On Tamagawa numbers of CM tori

In this article we investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter--Altug--Garcia--Gordon and of Guo--Sheu--Yu, respectively. We make a systematic study on Galois cohomology groups in a more general setting and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. Furthermore, we show that every (positive or negative) power of $2$ is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori.


Introduction
In his two fundamental papers [21,23] Takashi Ono investigated the arithmetic of algebraic tori.He introduced and explored the class number and Tamagawa number of T , which will be denoted by h(T ) and τ (T ) respectively (also see Section 2 for the definitions).One arithmetic significant of these invariants is that the class number h(G m,k ) is equal to the class number h k of the number field k, and the analytic class number formula for k can be reformulated by the simple statement τ (G m,k ) = 1.Thus, the class numbers of algebraic tori can be viewed as generalizations of class numbers of number fields, while Tamagawa numbers play a key role in the extension of analytic class number formulas.
Ono showed [23] that τ (T ) = |H 1 (k, X(T ))|/|X 1 (k, T )|, where X(T ) is the group of characters of T and X 1 (k, T ) is the Tate-Shafarevich group of T .Kottwitz [13] generalized Ono's formula to reductive groups and proved [14] the celebrated conjecture of Weil for the Tamagawa number of semi-simple simply connected groups.Ono constructed a 15-dimensional algebraic torus with Tamagawa number 1/4, showing that τ (T ) can be non-integral and conjectured in [22] that every positive rational number is equal to τ (T ) for some torus T .Ono's conjecture was proved by S. Katayama [10] for the number field case.For some later studies of class numbers and Tamagawa numbers of algebraic tori we refer to J.-M.Shyr [33, Theorem 1], S. Katayama [11], M. Morishita [19], C. González-Avilés [6,7] and M.-H.Tran [34] and references within.
In this article we are mainly concerned with the problem of computing the Tamagawa numbers of complex multiplication (CM) algebraic tori.CM tori are closely related to the arithmetic of CM abelian varieties and computing their Tamagawa numbers itself is a way of exploring the structure of CM fields.This problem directly contributes to recent works of Achter, Altug, Garcia and Gordon [1] and of J. Guo, N. Sheu and the fourth named author [8].In the former one the authors computed the size of an isogeny class of principally polarized abelian varieties over a finite field with commutative endomorphism algebra, and express the number in terms of a discriminant, the Tamagawa number, and the product of Frobenius local densities.In the latter one the authors computed formulas for certain CM abelian varieties and certain polarized abelian varieties over finite fields with commutative endomorphism algebras upon the results of [37].Using the class number formula for CM tori, they also computed the numbers of connected components of complex unitary Shimura varieties.In the appendix of [1], W.-W. Li and T. Rüd have obtained several initial results of the values of τ (T ).Our goals are to prove more cases of CM tori and to determine the range of the values of Tamagawa numbers of all CM tori.With a similar goal but using different methods, T. Rüd obtains several results along this direction [30].He provides an algorithm, among others, for giving precise lower bounds and determining possible Tamagawa numbers, and obtains the values τ (T ) for several other CM tori of lower dimension.
We shall describe our results towards computing Tamagawa numbers for a more general class of algebraic tori which include CM tori and then give more detailed results of CM tori.Let k be a global field and K := r i=1 K i be the product of finite separable field extensions K i of k.Let E := r i=1 E i , where each E i ⊂ K i is a subextension of K i .Denote by T K = i T Ki and T E = i T Ei the algebraic k-tori associated to the multiplicative groups of K and E, respectively, and let N K/E = i N Ki/Ei : T K → T E be the norm map.We write T K/E,1 for the kernel of N K/E and T K/E,k := N −1 K/E (G m,k ) for the preimage of the subtorus G m,k ֒→ T E via the diagonal embedding.Let L be the smallest splitting field of T K/E,k and let G = Gal(L/k).We let Λ := X(T K/E,k ) and Λ 1 := X(T K/E,1 ) be the character groups of T K/E,k and T K/E,1 , respectively.They fit in the following short exact sequence of G-lattices: Write H i := Gal(L/K i ), N i := Gal(L/E i ), and N ab i := N i /D( N i )H i , where D( N i ) denotes the commutator group of N i .When there is no confusion, for brevity we shall write H q (A) for H q (G, A) for any G-module A.
Let Ver G,Ni : G → N ab i denote the transfer from G into N ab i ; see Definition 3.6.For any abelian group H, the Pontryagin dual of H is denoted by H ∨ .By Ono's formula (cf.(2.6)) and the Poitou-Tate duality, we have τ (T ) = |H 1 (G, X(T ))|/|X 2 (G, X(T ))|, where X i (G, X(T )) is the ith Tate-Shafarevich group of X(T ).
Let D be the set of all decomposition groups of G. Denote by the restriction map for each G-module A. Put (1.2) H 2 (Z) ′ := {x ∈ H 2 (G, Z) : r 2 D,Z (x) ∈ Im(δ D )}, where δ D : D∈D H 1 (D, Λ 1 ) → D∈D H 2 (D, Z) is the connecting homomorphism induced from (1.1).The group H 2 (Z) ′ plays a similar role of a Selmer group.Theorem 1.1.Let the notation be as above.
(2) There is a canonical isomorphism where Ver G,Ni : G → N ab i is the transfer map.(3) Assume that K i /E i is cyclic with Galois group N i for all i.Then X 2 (Λ) ≃ H 2 (Z) ′ /Im(δ) and .
Using Theorem 1.1 (2), we give a different proof of a result of Li and Rüd [1, Proposition A.11] which does not rely on Kottwitz's formula; see Corollary 7.2.By (1.4), the ratio ( ) is a positive integer; this gives a simple upper bound of τ (T K/E,k ).Theorem 1.2.Suppose the following conditions hold: (a) for each 1 ≤ i ≤ r, the extension K i /k is Galois with Galois group G i , N i = Gal(K i /E i ) is cyclic and every decomposition group i=1 K i be a CM algebra over Q and E := K + the Q-subalgebra fixed by the canonical involution ι, and let T = T K/E,Q be the associated CM torus over Q.
Theorem 1.3.For any integer n, there exists a CM torus T over Q such that τ (T ) = 2 n .
Finally, we give a numerous of results of τ (T ) for Galois CM fields.
and T be the associated CM torus.
( (i) If g ab is even, then τ (T ) = 2; (ii) If g ab is odd, then there is a unique nonzero element ξ in the 2-torsion subgroup for all prime q|Q; 2, otherwise.
(5) Suppose the Galois extension K/Q has Galois group D n of order 2n.Then n is even and τ (T ) = 2.
We explain the idea of the proof of Theorem 1.3.First of all, it is rather difficult to construct a CM field such that the Tamagawa number τ (T ) of the associated algebraic torus T is small.Suppose that K/Q is Galois with Galois group G and let G 2 be a Sylow 2-group of G. T. Rüd implements a SAGE algorithm and has checked all 2-groups of order ≤ 128 and the first 29631 groups of order 256 for G 2 .Based on Rüd's result there is at most one case such that τ (T ) = 1/4; see [30,Section 6.4].To get around this, we construct an infinite family of "totally" linear disjoint Q 8 -CM fields {K i } for i ≥ 1 with τ (T i ) = 1/2, that is, the Galois group of the compositum of any finitely many of these Q 8 -CM fields K i is the product of the Galois groups Gal(K i /Q).The CM algebra K := r i=1 K i then satisfies the conditions in Theorem 1.2 and it follows that the CM torus T associated to K satisfies τ (T ) = 1/2 r .
We point out that the proof of Theorem 1.3 is actually quite tricky.First of all, it follows from [30] or Theorem 1.4 that Q 8 -CM fields are the simplest ones so that the associated CM tori T can have τ (T ) = 1/2.On the other hand, Theorem 1.2 requires a condition that every decomposition group of each CM field extension K i /Q is cyclic.Fortunately, for any Q 8 -CM field K i with CM torus T i , one has τ (T i ) = 1/2 if and only if this condition for K i /Q holds (see Proposition 6.7), so that we can apply Theorem 1.2 to the product of them.On other other hand, one may be wondering whether this condition is superfluous.For this question, we construct linearly disjoint Galois CM fields K 1 and K 2 such that where T 1 , T 2 and T are CM tori associated to K 1 , K 2 and K 1 × K 2 , respectively.This example shows that the cyclicity of decomposition groups of every K i /Q is not superfluous.Theorem 1.3 proves an analogous but more involved conjecture of Ono for CM tori.In the appendix Jianing Li and the fourth named author show that for any global field k and any positive rational number α, there exists an algebraic torus T over k with τ (T ) = α; see Theorem A.1.This extends the result of Katayama [10] and proves Ono's conjecture for global fields.
Though our original motivation of investigating Tamagawa numbers of CM tori comes from counting certain abelian varieties and exploring the structure of CM fields, the main part of the problem itself was to compute the Tate-Shafarevich group.We explain in Remark 7.10 how the Tate-Shafarevich group of a CM torus also comes into play in the theory of Shimura varieties of PEL-type.Tate-Shafarevich groups measure the failure of the local-global principle that is one of main interests in number theory and has been actively studied.Hürlimann [9] proved that the multiple norm principle holds for any etale k algebra K 1 × K 2 in which one component is cyclic and the other one is Galois.Prasad and Rapinchuk [27] settled the problem of the localglobal principle for embeddings of fields with involution into simple algebras with involution, where they also investigated the multiple norm principle.The multiple norm principle has been investigated further by Pollio and Rapinchuk [26,25], Demarche and D. Wei [4] and D. Wei and F. Xu [36].Bayer-Fluckiger, T.-Y. Lee and Parimala [2] studied the Tate-Shafarevich group of general multinorm one tori in which one of factors is a cyclic extension.They give a simple rule for determining the X-group in the case of products of extensions of prime degree p. T.-Y. Lee [17] computes explicitly the X-group for the cases where every factor is cyclic of degree p-power.
This article is organized as follows.Section 2 includes preliminaries and background on Tamagawa numbers of algebraic tori, and some known results of those of CM tori due to Li-Rüd and Guo-Sheu-Yu.Section 3 discusses transfer maps, their extensions and connection with class field theory.In Sections 4 and 5 we compute the Galois cohomology groups of character groups of a class of algebraic tori T K/E,k and T K/E,1 .Section 6 treats Galois CM tori and in Sections 7 ans 8 we determine the precise ranges of Tamagawa numbers of CM tori.In Section 9 we show there are infinitely many pairs of linearly disjoint Galois CM fields K 1 and K 2 satisfying (1.8).In the appendix Jianing Li and the fourth author prove Ono's conjecture for global fields.

2.
Preliminaries, background and some known results 2.1.Class numbers and Tamagawa numbers of algebraic tori.The cardinality of a set S will be denoted by |S| or #S.For any field k, let k be a fixed algebraic closure of k, let k sep be the separable closure of k in k and denote by Γ k := Gal(k sep /k) the Galois group of k.Let G m,k := Spec k[X, X −1 ] denote the multiplicative group associated to k with the usual multiplicative group law.Definition 2.1.(1) A connected linear algebraic group T over a field k is said to be an algebraic torus over k if there exists a finite field extension K/k such that there exists an isomorphism m,K of algebraic groups over K for some positive integer d.Then d is equal to the dimension of T .If T is an algebraic k-torus and K is a field extension of k that satisfies the above property, then K is called a splitting field of T .The smallest splitting field (which is unique and Galois, see below) is called the minimal splitting field of T .
(2) For any algebraic torus T over k, denote by X(T ) := Hom k sep (T ⊗ k k sep , G m,k sep ) the group of characters of T .It is a finite free Z-module of rank d together with a continuous action of the Galois group Γ k of k.
It is shown in [21, Proposition 1.2.1] (also see [38] for other proofs) that every algebraic k-torus splits over a finite separable field extension K/k.The action of Γ k on X(T ) gives a continuous representation which factors through a faithful representation of a finite quotient Gal(L/k) of Γ k .Here L is the fixed field of the kernel of r T and is the smallest splitting field of T .In particular, L is a finite Galois extension of k and X(T ) can be also regarded as a Gal(L/k)-module.
From the remainder of this article, k denotes a global field and a finite extension K/k will be separable, unless otherwise stated.For each place v of k, denote by k v the completion of k at v, and O v the ring of integers of k v if v is finite.For each finite place v, the group T (k v ) of k v -valued points of T contains a unique maximal open compact subgroup, which is denoted by T (O v ).For a non-empty finite set S of places of k containing all non-Archimedean places if k is a number field.Let A k be the adele ring of of k.Denote by U T,S = v∈S T (k v ) × v / ∈S T (O v ) the unit group with respect to S and let Cl S (T ) := T (A k )/T (k)U T,S be the S-class group of T .By a finiteness theorem of Borel [3], Cl S (T ) is a finite group and its cardinality is denoted by h S (T ), called the S-class number of T .If k is a number field and S = ∞ consists of all non-Archimedean places, we write U T := U T,∞ , Cl(T ) := T (A k )/T (k)U T the class group of T and call h(T ) := | Cl(T )| the class number of T .
It follows immediately from the definition that if K/k is a finite extension and as Gal(L/k)-modules, where L is any finite Galois extension of k over which the algebraic k-torus R K/k T K splits.We recall the definition of the Tamagawa number of an algebraic k-torus T .Fix a finite Galois splitting field extension K/k of T with Galois group G. Let χ T be the character of representation be the Artin L-function of the character χ T , where L v (s, K/k, χ T ) is the local Artin L-factor at v. It is well known (cf.[31]) that L(s, K/k, χ T ) has a pole of order a at s = 1, where a is the rank of the G-invariant sublattice X(T ) G .
Let ω be a nonzero invariant differential form on T of highest degree defined over k.To each place v, one associates a Haar measure ω v on T (k v ).Then the product of the Haar measures converges absolutely and defines a Haar measure on T (A k ).
Write (N) for the number field case and (F) for the function field case.For (N), let d k be the discriminant of k.For (F), k = F q (C) is the function field of a smooth projective geometrically connected curve over F q and let g(C) be the genus of C. Definition 2.2.Let T be an algebraic torus over a global field k and ω be a nonzero invariant differential form on T defined over k of highest degree.Then defines a Haar measure on T (A k ), which is called the Tamagawa measure on T (A k ), where ) , for (F), and a is the order of the pole of the Artin L-function L(s, K/k, χ T ) at s = 1.
Let ξ 1 , . . ., ξ a be a basis of X(T ) G .Define 1 denote the kernel of ξ; one has an isomorphism Let d × t := a i=1 dt i /t i be the canonical measure on R a + for (N) and d × t be the counting measure on (q Z ) a with measure (log q) a on each point for (F).Let ω 1 A,can be the unique Haar measure on By a well-known theorem of Borel and Harish-Chandra [24, Theorem 5.6], the quotient space T (A k ) 1 /T (k) has finite volume with respect to every Haar measure.In fact T (A k ) 1 /T (k) is the unique maximal compact subgroup of T (A k )/T (k), because R a + has no nontrivial compact subgroup.
Definition 2.3.Let T be an algebraic torus over a global field k.The Tamagawa number τ k (T ) of T is defined by (2.5) τ k (T ) := the volume of T (A k ) 1 /T (k) with respect to ω 1 A,can , where ω 1 A,can is the Haar measure on T (A k ) 1 defined in (2.4).
One has the following properties ([21, Theorem 3.5.1]cf.[23, Section 3.2, p. 57]): (i) For any two algebraic k-tori T and T ′ , one has (ii) For any finite extension K/k and any algebraic K-torus T K , one has Note that the last statement (iii) is equivalent to the analytic class number formula [16, VIII §2 Theorem 5, p. 161].

Values of Tamagawa numbers.
Theorem 2.4 (Ono's formula).Let K/k be a finite Galois extension with Galois group Γ, and T be an algebraic torus over k with splitting field K. Then is the Tate-Shafarevich group associated to H i (Γ, T ) for i ≥ 0 and w is a place of K over v.
According to Ono's formula, the Tamagawa number of any algebraic k-torus is a positive rational number.Ono constructed an infinite family of algebraic Q-tori T with τ (T ) = τ Q (T ) = 1/4, which particularly shows that τ (T ) needs not to be an integer.Ono conjectured [22] that every positive rational number can be realized as τ k (T ) for some algebraic k-torus T .Ono's conjecture was proved by S. Katayama [10] for the number field case.
For any abelian group G, the Pontryagin dual of G is defined to be (2.8) The Poitou-Tate duality ([24, Theorem 6.10] and [20, Theorem 8.6.8]) says that there is a natural isomorphism To simply the notation we shall often suppress the Galois group from Galois cohomology groups and write H 1 (X(T )) and X 2 (X(T )) for H 1 (Γ, X(T )) and X 2 (Γ, X(T )) etc., if there is no risk of confusion.
2.3.Some known results for CM tori.For the convenience of later generalization and investigation, we define here a more general class of k-tori as mentioned in Introduction.
For every commutative etale k-algebra K, denote by T K the algebraic k-torus whose group of R-valued points of T K for any commutative k-algebra R, is where R Ki/k is the Weil restriction of scalars from the kernel of the norm map N , and the preimage of G m,k in T K under N , where G m,k ֒→ T E is viewed as a subtorus of T E via the diagonal embedding.Then we have the following commutative diagram of algebraic k-tori in which each row is an exact sequence: (2.10) For the rest of this subsection we let k = Q and K = r i=1 K i be a CM algebra, where each K i is a CM field, with the canonical involution ι.The subalgebra K + ⊂ K fixed by ι is the product be the associated norm one CM torus and T K,Q := T K/K + ,Q the associated CM torus, respectively.As before, we have the following exact sequence of algebraic tori over Q and a commutative diagram similar to (2.10).
Proposition 2.6.Let K be a CM algebra and T = T K,Q the associated CM torus over Q.
(1) We have where r is the number of components of K and is the global norm index associated to T .
(2) We have where S K/K + is the finite set of rational primes p with some place v|p of K + ramified in K, and e T,p := Lemma 2.7.Let K and T be as in Proposition 2.6.
Proof.This is proved in [8, Lemma 4.7] (cf.[8, Section 5.1]) in the case where K is a CM field.The same proof using class field theory also proves the CM algebra case.
Proposition 2.8.Let K be a CM field and T the associated CM torus over Proof.See Propositions A.2 and A.12 of [1].

Transfer maps, corestriction maps and extensions
3.1.Transfer maps.Let G be a group and let H be a subgroup of G of finite index.Let X := G/H, and let ϕ : X → G be a section.If g ∈ G and x ∈ X, the elements ϕ(gx) and gϕ(x) belong to the same class of mod H; hence there exists a unique element h ϕ g,x ∈ H such that gϕ(x) = ϕ(gx)h ϕ g,x .Let Ver G,H (g) ∈ H ab be defined by: where D(H) = [H, H] is the commutator group of H and the product is computed in By [32,Theorem 7.1], the map Ver G,H : G → H ab is a group homomorphism and it does not depend on the choice of the section ϕ.This homomorphism is called the transfer of G into H ab (originally from the term "Verlagerung" in German).One may also view it as a homomorphism Ver G,H : G ab → H ab .
In the literature one also uses the right coset space X ′ := H\G but this does not effect the result.One can easily show that if Ver ′ G,H is the transfer map defined using X ′ , then Ver ′ G,H = Ver G,H .One has the following functorial property (see [32, p. 89]).
Lemma 3.2.Let G 1 and G 2 be groups, and let If G is a finite group and p is a prime, G p denotes a p-Sylow subgroup of G.
Proposition 3.3.Let G be a finite group and N ⊳ G be a cyclic normal subgroup of prime order p.Then the transfer Ver G,N : G ab → N ab is surjective if and only if G p is cyclic.
Proof.See Proposition 3.8 of [30].Note that the author assumes that the normal subgroup N is central in G throughout the whole paper; however, this assumption is not needed for this proposition in his proof.

Connection with corestriction maps.
Let H ⊂ G be a subgroup of finite index, and let A be a G-module.Let be the natural map of G-modules.Applying Galois cohomology H i (G, −) to the map f and by Shapiro's lemma, we obtain for each i ≥ 0 a morphism Proposition 3.4.Let H ⊂ G be a subgroup of finite index.Through the isomorphism (3.2) we have the following commutative diagram where func denotes the natural map induced from the inclusion A × k ֒→ A × K or H ֒→ G.When K/k is Galois and let L = K, the second diagram of (3.4) induces a homomorphism which is an isomorphism by class field theory.
3.4.Relative transfer maps.Let H ⊂ N ⊂ G be two subgroups of G of finite index.Let X := G/H and X := G/N with natural G-equivariant projections c : G → X and c : X → X.Let ϕ : X → G be a section, which induces a section ϕ : X → X.For each g ∈ G and x ∈ X, let n ϕ g,x be the unique element in N such that g ϕ(x) = ϕ(gx)n , Proof.(1) We choose the set of representatives S = {g j x i : i = 1, . . ., r, j = 0, . . ., f i − 1} of X = G/ N , which defines a section ϕ of the natural projection G → G/ N .Then the image of the element n ϕ g,x in H is given by One can show that the integer m(g, x i ) is independent of the choice of double coset representative in g x i H.
Then the map is given by the product of the maps Ver Gi,Ni : Proof.Let pr i : G → G i be the i-th projection.Then We obtain the following commutative diagrams (3.9) From this we see that Ver G/H,N/H : (G/H) ab → (N/H) ab does not depend on the choice of the Galois extension L/k.

Cohomology groups of algebraic tori
4.1.H 1 (Λ 1 ) and H 1 (Λ).Let K = r i=1 K i be a commutative etale k-algebra and E = r i=1 E i a k-subalgebra.Let T K/E,k and T K/E,1 be the k-tori defined in Subsection 2.3.Let L/k be a splitting Galois field extension for T K , and let G = Gal(L/k).Put Λ := X(T K/E,k ) and Λ 1 := X(T K/E,1 ), and then N ab i is the abelianization of the group N i .From the diagram (2.10), we obtain the commutative diagram of G-modules: We have (4.2) Ki/Ei G m,K i and Ki/Ei G m,K i ).By Shapiro's lemma, one has Ei ) = 0. From the exact sequence of algebraic E i -tori one has an exact sequence of N i -modules tori in (4.5) split over K i and hence (4.6) becomes an exact sequence of N i -modules and X(R Ki/Ei (G m,Ki )) ≃ Ind Ni 1 Z is an induced module.Taking Galois cohomology to the the lower exact sequence of (4.1), we have an exact sequence Proposition 4.1.Let the notation be as above.
(1) There is a canonical isomorphism where Ver G,Ni : G → N ab i is the transfer map.In particular, H 1 (G, Λ) ≃ Ker( Ver ∨ G,Ni ).Furthermore, the map in (4.8) factors as (4.9) where π Hi : G ab → (G/H i ) ab the map mod H i .
Proof.(1) Taking Galois cohomology of the upper exact sequence of (4.1), we have a long exact sequence of abelian groups: Using the relations (4.2) and by Shapiro's lemma, Ei ) and the exact sequence (4.10) becomes It is clear that the following sequence 2) for any group H, we rewrite (4.12) as follows:

Inf Res
Comparing (4.13) and (4.11), there is a unique isomorphism which fits the following commutative diagram (4.15)

Inf Res δ Res
This shows the first statement.
(2) The map ∆ in (4.1) is induced from the restriction of X(T E ) to the subtorus G m,k and therefore is given by Thus, by the definition, the induced map From the diagram (4.1), we obtain the following commutative diagram: (4.17) With the identification (4.14), we have δ = Cor • δ = Cor • Inf and the lower long exact sequence of (4.17) becomes (4.18) By Propositions 3.5 and 3.4, under the isomorphism (3.2) the map Cor • Inf : , where Ver ∨ G,Ni is the dual of the transfer Ver G,Ni : G ab → N ab i relative to H i .This proves (4.8).Then it follows from (4.18) that H 1 (Λ) ≃ Ker( r i=1 Ver ∨ G,Ni ).As the map Ver G,Ni : G ab → N ab i factors as i , the last assertion (4.9) follows.
Remark 4.2.In terms of class field theory, we have Let L 0 be the compositum of all Galois closures of K i over k; this is the minimal splitting field of the algebraic torus T K/E,k .One has L 0 ⊂ L and its Galois group G 0 := Gal(L 0 /k) is a quotient of G. Thus, we have the following commutative diagram: Taking the Pontryagin dual, the lower row gives From this, we see that the map Ver ∨ G,Ni in (4.8) has image contained in G ab,∨ 0 .It follows from (4.19) that this map is independent of the choice of the splitting field L, also cf.Remark 2.5.

X 2 (Λ).
Let D be the set of all decomposition groups of G.For any G-module A, denote by the restriction map to subgroups D in D. By definition, X i (A) = Ker r i D,A .We shall write r D and r D for r i D,A and r i D,A , respectively, if it is clear from the content.For the remainder of this section, we assume that the extension K i /E i is cyclic with Galois group N i for all i.Consider the following commutative diagram (4.21) Proof.It is obvious that j(X 2 (Λ)) ⊂ X 2 (Λ 1 ) and then we have a long exact sequence because X 2 (X(T )) does not depend on the choice of the splitting field.Since K i /E i is cyclic, by Chebotarev's density theorem, we have X 2 (N i , Λ 1 Ei ) = 0 for all i and X 2 (Λ 1 ) = 0. Therefore, one gets the 4-term exact sequence From this we obtain X 2 (Λ) ≃ H 2 (Z) ′ /Im(δ) and K/k G m,K )) = 0.This gives an alternative proof of the first statement X 2 (Λ 1 ) = 0 of Proposition 4.3.
In order to compute the groups H 2 (Z) ′ and X 2 (Λ), we describe the maps δ and δ D in the first commutative diagram of (4.21).Since Ei , it suffices to describe the following commutative diagram: (4.26) Using the commutative diagram (4.17), it factors as the following one: (4.27) Proposition 4.5.Assume that the extension K i /E i is cyclic and both the subgroups , where D i = D ∩ N i and D i is its image in N i .Under these identifications the first commutative diagram in (4.21) decomposes as the following: Proof.For any element g ∈ G and any N i -module X, let X g = X be the g N i g −1 -module defined by Since We now show Ei .The bottom row of (4.30) then expresses as Similar to (4.13) and (4.14), using the Inflation-Restriction exact sequence, we make the following identification This proves the proposition.
The following proposition gives a group-theoretic description of X 2 (Λ). .
Proof.We translate the commutative diagram (4.28) in terms of group theory.For each decomposition group D ∈ D, we have the following corresponding commutative diagram: (4.37) Here we ignore the multiplicity [G : . Thus, by Proposition 4.3, we obtain (4.36).This proves the proposition.

Computations of some product cases
We keep the notation in the previous section.In this section we consider the case where the extensions K i /k are all Galois with Galois group As before, we put T = T K/E,k , Λ := X(T ), T 1 = T K/E,1 and Λ 1 := X(T 1 ).
We remark that Condition (ii) is not needed in the proof of Lemma 5.1.Proposition 5.2.Let H ⊂ G be a normal subgroup of a finite group G, let A be a G-module, and let n ≥ 1 be a positive integer.If H q (H, A) = 0 for all 0 < q < n, then we have a 5-term long exact sequence Proof.This follows from the Hochschild-Serre spectral sequence Ei ) and there is a natural homomorphism v i : By (4.32), we get the long exact sequence Since Ind Ni Hi Z = Ind Ni 1 Z is an induced module, it follows from (5.3) that (5.4) Since the algebraic torus T Ki splits over K i , the Ei ).Using the same argument as (5.4), we get On the other hand, (5.6) conjugation and the action is trivial.Thus, we obtain an exact sequence (5.7) Corollary 5.4.Let the notation and assumptions be as in Proposition 5.3.Assume further that the orders of groups G i are mutually relatively prime.Then (In our notation the group Proof.In this case, the maps d i 's are all zero.
Remark 5.5.It is curious to know what the natural homomorphism v i :

. τ (T ).
In this subsection we shall further assume that each subgroup N i , besides being cyclic, is also normal in G i .This assumption simplifies the description of the group H 2 (Z) ′ through Mackey's formula.For each 1 ≤ i ≤ r let T i := T Ki/Ei,k , and let Λ i := X(T i ) be the character group of T i .Let D i be the decomposition subgroups of G i for each i.
(2) Assume that for any 1 ≤ i ≤ r and D ′ i ∈ D i , there exists a member D ∈ D such that pr i (D) = D ′ i and a section s i : Let D = D P ∈ D be a decomposition group associated to a prime P of L. By definition The projection map pr i : G → G i sends each element σ to its restriction σ| Ki to K i .Let P i denote the prime of K i below P and we have pr i (D P ) = D Pi .We show that pr i (D For the other inclusion, since pr i : D P → D Pi is surjective, for each y ∈ D Pi ∩ N i there exists an element x = (x i ) ∈ D P such that x i = y.Since x i ∈ N i , x also lies in N i .This proves the other inclusion.Therefore, The map Since each Ver D,D P i ∩Ni = Ver D P i ,D P i ∩Ni • pr i (cf.(3.6)), the above map factorizes into the following composition To prove the other inclusion, we must show that each f i satisfies condition (b) provided f satisfies condition (a), which we assume from now on.For 1 ≤ i ≤ r, let D ∈ D be a decomposition group of G over D ′ i and s i : for some h k ∈ Hom(D k , Q/Z).Pulling back to D ′ i via s i , we have where Proposition 5.7.With the notation and assumptions be as above.Suppose further that for each 1 ≤ i ≤ r, every decomposition group of G i is cyclic.Then we have τ (T ) = r i=1 τ (T i ).Proof.We shall show that the assumption in Lemma 5.6 (2) holds.Let D ′ i ∈ D i be a decomposition group of G i ; by our assumption D ′ i is cyclic and let a i be a generator.Let D be a cyclic subgroup of G generated by ãi = (1, . . ., 1, a i , 1, . . ., 1) with a i at the i-th place.Clearly D is the decomposition group of some prime and let s i : D ′ i → D be the section sending a i 6.3.Certain Galois extensions.Proposition 6.3.Assume the short exact sequence ( * ) splits.Let g ab := |G +ab | and Λ := X(T ).
(2) We have G ≃ ι × G + and G ab ≃ ι × G +ab .(i) Suppose g ab is even.As (6.1), K contains two distinct imaginary quadratic fields.Thus, n K = 1 and τ (T ) = 2. (ii) Suppose g ab is odd.Put Λ 1 = X(T K,1 ) and we have the following exact sequence We have H 1 (Λ 1 ) ≃ Z/2Z by Proposition 4.1(1), H 2 (Λ 1 ) = 0 by Proposition 5.3, and [2] be the unique nonzero element.Then This completes the proof of the proposition.Remark 6.4.When G is non-abelian and the short exact sequence ( * ) does not split, we do not know the value of τ (T ) in general.However, if the involution ι is non-trivial on the maximal abelian extension K ab over Q in K, then K ab is a CM abelian field and it follows from Proposition 6.2 that τ (T ) ∈ {1, 2}.
It remains to determine the Tamagawa number when G is non-abelian, ( * ) is non-split, and K ab is totally real.This includes the cases of G = Q 8 and the dihedral groups, for which we treat in the next subsections.6.4.Q 8 -extensions.The quaternion group Q 8 = {±1, ±i, ±j, ±k} is the group of 8 elements generated by i, j with usual relations i 2 = j 2 = −1 and ij = −ji = k.We have the following well-known properties of Q 8 .Lemma 6.5.
(1) The group Q 8 contains 6 elements of order 4, one element of order 2, and the identity.It does not has a subgroup isomorphic to Z/2Z × Z/2Z.Thus, every proper subgroup is cyclic.
(2) Every non-trivial subgroup contains {±1}, and only subgroup which maps onto Proposition 6.6.Let P and Q be two odd positive integers such that (6.2) for some integers a, b ∈ N. Assume that Q is not a square.Let K := Q(β) be the simple extension of Q generated by β which satisfies Then K is a Galois CM field with Galois group Q 8 with maximal totally real field The Galois group Gal(K/Q) is generated by τ 1 and τ 2 given by Moreover, for each prime ℓ, the decomposition group D ℓ is cyclic except when ℓ|Q.For ℓ|Q, one has Proof.Note that gcd(P, Q) = 1 and P is not a square.Then Q( √ P , √ Q) is a totally real biquadratic field and its Galois group is generated by σ 1 and σ 2 , where It is clear that α is a totally negative, and hence K is a CM field.The maximal totally real subfield and obtains It follows that K/Q is Galois.Let τ 1 , τ 2 ∈ Gal(K/Q) be defined as in (6.3).One easily computes and obtains the relations Denote by D + ℓ the decomposition group at ℓ in Proposition 6.7.Let K be a CM field which is Galois over Q with Galois group Q 8 .Then Proof.Put G = Gal(K/Q) ≃ Q 8 and N = ι .We shall show that (6.6) and (6.7) X 2 (Λ) = Z/2Z ⊕ Z/2Z, if every decomposition group of K/Q is cyclic; 0, otherwise.
If there is a finite place of K whose decomposition group is not cyclic, then we have X 2 (Λ) = 0. Therefore, τ (T K,Q ) = 2. On the other hand, suppose that every decomposition group of Proof.By Proposition 6.6, the cyclicity of all decomposition groups of K/Q is equivalent to the condition P q = 1 for all primes q | Q. Hence the assertion follows from Proposition 6.7.
6.5.The dihedral case.Let D n denote the dihedral group of order 2n.Lemma 6.9.Suppose the CM-field K is Galois over Q with Galois group G, and let Proof.This is due to Jiangwei Xue.Let L/Q be the class field corresponding to the open subgroup Suppose p is a rational prime unramified in K/Q such that the Artin symbol (p, K/Q) lies in S. Since p splits completely in the fixed field E = K Dp of the decomposition group D p = (p, K/Q) of G at p and ι ∩ D p = {1} (by the definition of S), one has K = K + E and that every prime v of K + lying above p splits in K, and therefore It follows from class field theory that p splits completely in L. Thus, the density of S is less than or equal to that of primes splitting completely in L. By the Chebotarev density theorem, one yields |S|/|G| ≤ 1/[L : Q].Therefore, n K ≤ |G|/|S| and the assertions then follow immediately.Proposition 6.10.Let K be a Galois CM field with group G = D n and T the associated CM torus over Q.Then n is even and τ (T ) = 2.
Proof.Write D n = t, s : t n = s 2 = 1, sts = t −1 .One easily sees that the center Z(D n ) = {x ∈ t : x 2 = 1} contains an element of order 2 if and only if n = 2m is even.Since ι is central of order 2 in D n , n is even.In this case |S| = 2m + 1 and n K = 1 by Lemma 6.9.Therefore, τ (T ) = 2. Remark 6.11.The criterion in Lemma 6.9 does not help to compute τ (T ) for the case G = Q 8 or G = Z/2 n Z.However, these cases have been treated in Propositions 6.7 and 2.8, respectively.

Some non-simple CM cases
Keep the notation in Section 2.3.Write N i := Gal(K i /K + i ) = ι i with involution ι i on K i /K + i and ι * i ∈ N ∨ i = Hom(N i , Q/Z) for the unique non-trivial element, that is ι * i (ι) = 1/2 mod Z. Fix a Galois splitting field L of T = T K,Q and put G = Gal(L/Q).For a number field F , denote by Proof.We first describe the transfer map Ver G,Ni : G → N i .We may regard the CM fields K i as subfields of Q, and put X i := G/H i = Σ Ki and X i := G/ N i = Σ K + i .Fix a section ϕ : X i → G such that the induced section ϕ : X i → X i has image Φ i .Since N i = N i /H i is abelian, modulo D( N i )H i is the same as modulo H i .Following the definition of Ver G,Ni , for each g ∈ G, where the element n ϕ g,x ∈ N i is defined in Section 3.4 and n i (g) := |{x ∈ X i : , and let φ ∈ Φ i and φ ∈ Φ i be the corresponding elements.Put φ ′ := ϕ(gx) and φ ′ = ϕ(gx), the image of φ ′ .Then g φ and φ ′ are elements lying over gx and we have g φ = φ ′ n ϕ g,x .So gφ = φ ′ if and only if n ϕ g,x ∈ H i .On the other hand, since φ ′ ∈ Φ i and two elements gφ and φ ′ are lying over the same element gx, we have gives the bijection of subsets: This shows the desired equality |Φ i (g)| = n i (g).
By Proposition 4.1 and Lemma 7.1, we give an independent proof of the following result of Li and Rüd [1,Proposition A.11].
Corollary 7.2.Let K and T be as in Lemma 7.1 and Λ = X(T ) be the character group of T .Then Proof.Indeed after making the identity Proof.By [29, Theorem 1.2.9], we have the following commutative diagram with row exact sequence (7.2) where A pro denotes the profinite completion of an abelian group A. It follows from the Poitou-Tate duality and class field theory that (7.3) ) is finite, it is equal to its profinite completion.This proves the proposition.
Remark 7.4.Proposition 7.3 gives a cohomological interpretation of the group A × /Q × N (T (A)).This reminds the main theorem  ( The following is one of the most important key to prove Theorem 7.8.(1) there is a quadratic extension of (2) Since ℓ = 2 and a ∈ Z × ℓ , we have Hence the assertion follows from Proposition 8.1 and the assumption b ∈ Z × ℓ .(3) Since ℓ is not equal to 2, we have Therefore the assertion follows from Proposition 8.1. ( Now we prove the equality K r ∩ L ′ = Q, which gives the desired assertion.(8.2).Therefore K r is equal to the compositum of E r and K r ∩ L ′ by the assumption (i).Consequently, there is an isomorphism is a prime number, and hence K r /Q is abelian.This contradicts the assumption (i), which implies the desired equality K r ∩ L ′ = Q.
Proposition 8.8.For any λ ∈ L, there is a CM field K containing K + λ such that K/Q is a Q 8 -extension.
Proof.From the definition of K λi for 1 ≤ i ≤ r, condition (a) holds.
Take a finite places w of K λi and v of K + λi satisfying w | v. Then Lemma 8.7 implies the cyclicity of the decomposition group of Gal(K + λi /Q) at v. Since K λi /Q is a Q 8 -extension, the decomposition group at w is cyclic by Lemma 6.5 (2).Therefore condition (c) holds.

Products of two linearly disjoint Galois CM fields
In this section we show the following result.This theorem shows that the conclusion of Proposition 5.7 is no longer true if one drops the cyclicity of decomposition groups of G i for all i.We shall use the notations in Section 5.2.In particular, L = K 1 K 2 , G := Gal(L/Q), G i := Gal(K i /Q), H i := Gal(L/K i ), N i := Gal(L/K + i ) and N i = Gal(K i /K + i ) for i = 1, 2. First, we give a sufficient condition on K = K 1 × K 2 for which Theorem 9.1 holds.Let C and C i be the sets of cyclic subgroups of G and G i respectively.Then put Proof.The proof is the same as Lemma 5.6.
Here H ab is viewed as a G-module by conjugation and also as a G/H-module, since H acts trivially on H ab .Proof.See [20, Proposition 1.5.9].3.3.Connection with class field theory.Let k ⊂ K ⊂ L be three global fields such that the extension L/k is Galois.Put G = Gal(L/k) and H = Gal(L/K) ⊂ G. Denote by C k and C K the idele class groups of k and K, respectively.The Artin map is a surjective homomorphism Art L/k : C k → G ab ; similarly we have Art L/K : C K → H ab .By class field theory we have the following commutative diagrams (3.4) Ver G, Ni/Hi = Ver G/Hi, Ni/Hi • pr i = Ver Gi,Ni • pr i , cf.Remark 3.7.Thus, the map (3.7) is equal to i Ver Gi,Ni • (pr i ) i .Since the map (pr i ) i : G → i G i is the identity, we show that the map (3.7) is equal to i Ver Gi,Ni .3.5.Connection with class field theory.Let k ⊂ E ⊂ K be three global fields.Let L/k be a finite Galois extension containing K with Galois group G = Gal(L/k).Let H = Gal(L/K) ⊂ N = Gal(L/E) be subgroups of G.The Artin map produces the following isomorphisms (cf.(3.4))

Λ 1
6) we have an exact sequence of N i -modules (Ei − −−− → 0 on which H i acts trivially.Taking the Galois cohomology H * (D i , −), we get an exact sequence(4.33)

Proposition 4 . 6 .
Let the notation and assumptions be as in Proposition 4.5.LetVer G,N = (Ver G,Ni ) i : G ab → i N i and Ver D,D = (Ver D,Di ) i : D ab → i D idenote the corresponding transfer maps, respectively.Then

Lemma 5 . 6 .
(1) The inclusion r consider the following two conditions: (a) For each D ∈ D, the restriction f | D of f to D lies in the image of the map r i=1 Ver ∨ D,Di .(b) For each 1 ≤ i ≤ r and D ′ i ∈ D i , the restriction f i | D ′ i to D ′ i lies in the image of the map Ver ∨ D ′ i ,D ′ i ∩Ni .It suffices to show that the condition (b) implies (a).

Lemma 7 . 1 .
Let K be a CM algebra and T = T K,Q the associated CM torus over Q.Then the map ab,∨ = Hom(G, Q/Z) sends each element r i=1 a i ι * i , for a i ∈ Z, to the element f = r i=1 a i f i , where f i ∈ Hom(G, Q/Z) is the function on G given by f i (g) = |Φ i (g)|/2 mod Z.

8 . 1 .
Construction of an effective family of Q 8 -CM fields 8.Existence of Q 8 -extensions of fields.Let k be a field of characteristic different from 2, and denote by Br(k) the Brauer group of k.Then we define a pairing ( , ) k : k × × k × → Br(k) by sending (a, b) ∈ k × × k × to the Brauer class of the quaternion algebra a, b k := k ⊕ kα ⊕ kβ ⊕ kαβ, where α 2 = a, β 2 = b and βα = −αβ.By definition, the pairing ( , ) k is symmetric, and the image of ( , ) k is contained in the 2-torsion group of Br(k).For a ∈ k × , put k a := k[T ]/(T 2 − a).We denote by N ka/k : k a → k the norm map of k a /k.Proposition 8.1 ([5, Proposition 1.1.7]).Let a, b ∈ k × .Then the following are equivalent:

Theorem 9 . 1 .
There are infinitely many CM algebras K = K 1 ×K 2 with linearly disjoint Galois CM fields K 1 and K 2 such that
denote the maximal abelian quotient of N which maps H to zero.Let Ver G,N/H (g) ∈ (N/H) ab Definition 3.6.Let H ⊂ N be two subgroups of G of finite index.The group homomorphism Ver G,N/H : G → (N/H) ab defined in (3.5) is called the transfer of G into (N/H) ab relative to H.By abuse of notation, we denote the induced map by Ver G,N/H : G ab → (N/H) ab .One can check directly that the map Ver G,N/H : G ab → (N/H) ab factors through π H : G ab → (G/H) ab , the map modulo H.We denote the induced map by ϕ g,x .Let (N/H) ab = N/D(N )H mod D(N )H.Proposition 3.5.(1)ThemapVer G,N/H : G → (N/H) ab does not depend on the choice of the section ϕ and it is a group homomorphism.(2)One has Ver G,N/H = π H • Ver, where π H : N ab → (N/H) ab is the morphism mod H. Proof.Clearly, the statement (1) follows from (2), because Ver does not depend on the choice of ϕ and is a group homomorphism.(2) By definition Ver(g) = x∈X n ϕ g,x mod D(N ), thus Ver G,N/H = π H • Ver.(3.6) Ver G/H,N/H : (G/H) ab → (N/H) ab .Remark 3.7.If H ⊳ G is a normal subgroup, then the induced map Ver G/H,N/H is the transfer map from G/H to N/H associated to the subgroup N/H ⊂ G/H of finite index.Lemma 3.8.Let H ⊳ N be two subgroups of finite index in G with H normal in N and cyclic quotient because we are only concerned with the image of the map Cor • Inf in (4.28) and this does not effect the result.Then Ver G,N : G ab → i N i and Ver D,D : D ab → i D i are the respective compositions.By Proposition 4.5, the map δ D corresponds to Ver ∨ D,D .From the second description of H 2 (Z) ′ in (4.22), we obtain (4.35).By Propositions 4.5 and 4.1, the map δ : H 20, Theorem 2.1.5,p. 82]) and [20, Proposition 2.1.3,p. 81].be as before.Suppose that each K i /k is Galois with group G i , and that Conditions (i) and (ii) are satisfied.Then H 2 and (7.1) follows from Proposition 4.1.Proposition 7.3.Let K be a CM algebra and T the associated CM torus over Q.Then there is an isomorphism A class field theory, where K/k is Galois with Galois group G. Lemma 7.5.Let K 1 be a CM field and K ′ 1 }, . . ., {ℓ r , ℓ ′ r } in L such that |{ℓ 1 , . . ., ℓ r , ℓ ′ 1 , . . ., ℓ ′ r }| = 2r.Proof.By the Dirichlet prime number theorem, there are r distinct prime numbers ℓ 1 , . . ., ℓ r which are congruent to 1 modulo 4.Moreover, the Dirichlet prime number theorem implies the existence of prime numbers ℓ ′ 1 , . . ., ℓ ′ r satisfying the following: 