Λ} Λ_{𝔮})

is equivalent to

(5.10) Fit_{Λ_{𝔮}}(U_{{v|ml},𝔮}^{∞}/Λ_{𝔮} · η_{m_{0}}) = Fit_{Λ_{𝔮}}(P_{{v|ml},𝔮}^{∞}) · Fit_{Λ_{𝔮}}(X_{{v|ml∞},𝔮}^{∞}/Λ_{𝔮} · σ)

if ψ_{𝔮} is even and to

(5.11)  θ_{m_{0}} · Fit_{Λ_{𝔮}}(U_{{v|ml},𝔮}^{∞}) = Fit_{Λ_{𝔮}}(P_{{v|ml},𝔮}^{∞}) · Fit_{Λ_{𝔮}}(X_{{v|ml∞},𝔮}^{∞})

if ψ_{𝔮} is odd. Here Fit_{Λ_{𝔮}}(M) denotes the first Fitting (or order) ideal of any finitely generated torsion Λ_{𝔮}-module M. Put

U^{∞} := \underleftarrow{lim}_{n} O_{L_{m_{0}l^{n}}}^{×} ⊗_{ℤ} ℤ_{l}, P^{∞} := \underleftarrow{lim}_{n} Pic(O_{L_{m_{0}l^{n}}}) ⊗_{ℤ} ℤ_{l}.

The exact sequences of Λ-modules

(5.12)  0 → U^{∞} → U_{{v|ml}}^{∞} → Y_{{v|l}}^{∞} → P^{∞} → P_{{v|ml}}^{∞} → 0

and

0 → X_{{v|m_{0}}}^{∞} → X_{{v|lm∞}}^{∞} → Y_{{v|l}}^{∞} ⊕ Y_{{v|∞}}^{∞} → 0

together with the fact that σ is a basis of Y_{{v|∞},𝔮}^{∞} for ψ_{𝔮} even (resp. U_{𝔮}^{∞} = ℤ_{l}(1)_{𝔮} and Y_{{v|∞},𝔮}^{∞} = 0 for ψ_{𝔮} odd), imply that the identities (5.10) and (5.11) are equivalent, respectively, to

(5.13)  Fit_{Λ_{𝔮}}(U_{𝔮}^{∞}/Λ_{𝔮} · η_{m_{0}}) = Fit_{Λ_{𝔮}}(P_{𝔮}^{∞}) · Fit_{Λ_{𝔮}}(X_{{v|m_{0}},𝔮}^{∞})

and

(5.14)  θ_{m_{0}} · Fit_{Λ_{q}}(ℤ_{l}(1)_{𝔮}) = Fit_{Λ_{𝔮}}(P_{𝔮}^{∞}) · Fit_{Λ_{𝔮}}(X_{{v|m_{0}},𝔮}^{∞}).

The following two Lemmas then finish the verification of these identities.

Lemma 5.4. (Main conjecture) For any regular height 1 prime 𝔮 of Λ let d | m_{0} be such that ψ_{𝔮} has conductor d or dℓ. Put ϵ = 0 or 1 according to whether ψ_{𝔮} ≠ 1 or ψ_{𝔮} = 1. Then

Fit_{Λ_{𝔮}}(U_{𝔮}^{∞}/Λ_{𝔮} · T^{ϵ} · η_{d}) = Fit_{Λ_{𝔮}}(P_{𝔮}^{∞})

if ψ_{𝔮} is even, and

θ_{d} · Fit_{Λ_{q}}(ℤ_{l}(1)_{𝔮}) = Fit_{Λ_{𝔮}}(P_{𝔮}^{∞})

if ψ_{𝔮} is odd.

Lemma 5.5. For d and ϵ as in Lemma 5.4 we have

Fit_{Λ_{𝔮}}(Λ_{𝔮} · T^{ϵ} · η_{d}/Λ_{𝔮} · η_{m_{0}}) = T^{−ϵ} ∏_{p|m_{0}, p∤d}(1 − Fr_{p}^{−1}) = Fit_{Λ_{𝔮}}(X_{{v|m_{0}},𝔮}^{∞})

if ψ_{𝔮} is even and

Λ_{𝔮} · θ_{m_{0}}/θ_{d} = ∏_{p|m_{0}, p∤d}(1 − Fr_{p}^{−1}) = Fit_{Λ_{𝔮}}(X_{{v|m_{0}},𝔮}^{∞})

if ψ_{𝔮} is odd. Here we view the Frobenius automorphism Fr_{p} ∈ G_{m_{0}l^{∞}}/I_{p} as an element of G_{m_{0}l^{∞}} ⊂ Λ using the fact that I_{p} ≅ G_{p^{ordp(m)}} is canonically a direct factor of G_{m_{0}l^{∞}}.

Proof of Lemma 5.4. Denote by C^{cyclo} the Λ-submodule of U^{∞} generated by T η_{1} and η_{n} for 1 ≠ n | m_{0}. Let ψ be an even character of G_{ℓm_{0}}. The main conjecture [41][Th. 3.1+ Rem. b) and c)] says that the characteristic ideal of the ℤ_{l}[ψ][[T]]-module (P^{∞})_{ψ} equals that of (U^{∞}/C^{cyclo})_{ψ} where for any Λ-module M we put M_{ψ} = M ⊗_{ℤ_{l}[G_{ℓm_{0}} ]} ℤ_{l}[ψ] = M ⊗_{Λ} ℤ_{l}[ψ][[T]]. For any height 1 prime ideal 𝔮 with ψ_{𝔮} = ψ the map Λ → Λ_{𝔮} factors through Λ → ℤ_{l}[ψ][[T]] and we deduce Fit_{Λ_{𝔮}}(U_{𝔮}^{∞}/C_{𝔮}^{cyclo}) = Fit_{Λ_{𝔮}}(P_{𝔮}^{∞}). It remains to show that C_{𝔮}^{cyclo} is generated by η_{d} over Λ_{𝔮} (or T η_{1} if ψ_{𝔮} = 1). This follows from the distribution (or Euler system) relations satisfied by the η_{n}. For d | m_{0} put

N_{d} = ( ∑_{τ∈Gal(L_{m_{0}} /L_{d})} τ) ∈ Λ.

Then for d | n | m_{0}

(5.15) N_{d} · η_{n} = [L_{m_{0}} : L_{n}]Norm_{L_{n}/L_{d}}η_{n} = [L_{m_{0}} : L_{n}] ( ∏_{p|n, p∤d}(1 − Fr_{p}^{−1})) η_{d}.

Since ψ(N_{d}) = [L_{m_{0}} : L_{d}] ≠ 0 the element N_{d} is a unit in Λ_{𝔮} and we have therefore expressed η_{n} as a Λ_{𝔮}-multiple of η_{d}. For d ∤ n on the other hand, ψ is a nontrivial character of Gal(L_{m_{0}}/L_{n}) and we have

(η_{n})_{𝔮} = [L_{m_{0}} : L_{n}]^{−1}(N_{n}η_{n})_{𝔮} = [L_{m_{0}} : L_{n}]^{−1}ψ(N_{n})η_{n,𝔮} = 0.

Let now ψ be an odd character of G_{ℓm_{0}} and ω the Teichmueller character. The main conjecture [41][Th. 3.2] then asserts that the characteristic ideal of the ℤ_{l}[ψ][[T]]-module (P^{∞})_{ψ} equals e_{ψ} \frac{1}{2}θ_{d} if ψ ≠ ω. For ψ = ω the statement of [41][Th. 3.2] is that both

\frac{1}{2}θ_{1} · Fit_{ℤ_{l}[[T]]}(ℤ_{l}(1)) = \frac{1}{2}θ_{1} · (γ − χ_{cyclo}(γ)) · ℤ_{l}[[T]]

and the characteristic ideal of (P^{∞})_{ω} are equal to ℤ_{l}[[T]].

We note that the main conjecture for an even character ψ is equivalent to the main conjecture for the odd character ωψ^{−1} by an argument involving duality and Kummer theory (see the proof of Th. 3.2 in [41]). □

Proof of Lemma 5.5. For any prime p | m_{0} the decomposition group D_{p} ⊆ G_{m_{0}l^{∞}} has finite index and the inertia subgroup I_{p} ⊆ D_{p} is finite. Moreover one has a direct product decomposition D_{p} = I_{p} × \overline{< Fr_{p} >}. We have an isomorphism of Λ-modules

Y_{{v|p}}^{∞} ≅ Ind_{D_{p}}^{G_{m_{0}l^{∞}}} ℤ_{l}

and an isomorphism of ℤ_{l}[[D_{p}]]-modules

ℤ_{l} ≅ ℤ_{l}[[D_{p}]]/ < g − 1|g ∈ I_{p}; 1 − Fr_{p}^{−1} > .

So if ψ_{𝔮}|_{I_{p}} ≠ 1, i.e. p | d, then Y_{{v|p},𝔮}^{∞} = 0 and if ψ_{𝔮}|_{I_{p}} = 1 the characteristic ideal of Y_{{v|p},𝔮}^{∞} is generated by 1 − Fr_{p}^{−1}. The exact sequence of Λ-modules

0 → X_{{v|m_{0}}}^{∞} → Y_{{v|m_{0}}}^{∞} → ℤ_{l} → 0

accounts for the term T^{ϵ}. This verifies the second equalities in the two displayed equations of Lemma 5.5.

The respective first equalities follow for even ψ_{𝔮} from the Euler system relations (5.15) with n = m_{0} together with the fact that N_{d} is a unit in Λ_{𝔮}. For odd ψ_{𝔮} we also have an Euler system relation

(5.16)  N_{d} · θ_{m_{0}} = [L_{m_{0}} : L_{d}] · ( ∏_{p|m_{0}, p∤d}(1 − Fr_{p}^{−1})) θ_{d}

which expresses θ_{m_{0}} as a Λ_{𝔮}-multiple of θ_{d} since N_{d} is a unit in Λ_{𝔮} (note here our convention that we view the element θ_{d} ∈ ℤ_{l}[[G_{dl^{∞}}]] as the element [L_{m_{0}} : L_{d}]^{−1}N_{d} · θ̃_{d} ∈ Λ where˜ denotes any lift). □

Analysis of singular primes. The singular primes of Λ are in bijection with the ℚ_{l}-rational characters of G_{ℓm_{0}} of order prime to l. For a singular height 1 prime 𝔮 we denote by ψ_{𝔮} the corresponding character with ℚ_{l}-rational idempotent e_{ψ_{𝔮}} ∈ Λ.

The following Lemma relates μ-invariants to localization at singular primes.

Lemma 5.6. Let M be a finitely generated torsion Λ-module, 𝔮 a singular prime with character ψ = ψ_{𝔮} and χ a ℚ̄_{l}-valued character of G_{ℓm_{0}} with prime to l-part ψ. The following are equivalent

(i) The μ-invariant of the ℤ_{l}[[T]]-module e_{ψ}M vanishes.

(ii) The μ-invariant of the ℤ_{l}[χ][[T]]-module

M_{χ} := M ⊗_{ℤ[G_{m_{0}ℓ}]} ℤ_{l}[χ] ≅ e_{ψ}M ⊗_{ℤ[G_{m_{0}ℓ}]} ℤ_{l}[χ]

vanishes. (iii) M_{𝔮} = 0.

Proof. It is well known that the μ-invariant in (i) (resp.(ii)) vanishes if and only if (e_{ψ}M)_{(l)} = 0 (resp. M_{χ,(π)} = 0) where π is a uniformizer of ℤ_{l}[χ]. Since 𝔮 is the radical of (l) in Λ_{𝔮} the map (e_{ψ}M)_{(l)} → (e_{ψ}M)_{𝔮} = M_{𝔮} is an isomorphism which gives the equivalence of (i) and (iii). The map M/𝔮M → M_{χ}/πM_{χ} is an isomorphism which gives the equivalence of (ii) and (iii) by Nakayama’s Lemma.

□

Unlike the case of regular primes, we not only need the equality of the μinvariant of an Iwasawa module and a p-adic L-function but the vanishing of both.

We now analyze the localization of Δ^{∞} at a singular prime 𝔮. By the theorem of Ferrero and Washington [75][Thm. 7.15] the μ-invariant of P^{∞} vanishes. The module X_{{v|ml}}^{∞} is finite free over ℤ_{l} and hence has vanishing μ-invariant. The surjection P^{∞} → P_{{v|ml}}^{∞} and the exact sequence

0 → X_{{v|ml}}^{∞} → X_{{v|ml∞}}^{∞} → Y_{{v|∞}}^{∞} → 0

then show that

H^{2}(Δ^{∞})_{𝔮} = Y_{{v|∞},𝔮}^{∞} ≅ (Λ_{𝔮}/(c − 1)) · σ

where c ∈ G_{ℓm_{0}} is the complex conjugation. Concerning H^{1}(Δ^{∞})_{𝔮} one shows that η_{m_{0}} is a generator using the fact that all graded pieces of the filtration

Λ · η_{m_{0}} ⊆ C^{cyclo} ⊆ U^{∞} ⊆ U_{{v|ml}}^{∞}

have vanishing μ-invariant. The quotient U_{{v|ml}}^{∞}/U^{∞} injects into the finite free ℤ_{l}-module Y_{{v|l}}^{∞} and hence has vanishing μ-invariant. The quotient U^{∞}/C^{cyclo} has vanishing μ-invariant by the main result of C. Greither’s appendix to this article. Finally, the quotient C_{𝔮}^{cyclo}/Λ_{𝔮} · η_{m_{0}} is zero by (5.15) and the fact that 1 − Fr_{p}^{−1} ∈ Λ_{𝔮}^{×} for p | m_{0}. This follows because Fr_{p} has infinite order in G_{m_{0}l^{∞}}: We have Fr_{p}^{−N} = (1+T)^{ν} for some integer N and 0 ≠ ν ∈ ℤ_{l}. So the image 1−(1+T)^{ν} of 1 − Fr_{p}^{−N} in Λ/𝔮 ≅ 𝔽_{l}(ψ_{𝔮})[[T]] is nonzero, and hence so is the image of 1 − Fr_{p}^{−1}.

Having established that η_{m_{0}} is a generator of H^{1}(Δ^{∞})_{𝔮} one notes that

H^{1}(Δ^{∞})_{𝔮} ≅ (Λ_{𝔮}/(c − 1)) · η_{m_{0}}

since the image of η_{m_{0}} in the scalar extension to Q(ψ) vanishes precisely for odd ψ. So for l ≠ 2 we conclude that H^{1}(Δ^{∞})_{𝔮} (resp. H^{2}(Δ^{∞})_{𝔮}) is free with basis η_{m_{0}} (resp. σ) if ψ_{𝔮} is even, whereas H^{i}(Δ^{∞})_{𝔮} = 0 for i = 1, 2 if ψ_{𝔮} is odd. This proves the 𝔮-part of Theorem 5.2 for ψ_{𝔮} even, and for ψ_{𝔮} odd it remains to remark that θ_{m_{0}} is a unit in Λ_{𝔮} (otherwise there would be a ℚ̄_{l}-character χ of G_{m_{0}ℓ} so that all coefficients of χ(θ_{m_{0}}) ∈ ℤ_{l}[χ][[T]] have positive l-adic valuation which contradicts [75][p.131]) .

For l = 2 we conclude that the cohomology modules H^{i}(Δ^{∞})_{𝔮} are not of finite projective dimension over Λ_{𝔮} and hence that we cannot pass to cohomology when computing Det_{Λ_{𝔮}}(Δ_{𝔮}^{∞}). However, the complex Δ_{𝔮}^{∞} represents a class in

Ext_{Λ_{𝔮}}^{2}(H^{2}(Δ^{∞})_{𝔮}, H^{1}(Δ^{∞})_{𝔮}) ≅ Ext_{Λ_{𝔮}}^{2}(Λ_{𝔮}/(c − 1), Λ_{𝔮}/(c − 1))

and we have Ext_{Λ_{𝔮}}^{i}(Λ_{𝔮}/(c − 1), Λ_{𝔮}/(c − 1)) ≅ Λ_{𝔮}/(2, c − 1) for even i ≥ 2 as can be seen from the projective resolution

⋯ → Λ_{𝔮} ⟶^{1−c} Λ_{𝔮} ⟶^{1+c} Λ_{𝔮} ⟶^{1−c} Λ_{𝔮} ⟶ Λ_{𝔮}/(c − 1) → 0.

The complex Δ_{𝔮}^{∞} can be constructed as the pushout

(5.17)

0 ⟶ (1 + c) · Λ_{𝔮} ⟶ Λ_{𝔮} ⟶^{1−c} Λ_{𝔮} ⟶ Λ_{𝔮}/(c − 1) ⟶ 0
μ↓  ↓  ∥  ∥
0 ⟶ Λ_{𝔮}/(c − 1) ⟶ Δ_{𝔮}^{∞,1} ⟶ Δ_{𝔮}^{∞,2} ⟶ Λ_{𝔮}/(c − 1) ⟶ 0

via some homomorphism μ, determined by μ(1 + c). This diagram induces a commutative diagram

Ext_{Λ_{𝔮}}^{i}((c + 1) · Λ_{𝔮}, Λ_{𝔮}/(c − 1)) ⟶ Ext_{Λ_{𝔮}}^{i+2}(Λ_{𝔮}/(c − 1), Λ_{𝔮}/(c − 1))
μ_{∗}↓  ∥
Ext_{Λ_{𝔮}}^{i}(Λ_{𝔮}/(c − 1), Λ_{𝔮}/(c − 1)) ⟶ Ext_{Λ_{𝔮}}^{i+2}(Λ_{𝔮}/(c − 1), Λ_{𝔮}/(c − 1))

where the horizontal maps are isomorphisms for large i since Δ_{𝔮}^{∞,1} and Δ_{𝔮}^{∞,2} have finite projective dimension (as Δ_{𝔮}^{∞} is perfect). It follows that μ_{∗} is an isomorphism. Since both the source and target of μ_{∗} are isomorphic to Λ_{𝔮}/(2, c − 1) for i even, μ_{∗} is given by multiplication with a unit in Λ_{𝔮} (this ring being local) and hence μ(1 + c) ∈ Λ_{𝔮}/(c − 1)^{×}. This means μ is an isomorphism and we conclude that Δ_{𝔮}^{∞} is quasi-isomorphic to the top row in (5.17). We may pick bases γ_{i} of Δ_{𝔮}^{∞,i} so that γ_{2} ↦ σ and (c + 1)γ_{1} = η_{m_{0}}. It remains to verify that γ_{1}^{−1} ⊗ γ_{2} = u · L for some u ∈ Λ_{𝔮}^{×}, and this we may check in Q(ψ) for all ψ ∈ Ĝ_{ℓm_{0}}^{ℚ_{l}} which induce ψ_{𝔮}. For ψ even we have in Q(ψ)

γ_{1}^{−1} ⊗ γ_{2} = (\frac{1}{2} · η_{m_{0}})^{−1} ⊗ σ = L

and for ψ odd we note that γ_{1}^{−1} ⊗ 2 · γ_{2} is the canonical basis of Δ^{∞} ⊗_{Λ} Q(ψ) arising from the fact that this complex is acyclic. We then have

γ_{1}^{−1} ⊗ γ_{2} = \frac{1}{2} · γ_{1}^{−1} ⊗ 2 · γ_{2} = u · θ_{m_{0}}^{−1} = u · L

where u := \frac{θ_{m_{0}}}{2} is a unit in Λ_{𝔮} by the remark in [75][p.131] already used in the case l ≠ 2. □

The descent argument for j = 0. We now indicate how Theorem 5.2 implies the l-part of Conjecture 3 for M = h^{0}(Spec(L_{m})) and 𝔄 = ℤ[G_{m}]. We have a ring homomorphism

(5.18)  Λ → ℤ_{l}[G_{m}] = 𝔄_{l} ⊆ A_{l} = ∏_{χ∈Ĝ_{m}^{ℚ_{l}}} ℚ_{l}(χ),

a canonical isomorphism of perfect complexes

Δ^{∞} ⊗_{Λ}^{𝕃} 𝔄_{l} ≅ Δ(L_{m})

and a canonical isomorphism of determinants

(Det_{Λ} Δ^{∞}) ⊗_{Λ} 𝔄_{l} ≅ Det_{𝔄_{l}} Δ(L_{m}).

Given Theorem 5.2, the element L ⊗ 1 is an 𝔄_{l}-basis of Det_{𝔄_{l}} Δ(L_{m}) under this isomorphism. It remains then to verify that the image of L⊗1 in Det_{A_{l}}(Δ(L_{m})⊗_{ℤ_{l}} ℚ_{l}) agrees with that of _{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}). Luckily, this is again a computation taking place over the algebra A_{l} which avoids any delicate analysis over the possibly non-regular ring 𝔄_{l}, and which can be performed character by character. Denote by

(5.19) ϕ : Det_{ℚ_{l}(χ)}(Δ(L_{m}) ⊗_{𝔄_{l}} ℚ_{l}(χ))

≅ \begin{cases}Det_{ℚ_{l}(χ)}^{−1}(O_{L_{m}}^{×} ⊗_{𝔄} ℚ_{l}(χ)) ⊗ Det_{ℚ_{l}(χ)}(X_{\{v|∞\}} ⊗_{𝔄} ℚ_{l}(χ)) & χ ≠ 1 even \\ ℚ_{l}(χ) & otherwise \end{cases}

the isomorphism induced by (5.2). Then in view of (5.1) and (5.3), for each χ ∈ Ĝ_{m}^{ℚ_{l}} we must show

(5.20)  ∏_{p|ml}(E_{p}^{#})^{−1}ϕ(L ⊗ 1) = \begin{cases}2[L_{m} : L_{f_{χ}}][1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m} & χ ≠ 1 even \\ (L(χ, 0)^{#})^{−1} & otherwise. \end{cases}

In the remainder of this section we verify the identity (5.20). We also denote by χ the composite ring homomorphism Λ → ℚ_{l}(χ) in (5.18), and by 𝔮_{χ} its kernel. Then 𝔮_{χ} is a regular prime of Λ and Λ_{𝔮_{χ}} is a discrete valuation ring with residue field ℚ_{l}(χ) and fraction field some direct factor of Q(Λ) (indexed in (5.8) by the character ψ of G_{ℓm_{0}} obtained by the unique decomposition χ = ψ × η where η is a character of Gal(L_{ml^{∞}}/L_{ℓm_{0}})). We may view L as a Λ_{𝔮_{χ}}-basis of (Det_{Λ} Δ^{∞})_{𝔮_{χ}}. The following Lemma is the key ingredient in the descent computation.

Lemma 5.7. Let R be a discrete valuation ring with fraction field F, residue field k and uniformiser ϖ. Suppose Δ is a perfect complex of R-modules so that the R-torsion subgroup of each H^{i}(Δ) is annihilated by ϖ. Define free R-modules M^{i} by the short exact sequence

(5.21)  0 → H^{i}(Δ)_{ϖ} → H^{i}(Δ) → M^{i} → 0.

Together with the exact sequences of k-vector spaces

(5.22)  0 → H^{i}(Δ)/ϖ → H^{i}(Δ ⊗_{R}^{𝕃} k) → H^{i+1}(Δ)_{ϖ} → 0

induced by the exact triangle in the derived category of R-modules

Δ ⟶^{ϖ} Δ → Δ ⊗_{R}^{𝕃} k → Δ[1]

we find an isomorphism

Det_{k} H^{i}(Δ ⊗_{R}^{𝕃} k) ≅ Det_{k}(H^{i}(Δ)/ϖ) ⊗_{k} Det_{k}(H^{i+1}(Δ)_{ϖ})
≅ Det_{k}(H^{i}(Δ)_{ϖ}) ⊗_{k} Det_{k}(M^{i}/ϖ) ⊗_{k} Det_{k}(H^{i+1}(Δ)_{ϖ})

and hence an isomorphism

ϕ_{ϖ} : Det_{k} (Δ ⊗_{R}^{𝕃} k) ≅ ⨂_{i∈ℤ} Det_{k}(M^{i}/ϖ)^{(−1)^{i}}.

For each i fix an R-basis β_{i} of Det_{R}(M^{i}). Let e ∈ ℤ be such that b_{ϖ} = ϖ^{e} ⨂_{i∈ℤ}(β_{i})^{(−1)^{i}} is an R-basis of

Det_{R} Δ ⊆ Det_{F} (Δ ⊗_{R} F) ≅ ⨂_{i∈ℤ} (Det_{F} (M^{i} ⊗_{R} F))^{(−1)^{i}} .

Then the image of b_{ϖ} ⊗ 1 under the isomorphism

(Det_{R} Δ) ⊗_{R} k ≅ Det_{k}(Δ ⊗_{R}^{𝕃} k) ⟶^{ϕ_{ϖ}} ⨂_{i∈ℤ} Det_{k}(M^{i}/ϖ)^{(−1)^{i}}

is given by ⨂_{i∈ℤ}(β̄_{i})^{(−1)^{i}}.

Remark. A change of uniformizer ϖ will change b_{ϖ} as well as the isomorphism ϕ_{ϖ} (unless e = 0 , e.g. if each H^{i}(Δ) is R-free). If M^{i} = 0 for all i then det_{R}(M^{i}) ≅ R canonically and we may take β_{i} = 1. In this case the Lemma recovers the statement of [18] [Lemma 8.1] where the condition that H^{i}(Δ) is annihilated by ϖ is called ”semisimplicity at zero” (the motivating example being a ℤ_{l}[[T]]-module whose localisation at the prime (T) is semisimple).

Proof. Suppose Δ^{•} is a bounded complex of finitely generated free R-modules quasi-isomorphic to Δ. Let λ_{i}^{(k)}, μ_{i}^{(l)} ∈ Δ^{i} (where k and l run through two index sets depending on i) be elements whose images under δ^{i} form an R-basis of im(δ^{i}) and so that δ^{i}(μ_{i}^{(l)}) is an R-basis of im(δ^{i}) ∩ ϖker(δ^{i+1}) = im(δ^{i}) ∩ ϖΔ^{i+1}. Let β_{i}^{(n)} ∈ Δ^{i} map to an R-basis of M^{i}. Then

δ^{i−1}(λ_{i−1}^{(k)}), \frac{δ^{i−1}(μ_{i−1}^{(l)})}{ϖ}, β_{i}^{(n)}

is an R-basis of ker(δ^{i}) (the cardinality of the set {μ_{i−1}^{(l)}} is just dim_{k} H^{i}(Δ)_{ϖ}) and

(5.23)  δ^{i−1}(λ_{i−1}^{(k)}), \frac{δ^{i−1}(μ_{i−1}^{(l)})}{ϖ}, β_{i}^{(n)}, μ_{i}^{(l)}, λ_{i}^{(k)}

is an R-basis of Δ^{i}. Set μ_{i} = ⋀_{l} μ_{i}^{(l)}, λ_{i} = ⋀_{k} λ_{i}^{(k)}, β_{i} = ⋀_{n} β_{i}^{(n)} and define γ_{i} ∈ Det_{F} (Δ^{i} ⊗_{R} F)^{(−1)^{i}} by

γ_{i} = ϖ^{(−1)^{i+1} dim_{k}(H^{i}(Δ)_{ϖ})}δ(λ_{i−1})^{(−1)^{i}} ∧ δ(μ_{i−1})^{(−1)^{i}} ∧ β_{i}^{(−1)^{i}} ∧ μ_{i}^{(−1)^{i}} ∧ λ_{i}^{(−1)^{i}}.

Then γ_{i} is an R-basis of (Det_{R} Δ^{i})^{(−1)^{i}} and

b = ⨂_{i∈ℤ} γ_{i} ↦ ϖ^{e} ⨂_{i∈ℤ} β_{i}^{(−1)^{i}}; e = ∑_{i∈ℤ}(−1)^{i+1} dim_{k}(H^{i}(Δ)_{ϖ})

under the isomorphism Det_{F} (Δ^{•} ⊗_{R} F) ≅ ⨂_{i∈ℤ} Det_{F} H^{i}(Δ^{•} ⊗_{R} F)^{(−1)^{i}}. Now

\overline{δ(λ_{i−1}^{(k)})}, \overline{\frac{δ(μ_{i−1}^{(l)})}{ϖ}}, \overline{b_{i}^{(n)}}, \overline{μ_{i}^{(l)}}, \overline{λ_{i}^{(k)}} is a k-basis of \overline{Δ}^{i} := Δ^{i} ⊗_{R} k = Δ^{i}/ϖ,

\overline{δ(λ_{i−1}^{(k)})}, \overline{\frac{δ(μ_{i−1}^{(l)})}{ϖ}}, \overline{β_{i}^{(n)}}, \overline{μ_{i}^{(l)}}

is a k-basis of ker(\overline{δ}^{i}) and the images of \overline{\frac{δ(μ_{i−1}^{(l)})}{ϖ}}, \overline{β_{i}^{(n)}}, \overline{μ_{i}^{(l)}} are a k-basis of H^{i}(Δ^{•}/ϖ). The isomorphism

(Det_{R} Δ^{•}) ⊗_{R} k = Det_{k}(Δ^{•} ⊗_{R} k) ≅ ⨂_{i∈ℤ} Det_{k} H^{i}(Δ^{•} ⊗_{R} k)^{(−1)^{i}}

sends b ⊗ 1 = b̄ = ⨂_{i∈ℤ} γ̄_{i} to

⨂_{i∈ℤ} (⋀_{l} \overline{\frac{δ(μ_{i−1}^{(l)})}{ϖ}})^{(−1)^{i}} ∧ \overline{β}_{i}^{(−1)^{i}} ∧ \overline{μ}_{i}^{(−1)^{i}}.

The third map in (5.22) arises as the connecting homomorphism in the short exact sequence of complexes

0 ⟶ Δ^{i} ⟶^{ϖ} Δ^{i} ⟶ Δ^{i}/ϖ ⟶ 0
δ^{i}↓  _{δ^{i}}↓  \overline{δ}^{i}↓
0 ⟶ Δ^{i+1} ⟶^{ϖ} Δ^{i+1} ⟶ Δ^{i+1}/ϖ ⟶ 0

and therefore sends \overline{μ_{i}^{(l)}} to \frac{δ^{i}(μ_{i}^{(l)})}{ϖ} ∈ H^{i+1}(Δ)_{ϖ} and to \overline{\frac{δ^{i}(μ_{i}^{(l)})}{ϖ}} in H^{i+1}(Δ)/ϖ. The construction of ϕ_{ϖ} via the sequences (5.22) and (5.21) then shows that ϕ_{ϖ}(\overline{b}) =

⨂_{i∈ℤ} \overline{β}_{i}^{(−1)^{i}}. We remark that the ordering of the terms in γ_{i} is adapted to the particular short exact sequences inducing the isomorphism between the determinant of a complex and that of its cohomology, as well as to the sequences (5.22) and (5.21) involved in ϕ_{ϖ}. A different ordering would induce signs in various steps of the above computation which of course would cancel out eventually. □

For any ℚ_{l}-rational character χ of G_{m} we apply this Lemma to

R = Λ_{𝔮_{χ}}, Δ = Δ_{𝔮_{χ}}^{∞}, ϖ = 1 − γ^{l^{n}}

where γ^{l^{n}} is a topological generator of Gal(L_{ml^{∞}}/L_{m_{1}}) with m_{1} = m if l | m and m_{1} = ℓm if l ∤ m. Then the cohomology of Δ ⊗_{R}^{𝕃} k = Δ(L_{m}) ⊗_{𝔄_{l}} ℚ_{l}(χ) is concentrated in degrees 1 and 2 and ϕ_{ϖ} is induced by the exact sequence of k = ℚ_{l}(χ)-vector spaces

(5.24)  0 → M^{1}/ϖ → H^{1}(Δ ⊗_{R}^{𝕃} k) ⟶^{β_{ϖ}} H^{2}(Δ ⊗_{R}^{𝕃} k) → M^{2}/ϖ → 0

where β_{ϖ} (a so called Bockstein map) is the composite

H^{1}(Δ ⊗_{R}^{𝕃} k) → H^{2}(Δ)_{ϖ} → H^{2}(Δ)/ϖ → H^{2}(Δ ⊗_{R}^{𝕃} k).

Note that ϖ depends only on L_{m} and not on χ; indeed one can construct a Bockstein map before localizing at 𝔮_{χ}, described in the following Lemma.

Lemma 5.8. Define for p | m_{0} the element c_{p} ∈ ℤ_{l} by γ^{c_{p}l^{n}} = Fr_{p}^{−f_{p}} where f_{p} ∈ ℤ is the inertial degree at p of L_{m}/ℚ (this is an identity in Λ modulo multiplication with Gal(L_{m_{1}}/L_{m})). Put c_{l} = log_{l}(χ_{cyclo}(γ^{l^{n}}))^{−1} ∈ ℚ_{l}. Then β_{ϖ} is induced by the map

H^{1}(Δ(L_{m})) ⊗ ℚ_{l} = O_{L_{m}}[\frac{1}{ml}]^{×} ⊗ ℚ_{l} → X_{\{v|ml∞\}} ⊗ ℚ_{l} = H^{2}(Δ(L_{m})) ⊗ ℚ_{l}

given by

u ↦ ∑_{p|m_{0}} c_{p} ∑_{v|p} ord_{v}(u) · v + c_{l} ∑_{v|l} Tr_{L_{m,v}/ℚ_{l}}(log_{l}(u_{v})) · v.

Remark. One can verify directly that this last map has image in X_{{v|ml∞}} ⊗ℚ_{l} and not only in Y_{{v|ml∞}} ⊗ ℚ_{l} (although this is of course also a consequence of Lemma 5.8). Denoting by Nu ∈ ℤ[\frac{1}{ml}]^{×} the norm of u we have ∑_{v|p} ord_{v}(u) = f_{p}^{−1} ord_{p}(Nu) and ∑_{v|l} Tr_{L_{m,v}/ℚ_{l}}(log_{l}(u_{v})) = log_{l}(Nu). The required identity ∑_{p|m_{0}} c_{p}f_{p}^{−1} ord_{p}(x) + c_{l} log_{l}(x) = 0 holds for x = −1, for x = l (since log_{l}(l) = 0) and for x = p (applying log_{l} χ_{cyclo} to the defining identity of c_{p} we find c_{p}c_{l}^{−1} = c_{p} log_{l} χ_{cyclo}(γ^{l^{n}}) = −f_{p} log_{l}(p)), hence for all x ∈ ℤ[\frac{1}{ml}]^{×}.

Proof. The computation of the cohomology of Δ(L_{m}) given above arises from an exact triangle

τ^{≤2}RΓ(ℤ[\frac{1}{ml}], T_{l}^{∗}(1)) → Δ(L_{m}) → Y_{\{v|∞\}} ⊗_{ℤ} ℤ_{l}[−2]

(where the truncation τ^{≤2} is only necessary for l = 2). Passing to the inverse limit we find that there is an exact triangle of Λ-modules

τ^{≤2}RΓ(ℤ[\frac{1}{ml}], T_{l}^{∗}(1)^{∞}) → Δ^{∞} → Y_{\{v|∞\}}^{∞}[−2]

which induces, after localisation at 𝔮_{χ}, a commutative diagram of Bockstein maps

H^{1}(O_{L_{m}}[\frac{1}{ml}], ℤ_{l}(1)) ⊗ ℚ_{l}(χ) ⟶ H^{1}(Δ(L_{m})) ⊗ ℚ_{l}(χ) ⟶  0
β′↓  ^{β_{ϖ}} ↓  ↓
H^{2}(O_{L_{m}}[\frac{1}{ml}], ℤ_{l}(1)) ⊗ ℚ_{l}(χ) ⟶ H^{2}(Δ(L_{m})) ⊗ ℚ_{l}(χ) ⟶ Y_{\{v|∞\}} ⊗ ℚ_{l}(χ).

This shows that the image of β_{ϖ}, has no components at the infinite places. In order to compute β′ we apply the following Lemma to G = π_{1}^{et}(Spec(O_{L_{m_{1}}} [\frac{1}{ml}])), Γ = Gal(L_{ml^{∞}}/L_{m_{1}}), γ_{0} = γ^{l^{n}} and M = ℤ_{l}(1).

Lemma 5.9. Let Γ be a free ℤ_{l}-module of rank 1 with generator γ_{0} and G → Γ a surjection of profinite groups. Denote by θ ∈ H^{1}(G, ℤ_{l}) = Hom(G, ℤ_{l}) the unique homomorphism factoring through Γ with θ(γ_{0}) = 1 and put Λ = ℤ_{l}[[Γ]]. For any continuous ℤ_{l}[[G]]-module M we have an exact triangle in the derived category of Λ-modules

RΓ(G, M ⊗ Λ) ⟶^{1−γ_{0}} RΓ(G, M ⊗ Λ) → RΓ(G, M ⊗ Λ) ⊗_{Λ}^{𝕃} ℤ_{l} ≅ RΓ(G, M).

Then the Bockstein map

β^{i} : H^{i}(G, M) → H^{i+1}(G, M)

arising from this triangle coincides with the cup product θ ∪ −.

Proof. See [63][Lemma 1.2] . □

This Lemma describes β_{ϖ} as being induced by cup product with θ over the field L_{m_{1}}. Now in case that m_{1} ≠ m, i.e. l ∤ m, θ ∈ Hom(Gal(L_{ml^{∞}}/L_{m_{1}}), ℤ_{l}) is the restriction of a (unique) θ ∈ Hom(Gal(L_{ml^{∞}}/L_{m}), ℤ_{l}), and the projection formula for the cup product shows that β_{ϖ} is induced by cup product

H^{1}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1)) = O_{L_{m}}[\frac{1}{ml}]^{×} ⊗ ℚ_{l} ⟶^{θ∪} X_{\{v|ml\}} ⊗ ℚ_{l} = H^{2}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1))

over L_{m}. For any place v of L_{m} we have a commutative diagram

H^{1}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1)) ⟶^{θ∪} H^{2}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1))
↓  ↓
H^{1}(L_{m,v}, ℚ_{l}(1)) ⟶^{res_{v}(θ)∪} H^{2}(L_{m,v}, ℚ_{l}(1)) ≅ ℚ_{l}

and for u ∈ O_{L_{m}}[\frac{1}{ml}]^{×} ⊗ ℚ_{l} the element res_{v}(θ) ∪ res_{v}(u) can be computed by local class field theory. For v ∤ l one finds (see [49][Ch.II, 1.4.2]) that

res_{v}(θ) ∪ res_{v}(u) = −θ(Fr_{v}) ord_{v}(u) = θ(Fr_{p}^{−f_{p}}) ord_{v}(u) = c_{p} ord_{v}(u)

and for v | l one has by [49][Ch. II, 1.4.5]

res_{v}(θ) ∪ res_{v}(u) = c_{l} log_{l}(χ_{cyclo}) ∪ u_{v} = c_{l}Tr_{L_{m,v}/ℚ_{l}}(log_{l}(u_{v})).
□

We verify the assumptions of Lemma 5.7 and describe the bases β_{1} and β_{2}. The torsion submodule of U_{{v|ml}}^{∞} is ℤ_{l}(1) and hence H^{1}(Δ) = U_{{v|ml},𝔮_{χ}}^{∞} = M^{1} is free of rank one over Λ_{𝔮_{χ}} if χ is even, and H^{1}(Δ) = M^{1} = 0 for χ odd. It remains to find a basis if χ is even. The vanishing of P_{𝔮_{χ}}^{∞} combined with Lemma 5.4 gives that U_{𝔮_{χ}}^{∞} = Λ_{𝔮_{χ}} · (1 − γ)^{ϵ} · η_{f_{χ,0}} where f_{χ,0} | m_{0} is the prime-to-l-part of f_{χ} and ϵ = 1 or 0 according to whether χ factors through the cyclotomic ℤ_{l}-extension of ℚ or not. In this latter case, we either have χ = 1 or the element 1 − χ(γ) is a unit in Λ_{𝔮_{χ}} and η_{1} = η_{f_{χ,0}} will in fact be a basis. Combined with (5.12) we find a (nonsplit) exact sequence

0 → U_{𝔮_{χ}}^{∞} → U_{{v|ml},𝔮_{χ}}^{∞} → Y_{{v|l},𝔮_{χ}}^{∞} → 0

where the last term is nonzero (and then isomorphic to the residue field ℚ_{l}(χ)) only for χ(l) = 1.

The exact sequence (5.12) together with Lemma 5.4 and Lemma 5.11 below for χ odd (resp. the vanishing of P_{q_{χ}}^{∞} for χ even) show that P_{{v|ml},𝔮_{χ}}^{∞} = 0 for any χ. Hence the Λ_{𝔮_{χ}}-torsion submodule of H^{2}(Δ) is X_{{v|ml},𝔮_{χ}}^{∞} and M^{2} is canonically isomorphic to Y_{{v|∞},𝔮_{χ}}^{∞} if m_{0} ≠ 1 and to X_{{v|l∞},𝔮_{χ}}^{∞} if m_{0} = 1. However, if χ ≠ 1 we have an isomorphism X_{{v|l∞},𝔮_{χ}}^{∞} = Y_{{v|l∞},𝔮_{χ}}^{∞} = Y_{{v|∞},𝔮_{χ}}^{∞}. Summarizing we have

(5.25)

M^{1}  M^{2}
χ even, χ(l) ≠ 1  Λ_{𝔮_{χ}} · η_{f_{χ,0}} Λ_{𝔮_{χ}} · σ
χ ≠ 1 even, χ(l) = 1 Λ_{𝔮_{χ}} · \frac{η_{f_{χ,0}}}{ϖ} Λ_{𝔮_{χ}} · σ
χ = 1, m_{0} ≠ 1  Λ_{𝔮_{χ}} · η_{1}  Λ_{𝔮_{χ}} · σ
χ = 1, m_{0} = 1  Λ_{𝔮_{χ}} · η_{1}  Λ_{𝔮_{χ}} · (σ − λ)
χ odd  0  0

where λ is the unique place of L_{l^{∞}} above l.

We now break down the discussion into the various cases listed in this table. In each case we use a basis of the ℚ_{l}(χ)-space O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ), the source of the maps β_{ϖ} and val, in order to understand the trivializations ϕ and ϕ_{ϖ}. Then we write down L in terms of β_{1} and β_{2} and apply Lemma 5.7.

The ℚ_{l}(χ)-spaces Y_{{v|p}} ⊗_{𝔄} ℚ_{l}(χ) are nonzero if and only if χ(p) = 1, in which case they are of dimension 1 with basis any fixed place v_{p} | p. For each such p let x_{p} ∈ L_{m} be an element with nontrivial divisor concentrated at v_{p}. The ℚ_{l}(χ)- space spanned by x_{p} is then mapped isomorphically to Y_{{v|p}} ⊗_{𝔄} ℚ_{l}(χ) under the valuation map val. Put J = {p | m_{0}, χ(p) = 1}, x_{J} = ⋀_{p∈J} x_{p}, v_{J} = ⋀_{p∈J} v_{p} and c_{χ} = ∏_{p∈J} c_{p}.

The case of even χ with χ(l) ≠ 1. The element β̄_{1} is the image of the norm compatible system

η_{f_{χ,0}} = (1 − ζ_{f_{χ,0}l^{ν}}) ∈ \underleftarrow{lim}_{ν} O_{L_{m_{0}l^{ν}}} [\frac{1}{ml}]^{×} ⊗_{ℤ} ℤ_{l}

in M^{1}/ϖ ⊆ O_{L}[\frac{1}{ml}]^{×} ⊗_{ℤ[G]} ℚ_{l}(χ) where L ⊂ L_{ml^{∞}} is any field so that χ factors through G = Gal(L/ℚ). We insist on taking L = L_{m} so that we have

β̄_{1} = \begin{cases}(1 − χ^{−1}(l))(1 − ζ_{f_{χ,0}}) = (1 − χ^{−1}(l))(1 − ζ_{f_{χ}}) & μ = 0 \\ (1 − ζ_{f_{χ,0}l^{μ}}) = [L_{m} : L_{m_{0}l^{μ′}}]^{−1}(1 − χ^{−1}(l))(1 − ζ_{f_{χ}}) & μ > 0. \end{cases}

where μ = ord_{l}(m), μ′ = ord_{l}(f_{χ}) and we recall the convention that χ^{−1}(l) = 0 if μ′ > 0. The element β̄_{1} generates the one-dimensional subspace M^{1}/ϖ ⊆ O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ) of universal norms which coincides with O_{L_{m}}^{×} ⊗_{𝔄} ℚ_{l}(χ). The set \{β̄_{1}\} ∪ \{x_{p}| p ∈ J\} is a basis of O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ) and we also have

β̄_{2} = σ̄ = σ_{m} ∈ Y_{{v|∞}} ⊗_{𝔄} ℚ_{l}(χ) = X_{{v|∞}} ⊗_{𝔄} ℚ_{l}(χ).

Hence by Lemma 5.8

ϕ ∘ ϕ_{ϖ}^{−1}(β̄_{1}^{−1} ⊗ β̄_{2})
= ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1} ⊗ β_{ϖ}(x_{J}) ∧ β̄_{2})
= c_{χ}ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1} ⊗ val(x_{J}) ∧ β̄_{2})
= c_{χ}[L_{m} : L_{m_{0}l^{μ′}}](1 − χ^{−1}(l))^{−1} ϕ ([1 − ζ_{f_{χ}}]^{−1} ∧ x_{J}^{−1} ⊗ val(x_{J}) ∧ σ_{m})
(5.26) = c_{χ}[L_{m} : L_{m_{0}l^{μ′}}](1 − χ^{−1}(l))^{−1} [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}.

As elements of (Det_{Λ} Δ^{∞})_{𝔮_{χ}} we have using (5.15)

L = 2 · η_{m_{0}}^{−1} ⊗ σ = 2 · [L_{m_{0}} : L_{f_{χ,0}}] ∏_{p|m_{0}, p∤f_{χ,0}} \frac{1}{1 − Fr_{p}^{−1}} · η_{f_{χ,0}}^{−1} ⊗ σ

(5.27)  = 2 · [L_{m_{0}} : L_{f_{χ,0}}] ∏_{\begin{matrix}p|m_{0}, p∤f_{χ,0} \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − Fr_{p}^{−1}} · ∏_{p∈J} \frac{ϖ}{1 − Fr_{p}^{−1}} · ϖ^{e}β_{1}^{−1} ⊗ β_{2}.

For p ∈ J we have Fr_{p}^{−f_{p}} = γ^{c_{p}l^{n}} and

(5.28)  χ(\frac{ϖ}{1 − Fr_{p}^{−1}} ) = χ (\frac{(1 + Fr_{p}^{−1} + ⋯ Fr_{p}^{−f_{p}−1})(1 − γ^{l^{n}})}{1 − γ^{c_{p}l^{n}}}) = \frac{f_{p}}{c_{p}}

since χ(p) = χ(γ^{l^{n}}) = 1. Hence we conclude from Lemma 5.7, (5.26) and the identity [L_{m} : L_{f_{χ}}] = [L_{m_{0}} : L_{f_{χ,0}}] · [L_{m} : L_{m_{0}l^{μ′}}] that

ϕ(L ⊗ 1) =2 · [L_{m_{0}} : L_{f_{χ,0}}] ∏_{\begin{matrix}p|m_{0} \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − χ^{−1}(p)} · ∏_{p∈J} \frac{f_{p}}{c_{p}} · ϕ ∘ ϕ_{ϖ}^{−1}(β̄_{1}^{−1} ⊗ β̄_{2})

=2 · [L_{m} : L_{f_{χ}}] ∏_{\begin{matrix}p|ml \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − χ^{−1}(p)} · ∏_{p∈J} f_{p} · [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}

=2 · [L_{m} : L_{f_{χ}}] ∏_{p|ml}(E_{p}^{#}) · [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}

which is the identity (5.20) to be shown.

The case of even χ ≠ 1 with χ(l) = 1. In this case the subspace M^{1}/ϖ of universal norms in O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ) does not coincide with O_{L_{m}}^{×} ⊗_{𝔄} ℚ_{l}(χ). A basis of this latter space is given by (1 − ζ_{f_{χ}}) and a basis of O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ) is {β̄_{1}, (1 − ζ_{f_{χ}})} ∪ {x_{p}| p ∈ J}. The next Lemma gives the necessary information about β̄_{1}.

Lemma 5.10. (Solomon) Let L/ℚ be an abelian extension of conductor d > 1 in which l splits completely, and denote by L^{(∞)}/L the cyclotomic ℤ_{l}-extension and by L^{(i)}/L the subfield of degree l^{i}. Then there is an element κ(L^{(∞)}, γ^{l^{n}}) = (κ(L^{(i)}, γ^{l^{n}}))_{i} ∈ \underleftarrow{lim}_{i} O_{L^{(i)}}[\frac{1}{l}] ⊗_{ℤ} ℤ_{l} so that (1 − γ^{l^{n}}) · κ(L^{(∞)}, γ^{l^{n}}) = N · η_{d} where N ∈ Λ is the norm from L_{dℓl^{n}} to L, and

ord_{w}(κ(L, γ^{l^{n}})) = −c_{l} log_{l}(σ_{w}(N_{L_{d}/L}(1 − ζ_{d}))

for any place w of L dividing l.

Proof. See [71]. □

We apply this Lemma to the splitting field L_{χ} of l in L_{f_{χ}}/ℚ in which case d = f_{χ} = f_{χ,0}. We then have β_{1} = ϖ^{−1}η_{f_{χ,0}} = N^{−1}κ(L_{χ}^{(∞)}, γ^{l^{n}}) and

[L_{l^{n}ℓf_{χ}} : L_{χ}]β̄_{1} = \overline{κ(L_{χ}^{(∞)}, γ^{l^{n}})} = κ(L_{χ}, γ^{l^{n}})

as elements of O_{L_{m_{1}}} [\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ). Taking norms from L_{m_{1}} to L_{m} we have

[L_{l^{μ}f_{χ}} : L_{χ}]β̄_{1} = \overline{κ(L_{χ}^{(∞)}, γ^{l^{n}})} = κ(L_{χ}, γ^{l^{n}})

in O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l}(χ) where μ = ord_{l}(m). For each place v | l of L_{m} denote by w the place of L_{χ} induced by v. By Lemma 5.10 we have

ord_{v}(κ(L_{χ}, γ^{l^{n}})) = |I_{l}| ord_{w}(κ(L_{χ}, γ^{l^{n}})) = −|I_{l}|c_{l} log_{l}(N_{L_{fχ} /L_{χ}}(1 − ζ_{f_{χ}})_{v})

and hence

Tr_{L_{m,v}/ℚ_{l}}(log_{l}(1 − ζ_{f_{χ}})_{v}) =[L_{m,v} : L_{f_{χ},v}] log_{l}(N_{L_{fχ} /L_{χ}}(1 − ζ_{f_{χ}})_{v})
=\frac{|D_{l}|}{[L_{f_{χ}} : L_{χ}]} log_{l}(N_{L_{fχ} /L_{χ}}(1 − ζ_{f_{χ}})_{v})
= −\frac{f_{l}}{c_{l} · [L_{f_{χ}} : L_{χ}]} ord_{v}(κ(L_{χ}, γ^{l^{n}}))

and therefore

β_{ϖ}(1 − ζ_{f_{χ}}) = c_{l} ∑_{v|l} Tr_{L_{m,v}/ℚ_{l}}(log_{l}(1 − ζ_{f_{χ}})_{v}) · v

= −\frac{f_{l}}{[L_{f_{χ}} : L_{χ}]} ∑_{v|l} ord_{v}(κ(L_{χ}, γ^{l^{n}})) · v.

Since [L_{l^{μ}f_{χ}} : L_{χ}]β̄_{1} = \overline{κ(L_{χ}^{(∞)}, γ^{l^{n}})} = κ(L_{χ}, γ^{l^{n}}) we conclude

β_{ϖ}(1 − ζ_{f_{χ}}) = −\frac{f_{l} · [L_{l^{μ}f_{χ}} : L_{χ}]}{[L_{f_{χ}} : L_{χ}]} val(β̄_{1}) = −f_{l}[L_{l^{μ}f_{χ}} : L_{f_{χ}}] val(β̄_{1}).

We are now in a position to compute ϕ ∘ ϕ_{ϖ}^{−1}.

ϕ ∘ ϕ_{ϖ}^{−1}(β̄_{1}^{−1} ⊗ β̄_{2})
= ϕ(β̄_{1}^{−1} ∧ [1 − ζ_{f_{χ}}]^{−1} ∧ x_{J}^{−1} ⊗ β_{ϖ}(x_{J}) ∧ β_{ϖ}(1 − ζ_{f_{χ}}) ∧ β̄_{2})
= − c_{χ}f_{l}[L_{l^{μ}f_{χ}} : L_{f_{χ}}] ϕ(β̄_{1}^{−1} ∧ [1 − ζ_{f_{χ}}]^{−1} ∧ x_{J}^{−1} ⊗ val(x_{J}) ∧ val(β̄_{1}) ∧ β̄_{2})
= c_{χ}f_{l}[L_{l^{μ}f_{χ}} : L_{f_{χ}}] ϕ([1 − ζ_{f_{χ}}]^{−1} ∧ β̄_{1}^{−1} ∧ x_{J}^{−1} ⊗ val(x_{J}) ∧ val(β̄_{1}) ∧ β̄_{2})
(5.29) = c_{χ}f_{l}[L_{l^{μ}f_{χ}} : L_{f_{χ}}] [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}.

The element L can be described as in (5.27) with only the power of ϖ changing. Combining this description with (5.28), (5.29), the identity [L_{m} : L_{f_{χ}}] = [L_{m_{0}} : L_{f_{χ,0}}][L_{l^{μ}f_{χ}} : L_{f_{χ}}] and Lemma 5.7 we obtain as before

ϕ(L ⊗ 1) =2 · [L_{m} : L_{f_{χ}}] ∏_{\begin{matrix}p|m_{0} \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − χ^{−1}(p)} · ∏_{p∈J∪\{l\}} f_{p} · [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}

=2 · [L_{m} : L_{f_{χ}}] ∏_{p|ml}(E_{p}^{#}) · [1 − ζ_{f_{χ}}]^{−1} ⊗ σ_{m}

which is the identity (5.20) to be shown.

The case of the trivial character. As in the discussion of the case of even χ with χ(l) ≠ 1 we first compute the element β̄_{1} ∈ O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l} (recall β_{1} = η_{1}) using the fact that N_{L_{ℓ}/ℚ}(1 − ζ_{ℓ}) = l. We have

β̄_{1} = \begin{cases}l & μ = 0 \\ (1 − ζ_{l^{μ}}) = [L_{m} : L_{m_{0}}]^{−1}l & μ > 0. \end{cases}

A basis of O_{L_{m}}[\frac{1}{ml}]^{×} ⊗_{𝔄} ℚ_{l} is given by \{β̄_{1}\} ∪ \{x_{p}| p ∈ J\} = \{β̄_{1}\} ∪ \{x_{p}| p | m_{0}\}. The map val is an isomorphism but β_{ϖ} is not. If m_{0} > 1 a lift of β̄_{2} = σ_{m} to X_{{v|ml∞}} is given by σ_{m} − v_{l} where v_{l} is some place of L_{m} dividing l, and this remains true in the case m_{0} = 1 where β_{2} = σ − λ. Moreover,

val(l) = ∑_{v|l} |I_{l}|v = \frac{[L_{m} : ℚ]}{f_{l}}v_{l} ↦ −\frac{[L_{m} : ℚ]}{f_{l}}(σ_{m} − v_{l}) = −\frac{[L_{m} : ℚ]}{f_{l}} β̄_{2}

and hence val(β̄_{1}) = −[L_{m_{0}} : ℚ]f_{l}^{−1}β̄_{2}. Therefore

ϕ ∘ ϕ_{ϖ}^{−1}(β̄_{1}^{−1} ⊗ β̄_{2}) =ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1} ⊗ β_{ϖ}(x_{J}) ∧ β̄_{2})
= − c_{χ} \frac{f_{l}}{[L_{m_{0}} : ℚ]}ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1} ⊗ val(x_{J}) ∧ val(β̄_{1}))
= − c_{χ} \frac{f_{l}}{[L_{m_{0}} : ℚ]}.

Again we have

L = 2 · [L_{m_{0}} : ℚ] ∏_{p|m_{0}} \frac{ϖ}{1 − Fr_{p}^{−1}} · ϖ^{e}β_{1}^{−1} ⊗ β_{2}

and

ϕ(L ⊗ 1) = − 2 · [L_{m_{0}} : ℚ] ∏_{p|m_{0}} \frac{f_{p}}{c_{p}} · c_{χ} · \frac{f_{l}}{[L_{m_{0}} : ℚ]}
= − 2 · ∏_{p|ml}(E_{p}^{#}) = ζ(0)^{−1} · ∏_{p|ml}(E_{p}^{#})

which is the identity (5.20).

The case of odd χ. In this case the maps β_{ϖ} and val are isomorphisms. The valuation map u ↦ val(u) = ∑_{v|ml} ord_{v}(u) · v has a diagonal matrix in the bases {x_{p}} and {v_{p}}. The matrix of the map β_{ϖ} on the other hand is upper triangular with diagonal terms

β_{ϖ}(x_{p}) = c_{p} val(x_{p}); β_{ϖ}(x_{l}) = c_{l} ∑_{v|l} Tr_{L_{m,v}/ℚ_{l}}(log_{l}(x_{l,v})) · v

where the term corresponding to p only occurs for χ(p) = 1. If χ(l) = 1 then we may pick x_{l} to lie in the splitting field of l in L_{m}/ℚ and we obtain

∑_{v|l} Tr_{L_{m,v}/ℚ_{l}}(log_{l}(x_{l,v})) · v = |D_{l}| ∑_{g∈G_{m}/D_{l}} log_{l}(σ_{v_{l}}(gx_{l})) · g^{−1}v_{l}

where σ_{v_{l}} is the embedding corresponding to v_{l} (so that we have σ_{v_{l}}(x_{l}) ∈ ℚ_{l}). The image of this element in Y_{{v|l}} ⊗_{𝔄} ℚ_{l}(χ) is

|D_{l}| ∑_{g∈G_{m}/D_{l}} log_{l}(σ_{v_{l}}(gx_{l})) · χ(g)^{−1}v_{l} = |D_{l}| ∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}v_{l}

where G_{χ} = Gal(L_{χ}/ℚ) with L_{χ} the fixed field of χ and x̃_{l} is the Norm of x_{l} into L_{χ}. We have val(x_{l}) = ord_{v_{l}}(x_{l}) · v_{l} = |I_{l}| ord_{w_{l}}(x̃_{l}) · v_{l} where w_{l} is the place of L_{χ} induced by v_{l} and |I_{l}| is the ramification degree of l in L_{m}/ℚ. Hence

β_{ϖ}(x_{l}) = c_{l} \frac{|D_{l}| ∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}}{|I_{l}| ord_{w_{l}}(x̃_{l})} val(x_{l})

and the map ϕ ∘ ϕ_{ϖ}^{−1} is just multiplication with

(5.30)  ϕ ∘ ϕ_{ϖ}^{−1} = c_{χ}c_{l}f_{l} \frac{∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}}{ord_{w_{l}}(x̃_{l})}

where the last three factors only occur if χ(l) = 1. We have β_{1} = β_{2} = 1 and in (Det_{Λ} Δ^{∞})_{𝔮_{χ}} using (5.16)

(5.31)  L =θ_{m_{0}}^{−1} = \frac{ϖ^{δ}}{θ_{f_{χ,0}}} ∏_{\begin{matrix}p|m_{0}, p∤f_{χ,0} \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − Fr_{p}^{−1}} · ∏_{p∈J} \frac{ϖ}{1 − Fr_{p}^{−1}} · ϖ^{e}.

where δ = 1, 0 according to whether χ(l) = 1 or not. If χ(l) ≠ 1 we have

(5.32)  χ(θ_{f_{χ,0}}) = L(χ^{−1}, 0)(1 − χ^{−1}(l))

using (5.6) and the fact that the g_{k} satisfy Euler-system relations.

Lemma 5.11. (Ferrero-Greenberg) If χ(l) = 1 then

(5.33)  χ (\frac{θ_{f_{χ,0}}}{ϖ} ) = L(χ^{−1}, 0)c_{l} \frac{∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}}{ord_{w_{l}}(x̃_{l})}

Proof. Consider the l-adic L-function

L_{l}(χ^{−1}ω, s) = (χχ̂_{cyclo}^{s})(θ_{f_{χ,0}})

where ω is the Teichmueller character, s is a variable in ℤ_{l}, χ̂_{cyclo} = χ_{cyclo}ω^{−1} : G_{ml^{∞}} → ℤ_{l}^{×}, and we extend continuous characters of G_{ml^{∞}} to algebra homomorphisms Λ → ℚ_{l} in the usual way. In [33] a formula is given for the derivative L′_{l}(χ^{−1}ω, 0). For a certain l-unit (Gaussian sum) γ_{1} ∈ L_{ℓf_{χ}} one has

L′_{l}(χ^{−1}ω, 0) = \frac{d}{ds}L_{l}(χ^{−1}ω, s)|_{s=0} = ∑_{g∈G_{fχ} /<l>} χ^{−1}(g) log_{l}(σ_{v_{l}}(gγ_{1})).

On the other hand, Stickelberger’s theorem says that

val(γ_{1}) = ∑_{g∈G_{fχ} /<l>} ord_{gv_{l}}(γ_{1}) · gv_{l} = ∑_{\begin{matrix}c=1 \\ (c,f_{χ})=1 \end{matrix}}^{f_{χ}} \frac{c}{f_{χ}} · τ_{c}^{−1}v_{l}

where val is the valuation map normalized for the field L_{f_{χ}}. After applying χ we find

val(γ_{1}) = ∑_{c=1}^{f_{χ}} \frac{c}{f_{χ}} χ^{−1}(c) · v_{l} = −L(χ^{−1}, 0) · v_{l}.

On the one-dimensional ℚ_{l}(χ)-space O_{L_{ℓfχ}} [\frac{1}{l}]^{×} ⊗_{ℤ[G_{ℓfχ}]} ℚ_{l}(χ) ≅ O_{L_{m}}[\frac{1}{l}] ⊗_{𝔄} ℚ_{l}(χ) we have the two ℚ_{l}(χ) · v_{l}-valued linear forms β_{ϖ} and val whose ratio can be computed by evaluating on either the element x_{l} of γ_{1}

\frac{∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}}{ord_{w_{l}}(x̃_{l})} = \frac{∑_{g∈G_{fχ} /<l>} χ^{−1}(g) log_{l}(σ_{v_{l}}(gγ_{1}))}{−L(χ^{−1}, 0)}.

Now the function h(s) := (χχ̂_{cyclo}^{s})(ϖ) = (χχ̂_{cyclo}^{s})(1 − γ^{l^{n}}) also has a first order zero at s = 0 with derivative h′(0) = − log_{l}(χ_{cyclo}(γ^{l^{n}})) = −c_{l}^{−1} and

χ (\frac{θ_{f_{χ,0}}}{ϖ} ) = \frac{(χχ̂_{cyclo}^{s})(g_{f_{χ,0}})}{(χχ̂_{cyclo}^{s})(ϖ)}|_{s=0} = \frac{L′_{l}(0, χ^{−1}ω)}{h′(0)}

= L(χ^{−1}, 0)c_{l} \frac{∑_{g∈G_{χ}} log_{l}(σ_{v_{l}}(gx̃_{l})) · χ(g)^{−1}}{ord_{w_{l}}(x̃_{l})} .

□

By Lemma 5.7 (ϖ^{e} is mapped to 1) together with (5.30)-(5.33) and (5.28) we find

ϕ(L ⊗ 1) = \frac{1}{L(χ^{−1}, 0)} ∏_{\begin{matrix}p|ml \\ χ(p)≠1 \end{matrix}} \frac{1}{1 − χ^{−1}(p)} · ∏_{\begin{matrix}p|ml \\ χ(p)=1 \end{matrix}} f_{p} = (L(χ, 0)^{#})^{−1} ∏_{p|ml} E_{p}^{#}

which is (5.20).

The descent argument for j < 0. In this section we again fix 1 < m ≢ 2 mod 4 and M = h^{0}(Spec(L_{m}))(j) with j < 0, and we shall prove Conjecture 3 for M and A = ℚ[G_{m}] in a way which is completely parallel to the case j = 0. Theorem 5.2 is the key ingredient and the remaining arguments are computational.

The sequence Mot_{∞} is the ℝ-dual (with contragredient G_{m}-action) of the isomorphism

K_{1−2j}(O_{L_{m}}) ⊗_{ℤ} ℝ ⟶^{−ρ_{∞}} (⨁_{τ∈T} ℂ/ℝ · (2πi)^{1−j})^{+}

where ρ_{∞} is the Beilinson regulator map, T = Hom(L_{m}, ℂ) and the ℝ-dual of this last space is identified with ker(α_{M}) = M_{B,ℝ}^{+} by taking invariants in the Gal(ℂ/ℝ)- equivariant perfect pairing

⨁_{τ∈T} ℝ · (2πi)^{j} × ⨁_{τ∈T} ℂ/ℝ · (2πi)^{1−j} → ⨁_{τ∈T} ℂ/2πi · ℝ ⟶^{Σ} ℝ

induced by multiplication. Note that this pairing identifies the ℚ-dual of M_{B} = ⨁_{τ∈T} ℚ · (2πi)^{j} with ⨁_{τ∈T} ℚ · (2πi)^{−j} ⊆ ⨁_{τ∈T} ℂ/ℝ · (2πi)^{1−j}. Defining

Y (j) := (⨁_{τ∈T} ℚ · (2πi)^{j})^{+}

we obtain an identification as in the case j = 0

Ξ(_{A}M)^{#} = Det_{A}^{−1}(K_{1−2j}(O_{L_{m}}) ⊗_{ℤ} ℚ) ⊗_{A} Det_{A} Y (−j)
= ∏_{\begin{matrix}χ(−1) \\ =(−1)^{j} \end{matrix}}(K_{1−2j}(O_{L_{m}}) ⊗_{𝔄} ℚ(χ))^{−1} ⊗_{ℚ(χ)} (Y (−j) ⊗_{𝔄} ℚ(χ)) × ∏_{\begin{matrix}χ(−1) \\ =(−1)^{j+1} \end{matrix}} ℚ(χ).

The formulas for L^{∗}(η, 0) in section 5.1 generalize to j ≤ 0 (see [75][Ch.5]).

L(η, j) = − \frac{B_{1−j,η}}{1 − j} := −\frac{f_{η}^{−j}}{1 − j} ∑_{a=1}^{f_{η}} B_{1−j}(\frac{a}{f_{η}})η(a) ∈ ℚ(η)

\frac{d}{ds}L(η, s)|_{s=j} =(−j)! (\frac{2πi}{f_{η}})^{j} \frac{1}{2} ∑_{a=1}^{f_{η}} Li_{1−j}(e^{2πia/f_{η}})η(a) if η(−1) = (−1)^{j}

Here B_{k} is the k-th Bernoulli polynomial and Li_{k}(x) = ∑_{n=1}^{∞} \frac{x^{n}}{n^{k}}.

Theorem 5.12. (Beilinson/Huber-Wildeshaus) For integers f | m, j ≤ 0 there is an element ξ_{f}(j) ∈ K_{1−2j}(O_{L_{f}}) ⊗_{ℤ} ℚ whose image under the regulator map is given by

−ρ_{∞}(ξ_{f}(j)) = (−j)!f^{−j} ∑_{a∈(ℤ/mℤ)^{×}} Li_{1−j}(e^{2πia/f})τ_{a}^{−1}(σ_{m})

and whose image in H^{1}(L_{m}, ℚ_{l}(1−j)) under the étale Chern class map ρ_{l}^{et} is given by

ρ_{l}^{et}(ξ_{f}(j)) = ( ∑_{α^{l^{r}} =ζ_{f}}(1 − α) ⊗ (α^{f})^{⊗(−j)})_{r}

Proof. See [46][Cor. 9.6, 9.7]. In the notation of [46] we have

ξ_{f}(j) = (−j)!f^{−j}ϵ_{1−j}(ζ_{f}).

In particular ξ_{f}(0) = 1 − ζ_{f}. □

We find that the image of e_{η}ξ_{f_{η}}(j) under ρ_{∞} is [L_{m} : L_{f_{η}}]·2·L′(η^{−1}, j)(2πi)^{−j} · σ_{m} and hence that _{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}) = (L^{∗}(_{A}M, 0)^{−1})^{#}_{A}ϑ_{∞}(1) has components

Aϑ_{∞}(L^{∗}(_{A}M, 0)^{−1})_{χ} = \begin{cases}2 · [L_{m} : L_{f_{χ}}][ξ_{f_{χ}}(j)]^{−1} ⊗ (2πi)^{−j} · σ_{m} & χ(−1) = (−1)^{j} \\ (L(χ, j)^{#})^{−1} & χ(−1) = (−1)^{j+1}. \end{cases}

Defining Δ(L_{m})(j) = RΓ_{c}(ℤ[\frac{1}{ml}], T_{l})^{∗}[−3] we have isomorphisms

H^{1}(Δ(L_{m})(j))_{ℚ_{l}} ≅ H^{1}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1 − j)) ⟵^{ρ_{l}^{et}} K_{1−2j}(O_{L_{m}}) ⊗_{ℤ} ℚ_{l}

H^{2}(Δ(L_{m})(j))_{ℚ_{l}} ≅ (⨁_{τ∈T} ℚ_{l}(−j))^{+} ≅ Y (−j) ⊗_{ℚ} ℚ_{l}

The isomorphism _{A}ϑ_{l} : Ξ(_{A}M)_{ℚ_{l}}^{#} ≅ Det_{A_{l}} Δ(L_{m})(j)_{ℚ_{l}} sends _{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}) to

∏_{p|ml} \frac{1}{1 − χ(p)^{−1}p^{−j}} · 2 · [L_{m} : L_{f_{χ}}][ξ_{f_{χ}}(j)]^{−1} ⊗ ζ_{l^{∞}}^{⊗−j} · σ_{m}  if χ(−1) = (−1)^{j} (5.34)

∏_{p|ml} \frac{1}{1 − χ(p)^{−1}p^{−j}} · (L(χ, j)^{#})^{−1}  if χ(−1) = (−1)^{j+1} (5.35)

where ζ_{l^{∞}} is the generator of ℚ_{l}(1) given by the inverse system (ζ_{l^{n+1}})_{n≥0}.

For j ∈ ℤ we denote by κ^{j} : G_{ml^{∞}} → Λ^{×} the character g ↦ χ_{cyclo}(g)^{j}g as well as the induced ring automorphism κ^{j} : Λ → Λ. If there is no risk of confusion we also denote by κ^{j} : Λ → 𝔄_{l} ⊆ A_{l} the composite of κ^{j} and the natural projection to 𝔄_{l} or A_{l}.

Lemma 5.13. a) For j ∈ ℤ there is a natural isomorphism

Δ^{∞} ⊗_{Λ,κ^{j}}^{𝕃} 𝔄_{l} ≅ Δ(L_{m})(j).

b) For j ∈ ℤ the image of an element

u = (u_{n})_{n≥0} ∈ \underleftarrow{lim}_{n} H^{1}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ(1)) ≅ U_{\{v|ml\}}^{∞} = H^{1}(Δ^{∞})

under the isomorphism H^{1}(Δ^{∞}) ⊗_{Λ,κ^{j}} A_{l} ≅ H^{1}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1 − j)) is given by

(5.36)  Tr_{L_{m_{0}l^{n}/Lm}} (u_{n} ∪ ζ_{l^{n}}^{⊗−j})_{n>>0}

c) For j ∈ ℤ the image of an element

s = (s_{n})_{n≥0} ∈ \underleftarrow{lim}_{n} ℤ/l^{n}ℤ[G_{m_{0}l^{n}}] · σ = Y_{\{v|∞\}}^{∞}

under the isomorphism Y_{{v|∞}}^{∞} ⊗_{Λ,κ^{j}} A_{l} ≅ H^{0}(Spec(L_{m} ⊗ ℝ), ℚ_{l}(−j)) is given by

(s_{n} ∪ ζ_{l^{n}}^{⊗−j})_{n≥0}

Proof. The automorphism κ^{j} is the inverse limit of similarly defined automorphisms κ^{j} of the rings Λ_{n} := ℤ/l^{n}ℤ[G_{m_{0}l^{n}}]. The sheaf F_{n} := f_{n,∗}f_{n}^{∗}ℤ/l^{n}ℤ (where f_{n} : Spec(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}]) → Spec(ℤ[\frac{1}{ml}]) is the natural map) is free of rank one over Λ_{n} with π_{1}(Spec(ℤ[\frac{1}{ml}]))-action given by the inverse of the natural projection G_{ℚ} → G_{m_{0}l^{n}}. There is a Λ_{n}-κ^{−j}-semilinear isomorphism tw^{j} : F_{n} → F_{n}(j). Shapiro’s lemma gives a commutative diagram of isomorphisms

(5.37)  \begin{matrix}RΓ_{c}(ℤ[\frac{1}{ml}], F_{n}) & ⟶^{tw^{j}} & RΓ_{c}(ℤ[\frac{1}{ml}], F_{n}(j)) \\ ↓ & & ↓ \\ RΓ_{c}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ) & ⟶^{∪ζ_{l^{n}}^{⊗j}} & RΓ_{c}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ(j)) \end{matrix}

where the horizontal arrows are Λ_{n}-κ^{−j}-semilinear. Taking the ℤ/l^{n}ℤ-dual of the lower row (with contragredient G_{m_{0}l^{n}}-action) we obtain a # ∘ κ^{−j} ∘ # = κ^{j}- semilinear isomorphism

RΓ_{c}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ(j))^{∗}[−3] → RΓ_{c}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ)^{∗}[−3].

After passage to the limit this gives a κ^{j}-semilinear isomorphism Δ^{∞} ≅ Δ^{∞}(j), i.e. a Λ-linear isomorphism Δ^{∞} ⊗_{Λ,κ^{j}} Λ ≅ Δ^{∞}(j). Part a) follows by tensoring over Λ with 𝔄_{l}. The ℤ/l^{n}ℤ-dual of the H^{2} of the inverse map in the lower row in (5.37) coincides with

H^{1}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ(1 − j)) ⟵^{∪ζ_{l^{n}}^{⊗−j}} H^{1}(O_{L_{m_{0}l^{n}}} [\frac{1}{ml}], ℤ/l^{n}ℤ(1))

by Poitou-Tate duality. This gives b). Similarly to the lower row in (5.37) we have a κ^{−j}-semilinear map

F_{n}^{c=1} = H^{0}(L_{m_{0}l^{n}} ⊗ ℝ, ℤ/l^{n}ℤ) ⟶^{∪ζ_{l^{n}}^{⊗j}} H^{0}(L_{m_{0}l^{n}} ⊗ ℝ, ℤ/l^{n}ℤ(j)) = F_{n}(j)^{c=1},

the ℤ/l^{n}ℤ-dual of which is the κ^{j}-semilinear isomorphism Λ_{n} · σ ← Λ_{n} · σ ∪ ζ_{l^{n}}^{⊗−j} given by cup product with ζ_{l^{n}}^{⊗−j}. Passing to the limit and tensoring over Λ with A_{l} we deduce c). □

By Lemma 5.13 a) we have an isomorphism of perfect complexes of 𝔄_{l}-modules

(Det_{Λ} Δ^{∞}) ⊗_{Λ,κ^{j}} 𝔄_{l} ≅ Det_{𝔄_{l}} Δ(L_{m})(j)

under which L ⊗ 1 is an 𝔄_{l}-basis of the right hand side by Theorem 5.2. For any χ ∈ Ĝ_{m}^{ℚ_{l}} we have the corresponding ring homomorphism

χκ^{j} : Λ → ℚ_{l}(χ)

whose kernel we denote by 𝔮_{χ,j}. This is a regular prime of Λ and we again apply Lemma 5.7 with R = Λ_{𝔮_{χ,j}}.

If χ(−1) = (−1)^{j+1} then ψ_{𝔮_{χ,j}}(−1) = χ(−1)ω(−1)^{j} = −1 i.e. ψ_{𝔮_{χ,j}} is odd. In this case Δ_{𝔮_{χ,j}}^{∞} is acyclic (there are no trivial zeros at the character χκ^{j}) and Lemma 5.7 applies with β_{i} = b = 1. The image of L in (Det_{Λ} Δ^{∞})_{𝔮_{χ,j}} is

L = θ_{m_{0}}^{−1} = θ_{f_{χ,0}}^{−1} ∏_{p|m_{0},p∤f_{χ,0}} \frac{1}{1 − Fr_{p}^{−1}}

and the image of L ⊗ 1 is

(5.38)  χκ^{j}(L) = ∏_{p|m_{0}} \frac{1}{1 − χ(p)^{−1}p^{−j}} L_{l}(χ^{−1}ω^{1−j}, j)^{−1}

= ∏_{p|ml} \frac{1}{1 − χ(p)^{−1}p^{−j}} L(χ^{−1}, j)^{−1}

where the last equality follows from (5.7). This value agrees with (5.35) which finishes the proof of Conjecture 3 in the case χ(−1) = (−1)^{j+1}.

If χ(−1) = (−1)^{j} then the character ψ_{𝔮_{χ,j}} is even and in order to apply Lemma 5.7 with R = Λ_{𝔮_{χ,j}} we must describe the bases β_{1} and β_{2} in this case. By Lemma 5.13 b) the image of η_{f_{χ,0}} ∈ H^{1}(Δ^{∞})_{𝔮_{χ,j}} in M^{1}/ϖ ≅ H^{1}(O_{L_{m}}[\frac{1}{ml}], ℚ_{l}(1 − j)) is given by the element described in Theorem 5.12, in particular it is nonzero. Hence β_{1} := η_{f_{χ,0}} is a Λ_{𝔮_{χ,j}}-basis of M_{1}, and by Lemma 5.4 the image of σ is then a basis of M^{2} = H^{2}(Δ^{∞})_{𝔮_{χ,j}}. By Lemma 5.13 c) the image of σ in M^{2}/ϖ is ζ_{l^{∞}}^{⊗−j} · σ_{m}. The computation showing that L ⊗ 1 equals the element in (5.34) is now exactly the same as in the case j = 0, χ even, χ(l) ≠ 1, J = ∅.

5.2. CM elliptic curves and the main conjecture for imaginary quadratic fields. Let K be an imaginary quadratic field, and ψ a Hecke character of K of infinity type (1, 0) (such a ψ always exists; see [77] for a nice discussion of such characters with minimal conductor and field of values). The field of values of ψ is a CM-field and we denote by ψ̄ the conjugate character. Any algebraic Hecke character of K is of the form Ψ = ψ^{k}ψ̄^{j}χ where j, k ∈ ℤ and χ is a finite order (Dirichlet) character of K.

The computations of the previous section should have analogues for motives M(Ψ) associated to any algebraic Hecke character Ψ of K, and all primes l but this has not been worked out in all cases. The analogue of the Iwasawa main conjecture (Theorem 5.2 above) is actually simpler since there is no distinction between even and odd characters: The l-adic L-function L is just given by an appropriate norm compatible system η_{m_{0}} of elliptic units. The method of Euler systems allows to prove this main conjecture (for l not dividing the number of roots so unity in the Hilbert class field of K a proof is given in [66]), and the analogue of the theorem of Ferrero and Washington on the vanishing of the μ-invariant is also known if l is split in K/ℚ [38].

The critical case. Recall that a motive M is called critical if ker(α_{M}) = coker(α_{M}) = 0. If the weight of M is different from −1 Conjecture Mot_{∞} then implies that H_{f}^{i}(M) = H_{f}^{i}(M^{∗}(1)) = 0 for all i. If M(Ψ) is critical and of negative weight (modulo replacing Ψ by Ψ̄ this is the range k < 0 and 0 ≤ j < −k) the relation of η_{m_{0}} to the leading coefficient L^{∗}(M(Ψ)) is given by an explicit reciprocity law due to Wiles (for k = −1, j = 0 [78]), Kato (for j = 0 [49][Thm. 2.1.7]) and Tsuji (in general [74]). Proofs of Conjecture 3 (for certain critical Ψ and certain l and A) can be found in [66][Thm. 11.1], [43], [42], [26], [24]. We quote here one result from [24] dealing with the classical Birch and Swinnerton-Dyer case k = −1, j = 0 but with emphasis on non-maximal orders.

Theorem 5.14. (Colwell) Let F/K be an abelian extension and E/F an elliptic curve with CM by K and so that the Weil-Restriction B = Res_{K}^{F}(E) is of CMtype. Assume rank_{ℤ} E(F) = 0. Then Conjecture 3 holds for M = h^{1}(B)(1), 𝔄 = End_{K}(B) and l > 3 any prime number not dividing the class number of K.

Note here that End_{F} (E) may be any order in K and that 𝔄 is usually a nonmaximal order in A = End_{K}(B) ⊗ ℚ.

The non-critical case. Here our knowledge is incomplete in one basic aspect: The ℚ-space H_{f}^{1}(M(Ψ)) is not known to be finite dimensional unless M(Ψ) is a direct summand of a motive discussed in our Example a) (i.e. k = j and we are dealing with M = h^{0}(Spec(L))(−j) where L/K is an abelian extension). One actually works with an explicit subspace of H_{f}^{1}(M(Ψ)) of the expected dimension (i.e. such that Conjecture Mot_{∞} holds). The construction of this space is due to Deninger [29] and the computation (in certain cases) of the étale regulator of its elements due to Kings. Conjecture Mot_{l} is not known for this space in all cases (this is equivalent to the vanishing of H^{2}(ℤ[\frac{1}{ml}], M_{l}) discussed in [53]). To illustrate we quote the main result from [53] corresponding to k < −1, j = k + 1.

Theorem 5.15. (Kings) Let E/K be an elliptic curve with CM by O_{K}. Then Conjecture 3 holds for M = h^{1}(E)(j) where j ≥ 2, 𝔄 = O_{K} and l > 3 is prime to the conductor of E and such that H^{2}(ℤ[\frac{1}{ml}], M_{l}) = 0.

It is quite likely that the arguments of Kings can be used to prove Conjecture 3 for M = h^{1}(E)(j) with j ≤ 0 without any assumption on a vanishing of H^{2}. For a generalisation of Kings’ method to some other Hecke characters of K see [3].

5.3. Adjoint motives of modular forms. Let f be a holomorphic newform on the upper half plane (of weight k ≥ 2, level N and some character), denote by E_{f} the number field generated by the Fourier-coefficients of f and by M(f) the motive associated to f (of rank 2 over E_{f}). Let S_{f} be the finite set of places λ of E_{f} which either divide Nk! or such that the G_{F} -representation on T_{λ}/λ is absolutely reducible, where F is the quadratic subfield of ℚ(ζ_{l}), l is the rational prime below λ, and T_{λ} ⊂ M(f)_{λ} is a G_{ℚ}-stable lattice.

Theorem 5.16. (Diamond, Flach, Guo [31]) Let M = Ad^{0}M(f) be the adjoint motive of M(f) consisting of all endomorphisms of trace 0 in Hom(M(f), M(f)). Then conjecture 3 holds for M or M(1), 𝔄 = O_{E_{f}} and any prime λ ∉ S_{f}.

The proof is rather different from the previous examples as it is not based on either Euler systems or an Iwasawa Main Conjecture. The key ingredient is the Taylor-Wiles method developed to show modularity of elliptic curves over ℚ.

For an integer N let Σ_{N} be the set of newforms of weight 2, level N and trivial character. For the adjoint of the motive M := h^{1}(X_{0}(N)) ≅ ∏_{f∈Σ_{N}} M(f) Theorem 5.16 implies Conjecture 3 with respect to the maximal order 𝔄 = ∏_{f∈Σ_{N}} O_{E_{f}} in the algebra A = ∏_{f∈Σ_{N}} E_{f} ≅ End(J_{0}(N)) ⊗ ℚ. This can be refined as follows.

Theorem 5.17. (Qiang Lin [58]) Let N be a prime number, M = Ad^{0}h^{1}(X_{0}(N)) and 𝔗_{N} the integral Hecke algebra of weight 2 and level N (which is known to coincide with End(Jac(X_{0}(N)))). Then Conjecture 3 holds for M or M(1), 𝔄 = 𝔗_{N} and primes l so that λ ∉ S_{f} for all λ | l and all f ∈ Σ_{N}.

This is proven by an explicit algebraic computation using Theorem 5.16 and the fact that 𝔄[\frac{1}{2}] is a local complete intersection ring (which follows from the Taylor-Wiles method).

5.4. Motives of modular forms. The results for motives M(f) associated to newforms are less complete than those discussed in the previous sections. For example, Conjecture 3 includes the conjecture of Birch and Swinnerton-Dyer (BSD) for (modular) elliptic curves E over ℚ. In the last two decades there has been quite some progress regarding this conjecture in the case ord_{s=1} L(E, s) ≤ 1 (but none for higher vanishing order). Both the Euler system of Heegner points, discovered by Kolyvagin [55], and the Euler system of K_{2}-elements, discovered by Kato [50], allow to prove upper bounds for the Tate-Shafarevich group which are related to L^{∗}(E, 1). Sometimes this suffices to prove equality, either because the upper bound is 1, as it is for all but finitely many l, or because it can be achieved by the construction of elements in the Tate-Shafarevich group (for example using the idea of visibility due to Mazur and Cremona [25]). This allows to verify the l-primary part of the BSD-conjecture for many curves E and primes l (but eventually only for E in a finite list of examples). The paper [2] contains an extension of visibility arguments to modular abelian varieties of dimension > 1, corresponding to weight 2 forms f with E_{f} ≠ ℚ.

We remark at this point that a strategy for proving Kato’s Iwasawa main conjecture [50][Conj. 17.6] for the cyclotomic deformation of motives M(f) at ordinary primes l has been outlined recently by Skinner and Urban. This main conjecture is an analogue of Theorem 5.2, and descent computations along the lines of those given after Lemma 5.7 allow to deduce the l-part of the conjecture of Birch and Swinnerton-Dyer if ord_{s=1} L(E, s) ≤ 1. These computations are straightforward if ord_{s=1} L(E, s) = 0 but require an l-adic analogue of the Gross-Zagier formula due to Perrin-Riou [61] if ord_{s=1} L(E, s) = 1.

In a similar vein, Bertolini and Darmon [8] study the main conjecture for the anticyclotomic deformation of motives M(f) (at least if f is attached to an elliptic curve E) with respect to an auxiliary imaginary quadratic field K and an ordinary prime l. This line of research might eventually lead to cases of the Birch and Swinnerton-Dyer conjecture for E over ring class fields H of K (and 𝔄 = ℤ[Gal(H/K)]).

The motive M = h^{1}(X_{0}(N))(1). A reader familiar with the usual formulation of the BSD-conjecture might not recognize it in Conjecture 3. In this section we illustrate how to go back and forth between the two formulations in the example M = h^{1}(X_{0}(N))(1), N prime, 𝔄 = 𝔗_{N} := End(J_{0}(N)) (the adjoint of which was considered in Theorem 5.17). We hope that such a reformulation can serve as a basis for numerical checks of the equivariant conjecture over 𝔗_{N}, in the spirit of the paper [2] by Stein and Agashe.

In any case the link between Conjecture 3 and its classical counterpart (i.e. the formulation originally used by Bloch and Kato in [10]) is given by an integral version

(5.39)  RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) → RΓ_{f}(ℚ, T_{l}) → ⨁_{p∈S} RΓ_{f}(ℚ_{p}, T_{l})

of the exact triangle (3.1). One has to ensure that all terms are perfect complexes of 𝔄_{l}-modules and there may not be a natural (or any) way to do so. If 𝔄 = ℤ however, or more generally if 𝔄 is maximal and hence a product of Dedekind rings, one can simply define RΓ_{f}(ℚ_{p}, T_{l}) by prescribing its cohomology to be the group H_{f}^{1}(ℚ_{p}, T_{l}) ⊆ H_{f}^{1}(ℚ_{p}, M_{l}) defined by Bloch and Kato in [10] (see the end of section 1.5 in [14]).

Coming back to our example where 𝔄 = 𝔗_{N} we first recall Mazur’s fundamental result [60] that for l ≠ 2 the module T_{l} := H^{1}(X_{0}(N)ℚ̄, ℤ_{l}(1)) is free of rank 2 over 𝔗_{N,l}. In particular the projectivity condition before Conjecture 3 is satisfied. Mazur also shows that D_{l} := H_{dR}^{1}(X_{0}(N)/ℤ_{l})(1) and Fil^{0} D_{l} = H^{0}(X_{0}(N)_{/ℤ_{l}}, Ω_{X_{0}(N)/ℤ_{l}}^{1}) are free of rank 2, resp. 1 over 𝔗_{N,l}. We shall only be able to exploit this in the ”good reduction case” where l ≠ N although the following arguments might be pushed to l = N. We have D_{l} := D_{cris}(M_{l}) = D_{dr}(M_{l}) and by [10][Lemma 4.5] there is a quasi-isomorphism (given by the vertical map)

Fil^{0} D_{l} ⟶^{1−Fr_{p}}  D_{l}
↓  ↓
D_{l}  ⟶^{(1−Fr_{p},π)} D_{l} ⊕ D_{l}/ Fil^{0} D_{l}.

The top row contains an obvious integral complex, and we may define a perfect 𝔗_{N,l}-complex RΓ_{f}(ℚ_{p}, T_{l}) as

RΓ_{f}(ℚ_{p}, T_{l}) = \begin{cases}T_{l}^{I_{p}} ⟶^{1−Fr_{p}} T_{l}^{I_{p}} & l ≠ p \\ Fil^{0} D_{l} ⟶^{1−Fr_{p}} D_{l} & l = p. \end{cases}

If we then define the complex RΓ_{f}(ℚ, T_{l}) by the exact triangle before (3.1) its cohomology is given as follows.

Lemma 5.18. If � (J_{0}(N)) is finite then

H_{f}^{0}(ℚ, T_{l}) = 0, H_{f}^{3}(ℚ, T_{l}) ≅ Hom_{ℤ}(J_{0}(N)(ℚ)_{l^{∞}}, ℚ_{l}/ℤ_{l})

and there are exact sequences of 𝔗_{N,l}-modules

0 → H_{f}^{1}(ℚ, T_{l}) → J_{0}(N)(ℚ) ⊗_{ℤ} ℤ_{l} → Φ_{N,l^{∞}} → H_{f}^{2}(ℚ, T_{l}) → H_{f}^{2}(ℚ, T_{l})^{BK} → 0

0 → � (J_{0}(N))_{l^{∞}} → H_{f}^{2}(F, T)^{BK} → Hom_{ℤ}(J_{0}(N)(ℚ), ℤ_{l}) → 0

where Φ_{N} is the group of components of the reduction of the Neron model of J_{0}(N) in characteristic N.

Proof. See the computations in [14][(1.35)-(1.37)] □

For simplicity we now put ourselves in a rank zero situation. The winding quotient π : J_{0}(N) → J is an abelian variety over ℚ, maximal (up to isogeny) with respect to the property that L(h^{1}(J), 1) ≠ 0. More precisely, if e ∈ H_{1}(X_{0}(N), ℚ)^{+} is the image of the path from i∞ to 0 in the upper half plane, ℑ_{e} ⊆ 𝔗_{N} is the annihilator of e and

ℑ = Ann_{𝔗_{N}} (Ann_{𝔗_{N}} (ℑ_{e})) ⊇ ℑ_{e}

is the saturation of ℑ_{e} then J = J_{0}(N)/ℑ_{e}J_{0}(N) = J_{0}(N)/ℑJ_{0}(N) and

H_{1}(J(ℂ), ℤ) ≅ H_{1}(J_{0}(N)(ℂ), ℤ)/ℑ · H_{1}(J_{0}(N)(ℂ), ℤ)

(after inverting 2). In particular, the homology of J is locally free (away from 2) over 𝔗 := 𝔗_{N}/ℑ and the conditions before Conjecture 3 are satisfied for J and 𝔄 = 𝔗 and l ≠ 2. The usual BSD-conjecture for J is studied for example in [1] and we shall make precise the conjecture over 𝔄 = 𝔗.

Mazur [60] shows that 𝔗_{N}[\frac{1}{2}] is Gorenstein, and we shall henceforth assume the same for 𝔗[\frac{1}{2}] (for many N it will simply be the case that ℑ[\frac{1}{2}] = (\frac{1+w}{2} ) where w is the Atkin Lehner involution, and 𝔗[\frac{1}{2}] will be a direct factor of 𝔗_{N}[\frac{1}{2}] and hence Gorenstein). Then the homology of the dual abelian variety Ĵ is also locally free over 𝔗. Denoting Neron models by /ℤ, it is known that the map

π^{∗} : H^{0}(J_{/ℤ}, Ω_{J/ℤ}^{1}) → H^{0}(J_{0}(N)_{/ℤ}, Ω_{J_{0}(N)/ℤ}^{1})[ℑ_{e}]

is injective and has 2-torsion cokernel (the cokernel may be regarded as a generalized Manin constant attach to J). Dualizing, we find that H^{1}(Ĵ_{/ℤ_{l}}, O) is free of rank 1 over 𝔗_{l} for l ≠ 2 and so this also holds for H^{0}(J_{/ℤ_{l}}, Ω_{J/ℤ_{l}}^{1}) and H_{dR}^{1}(J/ℤ_{l}) by our Gorenstein assumption. In summary, all of the above considerations apply to J or Ĵ in place of J_{0}(N).

For M = h^{1}(Ĵ)(1) we have A := 𝔗_{ℚ}-isomorphisms

M_{B} = H^{1}(Ĵ(ℂ), 2πi · ℚ) ≅ H^{1}(J(ℂ), ℚ)^{∗} ≅ H_{1}(J(ℂ), ℚ)
M_{dR}/ Fil^{0} M_{dR} ≅ H^{1}(Ĵ, OĴ) ≅ H^{0}(J, Ω_{J/ℚ}^{1})^{∗}
Ξ(_{A}M) = (H_{1}(J(ℂ), ℚ)^{+})^{−1} ⊗_{A} H^{0}(J, Ω_{J/ℚ}^{1})^{∗}

where ∗ denotes the ℚ-dual. The period isomorphism sends a path γ to the linear form ω ↦ ∫_{γ} ω on differentials, and α_{M} : H_{1}(J(ℂ), ℝ)^{+} ≅ H^{0}(J, Ω_{J/ℚ}^{1})_{ℝ}^{∗} is an isomorphism. By a well known fact in the theory of modular forms the ℤ-linear form a_{1} ∈ H^{0}(J_{0}(N)_{/ℤ}, Ω_{J_{0}(N)/ℤ}^{1})^{∗} given by

H^{0}(J_{0}(N)_{/ℤ}, Ω_{J_{0}(N)/ℤ}^{1}) ≅ S^{2}(Γ_{0}(N), ℤ) ∋ g ↦ a_{1}(g)

is in fact a 𝔗_{N}-basis of H^{0}(J_{0}(N)_{/ℤ}, Ω_{J_{0}(N)/ℤ}^{1})^{∗} and hence its image is a 𝔗-basis of

H^{0}(J_{0}(N)_{/ℤ}, Ω_{J_{0}(N)/ℤ}^{1})^{∗}/ℑ ≅ H^{0}(J_{/ℤ}, Ω_{J/ℤ}^{1})^{∗}

away from 2.

Lemma 5.19. With the notation introduced above we have

Aϑ_{∞}^{−1}(L^{∗}(_{A}M, 0)^{−1}) = e^{−1} ⊗ a_{1} ∈ Ξ(_{A}M)

Proof. By definition of _{A}ϑ_{∞} we have _{A}ϑ_{∞}(e^{−1} ⊗ α_{M}(e)) = 1. We can write α_{M}(e) = t · a_{1} with t = (t_{f}) ∈ 𝔗_{ℝ} ≅ ∏_{f∈Σ′_{N}} ℝ where Σ′_{N} is the subset of forms in Σ_{N} corresponding to J. Applying this identity to a newform f ∈ Σ′_{N} we find

L(f, 1) = ∫_{i∞}^{0} f(q)\frac{dq}{q} = ∫_{e} ω_{f} = α_{M}(e)(ω_{f}) = ta_{1}(ω_{f}) = t_{f}a_{1}(f) = t_{f}.

Hence t = L(_{A}M, 0) = L^{∗}(_{A}M, 0) and _{A}ϑ_{∞}(e^{−1} ⊗ a_{1}) = L^{∗}(_{A}M, 0)^{−1}. □

One verifies easily that a_{1} is in fact a 𝔗_{l}-basis of det_{𝔗_{l}} RΓ_{f}(ℚ_{p}, T_{l}) as defined above. However, e need not be a 𝔗_{l}-basis of H_{1}(J(ℂ), ℤ_{l})^{+}. So let 𝔟 be the invertible fractional 𝔗-ideal so that

(5.40)  𝔟 · e = H_{1}(J(ℂ), ℤ)^{+}

away from l = 2. The triangle (5.39) induces an isomorphism

Det_{𝔗_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l})

≅ Det_{𝔗_{l}} RΓ_{f}(ℚ, T_{l}) ⊗ Det_{𝔗_{l}}^{−1} RΓ_{f}(ℚ_{l}, T_{l}) ⊗ Det_{𝔗_{l}}^{−1} H_{1}(J(ℂ), ℤ)^{+}
≅ Det_{𝔗_{l}} RΓ_{f}(ℚ, T_{l}) ⊗ 𝔟^{−1}a_{1} ⊗ e^{−1}.

Now recall that since the rank of J(ℚ) is zero RΓ_{f}(ℚ, T_{l}) is a (perfect) 𝔗_{l}-complex with torsion cohomology and hence

Det_{𝔗_{l}} RΓ_{f}(ℚ, T_{l}) ⊆ Det_{𝔗_{ℚ_{l}}} RΓ_{f}(ℚ, M_{l}) ≅ 𝔗_{ℚ_{l}}

identfies with a fractional ideal. The statement that e^{−1} ⊗ a_{1} is a basis of the invertible 𝔗_{l}-module Det_{𝔗_{l}} RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) is equivalent to

(5.41)  Det_{𝔗_{l}} RΓ_{f}(ℚ, T_{l}) = 𝔟_{l}.

Summarizing we have

Proposition 5.1. Let N be a prime number and J the winding quotient for J_{0}(N). For any prime l ≠ 2, N there exists a perfect complex RΓ_{f}(ℚ, T_{l}) of 𝔗_{l}- modules with cohomology in degrees 1, 2, 3 given by the exact sequence

0 → H_{f}^{1}(ℚ, T_{l}) → J(ℚ)_{l^{∞}} → Φ_{l^{∞}} → H_{f}^{2}(ℚ, T_{l}) → � (J/ℚ)_{l^{∞}} → 0

and H_{f}^{3}(ℚ, T_{l}) ≅ Ĵ(ℚ)_{l^{∞}}. Moreover, the l-primary part of Conjecture 3 for M = h^{1}(Ĵ)(1) and 𝔄 = 𝔗 is equivalent to (5.41) where 𝔟 is defined in (5.40).

Taking norms from 𝔗 to ℤ we find that the usual BSD-conjecture for J is equivalent to the statement

\frac{|� (J/ℚ)||Φ|}{|J(ℚ)||Ĵ(ℚ)|} = N_{𝔗/ℤ}(𝔟)^{−1}

up to powers of 2 and N. The right hand side is the index of the ℤ-lattice 𝔗 · e in the lattice H_{1}(J(ℂ), ℤ)^{+} (in the generalized sense since the two are not contained in each other). We therefore (almost) recover Agashe’s formula [1][Eq. (1)]. The difference is that we work with the quotient H_{1}(J(ℂ), ℤ) of H_{1}(J_{0}(N)(ℂ), ℤ) rather than the submodule H_{e} used in [1]. So there is no analog of the term H^{+}/(I_{e}H)^{+} + H_{e}^{+} in our formula. Moreover, the factor c_{M} (resp. n) in the left hand side of [1][Eq. (1)] does not occur in our formula because we disregard l = 2 (resp. we do not introduce the Eisenstein ideal into our computation).

It is expected that the only primes l dividing |J(ℚ)|, |Ĵ(ℚ)| and |Φ| are those dividing the numerator n of (N − 1)/12. Assuming this and l ∤ n we find that � (J/ℚ)_{l^{∞}} is of finite projective dimension over 𝔗_{l} (and its fitting ideal should be 𝔟_{l}^{−1}). It would be interesting to look for examples of pairs (N, l) where 𝔟_{l} is nontrivial, 𝔗_{l} is nonmaximal and l ∤ n. In such a case ”being of finite projective dimension” would ascertain a nontrivial module theoretic property of � (J/ℚ).

Motives M(f)(n) for f of weight k. In the critical range 1 ≤ n ≤ k − 1 Conjecture 2 (apart from the vanishing of H_{f}^{1}(M)) is known and the results of Kato [50] yield upper bounds for the Selmer group similar to the Birch and Swinnerton-Dyer case discussed above. If k is even and the sign of the functional equation of L(f, s) is −1 a limit formula for L′(f, \frac{k}{2} ) has been proven by Zhang in [81], generalizing the Gross-Zagier formula in the case k = 2.

In the noncritical range n ≥ k one has results towards Conjecture 2 (the Beilinson conjecture), involving the construction of a subspace of H_{f}^{1}(M) whose image under the regulator is related to L(f, n) (see [68] for k = 2 and n ≥ 2. The case n ≥ k > 2 has been announced in [30] but has not yet appeared in print). For progress towards computation of the étale regulator of the elements constructed in [30] see [37].

5.5. Totally real fields and assorted results. The results of sections 5.1, 5.4 and 5.3 all have partial analogues for a totally real base field F in place of ℚ. In this section we fix such a field F.

Abelian extensions of F. The main conjecture of Iwasawa theory, proved by Wiles [79], together with the vanishing of the μ-invariant for totally real fields, very recently shown by Barsky [4], yields the following result. For a CM field L we denote by c the unique complex conjugation of L.

Theorem 5.20. (Wiles/Barsky) Let L/F be an abelian extension with group G and so that L is a CM-field. For j < 0 let M^{−} be the direct summand of h^{0}(Spec(L))(j) cut out by the rational idempotent associated to (1 + (−1)^{j}c)/2 of ℚ[G]. Then conjecture 3 holds for M^{−}, l ≠ 2, and 𝔄 = ℤ[G]/(1 − (−1)^{j}c) (one also needs to assume that the ray class field of F of conductor ℓ is a CM field to satisfy hypothesis H-0 of [4]).

Proof. (Sketch) The l-adic L-function interpolating the (critical) values of L-functions of finite order Hecke characters of F is given by an inverse limit of Stickelberger elements (as in [80][Eq. (1)]), similar to the elements θ_{m_{0}} discussed in section 5.1. However, there is no analogue of cyclotomic units (unless F/ℚ is abelian), and the generalization of Theorem 5.2 can only can be proven for Δ^{∞} ⊗_{Λ}^{𝕃} Λ/(c + 1). The analogue of the odd part of Lemma 5.4 is [79][Thm. 1.2, Thm. 1.4], and the vanishing of the μ-invariant of θ_{m_{0}} and hence of P^{∞} is proven recently in [4]. The descent arguments are then identical to those given in section 5.1. □

Remark. For j = 0 and (odd) characters χ such that χ(l) = 1 there is no analogue of Lemma 5.11 and therefore a descent along the lines indicated above is not possible. With the results of [80][Thm. 1.3] one might be able to deduce some cases of Conjecture 3 for motives h^{0}(Spec L)(χ) cut out by rational characters χ of G and 𝔄 a maximal order in ℚ(χ).

Note that the conditions j < 0 and χ(c) = (−1)^{j+1} ensure that M(χ) is not only critical but that in fact all the six ℚ-spaces involved in the definition of Ξ(M) vanish. This is the simplest possible situation of the Tamagawa number conjecture: the L-value is an algebraic number (in the coefficient field A).

Hilbert modular forms. Shouwu Zhang has recently generalized the formula of Gross-Zagier from X_{0}(N) to arbitrary Shimura curves over totally real fields F [82]. One therefore has a generalization of the Euler system of Heegner points from motives M(f) discussed in section 5.4 to motives attached to Hilbert modular forms f of parallel weight 2 with all the ensuing consequences. The work of Zhang probably represents the most significant advance concerning Conjectures 1-2 in recent years. For illustration we quote one result from [82].

Theorem 5.21. (Zhang) Let f be a Hilbert modular newform over F, of parallel weight 2 and level N. Assume that either [F : ℚ] is odd or that ord_{v}(N) = 1 for one place of F, and let A(f) be the abelian variety associated to f. Finally assume that ord_{s=1} L(f, s) ≤ 1. Then Conjectures 1 and 2 hold for M = h^{1}(A(f))(1) and A = E_{f}, the field generated by the Hecke eigenvalues of f. Moreover, � (A(f)) is finite or, equivalently, Conjecture Mot_{l} holds for all l.

For more details we refer to the papers by Zhang [82], [83] and Tian [73]. We remark that an analogue for Kato’s Euler system is still missing over totally real fields (and may be not expected).

Adjoint motives of Hilbert modular forms. The Taylor-Wiles method has been partially generalized to totally real fields [36] and may be expected to give instances of Conjecture 3, analogous to those in section 5.3. To a certain extent this has been worked out in the recent thesis of Dimitrov [32] (without the precise relation to a motivic period, however). In a slightly different direction, using Fujiwara’s work for Hilbert modular forms of CM-type, Hida [44] has recently shown many cases of the anticyclotomic main conjecture for CM-fields.

Hecke Characters. Conjectures 1 and 2 are known for critical motives M(Ψ) where Ψ is any Hecke character of any number field [9]. Using the work of Hida [44] mentioned in the previous paragraph it is quite likely that one can also obtain many instances of Conjecture 3 for such motives (if Ψ is a Hecke character of a CM-field). In the noncritical range Conjectures 1 and 2 are known for Tate motives h^{0}(Spec(L))(j) over any number field L and A = ℚ [12].

We refer to Ramakrishnan’s comprehensive survey article [64] for the state of affairs with regard to Conjectures 1 and 2 circa 1989, before Conjecture 3 was formulated.

Part 3. Determinant Functors: Some Algebra

In order to formulate an equivariant Tamagawa Number Conjecture over a not necessarily commutative semisimple algebra A we need to discuss some algebraic preliminairies. Recall that for a commutative ring R and finitely generated projective R-module P, the rank of P is a locally constant integer valued function rank_{R}(P) ∈ H^{0}(Spec(R), ℤ). The determinant of P is the invertible R-module

det_{R}(P) := ⋀^{rank_{R}(P)} P.

A short exact sequence

0 → P_{1} → P_{2} → P_{3} → 0

induces a (functorial) isomorphism

(5.42)  det_{R}(P_{2}) ≅ det_{R}(P_{1}) ⊗ det_{R}(P_{3}).

If P_{2} = P_{1} ⊕ P_{3}, and one does not specify which of P_{1} and P_{3} one views as the submodule and which as the quotient, there is a sign ambiguity in the isomorphism (5.42) (if the tensor product of invertible modules is endowed with its usual commutativity isomorphism x ⊗ y ↦ y ⊗ x). In order to avoid such ambiguity one has to retain the rank information and view the determinant as a functor

Det_{R} : (PrMod(R), is) → Inv(R), P ↦ Det_{R}(P) := (det_{R}(P), rank_{R}(P))

from the category of finitely generated projective R-modules PrMod(R), and isomorphism of such, to the category Inv(R) of graded invertible R-modules, and isomorphisms of such. The monoidal category Inv(R) has a modified commutativity constraint involving a sign depending on the grading (see [16][2.5] for more details). The functor Det_{R} can be extended to perfect complexes and isomorphisms of such, and there is an isomorphism (5.42) for short exact sequences of complexes. All computations in the last section are understood to be performed with Det rather than det. For example, the source of the map ϑ_{∞} is really the graded module (ℝ, 0).

We now indicate how a slightly more abstract point of view on the determinant functor leads to its generalization to non-commutative rings.

The category Inv(R) is an example of a so called Picard category: • All morphisms are isomorphism.

• There is a bifunctor (L, M) ↦ L ⊠ M with unit object 1, inverses, associativity and commutativity constraint.

Definition. Let R be any ring. A determinant functor is a Picard category P, a functor D : (PrMod(R), is) → P, and functorial isomorphisms D(P_{2}) ≅ D(P_{1}) ⊠ D(P_{3}) for short exact sequences, satisfying conditions as indicated in [16][2.3].

Theorem 5.22. (Deligne, [28]) For any ring R there is a universal determinant functor

D_{R} : (PrMod(R), is) → V (R).

V (R) is called the category of virtual objects of R.

This is a categorical version of the Grothendieck group K_{0}(R). Indeed, any Picard category P gives rise to two abelian groups π_{0}(P) and π_{1}(P), the group of isomorphism classes of objects of P with product induced by ⊠, and the automorphism group of the unit object 1 (or any other object). Deligne also shows that D_{R} induces isomorphisms

K_{0}(R) ⟶^{∼} π_{0}(V (R)) := \{ \begin{matrix}Isomorphism classes of objects \\ with product induced by ⊠ \end{matrix}
K_{1}(R) ⟶^{∼} π_{1}(V (R)) := Aut_{V (R)}(1)

If R is commutative we have a (monoidal) functor V (R) → Inv(R) by universality. This functor induces (split) surjections

K_{0}(R) ↠ Pic(R) ⊕ H^{0}(Spec(R), ℤ) = π_{0}(Inv(R))
K_{1}(R) ↠ R^{×} = π_{1}(Inv(R))

and V (R) → Inv(R) is an equivalence of categories if and only if both maps are isomorphisms. For a general commutative ring R this is rarely the case but it is true if R is a product of local rings. Examples are the rings R = A, A_{l} := A_{ℚ_{l}}, 𝔄_{l}, A_{ℝ} considered in the previous section.

6. Noncommutative Coefficients

In order to formulate Conjectures 1-3 in the commutative case it was necessary to take determinants over R = A, A_{l}, 𝔄_{l}, A_{ℝ}, and for these rings we have an equivalence of categories V (R) ≅ Inv(R). This suggests that the categories V (R) can be used to generalize Conjectures 1-3 to motives M with an action of an arbitrary (semisimple) algebra A, and ℤ-orders 𝔄 ⊆ A so that there exists a projective, G_{ℚ}-stable 𝔄_{l}-lattice T_{l} ⊆ M_{l}. This is indeed the case. More specifically:

• Ξ(_{A}M) is an object of V (A).

• _{A}ϑ_{∞} : 1_{V (A_{ℝ})} ≅ Ξ(_{A}M) ⊗_{A} A_{ℝ} is an isomorphism in V (A_{ℝ}). Here the tensor product V ⊗_{R} R′ for V an object of V (R) and R → R′ any ring extension has to be understood as the monoidal functor

− ⊗_{R} R′ : V (R) → V (R′)

induced by the exact functor

− ⊗_{R} R′ : PrMod(R) → PrMod(R′)

and the universal property of V (R).

• D_{𝔄_{l}}(RΓ_{c}(ℤ[\frac{1}{S}], T_{l})) is an object of V (𝔄_{l}).

• _{A}ϑ_{l} : Ξ(M) ⊗_{A} A_{l} ≅ D_{𝔄_{l}}(RΓ_{c}(ℤ[\frac{1}{S}], T_{l})) ⊗_{𝔄_{l}} A_{l} is an isomorphism in

V (A_{l}). An A-equivariant L-function L(_{A}M, s) can be defined as a meromorphic func-

tion with values in the center ζ(A_{ℂ}) of A_{ℂ} [16][Sec. 4]. Its vanishing order can be viewed as a locally constant function

r(_{A}M) ∈ H^{0}(Spec(ζ(A_{ℝ}), ℤ)

and for any finitely generated A-module P we define dim_{A} P ∈ H^{0}(Spec(ζ(A), ℤ) as its reduced rank ([16][2.6]).

Conjecture 1 (Final Version):

r(_{A}M) = dim_{A} H_{f}^{1}(M^{∗}(1)) − dim_{A} H_{f}^{0}(M^{∗}(1))

The leading coefficient L^{∗}(_{A}M) of L(_{A}M, s) is a unit in ζ(A_{ℝ}). There is a reduced norm map

Aut_{V (A_{ℝ})}(1) = K_{1}(A_{ℝ}) ⟶^{nr} ζ(A_{ℝ})^{×}

which is injective but not necessarily surjective (in the case where A_{ℝ} has quaternionic Wedderburn components the cokernel of nr is a group of exponent 2). One may pick μ ∈ ζ(A)^{×} so that

μL^{∗}(_{A}M) = nr(L_{μ,ℝ})

for a (unique) L_{μ,ℝ} ∈ K_{1}(A_{ℝ}). Then the following conjecture is independent of the choice of μ.

Conjecture 2 (Final Version): The composite morphism

1_{V (A_{ℝ})} ⟶^{L_{μ,ℝ}^{−1}} 1_{V (A_{ℝ})} ⟶^{_{A}ϑ_{∞}} Ξ(_{A}M) ⊗_{A} A_{ℝ}

in V (A_{ℝ}) is the image of a morphism 1_{V (A)} ⟶^{L_{μ}^{−1}} Ξ(_{A}M) in V (A) under the scalar extension functor − ⊗_{A} A_{ℝ}.

Now recall that for any prime number l the reduced norm map nr_{l} : K_{1}(A_{l}) → ζ(A_{l})^{×} is an isomorphism so that there is a unique element μ_{l} ∈ K_{1}(A_{l}) with nr_{l}(μ_{l}) = μ. Assuming Conjecture 2, the morphism

L_{l}^{−1} : 1_{V (A_{l})} ⟶^{μ_{l}} 1_{V (A_{l})} ⟶^{L_{μ}^{−1}⊗_{A}A_{l}} Ξ(_{A}M) ⊗_{A} A_{l}

is independent of the choice of μ.

Conjecture 3 (Final Version): The composite morphism

1_{V (A_{l})} ⟶^{L_{l}^{−1}} Ξ(_{A}M) ⊗_{A} A_{l} ⟶^{_{A}ϑ_{l}} D_{𝔄_{l}}(RΓ_{c}(ℤ[\frac{1}{S}], T_{l})) ⊗_{𝔄_{l}} A_{l}

in V (A_{l}) is the image of a morphism 1_{V (𝔄_{l})} → D_{𝔄_{l}}(RΓ_{c}(ℤ[\frac{1}{S}], T_{l})) in V (𝔄_{l}) under the scalar extension functor − ⊗_{𝔄_{l}} A_{l}.

This conjecture is only of interest for non-maximal orders 𝔄, in the sense that it is implied by Conjecture 3 for commutative coefficients (for certain motives related to M) if 𝔄 is maximal [16][Prop. 4.2].

A Reformulation. Let V (𝔄_{l}, ℚ_{l}) denote the Picard category whose objects are pairs (V, τ) where V is an object of V (𝔄_{l}) and τ : V ⊗_{𝔄_{l}} A_{l} ≅ 1_{V (A_{l})} is an isomorphism. Then π_{0}(V (𝔄_{l}, ℚ_{l})) is the usual relative K_{0}(𝔄_{l}, ℚ_{l}) [16][Prop. 2.5] and the object D_{𝔄_{l}}(RΓ_{c}(ℤ[\frac{1}{S}], T_{l})), together with the map in Conjecture 3 defines a class T Ω(M, 𝔄)_{l} ∈ K_{0}(𝔄_{l}, ℚ_{l}). Conjecture 3 is equivalent to the vanishing

T Ω(M, 𝔄)_{l} = 0

in K_{0}(𝔄_{l}, ℚ_{l}). Moreover, as shown in [16][Sec. 3], one can define a class T Ω(M, 𝔄) ∈ K_{0}(𝔄, ℝ) so that Conjecture 2 holds if and only if

T Ω(M, 𝔄) ∈ K_{0}(𝔄, ℚ) ⊆ K_{0}(𝔄, ℝ).

If this is the case then T Ω(M, 𝔄)_{l} coincides with the l-component of T Ω(M, 𝔄) under the decomposition K_{0}(𝔄, ℚ) ≅ ⨁_{l} K_{0}(𝔄_{l}, ℚ_{l}).

7. The Stark Conjectures

There is only one class of examples where Conjecture 3 with noncommutative coefficients has been considered so far, essentially those of our Example a) with j ≤ 0. Traditionally the motivation in this work is to investigate the ℤ[G]-structure of the unit group O_{L}^{×} (or of K_{1−2j}(O_{L}) for j < 0 [14], [23]).

The following theorem indicates how previous work is implied by Conjecture 3.

Theorem 7.1. Let L/K be a Galois extension of number fields with group G and put M = h^{0}(Spec(L)), A = ℚ[G]. Denote by 𝔐 a maximal order of A containing ℤ[G].

a) Conjecture 1 is true.

b) Conjecture 2 is equivalent to Stark’s Conjecture as given in [72][Ch. I, Conj. 5.1].

c) Conjecture 3 for 𝔄 = 𝔐 (and all l) is equivalent to the strong Stark Conjecture as formulated by Chinburg [20][Conj. 2.2].

d) Conjecture 3 for 𝔄 = ℤ[G] (and all l) implies Chinburg’s conjecture [21]

ω(L/K) + Ω(L/K, 3) = 0.

Proof. We refer to [17] for details of this theorem. The key fact is the existence of a perfect complex of ℤ[G]-modules Ψ which underlies both RΓ_{c}(ℤ[\frac{1}{S}], T_{l}) and Ξ(_{A}M), and which is quasi-isomorphic to a complex used in Chinburg’s work (a so called Tate-sequence). The proof of this quasi-isomorphism is rather involved and given in [15]. The implication in d) follows because the image of T Ω(M, 𝔄) under the natural map K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[G]) is ω(L/K) + Ω(L/K, 3). □

The supply of known cases of Conjecture 3 for M = h^{0}(Spec(L)) and noncommutative 𝔄 = ℤ[G] is rather scarce.

Theorem 7.2. (Burns-Flach/Chinburg) Let l be any rational prime such that l ≡ 1 mod 12 and also (\frac{l}{7}) = −(\frac{l}{5}) = 1. There is a unique extension L/ℚ with Galois group the quaternion group Q_{8} of order 8, ramified precisely at {3, 5, 7, l, ∞} and containing ℚ(\sqrt{21}, \sqrt{5}) [22][Prop. 4.1.3]. Then Conjecture 3 holds for M = h^{0}(Spec(L)) and 𝔄 = ℤ[Q_{8}].

Proof. Put V = Gal(ℚ(\sqrt{21}, \sqrt{5})/ℚ). One verifies that the kernel of the homomorphism

K_{0}(ℤ[Q_{8}], ℝ) → K_{0}(𝔐, ℝ) × K_{0}(ℤ[Q_{8}]) × K_{0}(ℤ[V], ℝ)

given by the natural maps to each factor, is trivial (although the kernel of each map to a pair of factors is not) [17][Lemma 4]. The image of T Ω(M, ℤ[Q_{8}]) in K_{0}(𝔐, ℝ) is trivial because of Theorem 7.1 c) and the fact that the Strong Stark Conjecture is known for groups all of whose characters are rational (such as Q_{8}) [72][Ch.II, Th. 6.8]. The image of T Ω(M, ℤ[Q_{8}]) in K_{0}(ℤ[Q_{8}]) is trivial because of Theorem 7.1 d) and computations of Chinburg showing the vanishing of ω(L/ℚ) + Ω(L/ℚ, 3) [22]. Finally the image in K_{0}(ℤ[V], ℝ) is trivial by functoriality [16][Prop. 4.1b)] and Theorem 5.1. For a nice concrete interpretation of this last vanishing see [52]. □

One may ask whether there is hope of verifying Conjecture 3, or even just Conjecture 2, in genuine non-abelian cases, for example where G = Gal(L/ℚ) is isomorphic to the alternating group A_{5}. This group has five irreducible characters χ_{1}, χ_{3}, χ̄_{3}, χ_{4}, χ_{5} where χ_{3} takes values in ℚ(\sqrt{5}) and the other characters are rational. For rational characters χ the strong Stark conjecture for L^{∗}(χ, 0) is known by [72][Ch.II, Th. 6.8]. For χ_{3} one is in the favorable situation where ord_{s=0} L(χ_{3}, s) = 1 (if L is imaginary) and one might be able to verify the Stark conjecture numerically (see [67] for an example of such verifications).

Conjecture 3 for 𝔄 = ℤ[G], on the other hand cannot be checked character by character (but rather entails some relationship among all L^{∗}(χ_{i}, 0)). A numerical verification would possibly involve a computation of the full group of units of L with its ℤ[G]-structure, and seems currently out of reach. We remark that Bley [11] has verified Conjecture 3 numerically for certain cyclic extensions L/ℚ(\sqrt{5}) of order 3 and 5.

Beyond numerical verifications, one has to prove limit formulas such as the one in section 5.1 relating Dirichlet L-series and cyclotomic units. There is now a large supply of extensions L/ℚ with group A_{5} where it is known that L(χ_{i}, s) coincides with the L-function attached to an automorphic form on GL_{i}(ℚ) for all i (combine [19] with arguments as in [65] and the recent preprint [51]). This does not seem to help in proving any relationship between the leading coefficient L^{∗}(χ_{i}, 0) and units in L, however.

For more information on the Stark conjectures we refer to the article of David Burns in this volume.

APPENDIX: On the vanishing of μ-invariants by C. Greither

The purpose of this appendix is to give a proof that the μ-invariant of an often-used Iwasawa module (the one referred to as “limit of units modulo limit of cyclotomic units”) is zero for the cyclotomic p-tower over every absolutely abelian field K and for every prime p. The stress lies on the two occurrences of “every”: for odd p this seems to be well-known, and for p = 2 and K a full cyclotomic field, the result may be extracted from work of Kuz’min, see below. So if there is anything really new here, it is the case p = 2 and K not a full cyclotomic field. But for the reader’s convenience, and since it does not cost anything extra, we shall give a unified argument for all p and K, with the only restriction that K is assumed imaginary; we will provide a few comments on the real case at the end. The author would like to use the opportunity to point out the following: In the author’s paper [41], Theorem 3.1 and the subsequent remark c) claim an equality of characteristic ideals, which would imply the desired vanishing of the μ-invariant. But this implication does not stand since the argument in loc. cit. only yields that the characteristic ideals are equal up to a power of p. Thanks to Annette Huber and Matthias Flach, who (independently) noticed this.

Let us define our objects precisely before we state the result. Let p be a prime number (we repeat that p = 2 is permitted), fixed in the sequel. Let us also fix an absolutely abelian field K with Galois group G over the rationals. Let K_{∞} be the cyclotomic ℤ_{p}-extension of K; write Γ for Gal(K_{∞}/K) and Γ_{n} for Gal(K_{∞}/K_{n}) where K_{n} is the n-th layer of K_{∞}. We subsume p-adification in our notation by writing E_{n} for ℤ_{p} ⊗ O_{K_{n}}^{∗} and C_{n} for ℤ_{p} ⊗ Sinn_{K_{n}} where Sinn_{L} denotes the group of circular units in Sinnott’s sense, for any abelian field L. Let E (respectively C) denote the limit of the projective system (E_{n})_{n} resp. (C_{n})_{n} via the norm maps. (These are often written E_{∞} and C_{∞}.) Finally we put Λ = ℤ_{p}[[Γ]] as customary. The result reads:

Theorem. Under the above assumption that K is absolutely abelian and imaginary, and with the notation just introduced, the Λ-module E/C has μ-invariant zero.

Apart from a few short remarks at the end, the rest of this appendix is occupied by the proof of this result. The essential input comes from Sinnott’s work: from his main theorem [70][p.182] we immediately deduce the existence of a constant c_{K} depending only on K (not on n) such that

[E_{n} : C_{n}] = c_{K} · h(K_{n}^{+})

for all sufficiently large n ∈ ℕ. This together with the Ferrero-Washington theorem implies that there exists another constant λ = λ_{K} such that the index [E_{n} : C_{n}] is O(p^{λn}) as a function of n. This does not directly imply the nullity of μ(E/C), since there certainly exists a projective system (X_{n})_{n} of ℤ_{p}[Γ/Γ_{n}]-modules with quite slow growth and such that the limit is Λ/pΛ: take X_{n} = Λ/(p, T^{n}). The point is that the “natural” system (E_{n}/C_{n})_{n} cannot misbehave in that way, as we will show.

We will use an ad hoc notion. A family (not necessarily a projective system) (A_{n})_{n} of finite ℤ_{p}-modules is called tame if the following holds: There exist positive integers c_{1} and c_{2} and submodules A′_{n} ⊂ A_{n} requiring at most c_{2} ℤ_{p}-generators each, such that the indices [A_{n} : A′_{n}] are O(p^{c_{1}n}) as a function of n. We observe: Bounded families are tame; and if U_{n} ⊂ A_{n} for all n, then the family (A_{n})_{n} is tame iff both families (U_{n})_{n} and (A_{n}/U_{n})_{n} are tame.

These observations easily lead to the following consequence: If the Λ-module X has μ(X) > 0, then the family of Γ_{n}-coinvariants (X_{Γ_{n}})_{n} will not be tame. We therefore just need to show that the family ((E/C)_{Γ_{n}})_{n} is tame.

Let i_{n} be the canonical map E_{Γ_{n}} → E_{n} and let B_{n} stand for its kernel. Write B′_{n} for the image of B_{n} in (E/C)_{Γ_{n}}. Then the induced map j_{n} : ((E/C)_{Γ_{n}})/B′_{n} → E_{n}/C′_{n} is injective, where C′_{n} is the image of the n-th projection C → C_{n}. Alternatively, C′_{n} can be described as the subgroup of stable norms inside C_{n}, more explicitly: C′_{n} is the set of all x ∈ C_{n}, which are in the image of C_{m} → C_{n} for all m ≥ n.

Assume we can establish the following two facts:

(1) (E_{n}/C′_{n})_{n} is tame; and

(2) (B′_{n})_{n} is tame.

Then from the injectivity of j_{n} and the abovementioned simple properties of tame families we get tameness of (E/C)_{Γ_{n}}, and we will be done. Fact (1) is, obviously, a consequence of the tameness of (E_{n}/C_{n})_{n} (which we know to hold) and another fact:

(3) (C_{n}/C′_{n})_{n} is tame.

It thus remains to establish (2) and (3); in fact we will show tameness with the parameter c_{1} set to zero, that is, the number of required ℤ_{p}-generators is bounded in both families.

Proof of Fact (3): One has to recall the construction of Sinnott units. Consider the following group D_{n} of p-adified Sinnott circular numbers: D_{n} is generated over ℤ_{p}[G] by all elements z_{d} = N_{ℚ(d)/ℚ(d)∩K_{n}}(1 − ζ_{d}), where ℚ(d) is short for ℚ(ζ_{d}) and d ≠ 1 divides the conductor f_{n} of K_{n} and (d, f_{n}/d) = 1. Then C_{n} = D_{n} ∩ E_{n}, by [57][Prop. 1] and an easy extra argument to eliminate those d which are not coprime to f_{n}/d. (For p = 2, one also has to put −1 into D_{n}.) Since a Sinnott circular number as above can only be a non-unit at places dividing the conductor of K_{n}, and the valuations at all places above the same rational place are the same, the intersection of the stable norms D′_{n} with C_{n} gives the stable norms C′_{n}. It therefore suffices to show tameness of (D_{n}/D′_{n})_{n}.

There is some n_{0} such the conductor of K_{n} is exactly divisible by p^{n+n_{0}+1} for all large n. Look at divisors d of f_{n} which are divisible by p, and note that this forces d to be exactly divisible by p^{n+n_{0}+1} since we only consider d which are coprime to f_{n}/d. Using Galois theory one can check that for n large enough, the degree of the field ℚ(pd) ∩ K_{n+1} over ℚ is exactly p times the degree of ℚ(d) ∩ K_{n}. If n is large enough to make this happen, then the obvious fact that 1 − ζ_{d} is the norm of 1−ζ_{pd} in the degree p extension ℚ(pd)/ℚ(d), implies by chasing a diagram of fields that z_{d} is the norm of z_{pd} in K_{n+1}/K_{n}.

Therefore for any large n, all z_{d} with d divisible by p are stable norms. There remain only those z_{d} with d prime to p, so d must divide cond(K), so z_{d} is actually in K, and a unit outside cond(K). This gives an a priori bound of the number of ℤ_{p}-generators required for the quotient D_{n}/D′_{n}, and Fact (3) is proved.

Proof of Fact (2): We will find a bound c_{2} so that all B_{n} are c_{2}-generated over ℤ_{p}. One easily sees that B_{n} = proj lim_{m≥n} H^{1}(Γ_{m,n}, E_{m}), with Γ_{m,n} = Gal(K_{m}/K_{n}). If E′_{m} denotes ℤ_{p} ⊗ O_{K_{m}}[1/p]^{∗} (the p-adified p-units in K_{m}), then we have short exact sequences

1 → E_{m} → E′_{m} → V_{m} → 0,

where V_{m} is defined by the sequence. Then the ℤ_{p}-rank of the free ℤ_{p}-module V_{m} equals the number of p-adic places of K_{m} and is therefore bounded as m → ∞, by c′ say. Moreover the induced maps V_{m+1} → V_{m} are all monic, so V = proj lim V_{m} is a free ℤ_{p}-module of rank at most c′ as well.

By an old result of Iwasawa [47], the orders of H^{1}(Γ_{m,n}, E′_{m}) are bounded independently of m and n, and in particular all these groups are c′′-generated for some c′′. Looking at the cohomology sequence coming from the short exact sequence above we therefore see that we will be done, with c_{2} = c′ + c′′, as soon as we can show that the modules proj lim_{m≥n} H^{0}(Γ_{m,n}, V_{m}) are d′-generated over ℤ_{p} for all large n. But this is clear since proj lim_{m≥n} H^{0}(Γ_{m,n}, V_{m}) ≅ H^{0}(Γ_{n}, V), and V is c′-generated.

This finishes the proof of Fact (2), and the proof of the theorem.

Remarks: (a) The case p = 2 and K a full cyclotomic field can be deduced from a result of Kuz’min [56, Thm. 3.1]; a little argument is necessary for which we refer to [62, p.77].

(b) One can show the following (we do not give the proofs, which use the same methods). If one takes K a real abelian field, and retains the definition of C_{n} and C as above, the result is no longer true for p = 2: instead one gets μ(E/C) = [K : ℚ]. If one replaces Sinnott units by modified Sinnott units, i.e. by the groups denoted C_{1,n} by Sinnott [70, p.182], and defines C̃ = proj lim(ℤ_{2} ⊗ C_{1,n}), then μ(E/C̃) turns out to be zero. One can even show that C̃/C is isomorphic to Λ/(2)^{[K:ℚ]}. By definition, C_{1,n} is the group of all units of K_{n} whose squares are Sinnott units, and the main point is that in a precise sense, almost all Sinnott units are squares for real fields K.

(c) If K is of odd degree over ℚ, then μ(E/C̃) = 0 already essentially follows from work of Gillard [39] and, again, the theorem of Ferrero-Washington.

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Dept. of Math., Caltech 253-37, Pasadena, CA 91125, USA

    E-mail address: flach@its.caltech.edu