On Tate’s refinement for a conjecture of Gross and its generalization

de Bordeaux 16(2004), 457–486

On Tate’s refinement for a conjecture of Gross and its generalization

parNoboru AOKI

R´esum´e. Nous ´etudions un raffinement d`u `a Tate de la conjecture de Gross sur les valeurs de fonctionsLab´eliennes ens= 0 et for- mulons sa g´en´eralisation `a une extension cyclique abitraire. Nous prouvons que notre conjecture g´en´eralis´ee est vraie dans le cas des corps de nombres. Cela entraine en particulier que le raffinement de Tate est vrai pour tout corps de nombres.

Abstract. We study Tate’s refinement for a conjecture of Gross on the values of abelian L-function at s = 0 and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.

1. Introduction

In [9] Gross proposed a conjecture which predicts a relation between two arithmetic objects, the Stickelberger element and the Gross regulator, both of which are defined for any data (K/k, S, T), whereK/k is an abelian ex- tension of global fields andS, T are finite, non-empty subsets of the places of k satisfying certain conditions. In the same paper, he proved the con- jecture in the case of unramified extensions of number fields, and obtained a partial result when the extension is cyclic of prime degree. Since then the conjecture has been verified to be true in several cases (see Proposition 2.2 below), but yet it remains to be proved in general. Taking a close look at the conjecture, however, one can easily notice that it becomes trivial in some cases. For example, ifS contains a place which completely splits in K, then both the Stickelberger element and the Gross regulator are zero, and the conjecture is trivially true. Besides such a trivial case, there are still some cases where the conjecture becomes trivial. As observed by Tate [22], this is the case ifK/kis a cyclic extension whose degree is a power of a prime numberl and if at least one place of S “almost splits completely”

Manuscrit re¸cu le 6 juin 2003.

in K/k (see Section 3 for the definition) and another place in S splits in K/k. He then proposed a refined conjecture in that case.

The purpose of this paper is to study Tate’s refined conjecture from a cohomological view point and to generalize it to arbitrary cyclic extensions.

(In a forthcoming paper [2], a further generalization of the conjecture will be given.) We will prove that a weak congruence holds for any cyclic l- extension (Theorem 3.3), which implies Tate’s refined conjecture when k is a number field (Theorem 3.4). Piecing the congruences together for all primes l, we will also obtain a weak congruence for arbitrary cyclic extensions (Theorem 4.2), which is a partial result in the direction of our conjecture. In particular it shows that the generalized conjecture (and hence the Gross conjecture) is true for arbitrary cyclic extensions of number fields (Theorem 4.3 and Corollary 4.4). In the last section, using the results above, we will give a new proof of the Gross conjecture for arbitrary abelian extensionsK/Q(Theorem 10.1), which simplifies our previous proof in [1].

The main idea of the proof consists of two ingredients: one is an inter- pretation of the Gross regulator map in terms of Galois cohomology, and the other is genus theory for cyclic extensionsK/k. Here by genus theory we mean a formula (Theorem 7.1) for the (S, T)-ambiguous class number ofK/k, and it will play an important role when we relate the Stickelberger element to the Gross regulator in the proof of Theorem 4.2. The idea to use genus theory can be already found in the paper of Gross [9], where he implicitly used it to prove a weak congruence in the case of cyclic ex- tensions of prime degree. Thus our proof may be regarded as a natural generalization of his.

Acknowledgements I would like to thank Joongul Lee and Ki-Seng Tan for reading the manuscript very carefully and making a number of helpful suggestions. Especially I am considerably indebted to Lee for the treatment of the case “m0 >0” in Theorem 9.2. I am very grateful to David Burns for letting me know of recent work by himself [3] and by Anthony Hayward [11] both of which are closely related to this article. I also wish to express my gratitude to John Tate for many valuable comments and suggestions.

2. The Gross conjecture

In this section we briefly recall the Gross conjecture. Let k be a global field, and let S be any finite, non-empty set of places ofk which contains all the archimedean places if k is a number field. Let OS be the ring of S-integers ofk and consider the S-zeta function

ζ_S(s) = X

a⊆OS

(Na)^−s,

where the summation is over the ideals a of OS. It is well known that this series converges for <(s) >1 and has a meromorphic continuation to the s-plane, with only a simple pole at s = 1. The analytic class number formula forksays that the Taylor expansion of ζ_S(s) at s= 0 begins:

ζ_S(s) =−h_SR_S

w_S sⁿ+O(sⁿ⁺¹),

whereh_Sis the class number ofOS,R_Sis theS-regulator,w_Sis the number of root of unity in theS-unit group U_S =O_S^× and n=|S| −1. To achieve a formula with no denominator, following Gross, we introduce a slightly modified zeta function. LetT be a finite set of places ofkwhich is disjoint fromS, and define the (S, T)-zeta function

ζ_S,T(s) = Y

v∈T

(1−N v^1−s)·ζ_S(s),

whereN vis the cardinality of the residue field ofv. To describe the corre- sponding formula forζ_S,T(s), we define the (S, T)-unit group of kby

U_S,T ={u∈U_S |u≡1 (modv) for allv∈T}.

Clearly the index (US :US,T) is finite, and is a divisor of Q

q∈T(N v−1).

Define the (S, T)-class number h=hS,T by the formula

(1) h=h_S·

Q

v∈T(N v−1) (U_S :U_S,T) . Then the Taylor expansion ofζ_S,T(s) at s= 0 begins:

(2) ζS,T(s) = (−1)^|T|−1hRS,T

w sⁿ+O(sⁿ⁺¹),

where RS,T is the (S, T)-regulator (see below for the definition) and w is the number of root of unity inU_S,T. We henceforth assume thatT is chosen so that US,T is torsion-free. Then from (2) we obtain a formula without a denominator:

(3) ζS,T(s) = (−1)^|T|−1hRS,Tsⁿ+O(sⁿ⁺¹).

To define the (S, T)-regulator, letY_S be the free abelian group generated by the places of S and

XS = (

X

v∈S

av·v∈YS

X

v∈S

av = 0 )

the subgroup of elements of degree zero in YS. Then XS ∼=Zⁿ. Since we are assuming thatU_S,T is torsion-free, we have an isomorphismU_S,T ∼=Zⁿ.

Let hu₁, . . . , uni and hx₁, . . . , xni be Z-bases of U_S,T and X_S respectively.

Let det_R(λ) be the determinant of the map

λ_R,S,T :U_S,T −→R⊗X_S, u7→X

v∈S

log||u||_v⊗v.

taken with respect to Z-bases of U_S,T and X_S chosen above. The (S, T)- regulator RS,T is by definition the absolute value of det_R(λ). By (3) we have

(4) ζS,T(s) =±h·det_R(λ)sⁿ+O(sⁿ⁺¹),

where the sign of course depends on the choice of bases ofU_S,T and X_S. The Gross conjecture predicts that there is an analogous congruence relation if we replaceζ_S,T(s) and det_R(λ) in (4) by the Stickelberger element and the Gross regulator respectively. We give a brief review of the definition of these two objects.

First we define the Stickelberger element. Let K/k be a finite abelian extension which is unramified outside S and G the Galois group of K/k.

For any complex valued character χ of G, we define the (S, T)-L function associated to χto be the infinite product

L_S,T(χ, s) = Y

v∈T

(1−χ(F r_v)N(v)^1−s)Y

v6∈S

(1−χ(F r_v)N(v)^−s)⁻¹, where F r_v ∈ G denotes the Frobenius element at v. Then there exists a unique element θG = θG,S,T ∈ C[G] such that χ(θG) = LS,T(χ,0) for any χ. Gross [9] showed that θ_G is in Z[G] using integrality properties proved independently by Deligne-Ribet [7] and Barsky-Cassou-Nogu`es [5].

Next, to define the Gross regulator, consider the map λ=λ_G,S,T :U_S,T −→G⊗X_S, u7→X

v∈S

rv(u)⊗v,

where rv : k^×_v −→ G denotes the local reciprocity map. Let (gij) be the n×nmatrix representingλwith respect to the bases{u_i}and{x_j}chosen above, namely:

λ(ui) =

n

X

j=1

gij⊗xj (i= 1, . . . , n).

LetZ[G] be the integral group ring of Gand I_G the augmentation ideal of Z[G]. Then there is an isomorphism G ∼= IG/I_G², g 7→ g−1. Using this isomorphism we can view the matrix (gij −1) with entries in IG/I_G² as a matrix representingλ. Define the Gross regulator by

detG(λ) =X

σ

sgn(σ)(g_1,σ(1)−1)· · ·(g_n,σ(n)−1) ∈I_Gⁿ/I_Gⁿ⁺¹, where the sum is over the permutations σ of {1, . . . , n}.

We are now in a position to state the conjecture of Gross.

Conjecture 2.1. Let the notation be as above. Then θ_G ∈ I_Gⁿ and the following congruence holds:

θG ≡ ±hdet_G(λ) (modI_Gⁿ⁺¹), where the sign is chosen in a way consistent with (4).

In the following we have a list of the cases where Conjecture 2.1 is proved.

Proposition 2.2. Conjecture 2.1 is true in the following cases:

(i) n= 0.

(ii) k is a number field and S is the set of the archimedean places.

(iii) K is a quadratic extension of k.

(iv) k is a function field and n= 1.

(v) k=Q.

(vi) K/k is an abelian p-extension of function fields of characteristicp.

(vii) K/k is an abelian l-extension, where l is a prime number different from the characteristic of k and divides neither the class number of k nor the number of roots of unity in K.

(viii) K/k is an abelian extension of a rational function fieldkover a finite field and |S| ≤3.

Proof. In the case of (i) Conjecture 2.1 is an immediate consequence of (4).

In both cases (ii) and (iii) the conjecture was proved by Gross in [9]. Case (iv) is a consequence of the work of Hayes [10] proving a refined version of the Stark conjecture. Case (v) was treated in our previous paper [1]. In the cases (vi), (vii) and (viii) the conjecture was proved by Tan [19], Lee

[15] and Reid [16], respectively.

As mentioned in the introduction, Gross [9] proved a weak congruence whenK/k is a cyclic extension of a prime degree. Here we give the precise statement because our main results (Theorem 3.3 and Theorem 4.2) may be viewed as a generalization of it to arbitrary cyclic extensions.

Proposition 2.3. Suppose K/k is a cyclic extension of prime degree l.

Then there is a constantc∈(Z/lZ)^× such that θ_G ≡c·hdet_G(λ) (modI_Gⁿ⁺¹).

Proof. See [9, Proposition 6.15].

3. Tate’s refinement for the Gross conjecture

In this section we give the precise statement of Tate’s refinement for the Gross conjecture.

First, we assume thatGis an arbitrary finite abelian group. Choose and fix a placev₀∈S and setS₁ =S\ {v₀}={v₁, . . . , v_n}. Then, as aZ-basis ofXS, we can take {v₁−v0, . . . , vn−v0}. In this case we have

λ(u) =

n

X

j=1

r_vj(u)⊗(v_j −v₀).

Choosing aZ-basis{u₁, . . . , un} ofUS,T, we define

R_G=R_G,S,T := det (rvi(uj)−1)_1≤i,j≤n∈Z[G].

It is clear from the definition thatR_G∈I_GⁿandR_G≡detG(λ) (mod I_Gⁿ⁺¹).

Now, suppose thatG is a cyclic group of degreel^m, a power of a prime number l. For each v ∈S, let G_v denote the decomposition group of v in G. We fix the ordering of the elements ofS={v₀, v1, . . . , vn} so that

G_v0 ⊇G_v1 ⊇ · · · ⊇G_vn.

Letl^mi = (G:G_vi) for i= 0, . . . , n. Thusm₀≤m₁≤. . .≤m_n≤m. Let N =l^m0 +· · ·+l^mn−1 =|S(K)| −l^mn.

Clearly N ≥n, and N =nif and only if m0 =· · ·=mn−1 = 0.

If m_n = m, that is, v_n splits completely in K/k, then θ_G = R_G = 0.

Hence Conjecture 2.1 trivially holds. Let us consider the second simplest case m_n =m−1. Following Tate, we say that the place v_n almost splits completely inK ifm_n=m−1. Tate [22] proved the following.

Theorem 3.1. Assume that mn = m−1. Then θG ∈ I_G^N and R_G ∈ I_G^N−lm0+1. Moreover, the image of R_G in I_G^N−lm0+1/I_G^N−lm0+2 is, up to the sign, independent of the choice of the basis of US,T and the choice ofv0.

Since N ≥ n, this theorem, in particular, shows that θ_G ∈ I_Gⁿ. Let us consider the case where m0 > 0. In this case we have N > n and hence Theorem 3.1 also shows that θ_G ∈ I_Gⁿ⁺¹. Moreover, one can show that hR_G ≡ 0 (modI_Gⁿ⁺¹) (see Theorem 9.2, (i)). Therefore Conjecture 2.1 is trivially true if m0 > 0. On the other hand, ifm0 = 0, then both θ_G and R_G are in the same idealI_G^N. Therefore it is meaningful to compare them in the quotient group I_G^N/I_G^N+1. Based on these facts, Tate [22] proposed a refinement for the Gross conjecture.

Conjecture 3.2. Assume that m₀ = 0 and m_n=m−1. Then θG≡ ±hR_G (modI_G^N+1),

where the sign is chosen in a way consistent with (4).

Obviously Conjecture 3.2 implies Conjecture 2.1 sinceN ≥nand det_G(λ_S,T)≡ R_G (modI_Gⁿ⁺¹).

Now, we can state one of our main results.

Theorem 3.3. Let the notation and assumptions be as in Conjecture 3.2.

Then there exists an integer cprime to l such that θG≡c·hR_G (modI_G^N+1).

In particular, if l= 2, then Conjecture 3.2 is true.

We will give the proof of this theorem (more precisely, of Theorem 9.1 which is equivalent to Theorem 3.3) in Section 9. Here we only note that the last assertion immediately follows from the first one. To see this we have only to observe thatI_G^N/I_G^N+1∼=Z/|G|ZsinceGis a cyclic group and that bothθG and R_G are killed byl inI_G^N/I_G^N+1 under the condition that m_n =m−1 (see Proposition 5.4). Therefore, if l= 2, then chR_G ≡hR_G (modI_G^N+1). Thus Conjecture 3.2 is true.

As a special case of Theorem 3.3 we have the following.

Corollary 3.4. If K/k is a cyclic l-extension of number fields, then Con- jecture 3.2 is true.

Proof. Actually, Conjecture 3.2 is non-trivial only whenl= 2 in the number field case sinceI_G^N/I_G^N+1is anl-group and bothθ_GandR_Gare killed by 2 in I_G^N/I_G^N+1. Thus Corollary 3.4 is a direct consequence of Theorem 3.3.

Remark 3.5. As remarked by Lee [14], one can not drop the condition m0 = 0 from the conditions in Conjecture 3.2. Indeed, he showed that there are infinitely many cyclicl-extensionsK/k for which θG6∈I_G^N+1 but hdet_G(λ)∈I_G^N+1. In the next section, we will generalize Conjecture 3.2 to arbitrary cyclic extensions in order to remove the restriction onm0.

4. A generalization of Tate’s conjecture

In this sectionGwill be an arbitrary cyclic group. For any v∈S, let IG(v) = Ker(Z[G]−→Z[G/Gv])

be the kernel of the canonical surjection Z[G]−→Z[G/Gv]. Let us choose and fix a placev₀ ∈S. LetS₁ =S\ {v₀} and consider the ideal

IG(S1) =

n

Y

i=1

IG(vi)

inZ[G]. Ifσ is a generator of G, then G_vi is generated by σ_vi :=σ^(G:Gvi) andI_G(S₁) is a principal ideal generated by (σ_v1−1)· · ·(σ_vn−1). Since the entries of thei-th row of the matrix (rvi(uj)−1)i,j are in the idealIG(vi), its determinant R_G belongs to the ideal I_G(S₁). Moreover, the image of R_G in the quotient groupIG(S1)/IGIG(S1) is, up to the sign, independent of the choice of the basis ofU_S,T.

Given a finite abelian extensionK/kof global fields and finite setsS, T of places ofksuch thatS∩T =∅, we call the triple (K/k, S, T) anadmissible dataif the following conditions are satisfied:

(i) S contains the places ofkramifying inKand the archimedean places of k.

(ii) U_K,S,T is torsion-free.

In [9] Gross gave a sufficient condition forU_S,T to be torsion-free:

• In the function field case U_S,T is torsion-free ifT is non-empty.

• In the number field caseU_S,T is torsion-free ifT contains either primes of different residue characteristics or a primevwhose absolute ramifi- cation indexev is strictly less thanp−1, wherepis the characteristic of Fv.

It is worthwhile to note that each condition above also ensures thatU_K,S,T is torsion-free for any finite abelian extension unramified outside S.

We propose the following conjecture which will turn out to be equivalent to Tate’s refinement whenK/kis a cyclicl-extension such thatm0 = 0 and m_n=m−1.

Conjecture 4.1. Let (K/k, S, T) be an admissible data such that G = Gal(K/k) is a cyclic group. Then

θ_G≡ ±hR_G (modI_GI_G(S₁)), where the sign is chosen in a way consistent with (4).

It is very likely that the congruence in the conjecture holds for any finite abelian extension K/k. In a forthcoming paper [2], this will be discussed in some detail and some evidence will be given.

Iflis a prime number dividing|G|, we denote byG_lthel-Sylow subgroup ofG. For eachv ∈S, letG_v be the decomposition group ofvinG, and put G_v,l = Gv∩G_l. Thus G_v,l is the l-Sylow subgroup of Gv. Now, consider the following condition:

(5)

There exists a place v_n∈S₁ such that|G_vn,l| ≤l for any prime divisor lof |G|.

In other words, this requires that there exists a placev_n inS which either splits completely or almost splits completely in the maximall-extension of kcontained inKfor each prime divisor lof|G|. Although this condition is very restrictive in the function field case, it is always satisfied in the number field case since|G_v|= 1 or 2 for any archimedean placev of k.

Now, we can state our main result, which may be viewed as a partial answer to Conjecture 4.1.

Theorem 4.2. Suppose condition (5) holds. Then θ_G belongs to I_G(S1) and there exists an integer c prime to|G|such that

θG≡c·hR_G (mod IGIG(S1)).

In the next section we will see that in order to prove Theorem 4.2 it suffices to prove it when Gis a cyclic group of a prime power order.

In the case of number fields, Theorem 4.2 gives a complete answer to Conjecture 4.1.

Theorem 4.3. If K/k is a cyclic extension of number fields, then we have θ_G≡hR_G (modI_GI_G(S₁)).

Proof. As we have remarked above, condition (5) is always satisfied in the number field case. As is well known, if k is not totally real or K is not a totally imaginary, then Conjecture 2.1 is trivial. Suppose K is a totally imaginary extension of a totally real fieldk. Then, as we will see later (see Proposition 5.4), the quotient groupI_G/I_GI_G(S₁) is a cyclic group of order 2. Thus Theorem 4.3 immediately follows from Theorem 4.2.

Since Conjecture 3.2 implies Conjecture 2.1, Theorem 4.3 proves the following.

Corollary 4.4. If K/k is a cyclic extension of number fields, then Con- jecture 2.1 is true.

5. Stickelberger elements for cyclic l-extensions

Throughout this section we will assume that G is a cyclic l-group and that (K/k, S, T) is an admissible data. In particular U_K,S,T is torsion-free.

In order to state Theorem 5.1 below, letF be the intermediate field ofK/k with [K:F] =l and put

h^∗_K,S,T =h_K,S,T/h_F,S,T.

If at least one place in S does not split completely in K/k, then h^∗_K,S,T is an integer by [17, Lemma 3.4]. In particular, ifm_n=m−1, thenh^∗_K,S,T is an integer.

Theorem 5.1. Suppose mn=m−1. Then θ_G≡0 (modI_G(S₁)).

Moreover, the following assertions hold.

(i) If m0>0, then θG≡0 (mod IGIG(S1)).

(ii) If m₀ = 0, then θ_G ≡ 0 (modI_GI_G(S₁)) if and only if h^∗_K,S,T ≡ 0 (mod l).

Although both the first statement and (i) of the second statement follows from Tate’s theorem (Theorem 3.1) in view of Proposition 5.4, we will give a proof for the completeness. We begin with a lemma.

Lemma 5.2. Let M be an intermediate field of K/k such that |S(K)| =

|S(M)|. Then U_K,S,T =U_M,S,T.

Proof. Letu be any element ofU_K,S,T. For any σ∈G,u^σ−1 is also an ele- ment ofU_K,S,T since U_K,S,T isG-stable. Moreover the following argument shows thatu^σ−1 is a root of unity. Indeed, the assumption of the lemma im- plies thatUK,S,T/UM,S,T is a finite group. It follows that there exists a pos- itive integerm such thatu^m ∈UM,S,T. Therefore (u^σ−1)^m = (u^m)^σ−1= 1.

Thus u^σ−1 is an m-th root of unity. However, since UK,S,T is torsion-free, this shows thatu^σ−1 = 1 for anyσ ∈Gal(K/M), henceu∈M^×. The asser- tion of the lemma then follows from the fact thatU_K,S,T∩M^×=U_M,S,T. Proposition 5.3. Assume that m_n=m−1. Then

Y

χ

χ(θG) =±l^|S(K)|−1h^∗_K,S,T, where χ runs through the faithful characters of G.

Proof. Let F be as above. Then the assumption on Gv’s implies that

|S(K)|=|S(F)|. This, in particular, implies that ords=0ζ_K,S,T(s) = ords=0ζ_F,S,T(s).

Further we have

(6) lim

s→0

ζ_K,S,T(s) ζ_F,S,T(s) =Y

χ

L_S,T(χ,0) =Y

χ

χ(θ_G),

whereχ runs through the faithful characters ofG. On the other hand, by (3), we have

(7) lim

s→0

ζ_K,S,T(s)

ζ_F,S,T(s) =±h_K,S,TR_K,S,T h_F,S,TR_F,S,T . Gross [9, (6.4), (6.5)] showed that

R_K,S,T RF,S,T

= l^|S(K)|−1 (UK,S,T :UF,S,T).

The denominator of the right hand side is 1 by Lemma 5.2, hence R_K,S,T

RF,S,T

=l^|S(K)|−1.

The assertion of the proposition then immediately follows from (6) and

(7).

Proposition 5.4. Assume that mn=m−1. Let ρ be an element ofG of order l. Then we have an equality

(8) I_G(S1) =I_G^N−lm0+1∩(ρ−1)Z[G]

and an isomorophism

(9) I_G(S₁)/I_GI_G(S₁)∼=Z/lZ.

Proof. See [22].

Corollary 5.5. If K/kis a cyclic l-extension such that m₀ = 0 andm_n= m−1, then Conjecture 3.2 is equivalent to Conjecture 4.1

Proof. If m₀ = 0, then from (8) we have

I_G(S₁) =I_G^N ∩(ρ−1)Z[G], I_GI_G(S₁) =I_G^N+1∩(ρ−1)Z[G].

Since bothθ_G and R_G belong to (ρ−1)Z[G], this shows that Conjecture

3.2 is equivalent to Conjecture 4.1.

Proof of Theorem 5.1. LetN_Q(ζlm)/Q denote the norm map fromQ(ζl^m) to Q. Then by Proposition 5.3 we have

ord_l N_Q(ζlm)/Q(χ(θ_G))

=|S(K)| −1 +ν,

where ν = ord_l(h^∗_K,S,T). Since l completely ramifies in Q(ζ_l^m), it follows from this that

ord_l(χ(θ_G)) =|S(K)| −1 +ν.

Since |S(K)| −1 =N +l^m−1−1 and θ_G ∈(ρ−1)Z[G], from Proposition 5.3 and the lemma of [22] we deduce that

(10) θG ∈I_G^N+ν\I_G^N+ν+1.

ThusθG∈IG(S1) by Proposition 5.4. This proves the first statement.

Ifm₀ >0, then l^m0 >1, whence

IGIG(S1)⊇I_G^N ∩(ρ−1)Z[G]

by Proposition 5.4. Therefore θG ∈IGIG(S1) by Theorem 3.1. Ifm0 = 0, then by the same proposition we have

(11) I_GI_G(S₁) =I_G^N+1∩(ρ−1)Z[G].

Therefore from (10) and (11) we deduce that θG ∈IGIG(S1) if and only if h^∗_K,S,T ≡0 (mod l). This completes the proof. 2

6. Reduction to cyclic l-extensions

We wish to reduce Theorem 4.2 to the case of cyclic l-extensions. For that purpose we consider the ring homomorphism Ψ_l : Z[G] −→ Z[G_l] induced from the canonical surjectionG−→G_l. Let

Ψ =⊕Ψ_l :Z[G]−→M

l

Z[Gl],

wherelruns through the prime numbers dividing|G|. For any placev∈S₁, we have Ψ_l(IG(v)) =IG_l(v) and hence Ψ_l(IG(S1)) =IG_l(S1). If ∗ denotes eitherv orS₁, then we define the map

Ψ∗ :IG(∗)−→M

l

IGl(∗) to be the restriction of Ψ toI_G(∗).

Proposition 6.1. The following assertions hold for the mapsΨ_v andΨ_S1. (i) Both Ψv andΨ_S1 are surjective.

(ii) Ker(Ψ_v) = Ker(Ψ_S1).

(iii) The maps Ψv and ΨS1 respectively induce isomorphisms I_G(v)/I_G(S₁)∼=M

l

I_Gl(v)/I_Gl(S₁), I_G(S₁)/I_GI_G(S₁)∼=M

l

I_Gl(S₁)/I_GlI_Gl(S₁), where l runs through the prime numbers dividing |G|.

Before giving the proof of this proposition, we show how the proof of Theorem 4.2 can be reduced to the case of cyclicl-extensions. We continue to use the notation above and assume that condition (5) holds. Then θG

belongs toI_G(v_n). Suppose that Theorem 6.1 is true for anyG_l, that is, for each prime divisorlof|G|,θGlbelongs toIGl(S1) and we have a congruence (12) θ_Gl≡c_l·hR_Gl (modI_GlI_Gl(S₁))

for some integer c_l prime to l. Since Ψ_l(θ_G) = θ_Gl, Proposition 6.1, (iii) shows that θG belongs to IG(S1). Let c be an integer such that c ≡ c_l (modl) for any primeldividing|G|. By Proposition 5.4,I_Gl(S₁)/I_GlI_Gl(S₁) is trivial or isomorphic toZ/lZ according as|G_v,l|= 1 or l. It follows that

Ψl(c·hR_G)≡cl·hR_Gl (modIGlIGl(S1))

for all ldividing|G|. Then Proposition 6.1, (iv) and (12) shows that θ_G≡c·hR_G (mod I_GI_G(S₁)),

as desired. 2

To state the following lemma, for any subgroup H of G, we denote by I_H the kernel of the natural surjection Z[G]→Z[G/H].

Lemma 6.2. Let Ga finite abelian group and G₁, G₂ subgroups ofG such thatGCD(|G₁|,|G₂|) = 1. Then

I_G1I_G2 ⊆I_G^t₁I_G2 +I_G1I_G^t₂. for any positive integert.

Proof. We have only to show that

(13) (g1−1)(g2−1)∈I_G^t₁I_G2 +I_G1I_G^t₂

for anyg_i ∈G_i and any positive integer t. First we note that|G_i|(g_i−1)∈ I_G²

ifori= 1,2. By induction ontone can easily see that|G_i|^t(g_i−1)∈I_G^t+1

i

for any positive integer t. Since GCD(|G₁|,|G₂|) = 1, there exist integers a1, a2 such thata1|G₁|^t+a2|G₂|^t= 1. Then the identity

(g₁−1)(g₂−1) =a₁|G₁|^t(g₁−1)(g₂−1) +a₂|G₂|^t(g₁−1)(g₂−1) shows that equation (13) holds, completing the proof.

Proof of Proposition 6.1. It is clear from the definition that (14) Ψ_l(I_Gv,l) =I_Gl(v)

for anyv∈S and for any primeldividing|G|. SinceI_G(v) is generated by I_Gv,l aslranges over the prime divisors of|G|, it follows from (14) that Ψ_v is surjective. From (14) and the definition of Ψ_l we also deduce that

Ψ_l Y

v∈S

I_Gv,l0

!

=

(I_Gl(S₁) ifl=l⁰,

0 otherwise.

Since I_G(S₁) contains Q

v∈S₁I_Gv,l for all l, we conclude that the map Ψ_S1 is also surjective.

To prove (ii) note that the ideal Ker(Ψ_v) is generated by the products I_Gv,lI_G

v,l0 of two ideals I_Gv,l and I_G

v,l0 as l, l⁰ runs through distinct prime divisors of|G|. By Lemma 6.2 we have an inclusion

IGv,lIG_v,l0 ⊆I_G^t_v,lIG_v,l0 +IGv,lI_G^t

v,l0

for any positive integer t. Since I_G^t

v,l ⊆ I_G(S₁) for any t ≥ n, this shows thatIG_v,lIG_v,l0 ⊆IG(S1). Therefore Ker(Ψv)⊆IG(S1), whence Ker(ΨS1) = Ker(Ψ_v).

To prove the first isomorphism of (iii), for ∗=v orS1 let P_G(∗) =M

l

I_Gl(∗).

SinceP_G(v)/P_G(S1) is isomorphic toL

lI_Gl(v)/I_Gl(S1), Ψv induces a map Ψ_v :I_G(v)/I_G(S₁)−→P_G(v)/P_G(S₁).

We have to show that Ψ_v is an isomorphism. To this end, consider the commutative diagram

0 −−−−→ I_G(S₁) −−−−→ I_G(v) −−−−→ I_G(v)/I_G(S₁) −−−−→ 0



 y^ΨS1



 y^Ψv



 y^Ψv

0 −−−−→ P_G(S₁) −−−−→ P_G(v) −−−−→ P_G(v)/P_G(S₁) −−−−→ 0.

Since both ΨS1 and Ψv are surjective, this diagram shows that Ψv is also surjective. Moreover by the snake lemma we have an exact sequence

0 −−−−→ Ker(Ψ_S1) −−−−→ Ker(Ψv) −−−−→ Ker(Ψv) −−−−→ 0.

Then the equality Ker(Ψ_S1) = Ker(Ψ_v) shows that Ker(Ψ_v) = 0, whence Ψv is an isomorphism.

In order to prove the second isomorphism of (iii), let Q_G(S₁) =M

l

I_GlI_Gl(S₁).

Then QG(S1) is a subgroup of PG(S1) and PG(S1)/QG(S1) is isomorphic toL

lI_Gl(S₁)/I_GlI_Gl(S₁). Hence Ψ_S1 induces a map Ψ_S1 :I_G(S₁)/I_GI_G(S₁)−→P_G(S₁)/Q_G(S₁).

To show that ΨS1 is an isomorphism, we consider the commutative diagram 0 −−−−→ I_GI_G(S₁) −−−−→ I_G(S₁) −−−−→ I_G(S₁)/I_GI_G(S₁) −−−−→ 0



 y

ϕS1



 y^ΨS1



 y^ΨS1

0 −−−−→ Q_G(S₁) −−−−→ P_G(S₁) −−−−→ P_G(S₁)/Q_G(S₁) −−−−→ 0, where ϕS1 denotes the restriction of ΨS1 to IGIG(S1). As we have seen above, Ψ_S1 is surjective. Moreover, quite similarly as in the proof of the surjectivity of Ψ_S1, one can prove thatϕ_S1 is also surjective. Hence, by the snake lemma again, ΨS1 is also surjective and we obtain an exact sequence (15) 0 −−−−→ Ker(ϕ_S1) −−−−→ Ker(Ψ_S1) −−−−→ Ker(Ψ_S1) −−−−→ 0.

We wish to show that Ker(ΨS1) = 0. To see this note that in the proof of (i) we have actually proved that Ker(Ψ_v) is contained in the ideal I_GI_G(S₁).

Therefore Ker(ΨS1) is also contained in the same ideal, whence Ker(ΨS1) = Ker(ϕS1). Thus from (15) we deduce that Ker(ΨS1) = 0, as desired. This

completes the proof of Proposition 6.1. 2

7. The (S,T)-ambiguous class number

LetS, T be any finite sets of places ofksuch thatS∩T =∅. (We do not impose any other condition on S and T.) Let J_k be the id`ele group of k.

Ifv is a place ofk, then we denote by k_v and Ov the completion ofk and the integer ring of k at v, respectively. We define the (S, T)-id`ele group J_S,T =J_k,S,T of kto be the subgroup

J_S,T = Y

v∈S

k_v^××Y

v∈T

O_v,1^× × Y

v6∈S∪T

Ov^×

of Jk, where forv ∈T we putO_v,1^× ={u∈ O_v^× | u≡1 (mod v)}. Clearly we have

(16) US,T =k^×∩JS,T.

The (S, T)-id`ele class group CS,T = Ck,S,T is defined to be the quotient group

C_S,T =J_S,T/U_S,T.

LetC_k =J_k/k^× be the id`ele class group ofk. It follows from (16) that the inclusion J_S,T ,→ J_k induces an injective map C_S,T ,→ C_k. We define the (S, T)-ideal class group ClS,T =Cl_k,S,T of k by

ClS,T =C_k/CS,T.

Then Cl_S,T is isomorphic to the ray class group J_k/k^×J_S,T corresponding to the subgroupk^×JS,T of Jk. To see this, note that

Ker(J_k/k^×−→J_k/k^×J_S,T) =k^×J_S,T/k^×.

By (16) we have an isomorphismk^×JS,T/k^×∼=JS,T/US,T, whence J_k/k^×JS,T ∼= Jk/k^×

J_S,T/U_S,T

∼=C_k/CS,T =ClS,T.

Lethbe the (S, T)-class number defined in (1). This naming will be justified if we show that h = |Cl_S,T|. To show this, note that we have an exact sequence

(17) 0−→U_S,T −→J_S,T −→C_k−→Cl_S,T −→0.

For simplicity we letCl_S :=Cl_S,∅ (the S-ideal class group ofk) and J_S = JS,∅ (the S-id`ele group ofk), respectively. Then, as a special case of (17), we have an exact sequence

(18) 0−→U_S −→J_S −→C_k−→Cl_S−→0.

For any v ∈T, letFv be the residue field of v. Then two exact sequences (17) and (18) fit into the following commutative diagram:

0 0 0



 y



 y



 y

U_S,T U_S Q

v∈T F^×_v



 y



 y



 y 0 −−−−→ JS,T −−−−→ JS −−−−→ Q

v∈T F^×_v −−−−→ 0



 y



 y



 y

0 −−−−→ C_k −−−−→ C_k −−−−→ 0 −−−−→ 0



 y



 y



 y

Cl_S,T Cl_S 0



 y



 y

0 0

Applying the snake lemma to this diagram, we obtain an exact sequence 0−→U_S,T −→U_S −→ Y

v∈T

F^×_v −→Cl_S,T −→Cl_S −→0.

Sinceh_S =|Cl_S|, it follows that

|Cl_S,T|=h_S· Q

v∈T(N v−1) (US :US,T) . Thereforeh=|Cl_S,T|, as desired.

Now, let K/k be a finite Galois extension with the Galois group G.

We define UK,S,T, JK,S,T, CK,S,T, etc. similarly as above. Then, taking H^·(G,−) of the exact sequence

(19) 0 −−−−→ U_K,S,T −−−−→^α J_K,S,T −−−−→^β C_K,S,T −−−−→ 0 ofG-modules, we obtain the long exact sequence

(20)

0 −−−−→ US,T

α0

−−−−→ JS,T

β0

−−−−→ CK,S,TG

−−−−→ H¹(G, U_K,S,T) −−−−→^α1 H¹(G, J_K,S,T) −−−−→^β1 H¹(G, C_K,S,T)

−−−−→ H²(G, U_K,S,T) −−−−→^α2 H²(G, J_K,S,T) −−−−→^β2 H²(G, C_K,S,T)

−−−−→ · · ·.

The next formula will play a key role in the proof of Theorem 4.2.

Theorem 7.1. Suppose Gis cyclic. Then

ClK,S,TG

= h|Coker(α₂)|

|G| .

Proof. Since H¹(G, C_K) = 0, taking cohomology of the short exact se- quence

0−→CK,S,T −→CK −→ClK,S,T −→0 yields the exact sequence

0−→C_K,S,T^G −→C_K^G −→Cl_K,S,T^G−→H¹(G, C_K,S,T)−→0.

Noticing thatC_K^G =C_k, we obtain

(21) |Cl_K,S,T^G|=|H¹(G, C_K,S,T)| · |C_k/C_K,S,T^G|.

For anyG-moduleM, letQ(M) denote the Herbrand quotient:

Q(M) = |H²(G, M)|

|H¹(G, M)|. From the exact sequence (20) we deduce that (22) Q(J_K,S,T)

Q(U_K,S,T)· |H¹(G, C_K,S,T)|=|C_K,S,T^G/C_S,T| · |Coker(α₂)|.

By class field theory we know that Q(CK) = |G|. Moreover, we have Q(Cl_K,S,T) = 1 since Cl_K,S,T is a finite group. Therefore, from the exact sequence

0−→CK,S,T −→CK −→ClK,S,T −→0

we obtain Q(C_K,S,T) = |G|. Since Q(J_K,S,T) = Q(U_K,S,T)Q(C_K,S,T), it follows thatQ(JK,S,T)/Q(UK,S,T) =|G|. Therefore by (22) we have

|H¹(G, C_K,S,T)|= |C_K,S,T^G/C_S,T| · |Coker(α₂)|

|G| .

Substituting this into (21), we obtain

|Cl_K,S,T^G|= |C_k/CK,S,TG| · |C_K,S,T^G/CS,T| · |Coker(α₂)|

|G|

= |C_k/CS,T| · |Coker(α₂)|

|G| .

Since|C_k/CS,T|=|Cl_S,T|=h, this proves the theorem.

Remark 7.2. IfS is the set of archimedean placesS∞ and T =∅, thenh is the usual class numberhkofkand Theorem 7.1 reduces to a well known formula for the ambiguous class number for the cyclic extensionK/k:

(23) |Cl_K^G|= hke(K/k)

|G|(E_k:N_K/kK^×∩E_k),

whereh_k=|Cl_k|,e(K/k) andE_kdenote the class number ofk, the product of ramification indices in K/k of the places of k and the unit group of k, respectively. For this formula we refer the reader to [12, Lemma 4.1]. We should also remark that Federer [8] obtained a similar formula for|Cl_K,S^G| whenS is arbitrary. Thus our formula may be viewed as a generalization of those formulae. To see that (23) is a special case of Theorem 7.1, it suffices to prove the formula

(24) |Coker(α₂)|= e(K/k) (E_k:N_K/kK^×∩E_k) in the case ofS =S∞, T =∅. To prove this note that

H²(G, JK,S∞)∼=Y

v

Ov^×/N_Kw/kv(Ow^×)∼=Y

v

Z/evZ,

where v runs through the places of k. Hence |H²(G, JK,S∞)| = e(K/k).

Moreover, we have H²(G, UK,S∞) ∼= E_k/N_K/kEK, where EK = UK,S∞ is the unit group of K. Therefore (24) follows if we prove the formula

Ker(α2) = N_K/kK^×∩E_k N_K/kE_K .

But this is an easy consequence of the Hasse’s norm theorem asserting the injectivity of the natural map

k^×/N_K/kK^× −→Y

v

k^×_v/N_Kw/kvK_w^×. 8. Cohomological interpretation of λ

Throughout this section we will assume that (K/k, S, T) is an admissible data such thatG = Gal(K/k) is a finite cyclic extension. In Section 1 we have definedλto be a map fromUS,T toG⊗X_S. However, in order to give a cohomological interpretation of λ, it seems natural to replace the target groupG⊗XS with a subgroup XG,S defined below. To begin with, we let

YG,S =M

v∈S

Gv.

We will regard Y_G,S as a subgroup of G⊗Y_S via the natural injection sending (. . . , g_v, . . .)v∈S to P

v∈Sg_v⊗v. Next we define a subgroup X_G,S ofY_G,S by the exact sequence

0−→XG,S −→YG,S −→DS −→0,

whereDS is the subgroup ofGgenerated byGv for allv∈S and the map Y_G,S −→ D_S is defined by sending (. . . , gv, . . .)v∈S to Q

v∈Sgv. Then the image of λis contained inX_G,S. We will hereafter regard λas a map

λ:U_S,T −→X_G,S.

For example Coker(λ) will stand for the quotient group X_G,S/Im(λ).

We now wish to reveal a connection between the Gross regulator map λand the (S, T)-ambiguous class number formula (Theorem 7.1). To this end, we start with studying H²(G, J_K,S,T). For each place v of k, choose, once and for all, a place wof K lying above v. Then by Schapiro’s lemma we have an isomorphism

(25) H²(G, J_K)∼=M

v

H²(G_v, K_w^×), wherev runs through the places ofk. Let

pr_S :H²(G, J_K)−→M

v∈S

H²(G_v, K_w^×).

be the projection to theS-part in the right hand side of (25). Similarly we have an isomorphism

H²(G, JK,S,T)∼=M

v∈S

H²(Gv, K_w^×)⊕M

v∈T

H²(Gv,O_w,1^× )⊕ M

v6∈S∪T

H²(Gv,O_w^×).

Lemma 8.1. BothH²(Gv,O_w^×) andH²(Gv,O_w,1^× ) vanish whenever v6∈S.

Proof. First, for anyv we haveH²(G_v,Ow^×)∼=Z/e_vZ, wheree_v denotes the ramification index ofvin the extensionK/k. Therefore, ifv6∈S, thenK/k is unramified atv, and soH²(G_v,Ow^×) = 0. Next, to see thatH²(G_v,O_w,1^× ) also vanishes for anyv6∈S, we consider the long exact sequence

(26) · · · −→H¹(Gv,F^×_w)−→H²(Gv,O_w,1^× )−→H²(Gv,O_w^×)−→ · · · obtained from the short exact sequence

0−→O_w,1^× −→Ow^× −→F^×w −→0.

By Hilbert’s theorem 90 we have H¹(Gv,F^×w) = 0. From this and (26) it follows thatH²(Gv,O_w,1^× ) = 0 for anyv6∈S. This completes the proof.

By this lemma we may viewH²(G, J_K,S,T) as a subgroup ofH²(G, J_K).

Thus, restricting the map pr_S toH²(G, J_K,S,T), we obtain an isomorphism pr_S :H²(G, J_K,S,T)−→M

v∈S

H²(Gv, K_w^×).

Recall that the local invariant map

inv_v :H²(G_v, K_w^×)−→ 1

|G_v|Z/Z is an isomorphism for any placev. Let

YG,S =M

v∈S

1

|G_v|Z/Z.

Then the map

invS:= (⊕_v∈Sinvv)◦pr_S :H²(G, JK,S,T)−→YG,S

is an isomorphism. In order to study the image of Im(α2)⊆H²(G, JK,S,T) under the map inv_S, let

s:YG,S −→Q/Z, (. . . , y_v, . . .)v∈S 7→X

v∈S

y_v

be the summation map. If we denote byDS the subgroup of Ggenerated by G_v for all v∈S, then Im(s) = _|D¹

S|Z/Z. We define XG,S by the exact sequence

(27) 0 −−−−→ XG,S −−−−→ YG,S

−−−−→s _|D¹

S|Z/Z −−−−→ 0.

Lemma 8.2. inv_S(Im(α₂))⊆XG,S.

Proof. By class field theory we have an isomorphism inv_K/k:H²(G, CK)−→ 1

|G|Z/Z.

Let inv_K/k,S :H²(G, C_K,S,T)−→ _|G|¹ Z/Zbe the composite map of inv_S and the natural map H²(G, C_K,S,T)−→ H²(G, C_K). Then we have a commu- tative diagram

H²(G, U_K,S,T) −−−−→^α2 H²(G, J_K,S,T) −−−−→^β2 H²(G, C_K,S,T)



 y^invS



 y^invK/k,S 0 −−−−→ XG,S −−−−→ YG,S s

−−−−→ _|G|¹ Z/Z,

from which the assertion of the lemma follows.

Now, the connecting homomorphism

δ: Hom(G,Q/Z)−→H²(G,Z) obtained from the short exact sequence

0−→Z−→Q−→Q/Z−→0.

is an isomorphism since Hⁱ(G,Q) = 0 for i >0. For any G-moduleM we consider the cup product

∪:H⁰(G, M)×H²(G,Z)−→H²(G, M).

Choose and fix a faithful additive characterψ∈Hom(G,Q/Z) ofG. Then δ(ψ)∈H²(G,Z) defines an homomorphism

∪δ(ψ) :M^G=H⁰(G, M)−→H²(G, M), x7→x∪δ(ψ).

Lemma 8.3. If M is torsion free, then the map ∪δ(ψ) is surjective.