Number fields with prescribed norms

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.


Introduction
Let k be a number field.In this paper we are interested in the images of the norm maps N K/k : K * → k * for finite field extensions K/k.Specifically, given an element α ∈ k * and a finite group G, does there exist an extension K/k with Galois group G such that α is a norm from K? We are able to answer this question positively if one restricts to abelian extensions of k.Furthermore, in the abelian setting, we prove the existence of such an extension from which a given finite set of elements of k * are norms.Theorem 1.1.Let k be a number field, G a finite abelian group and A ⊂ k * a finitely generated subgroup.Then there exists an abelian extension K/k with Galois group G such that every element of A is a norm from K.
As an application, we obtain the following corollary.Corollary 1.2.Let k be a number field, G a finite abelian group and S a finite set of places of k.Then there exists an abelian extension K/k with Galois group G such that every S-unit of k is a norm from K.
We prove Theorem 1.1 by counting the collection of abelian extensions under consideration; we obtain an asymptotic formula for the number of such extensions of bounded conductor, and show explicitly that the leading constant in this formula is non-zero.In particular, we prove the existence of infinitely many extensions with the desired properties.The strategy of proving existence via counting is widely used in analytic number theory, for example in the context of the Hardy-Littlewood circle method.Our proof of Theorem 1.1 seems to be the first case where it is implemented for number fields.Our methods even allow us to prove existence of such an extension K/k which satisfies any finite collection of admissible local conditions (Corollary 4.12).
Before we can explain these more general results, we must introduce some notation.Fix a choice of algebraic closure k of k and let G be a finite abelian group.By a G-extension of k, we mean a surjective continuous homomorphism ϕ : Gal(k/k) → G.This corresponds to choosing an extension k ⊂ K ⊂ k together with an isomorphism Gal(K/k) ∼ = G.Keeping track of the isomorphism with G simplifies the set-up and the counting.It has no qualitative effect on the results; forgetting the choice of isomorphism merely scales all the counting results by | Aut(G)|.We write G-ext(k) for the set of all G-extensions of k.Given ϕ ∈ Gext(k), we write K ϕ for the corresponding number field, and Φ(ϕ) for the norm of the conductor of K ϕ (viewed as an ideal of k).Moreover, we write A * Kϕ for the ideles of the number field K ϕ .We are interested in the counting functions ) The first counts all G-extensions ϕ of k of bounded conductor, the second counts only those for which every element of A is everywhere locally a norm, the third only those for which every element of A is a global norm.
An asymptotic formula for N(k, G, B) was first obtained by Wood in [49], building on numerous special cases.In this paper we obtain asymptotic formulae for the other counting functions.Our formulae are stated in terms of the invariant ̟(k, G, A) which we now define.Definition 1.3.Let k be a number field, G a finite abelian group, and A ⊂ k * a finitely generated subgroup.For d ∈ Z ≥1 , let k d = k(µ d , d √ A).We define where |g| denotes the order of g in G and id G ∈ G is the identity element.
Theorem 1.4.Let k be a number field, G a non-trivial finite abelian group, and A ⊂ k * a finitely generated subgroup.Then as B → ∞, for some c k,G,A > 0.
This theorem gives an asymptotic formula for the number of G-extensions from which every element of A is a global norm.It is natural to ask how the number of such extensions compares with the total number N(k, G, B) of G-extensions of k of conductor bounded by B. We observe that N(k, G, B) = N glob (k, G, {1}, B) and note that in this case the formula of Theorem 1.4 agrees with [49,Thm. 3.1].
The next theorem generalises this observation.It says that, unless we are in a very special case, for 100% of G-extensions of k not all elements of A are norms.
Theorem 1.6.Let k be a number field, G a non-trivial finite abelian group of exponent e, and A ⊂ k * a finitely generated subgroup.Then the following are equivalent: (1) v for all but finitely many places v of k.There is a nice cohomological way to interpret the condition (3) in Theorem 1.6 via certain Tate-Shafarevich groups (see §4. 6).Together with some class field theory, this will allow us to deduce the following result.
Corollary 1.7.Let A ⊂ k * be a finitely generated subgroup and let e be the exponent of G. Then the limit (i) only depends on the image Ak * e of A in k * /k * e ; (ii) equals one if A ⊂ k * e ; (iii) is zero for all but finitely many finite subgroups Ak * e ⊂ k * /k * e ; (iv) is zero for all finitely generated subgroups A ⊂ k * e if and only if the extension k(µ 2 r )/k is cyclic, where 2 r is the largest power of 2 dividing e.
Condition (iv) holds for example if 8 ∤ e or µ e ⊂ k * .Our next result shows that if G is cyclic then in order to have for some choice of A, the field k must have more than one prime lying above 2.
Theorem 1.8.Let k be a number field, let A ⊂ k * be a finitely generated subgroup, and let G be a finite cyclic group.Suppose that k has only one prime lying above 2. Then the following are equivalent: (1) > 0; (2) every element of A is a global norm from every G-extension of k.
A necessary condition for an element of k to be a global norm is that it is a norm everywhere locally.However, this is not a sufficient condition in general due to possible failures of the Hasse norm principle (HNP).Nevertheless, to prove Theorem 1.4, we reduce to the case of everywhere local norms via the following theorem, which shows that, when ordered by conductor, "most" abelian extensions satisfy the Hasse norm principle.Theorem 1.9.Let k be a number field, G a finite abelian group, and A ⊂ k * a finitely generated subgroup.Then In particular Theorem 1.9 implies that lim Theorem 1.4 can thus be proved via an asymptotic formula for N loc (k, G, A, B), which we obtain in Theorem 4.1.We prove Theorem 1.9 using a purely local criterion for failure of the Hasse norm principle (Proposition 4.2).Taking A = {1} in Theorem 1.9, we obtain the following result.
Corollary 1.10.Let k be a number field and G a finite abelian group.Then 100% of G-extensions of k, ordered by conductor, satisfy the Hasse norm principle.
Corollary 1.10 stands in stark contrast to the results of [20], where a dichotomy occurs when counting by discriminant: in op.cit.we showed that for certain finite abelian groups G a positive proportion of G-extensions can fail the Hasse norm principle, when ordered by discriminant.This contrasting behaviour illustrates the fact, already observed by Wood in [49], that counting by conductor often leads to more natural statements than counting by discriminant.In fact, after seeing the results we obtained in [20] when counting extensions ordered by discriminant, Wood remarked that the dichotomy we had observed should disappear when ordering by conductor, and conjectured the statement of Corollary 1.10.
There are two reasons why it seems quite difficult to prove Theorem 1.1 when counting by discriminant, rather than conductor.Firstly, the condition that every element of A is a norm everywhere locally may be only rarely satisfied and, in the setting of [20,Thm. 1.4] where a positive proportion of G-extensions fail the Hasse norm principle, it becomes challenging to show the existence of a Gextension for which every element of A is a norm everywhere locally and the Hasse norm principle holds.Secondly, the leading constant obtained when counting by discriminant is very complicated, with potential for further cancellation, so it is difficult to prove its positivity, whereas when counting by conductor we have a simple criterion for positivity of the leading constant (see Theorem 3.1).
The counting techniques employed in this paper are fairly robust and enable us to prove a strengthening of Theorem 1.1 in which we impose local conditions at finitely many places.See Theorem 3.1 and Corollary 4.11 for precise statements.
Our work on the statistical behaviour of the Hasse norm principle brings together two major areas of modern number theory: namely, counting within families of number fields, and the quantitative study of the failure of local-global principles.Notable recent papers on the statistics of number fields include [1], [2], [4], [5], [18], [21], [27], [33] and [50].Some significant contributions to the study of local-global principles in families include [3], [6], [7], [8], [19], [29] and [30].For a summary of recent progress on counting failures of the Hasse principle, see [9].More specifically, the statistical behaviour of the Hasse norm principle is examined in [10], [31] and [35].In particular, in [35] Rome obtains an asymptotic formula for the number of biquadratic extensions of Q (ordered by discriminant) which fail the Hasse norm principle.Obtaining asymptotic formulae for the number of such failures for other classes of field extensions would seem to be an interesting problem.
Below, we give some examples illustrating our results in a variety of settings to demonstrate the wide range of phenomena manifested by norms in extensions of number fields.
(1) Take G = Z/nZ with 8 ∤ n and α ∈ k * not an nth power.Then Corollary 1.7 implies that for 100% of all Z/nZ-extensions of k ordered by conductor, α is not a norm.In the special case n = 2 of quadratic extensions, this result can be proved using standard techniques in analytic number theory; all other cases are new.(2) Take k = Q, α = 16 and G = Z/8Z.As is well known, 16 is an 8th power in Q * p for all odd primes p and in R * .It therefore follows from Theorems 1.6 and 1.8 that 16 is a norm from every Z/8Z-extension K/Q, despite not being an 8th power in Q.
), α = 16 and G = Z/8Z.Then, as above, we see that 16 is locally an 8th power at all places v such that v ∤ 2. Hence 16 is a local norm from all Z/8Z-extensions of k at all places v ∤ 2. However, let p, q be the two primes of k above 2.By [32, Thm.9.2.8] there exists a Z/8Zextension F/k such that F p /k p is unramified of degree 8. Therefore, 16 is not a local norm from F p /k p , and consequently not a global norm from F/k.Given the existence of one such an extension, an application of [49,Cor. 1.7] (or Theorem 3.1) yields the existence of a positive proportion of Z/8Z-extensions K/k which are unramified of degree 8 over p, thus the limit (1.2) is positive but not equal to 1 in this case.
Let us explain in more detail why [32, Thm.9.2.8] applies here but not in the previous example.Recall that a place v of a number field L is said to split (or decompose) in an extension M/L if there exist at least two distinct places of M above v.All places of Q apart from 2 split in the non-cyclic extension Q(µ 8 )/Q, so that (Q, 8, Ω Q \ {2}) is a so-called special case and [32, Thm.9.2.8] does not apply in example (2).However, in example (3), q is non-split in k(µ 8 )/k: both p and q are totally ramified in k(µ 8 )/k, since 2 is split in k/Q and totally ramified in Q(µ 8 )/Q.Therefore, (k, 8, Ω k \ {p}) is not a special case and [32, Thm.9.2.8] can be applied in example (3).(4) Take k = Q, α = 5 2 and G = (Z/2Z) 2 .A simple argument (cf.Lemma 4.4) shows that 5 2 is a norm everywhere locally from every biquadratic extension of Q.By Theorem 1.9, it is thus a global norm from 100% of biquadratic extensions of Q ordered by conductor.However, 5 2 is not a global norm from ) (failure of the Hasse norm principle [11,p. 360,Exercise 5.3]).Therefore, it is not true that 5 2 is a global norm from every biquadratic extension of Q. Remark 1.12.A simple application of local class field theory (Lemma 4.4) shows that every element of k * e is everywhere locally a norm from every Gextension of k, where e denotes the exponent of G. Using this, one can show that in our results, the assumption that A is a finitely generated subgroup of k * can be replaced by the weaker assumption that the image of A in k * /k * e is finite.We have chosen to make the stronger assumption as it simplifies the exposition and some technical aspects of the proofs.
We finish with a simple example which solves the problem analogous to Theorem 1.1 for field extensions of degree n with maximal Galois group.
Example 1.13.Let α ∈ Q * and n ≥ 3. Then the polynomial has Galois group S n over Q(t) for all but finitely many c ∈ Q (see [26,Satz 1]).Therefore, Hilbert's irreducibility theorem implies that for infinitely many specialisations t ∈ Q, the Galois group is S n , and α is clearly a norm from such an extension, being the product of the roots of the defining polynomial.
1.1.Methodology and structure of the paper.In §2 we recall some of the theory of frobenian functions from Serre's book [41, §3.3], in order to help analyse the Dirichlet series which arise in this paper.
In §3 we prove our main technical result, Theorem 3.1.This is a general theorem for counting abelian extensions with local conditions imposed.To prove this we study the analytic properties of the Dirichlet series corresponding to our counting functions.We achieve this with the help of the harmonic analysis techniques developed in our earlier paper [20].In our case, however, the analysis is more difficult as the singularities of our Dirichlet series will be branch point singularities, rather than poles, in general; this is reflected in the fact that ̟(k, G, A) in Theorem 1.4 can be a non-integral rational number.This section is the technical heart of the paper and is dedicated to the proof of Theorem 3.1.
Let us emphasise once more that we prove Theorem 1.1 by first counting the extensions of interest and then showing that the leading constant obtained is positive.Our situation presents an interesting difficulty, however: the leading constant we obtain is not an Euler product but a sum of Euler products and, in general, cancellation within these sums may occur for some choices of local conditions.For example, a famous theorem of Wang [47] says that there is no Z/8Z-extension of Q which realises the unramified extension of Q 2 of degree 8; in this case Wright observed in [51, p. 48] that the Euler products appearing in the leading constant cancel out.We have to carefully analyse these sums of Euler products and explicitly show that no cancellation occurs in our case.
In §4, we prove the major results stated in the introduction via suitable applications of Theorem 3.1 combined with Galois-cohomological techniques.At the end of §4 we also give a generalisation of Theorem 1.1 which allows one to impose local conditions on the abelian extension K/k at finitely many places.
The appendix (by Yonatan Harpaz and Olivier Wittenberg) contains a purely geometric proof of Theorem 1.1.It uses descent and a version of the fibration method developed in [24] to show that the Brauer-Manin obstruction controls the failure of weak approximation on a certain auxiliary variety.The existence of the required abelian extension is then shown using a version of Hilbert's irreducibility theorem due to Ekedahl [17] (see also [42, § §3.5-3.6]).
1.2.Notation and conventions.We fix a number field k throughout the paper and use the following notation: the residue field at a finite place v q v the cardinality of the residue field at a finite place v ζ k (s) the Dedekind zeta function of k.
For locally compact abelian groups A and B, we use the following notation: All finite groups are viewed as topological groups with the discrete topology.
For a place v of k, a finite abelian group G, and χ ∈ Hom(k * v , G), we denote by Φ v (χ v ) the reciprocal of the v-adic norm of the conductor of Ker χ v .For every χ ∈ Hom(A * /k * , G), we let Φ(χ) be the reciprocal of the idelic norm of the conductor of the kernel of χ; this equals the norm Φ(ϕ) of the conductor of the sub-G-extension ϕ corresponding to χ via the global Artin map.
Let K/k be an extension of number fields and α ∈ k * .We say that α is a

Frobenian functions
For the proofs of our main results, we will require some of the theory of frobenian functions, as can be found in Serre's book [41, §3.3].Recall that a class function on a group is a function which is constant on conjugacy classes.Definition 2.1.Let k be a number field and ρ : Ω k → C a function on the set of places of k.Let S be a finite set of places of k.We say that ρ is S-frobenian if there exist (a) a finite Galois extension K/k, with Galois group Γ, such that S contains all places which ramify in K/k, and (b) a class function ϕ : Γ → C, such that for all v ∈ S we have where Frob v ∈ Γ denotes a Frobenius element of v.We say that ρ is frobenian if it is S-frobenian for some S. A subset of Ω k is called (S-)frobenian if its indicator function is (S-)frobenian.
In Definition 2.1, we adopt a common abuse of notation (see [41, §3.2.1]), and denote by Frob v ∈ Γ the choice of some element of the Frobenius conjugacy class at v; note that ϕ(Frob v ) is well defined as ϕ is a class function.
We define the mean of ρ to be ] be a (not necessarily irreducible) polynomial.Then the set {v ∈ Ω k : f (x) has a root in k v } is frobenian.Indeed, take K to be the splitting field of f .Then for a place v which is unramified in K, the polynomial f has a root in k v if and only if Frob v acts with a fixed point on the roots of f over k; the set of such elements is a conjugacy invariant subset of the Galois group Γ.
We require the following result on the zeta function of a frobenian function.Throughout the paper, we write q v for the size of the residue field at a finite place v.Moreover, for any place v, let ζ k,v (s) be the Euler factor of ζ k (s) at v if v is non-archimedean, and ζ k,v (s) = 1 otherwise.

Proposition 2.3. Let S be a finite set of places of k containing all archimedean places and let ρ be an S-frobenian function. Assume that |ρ(v)| < q v holds for all v /
∈ S. Then the Euler product has the form for some c = c ρ > 0, and satisfies in this region the bound and the limit in (2.5) is non-zero.
Proof.First, note that the Euler factors 1 + ρ(v)q −s v are holomorphic on C and non-zero for Re s ≥ 1, as |ρ(v)| < q v by assumption.Next, recall that the irreducible characters of a finite group Γ form a basis for the space of complex class functions of Γ [22,Prop. 2.30].In particular, if ϕ : Γ → C is the class function associated to ρ, then we may write where λ χ ∈ C and the sum runs over the irreducible characters of Γ.For Re s > 1, we find that where L(χ, s) denotes the Artin L-function of χ and G 1 (s) is a holomorphic function with absolutely convergent Euler product on Re s > 1/2, which is nonzero on Re s ≥ 1.
For the trivial character χ = ½, we have By the Brauer induction theorem [11, Thm.VIII.7, p. 225], we may decompose each remaining L(χ, s) as a product of Z-powers of Hecke L-functions of nontrivial Hecke characters of subfields of K. Hence, we assume from now on that each L(χ, s) is an entire Hecke L-function (for some possibly different number field).By [28,Thm. 5.35], L(χ, s) respects a zero-free region of the form (2.3), for some c < 1/4 that may depend on χ.Since there are only finitely many characters to consider, we can find a constant c that works for all of them.Decreasing c further, we obtain a bound valid in the region (2.3) (cf.[38, p.230]).Using this bound and the fact that |G 1 (s)| ≪ ρ 1 in Re s ≥ 3/4 due to the absolute convergence of its Euler product, it is simple to verify that G(s) satisfies (2.4).
To verify (2.5), we start with the following fact, which is well known at least in the classical case of Dirichlet L-functions: for non-trivial χ, the Euler product of L(χ, s) converges for s = 1 and takes the value L(χ, 1).To see this, observe that log L(χ, s) can be defined for Re s > 1 as a Dirichlet series, use the prime number theorem for L(χ, s) (see [28,Thm. 5.13]) and partial summation to verify that this Dirichlet series converges for s = 1, and apply Abel's theorem.
Since G 1 (s) has an absolutely convergent Euler product for Re s > 1/2, this shows that the Euler product of ζ k (s) −m(ρ) F (s) = G(s) does indeed converge at s = 1 and takes the value Recalling our assumption that |ρ(v)| < q v , it is clear that the right-hand side of (2.5) is non-zero.

Remark 2.4.
(1) Note that frobenian functions are bounded; thus the condition |ρ(v)| < q v in Proposition 2.3 is always satisfied for all but finitely many v.

Counting with local conditions
All of the main counting results in this paper are obtained from a more general counting result, which we present in this section.To state this result we require some notation.

Statement of the result.
Let G be a finite abelian group, let F be a field and F a separable closure of F .We define a sub-G-extension of F to be a continuous homomorphism Gal( F /F ) → G.A sub-G-extension corresponds to a pair (L/F, ψ), where L/F is a Galois extension inside F and ψ is an injective homomorphism Gal(L/F ) → G.
For each place v of the number field k, we fix an algebraic closure kv and compatible embeddings k ֒→ k ֒→ kv and k ֒→ k v ֒→ kv .
Hence, a sub-G-extension ϕ of k induces a sub-G-extension ϕ v of k v at every place v.For each place v of k, let Λ v be a set of sub-G-extensions of k v .For Λ := (Λ v ) v∈Ω k we are interested in the function which counts those G-extensions of k of bounded conductor which satisfy the local conditions imposed by Λ at all places v. (Here Φ is as in §1.2.) In general, it is difficult to say anything about the counting function given in (3.1), especially when there are infinitely many local conditions imposed.Even in the case when one imposes finitely many conditions, the set being counted may be empty, as explained in §1.1.Our main technical result imposes arbitrary conditions at finitely many places, but at the remaining places we only impose those conditions which force every element of A to be a local norm.

there exists a sub-G-extension of k which realises the given local conditions for all places v.
The leading constant c k,G,Λ in this theorem is given by a finite sum of Euler products (see Theorem 3.22 for an explicit expression).Our condition for positivity is only the existence of some sub-G-extension of k which realises the given local conditions; we do not require the existence of a genuine G-extension of k, so we do not need to assume that the set of G-extensions being counted is non-empty to deduce the positivity of the constant.This means that one need only look for an extension with possibly smaller Galois group to prove positivity of the constant; we use this trick to great effect when proving Theorem 1.1.
We illustrate how one applies Theorem 3.1 in some simple cases.Firstly, one counts the total number of G-extensions of k by applying Theorem 3.1 with A = {1} and no local conditions, i.e. taking Λ v to be the set of all sub-Gextensions of k v for all places v.These local conditions are realised by the sub-G-extension given by the trivial extension k/k.For a more interesting example, consider the case A = {1} and the trivial local conditions Λ v = {½} for v ∈ S, which are again realised by the trivial extension k/k.This gives the following corollary.(Note that we do not need to avoid the places above 2.)

Corollary 3.2. Let S be a finite set of places. Then a positive proportion of G-extensions of k, ordered by conductor, are completely split at all places in S.
The rest of this section is dedicated to the proof of Theorem 3.1.All implied constants in the O and ≪ notation are allowed to depend on k, G, A and Λ.

The set of places S.
To prove Theorem 3.1, we are free to increase the size of S if we wish.Henceforth, we will assume that S contains all archimedean places of k and all places of k lying above the primes p ≤ |G|, that A ⊂ O * S , and that O S has trivial class group.
The reader should note that many of the formulae which follow are only valid for finite sets of places S which satisfy these conditions.For example, in the case where k = Q, G = Z/8Z, A = {1}, S = ∅, the expression for the leading constant in Theorem 3.22 does not hold.To compute c k,G,Λ in this instance, we may take S = {∞, 2, 3, 5} instead.

Dirichlet series.
To prove Theorem 3.1 we study the associated Dirichlet series and G 2 have coprime order, and µ((Z/pZ) n ) = (−1) n p n(n−1)/2 for a prime p and n ∈ Z ≥0 .Let f be a function on the subgroups of G.For subgroups H ⊂ G, we consider the function where the sum runs over all subgroups J ⊂ H.The Möbius inversion formula for finite abelian groups [16] states that Proof.Sorting the sub-H-extensions ϕ : Gal( k/k) → H by their images, we get Call the right-hand side g(H) and apply Möbius inversion (3.3).
We now consider the contribution to F Λ (s) of each subgroup H in turn.The contribution from H = {1} is either 0 or 1.From now on we focus on the contributions of the non-trivial subgroups H.
Hence, in our analysis of F Λ (s) we can now focus on the inner sums Our counting problem fits very well within the class-field-theoretic framework.
For each place v ∈ Ω k , we use local class field theory (specifically, the local Artin ).Thus, we consider Λ v as a subset of Hom(k * v , H).By the compatibility of local and global class field theory, we still have Proof.Let ϕ v be the sub-G-extension of k v associated to χ v .By local class field theory we have Ker χ v = N Kϕ v /kv K * ϕv , where K ϕv is the extension field of k v associated to ϕ v .However, as v / ∈ S, by assumption in Theorem 3.1 we have f Λv (χ v ) = 1 if and only if every element of A is a local norm from K ϕv ; the result follows.

Harmonic analysis. To deal with the sums
we shall use a version of the Poisson summation formula from harmonic analysis.The theory relevant to us was worked out in detail in [20, §3] when counting by discriminant.The same theory transfers almost verbatim to show the validity of the Poisson summation formula for counting by conductor.
However, for the purposes of Theorem 3.1, our case is special enough that we merely require a simplified version of the Poisson summation formula that can be proved using only character orthogonality for finite abelian groups.We may therefore forego some of the general theory from [20, §3] and proceed in a more explicit manner.We first recall the set-up for the harmonic analysis.
∈ S, these local functions take only the value 1 on the unramified elements by our choice of S and Lemma 3.5, and thus f Λ /Φ s extends to a well-defined and continuous function on Hom(A * , H).We define its Fourier transform to be For Re s ≫ 1, the global Fourier transform exists and defines a holomorphic function in this domain, and there is an Euler product decomposition Proof.We prove the result when v is non-archimedean, the case of archimedean v being analogous.By our choice of measures, we have This finite sum clearly defines a holomorphic function on C. If Re s ≥ 0 then the sum is ≪ k,H 1, since every summand is bounded absolutely and the number of summands is ≪ k,H 1.For the last part, we have For v ∈ S the set Λ v is non-empty by assumption.For v / ∈ S the set Λ v is again non-empty, as it always contains the trivial homomorphism k * v → H by Lemma 3.5.For s ∈ R, we therefore obtain a finite non-empty sum of positive real numbers, which is positive.Now let v be non-archimedean.Choosing a uniformiser of with Z and gives a splitting of the exact sequence This implies that the sequence since ψ v is unramified and hence Φ( We use the criterion from Lemma 3.5.We have this also shows the second assertion.
In the statement of the following lemma, note that the natural map v ⊗ H ∧ is injective, as the sequence (3.7) is split exact.Therefore, we may naturally view Proof.From (3.8) and Lemma 3.7 we have

Now character orthogonality gives
Note that the group O * S ⊗ H ∧ is finite by Dirichlet's S-unit theorem; in particular the right-hand sum is finite.
Recall that we have normalised our Haar measures on Hom(k * v , H) to be |H| −1 times the counting measure for non-archimedean v, and equal to the counting measure for archimedean v.We let S f be the set of non-archimedean places in S. Now Lemma 3.8 and (3.5) give where We now change the order of summation in the right-hand sum of (3.10) to obtain As  H).Thus, we may apply character orthogonality to find that We therefore obtain Moreover Proof.An element χ v ∈ Hom(O * v , H) is unramified if and only if it is trivial.Furthermore, since v / ∈ S and our assumptions on S in §3.2, the ramification is tame and hence for non-trivial characters χ v ∈ Hom(O * v , H), we have Φ v (χ v ) = q v .Therefore, by Lemmas 3.5 and 3.8 we have We claim that the natural map is an isomorphism.To see this, recall that Hensel's lemma yields a split short exact sequence The kernel of a continuous homomorphism 1 + p v → H contains 1 + p n v for some n ∈ N, and the successive quotients in the filtration Inputting this into (3.11), the result follows.
To study the analytic behaviour of the global Fourier transforms f Λ,H (x; s), we use the theory of frobenian functions from §2. (2) Let m be the largest divisor of

Lemma 3.11. Let e be the exponent of H. Consider a function d
Note that the group A/A e is finite.Moreover, our slight abuse of notation is harmless, as whether or not the polynomial appearing in (3.13) has a root is independent of the choice of representative of each element of A/A e .
Proof.To prove this we choose a presentation of H ∧ .We then work coordinatewise on H ∧ , using the fact that the intersection of finitely many frobenian sets is frobenian.Thus, we reduce to the case H ∧ = Z/eZ.Here we have O * S ⊗ H ∧ = O * S /O * e S .For x ∈ O * S , we have to show that the set is S-frobenian.However, we have v for some α v ∈ A v (depending on v).We find that the set in question is the set of places v such that the equation α∈A/A e (t e − xα) = 0 has a solution in k v ; this set is frobenian (see Example 2.2).As x is an S-unit, it is easily seen that this is S-frobenian for our choice of S in §3.2.
Proof.The product or sum of two S-frobenian functions is clearly S-frobenian (in Definition 2.1 one takes the compositum of the relevant field extensions).Moreover, the complement of a S-frobenian set is S-frobenian.The result therefore follows from Lemmas 3.11 and 3.12.
Definition 3.14.We denote by ̟(k, H, A, x) the mean of the S-frobenian function described in Corollary 3.13.
We now compare ̟(k, H, A, x) with ̟(k, H, A), as defined in Definition 1.3.̟(k, H, A, x) ≤ ̟(k, H, A, 1), the first assertion follows immediately from the second.So let us prove the second assertion.By Corollary 3.13 and Lemma 3.11, we see that ̟(k, H, A, 1) is the mean of an S-frobenian function ρ with ρ

Proof. As clearly
With the notation of the proof of Lemma 3.11, the corresponding class function on Gal(k e /k) is given by σ where G(x; s) is holomorphic in the region (2.3), for some c > 0, and satisfies (2.4).Moreover, we have In the case x = 1, this limit is non-zero.
Proof.We consider the Euler product expansion of f Λ,H (x; s) from (3.5), where Euler factors at v / ∈ S were determined in Lemma 3.10.By Corollary 3.13 and our assumptions on S, we may apply Proposition 2.3 to obtain with a function H(x; s) that is holomorphic in a region (2.3) and satisfies the bound (2.4).By Lemma 3.6, we may multiply H(x; s) by the Euler factors f Λv,H (x v ; s) for v ∈ S while still preserving these properties (possibly for a smaller c > 0 in (2.3)).Finally, the explicit form of the limit follows from (2.5) which, together with Lemma 3.6, also shows that the limit is non-zero if x = 1.
3.6.The asymptotic formula in Theorem 3.1.We now bring all our tools together to prove the first part of Theorem 3.1.Recall from Lemma 3.4 that we performed a Möbius inversion to obtain a sum over the subgroups H of G.Moreover, in Proposition 3.9 we used Poisson summation to understand the inner sums from Lemma 3.4.In summary, where  ̟(k, H, A) < ̟(k, G, A).
Proof.Follows immediately from Definition 1.3.
We are now finally in the position to prove the required asymptotic formula.
This proves the asymptotic formula in Theorem 3.1.Next, we study the leading constant.

3.7.
Formula for the leading constant.To calculate the leading constant, we first need to understand exactly which elements of O * S ⊗ H ∧ give rise to the leading singularity in the Poisson sum (Proposition 3.9).
for all but finitely many v}.
Then X (k, G, A) is finite and Proof.It is enough to prove the result for G ∧ a cyclic group of prime power order.Henceforth, let G ∧ = Z/qZ, where q = p r is a prime power.We view X (k, G, A) as a subgroup of k * /k * q .First, we claim that the image of X (k, G, A) in k(µ q ) * /k(µ q ) * q is equal to the image of A. One containment is clear, as A ⊗ G ∧ ⊂ X (k, G, A).For the other, let K = k(µ q , q √ A), so K = k q in the notation of Definition 1.3.Let K v be the completion of k at a choice of place of K above v.The image of X (k, G, A) in k(µ q ) * /k(µ q ) * q is contained in the following set: {x ∈ k(µ q ) * /k(µ q ) * q : x v ∈ K * q v for all but finitely many v} .As µ q ⊂ K, an application of the Chebotarev density theorem shows that this set equals (k(µ q ) * ∩ K * q )/k(µ q ) * q (this also follows from Lemma 4.9).On the other hand, Kummer theory shows that (k(µ q ) * ∩ K * q )/k(µ q ) * q is equal to the image of A in k(µ q ) * /k(µ q ) * q , and the claim is proved.In particular, the image X (k, G, A) in k(µ q ) * /k(µ q ) * q is finite, as A is finitely generated.
We now show that X (k, G, A) ⊂ O * S ⊗ G ∧ ; the rest follows from the fact that our condition is S-frobenian (see Lemma 3.12).Let x ∈ k * be such that its image in k * /k * q is in X (k, G, A).By the argument above, the image of x in k(µ q ) * /k(µ q ) * q is in (k(µ q ) * ∩ K * q )/k(µ q ) * q .In particular, x = y q for some y ∈ K * .By our assumptions in §3.2 that A ⊂ O * S and that S includes all primes dividing |G|, the extension K/k is unramified at all v / ∈ S. Therefore, for all v / ∈ S, the valuation ord v (x) = ord v (y q ) is divisible by q.Consequently, the fractional ideal xO S is the qth power of some fractional ideal I of O S .By our assumption in §3.2 that O S has trivial class group, I = zO S for some z ∈ k * .Therefore, x = uz q for some u ∈ O * S .This completes the proof.Lemma 3.21.
| − 1 as this group always contains the trivial homomorphism.The result now follows from the fact that the function in Corollary 3.13 is S-frobenian.
These lemmas show that the leading singularity comes from finitely many terms which are independent of S and our choice of local conditions for v ∈ S.This makes applications much easier when one is varying S (we require such applications for the proof of Theorem 1.9).Theorem 3.22.Retain the assumptions of Theorem 3.1 and the additional assumptions on the finite set of places S from §3.2.Let X (k, G, A) be as in Lemma 3.20 and let S f be the set of non-archimedean places in S.
, where the product over v / ∈ S is non-zero.
Proof.From Proposition 3.16, Proposition 3.19, and Lemma 3.21, we get the leading constant We have f Λv,G (x v ; 1) = f Λv,G (1; 1) for x ∈ X (k, G, A) and v / ∈ S by Lemma 3.21, and these factors are non-zero by Lemma 3.6.The explicit expressions for v / ∈ S follow from Lemma 3.8.For v ∈ S, we simply apply directly the definition of the local Fourier transforms from §3.4.1 (see (3.6) for a formula in the non-archimedean case) and change the order of summation.
Note that the expression for c k,G,Λ is independent of S, for any S which satisfies the assumptions of §3.2.Remark 3.23.In the special case A = {1}, our constant agrees with the constant which Wood obtains in [49,Thm. 3.1], up to the factor (Res s=1 ζ k (s)) ̟(k,G,A) .This factor is missing from Wood's paper: in the proof of [49,Thm. 3.1], she mistakenly uses the equality lim s→1 (s − 1)ζ K (s) = 1, which holds for K = Q but does not hold in general (the residue is given by the analytic class number formula).Thus the right-hand side of [49,Thm. 3.1] should contain an additional factor of (Res ).An examination of the proof of Lemma 3.20 gives the bounds The following examples show that either bound can be sharp.
An example where both bounds coincide is given by taking A = {1} and G ∧ = Z/2Z.One easily sees that in this case X (k, G, {1}) is trivial.

Positivity of the leading constant.
To finish the proof of Theorem 3.1, we need to show that c k,G,Λ > 0 if there exists some sub-G-extension which realises all the given local conditions.It suffices to consider the contributions from v ∈ S to the explicit expression given in Theorem 3.22, as the factors at v / ∈ S are clearly non-zero.By character orthogonality we have In particular, this sum is non-negative for all χ ∈ Hom( v∈S k * v , G).Hence, it suffices to show the existence of some χ such that this sum is non-zero.However, we have assumed the existence of a sub-G-extension ϕ which realises all the local conditions.Let ψ : A * /k * → G be the associated homomorphism coming from class field theory.Note It therefore suffices to show that However, for x ∈ X (k, G, A) we have x v ∈ A v ⊗G ∧ for all v / ∈ S, by Lemma 3.20.Moreover, by assumption every element of A is a local norm from K ϕ for all v / ∈ S, thus A v ⊂ Ker ψ v for all v / ∈ S by Lemma 3.5.The claim (3.16) follows, which completes the proof of Theorem 3.1.

Proof of results
We now apply Theorem 3.1 in various ways to prove the results from the introduction.
4.1.Asymptotic formula for everywhere local norms.We first derive an asymptotic formula for N loc (k, G, A, B) (see (1.1)) using Theorem 3.1.Theorem 4.1.We have Proof.For all v ∈ Ω k , let Λ v be the set of sub-G-extensions of k v corresponding to those extensions L/k v for which every element of A is a local norm from L/k v .Thus, in this setting Λ = (Λ v ) v∈Ω k is determined by A. We clearly have k, G, Λ, B).It therefore suffices to show that the leading constant in Theorem 3.1 is positive.To do so, we need to exhibit some sub-G-extension of k for which every element of A is everywhere locally a norm.However, the trivial extension k/k is such an extension.4.2.Proof of Theorem 1.9.As cyclic extensions always satisfy the Hasse norm principle, we may assume that G is non-cyclic.We use the following criterion for failure of the Hasse norm principle in the abelian setting, which was originally pointed out to us by Melanie Matchett Wood.(We use the notation from §3.1.)Proposition 4.2.Let ϕ be a G-extension of k.Then ϕ fails the Hasse norm principle if and only if there exists a proper subgroup Υ ⊂ ∧ 2 (G) that contains the image of the natural map Proof.Let K be the number field determined by ϕ.Recall that the failure of the Hasse norm principle is measured by the Tate-Shafarevich group where R 1 K/k G m denotes the associated norm 1 torus, see [34, §6.3].This group is finite by [34, Prop.6.9].As K/k is Galois, a theorem of Tate [34,Thm. 6.11] (see also [36,Ex. 5.6]) implies that there is an exact sequence However, as G is abelian, we have a well-known canonical isomorphism (see e.g.[20,Lem. 6.4]).Using this and applying Hom(•, Q/Z), we therefore obtain the exact sequence Thus, failure of the Hasse norm principle is equivalent to the first map in (4.1) failing to be surjective.
Therefore, to prove Theorem 1.9, it suffices to show the following.
Theorem 4.3.Let Υ ⊂ ∧ 2 (G) be a proper subgroup.Then Note that in Theorem 4.3, and henceforth, we abuse notation by writing ∧ 2 (Im ϕ v ) ⊂ Υ to mean that the image of the natural map ∧ 2 (Im ϕ v ) → ∧ 2 (G) is contained in Υ, despite the fact that this map is not injective in general.
We prove Theorem 4.3 via an application of Theorem 3.1.Note, however, that one cannot apply Theorem 3.1 directly, as the local conditions imposed at the infinitely many places will not be with the assumptions of Theorem 3.1.We therefore apply Theorem 3.1 to a suitable finite set of places, which we then allow to increase.4.2.1.Proof of Theorem 4.3.Let S 0 be a finite set of places of k satisfying the conditions of §3.2, which we consider as being fixed.Let T be a finite set of places of k which is disjoint from S 0 .Eventually, we will consider what happens as T increases.Let S = S 0 ∪ T .
We consider the local conditions Λ v given by We denote the collection of such conditions by Λ T .Note that we clearly have for all B. Applying Theorem 3.1 gives where c k,G,A,loc > 0 by Theorem 4.1.To prove Theorem 4.3 it therefore suffices to show that lim where as explained we consider S 0 as fixed and T as increasing and disjoint from S 0 .We do this using the explicit expression for the leading constant given in Theorem 3.22.We let e be the exponent of G.We require the following elementary observation.Proof.Let K be an extension of k with Galois group isomorphic to a subgroup of G and v a place of k such that α ∈ k * e v .Let K v be the completion of k at a choice of place of K above v.Then local class field theory yields Now G has exponent e, whereby the group k * v / N Kv/kv K * v has exponent dividing e.It follows that an eth power in k * v is a local norm.
We now obtain the following bounds.
Proof.The factors in Theorem 3.22 cancel out in the quotient c k,G,Λ T /c k,G,A,loc , except those at places v ∈ S. By Lemma 3.20, we have (this statement holds for any set of places satisfying the assumptions of §3.2).Moreover, for v ∈ T any element of A v is a local norm at v by our choice of Λ v ; it follows that χ v , x v = 1 for χ v ∈ Λ v as in Theorem 3.22, hence Therefore, we can split off Euler factors for all v ∈ T from the term involving S, while the remaining sum over Hom( v∈S 0 k * v , G) is the same in c k,G,Λ T and c k,G,A,loc .We have obtained the equality .
The quotient of each local factor is at most 1, so to obtain an upper bound we may just consider those places v ∈ T which are completely split in k e /k.For such places every element of A is an eth power in k * v , hence the condition that they are local norms is automatic by Lemma 4.4.The result follows.
We will make use of the following fact from [20,Lem. 6.9].Here, we use the term bicyclic for a non-cyclic group that is a direct sum of two cyclic groups.Lemma 4.6.Let G be a finite abelian non-cyclic group.Then there exists a finite collection of bicyclic subgroups As Υ ⊂ ∧ 2 (G) is a proper subgroup, there exists some i such that ∧ 2 (G i ) ⊂ Υ. Fix this i and write G i ∼ = Z/nZ × Z/mZ where n, m | e.Let v ∈ T be a place of k which is completely split in the extension k(µ e , e √ A).There exists a G i -extension of k v : simply adjoin an nth root of a uniformiser to the unique unramified extension of k v of degree m.Thus, by local class field theory, there Recall that tamely ramified χ v have conductor q v and there are |G| unramified G-characters.Using Lemma 4.5, it follows that However this diverges to 0 as diverges by the Chebotarev density theorem.This proves (4.2) and completes the proof of Theorem 4.3, hence the proof of Theorem 1.9.
Unfortunately the second part of the statement [20, Lem.6.9] is false (this claims that if the exponent of ∧ 2 (G) divides a prime p, then all the G i may chosen isomorphic to (Z/pZ) 2 ).A counterexample is given by the group G = Z/2Z × Z/4Z and the subgroup G 1 = Z/2Z × Z/2Z; here the induced map is trivial.This mistake in [20,Lem. 6.9] has various consequences for [20] which will be addressed in a forthcoming corrigendum.
v and α is a unit, its image in the residue field lies in F * d v .However, as d = gcd(e, q v − 1), we have The result therefore follows from Hensel's lemma.
Hence, the remaining implication (2)⇒(3) in Theorem 1.6 follows immediately from (4.3) and Lemma 4.8.4.6.Proof of Corollary 1.7.Let α ∈ A and consider αβ e for some β ∈ k * .By Lemma 4.4, we see that β e is a norm everywhere locally from all G-extensions of k.It follows that αβ e is a norm everywhere locally from a given G-extension if and only if α is a norm everywhere locally.Part (i) now follows from Theorem 1.9 and Theorem 4.1.Part (ii) also follows from Lemma 4.4 and Theorem 1.9.
For (iii) and (iv), we use the ω-version of Tate-Shafarevich groups.Namely, for a finite abelian group scheme M over k we let ) for all but finitely many v}.
By Kummer theory we have H 1 (L, µ e ) = L * /L * e for any field L of characteristic 0. Therefore Part (iii) of Theorem 1.6 is equivalent to The key observation is now the following.Lemma 4.9.Let k be a number field, let e ∈ Z ≥1 and let 2 r be the largest power of 2 dividing e.Then X ω (k, µ e ) = 0, unless the extension k(µ 2 r )/k is non-cyclic, where we have X ω (k, µ e ) ∼ = Z/2Z.As Br(V ⊗ K L) = 0 and Br(L) ։ Br(V ⊗ K L), the Hochschild-Serre spectral sequence provides an embedding of Coker(Br(K) → Br(V )) into the kernel of the restriction map H On the other hand, the exact sequence (see [44, p. 130]) embeds this group into H 2 (Gal(L/K), T ).
As f is smooth and surjective, we have f * Br(k(B)) ∩ Br(X ′ ) ⊆ f * Br(B) as subgroups of Br(k(X)).It follows that the pull-back map is injective as well.Thanks to this injectivity, we now see that in order to prove Proposition A.6, we may assume that k is algebraically closed.
The generic fibre of the natural map X → (T /G when k is algebraically closed, as we have seen in the proof of Proposition A.5.In addition, the unramified Brauer group of (T 1 × • • • × T 1 )/G vanishes when k is algebraically closed, by Saltman's formula [15,Thm. 8.7] and by the next lemma.Hence Br(X ′ ) = 0 in this case.
) vanish, this follows from the injectivity of the product of restriction maps It is a general fact, valid for an arbitrary finite group G, that the kernel of (A.This completes the proof of Proposition A.6. A.5. Nonabelian Galois groups.The descent-fibration argument described in §A.2 and §A.3 is modelled after a similar argument appearing in [24], profiting in addition from the favourable circumstance of G being abelian.In general, the inductive argument of [24] is constructed to handle also nonabelian groups, as long as they admit a suitable filtration into normal subgroups whose successive quotients are cyclic; such groups are also known as supersolvable.Though the variety X considered here is more complicated than the one considered in [24], the argument of loc.cit.can be adapted to yield the statement of Theorem 1.1 for any supersolvable G, see [25].Interestingly enough, though, it turns out that the stronger claim appearing in Corollary 4.12 does not hold for a general nonabelian group G, even when G is supersolvable (indeed, even when G is a 2-group).This is due to the fact that the variety X may contain unramified Brauer classes which are not vertical with respect to the projection f : X → B, and which can obstruct the weak approximation of local points on X, even when those local points lie over a rational point of B. (Such Brauer classes do not exist in the abelian case; see Proposition A.6.) Let us now illustrate how one can construct a nonabelian example where exactly this happens.
We shall say that a group H is weakly bicyclic if it is an extension of a cyclic group by a cyclic group.We note that if K/k is a Galois extension with Galois group G then the decomposition subgroups H v ⊆ G are weakly bicyclic at every finite place v which does not divide the order of G. Given a group G, we shall denote by B G the set of weakly bicyclic subgroups of G (a notation compatible with Lemma A.8 when G is abelian).Proposition A.9. Let G be a finite 2-group satisfying the following properties: (i) G has exponent ≤ 16.
(ii) The abelianization G ab has exponent 2 and is generated by images of elements of G of order 2. (iii) There exists an element ϕ ∈ H 2 (G, Z/2Z) whose restriction to every cyclic subgroup of G of order 16 vanishes, whose restriction to at least one cyclic subgroup of G of order 8 does not vanish, and whose image by the natural map δ : H 2 (G, Z/2Z) → H 2 (G, Q/Z) belongs to, and spans, the kernel of the product of restriction maps H 2 (G, Q/Z) → H∈B G H 2 (H, Q/Z).Let H ⊆ G be a cyclic subgroup of order 8 on which ϕ does not vanish.Then: (1) There exist G-extensions K/Q which are unramified at 2 and whose decomposition groups at 2 are conjugate to H. (2) For every G-extension K/Q as in (1), the element 256 ∈ Q * is a local norm from K at every place of Q, but not a global norm from K. In particular, the statement of Corollary 4.12 does not hold for G with k = Q, S = {2} and A ⊂ k * the subgroup generated by 256.
The proof of Proposition A.9 requires a bit of preparation.In the next lemma, we denote by Br nr (B), Br 1 (B), Br 1,nr (B), Br 0 (B) the subgroups of Br(B) consisting, respectively, of unramified, algebraic, algebraic unramified, constant classes.Condition (iii) implies that Br nr (B) = Br 1,nr (B).For every subgroup H ⊆ G, the Hochschild-Serre spectral sequences for the H-coverings π H : SL n → SL n /H and SL n, Q → SL n, Q/H , together with the inclusion of roots of unity µ ∞ ⊆ Q * , give rise to a commutative diagram Ker Br(SL n /H) → Br(SL n ) where Γ Q = Gal( Q/Q) is the absolute Galois group of Q.The horizontal arrows between the first two columns are isomorphisms since Pic(SL n ) = Pic(SL n, Q) = 0, and the bottom right horizontal map is an isomorphism since Q * /µ ∞ is uniquely divisible.In addition, the rightmost vertical map is surjective: indeed, this map fits in the middle of the commutative diagram with exact rows 0 / / Ext determined by the universal coefficient theorem, where Ext 1 (H 1 (H), µ ∞ ) = 0 since µ ∞ is a divisible group.We now fix a β ∈ Br nr (B) and aim to show that β is algebraic.By adding to β a constant class, we may assume that β(π G (1)) = 0.As SL n is rational over Q, we have Br nr (SL n ) = Br 0 (SL n ), and so π * G β = 0. Considering the diagram (A.7) for G = H and using the surjectivity of its right vertical map, we find β G ∈ H 2 (G, µ 2 ) whose eventual image in Br(BQ) is the same as the image of β.Now by Bogomolov's formula (see, e.g., [15,Thm. 7.1]), the group Br nr (SL n, Q/H ) vanishes whenever H is weakly bicyclic, and so by the naturality of (A.7), the image of β G in H 2 (H, µ ∞ ) vanishes for every H ∈ B G .Since µ ∞ ∼ = Q/Z as abelian groups via a choice of a compatible system of roots of unity, Condition (iii) implies that the image of β G in H 2 (G, µ ∞ ) is either 0 or the image of ϕ ∈ H 2 (G, Z/2) = H 2 (G, µ 2 ) under the natural map H 2 (G, µ 2 ) → H 2 (G, µ ∞ ).By possibly amending the choice of β G , we may assume that β G ∈ {0, ϕ}.We then write β 1 ∈ Br(B) for the image of β G , and set β 2 := β −β 1 .By construction, β 1 (and hence also β 2 ) vanishes when pulled back to SL n , and β 2 also vanishes when pulled back to BQ.In particular, β 2 ∈ Ker(Br 1 (B) → Br 1 (SL n )).
Let now H ⊆ G be a cyclic subgroup of order 8 on which ϕ does not vanish.As β is unramified, there exists a prime p 0 such that β evaluates trivially on B(Q p ) for all p > p 0 .Choose p > p 0 such that there exists a cyclic extension L/Q p of degree 4 that does not extend to a cyclic extension of degree 8 (any p such that −1 is a square but not a 4th power modulo p will do).Embed Gal(L/Q p ) into H.The image of the class of L/Q p by the resulting map H 1 (Q p , Gal(L/Q p )) → H 1 (Q p , G) is the class of the torsor π −1 G (b) for some point b ∈ B(Q p ) (see [23, §1.2]), which we fix.
For any place v of Q, let K v denote the completion of K at a place of K dividing v.The corresponding decomposition group D v ⊆ G is weakly bicyclic since G is a 2-group and K 2 /Q 2 is unramified.Letting Q v ⊆ K ϕ Dv ⊆ K v denote the intermediate cyclic extension determined by ϕ Dv ∈ H 1 (D v , Q/Z), a direct computation now reveals that x * v P = (16, K ϕ Dv /Q v ) ∈ Br(Q v ).Since ϕ is assumed to vanish on every cyclic subgroup of order 16, the class ϕ Dv becomes divisible by 2 when restricted to every such subgroup.Since the exponent of G divides 16, it follows that 8 ϕ Dv ∈ H 1 (D v , Q/Z) = Hom(D v , Q/Z) vanishes when restricted to any cyclic subgroup of D v , and hence vanishes; in other words, the degree of the extension K ϕ Dv /Q v divides 8. On the other hand, as D 2 is cyclic of order 8 and ϕ does not vanish when restricted to D 2 we have that ϕ 2 ∈ H 1 (D 2 , Q/Z) ∼ = Z/8Z is not divisible by 2 and so K ϕ D 2 = K 2 .We conclude that inv v (x * v P) = 0 for all v = 2 (recall that 16 is an 8th power at such v) while inv 2 (x * 2 P) = 1/2 ∈ Q/Z.We shall now construct a 2-group G satisfying the conditions of Proposition A.9. Let N be the group generated by 4 generators x, y, z + , z − under the following relations: (1) x 16 = y 16 = z 8 + = z 8 − = 1; (2) each of z + , z − commutes with each of x, y, z + , z − ; (3) [x, y] = z + z − .In particular, N is a central extension of the bicyclic group Z/16Z x, y by the bicyclic group Z/8Z z + , z − .Let σ : N → N be the involution given by σ(x) = x −1 , σ(y) = y −1 , σ(z + ) = z − and σ(z − ) = z + .We define G := N ⋊ Z/2Z σ to be the associated semi-direct product and view σ as an element of G.
It is straightforward that G satisfies Conditions (i) and (ii) of Proposition A.9. Let us now construct an element ϕ ∈ H 2 (G, Z/2Z) satisfying Condition (iii).The homomorphism ρ : N → Z/8Z which sends x, y to 0, z + to 1 and z − to −1 intertwines the action of σ with the action of −1 : Z/8Z → Z/8Z.Consequently, it induces a homomorphism ρ ′ : G = N ⋊ Z/2Z → Z/8Z ⋊ Z/2Z =: D 8 to the dihedral group of order 16.Consider the short exact sequence where D 16 := Z/16Z ⋊ Z/2Z is the dihedral group of order 32 and the map q is induced by the surjective map Z/16Z → Z/8Z.Let ϕ D 8 ∈ H 2 (D 8 , Z/2Z) be the element classifying the central extension (A.8) and let ϕ := (ρ ′ ) * ϕ D 8 ∈ H 2 (G, Z/2Z).We leave it to the reader to verify that ϕ has the desired properties.

Theorem 3 . 1 .
Let k be a number field, G a non-trivial finite abelian group, and A ⊂ k * a finitely generated subgroup.Let S be a finite set of places of k and for is a union of conjugacy classes, since each Gal(k e /k d ) is normal in Gal(k e /k).The sets Σ d for d | e form a partition of Gal(k e /k).Let ϕ : Gal(k e /k) → C be the class function that takes the constant value d on Σ d , for all d | e.We claim that d A,H (v) = ϕ(Frob v ) for all v / ∈ S, so in particular it is S-frobenian.Note that Frob v ∈ Σ d if and only if d is the largest divisor of e such that v splits completely in k d /k.Equivalently, d is the largest divisor of e such that d | q v − 1 and x d − α has a root in k v for all α ∈ A. By Hensel's lemma, this is equivalent to d = d A,H (v), and thus ϕ(Frob v ) = d A,H (v), as desired.

Lemma 4 . 4 .
Let α ∈ k * .If v is such that α ∈ k * e v ,then α is a local norm at v from every sub-G-extension of k.

Lemma 4 . 8 .
Let e ∈ Z ≥1 , let α ∈ k * , and let v be a place of k such that e, α ∈ O * v .Let d = gcd(e,

Lemma A. 8 .
Let B G denote the set of subgroups of G generated by two elements.For any finite abelian group G and for M a field which contains d distinct dth roots of unity and A ⊂ F * is a finitely generated subgroup, then we denote by F ( d √ A) the splitting field of the polynomials x d − α, where α runs over a set of generators of A.For a subgroup A ⊂ k * and a place v of k, we denote by A v the image of A in this Dirichlet series defines a holomorphic function on Re s > 1. (This follows from [49, Lem.2.10], but also from the analysis later in this paper.) 3.3.1.Möbius inversion.Recall that a G-extension of k is a surjective continuous homomorphism ϕ : Gal( k/k) → G.The condition that ϕ be surjective is difficult to deal with, hence we perform a Möbius inversion to remove it.Let µ be the Möbius function on isomorphism classes of finite abelian groups.That is, µ(G) = 0 if G has a cyclic subgroup of order p n with p a prime and n 3.3.2.Class field theory.Via global class field theory, we make the identification Hom(Gal( k/k), H) = Hom(A * /k * , H).(3.4)The canonical isomorphism (3.4) is induced by the global Artin map A , H), let Φ(χ) be the reciprocal of the idelic norm of the conductor of the kernel of χ, which is precisely the norm of the conductor of the sub-H-extension corresponding to χ. Together with Lemma 3.3, this discussion shows the following: * /k * → Gal(k ab /k).Using this isomorphism, we consider f Λ now as a function on Hom(A * /k * , H).For every χ ∈ Hom(A * /k * 3.4.1.Fourier transforms.The group Hom(A * /k * , H) is locally compact.Its Pontryagin dual is naturally identified with A * /k * ⊗ H ∧ (see [20, §3.1]).We denote the associated pairing by •, • : Hom(A * /k * , H) × (A * /k * ⊗ H ∧ ) → S 1 .Similarly, the Pontryagin dual of Hom(k * v , H) is naturally identified with k * v ⊗H ∧ , and we also denote the relevant Pontryagin pairing by •, • .For each place v, we equip the finite group Hom(k * v , H) with the unique Haar measure dχ v such that vol(Hom(k * v /O * v , H)) = 1.If v is non-archimedean, this is |H| −1 times the counting measure; for archimedean v, recalling our convention that O v = k v , we obtain the counting measure.The product of these measures yields a well-defined measure dχ on Hom(A * , H).We say that an element of Hom(k * v , H) is unramified if it lies in the subgroup Hom(k * v /O * v , H), i.e. if it is trivial on O * v , and that it is tamely ramified if it is ramified and trivial on 1 Poisson summation.We now prove the version of Poisson summation that we will require.In the statement, we view O * S ⊗ H ∧ as a subgroup of k * ⊗ H ∧ as follows: we have the exact sequence 0 → O * S → k * → P (O S ) → 0 (3.9)where P (O S ) denotes the group of non-zero principal fractional ideals of O S .Since P (O S ) is a free abelian group, we have Tor(P (O S ), H ∧ ) = 0. Therefore applying (•) ⊗ H ∧ to (3.9) we find that the map O * S ⊗ H ∧ → k * ⊗ H ∧ is injective, as required.For Re s > 1 the Fourier transform f Λ,H (•; s) exists and defines a holomorphic function on this domain.Moreover, we have the Poisson formula naturally identified with the Pontryagin dual of Hom(k * v /O * v , H).The result now follows on noting that k * v /O * v ∼ = Z and hence | Hom(k * v /O * v , H)| = |H|.3.4.3.χ∈Hom(A * /k * ,H) A * S and A * S /O * S are locally compact groups and their subgroups of nth powers are closed, an application of [20, Lem.3.2] gives canonical isomorphisms of abelian groups Hom(A * S , H) ∼ = (A * S ⊗ H ∧ ) ∧ and Hom(A * * S ⊗ H ∧ ) ∧ .Therefore, we can view an element χ ∈ Hom(A * S , H) as a character of A * S ⊗ H ∧ .It is easily seen that χ induces the trivial character on O * S ⊗ H ∧ if and only if χ ∈ Hom(A * S /O * S , , as O S has trivial class group, the natural map A * S /O * S → A * /k * is an isomorphism [49, Lem.2.8].The result now easily follows.3.5.Analytic continuation of the Fourier transforms.We now use the Poisson formula to study the analytic behaviour of the Dirichlet series under consideration.To do so, we shall calculate explicitly the local Fourier transforms for v / ∈ S. Fix some subgroup H of G.By a slight abuse of notation, for Therefore, any continuous homomorphism 1 + p v → H is trivial.It follows that Hom(F * v , H) = Hom(O * v , H).Moreover, A mod v lies in the kernel of a homomorphism F * v → H if and only if A v lies in the kernel of the induced homomorphism O * [39,39, Prop.IV.2.6]).Consequently, the quotient (1 + p v )/(1 + p n v ) has order a power of q v .Now recall that we assumed in §3.2 that gcd(q v , H) = 1.
Let H ⊂ G be a subgroup, let x ∈ O * S ⊗ H ∧ , and let a n (H, x) be the Dirichlet coefficient from(3.15).Then n (H, x) ∈ C. Lemma 3.17.
[32, by Theorem 1.6 there exists a cofinite set of places T ⊂ Ω k such that A ⊂ k * e Then at all places v = p, the cyclic algebra (χ, α) over k has local invariant zero, because α is a local norm at v byLemma 4.4.Now the Albert-Brauer-Hasse-Noether Theorem[32, Thm.8.1.17]showsthat(χ, α) has local invariant zero at p, meaning that α is also a local norm at p. Therefore, all elements of A are everywhere local norms from all G-extensions of k.But G is cyclic, hence every G-extension satisfies the Hasse norm principle; (2) now follows.Let T be a torus over a field K, with character group T , split by a finite Galois extension L/K.For any smooth and proper variety V over K containing a torsor under T as a dense open subset, there is a canonical embedding Coker(Br(K) → Br(V )) ֒→ H 2 (Gal(L/K), T ).Proof.Let L denote a separable closure of L and V 0 the open subset in question.
[32,r all v ∈ T .By[32, Thm.9.1.11],Kerk*/k* e → v∈T k * v /k * e v = Ker k * /k * e → vso we may assume that T contains all v ∤ 2. Let p be the unique prime of k lying above 2. Let χ : Gal( k/k) → G be a G-extension and let α ∈ A.4.8.Variants of Theorems 1.1 and 1.4.We finish with some variants of our results, which allow one to impose local conditions at finitely many places.Our first result is a variant of Theorem 1.4, and follows immediately from Theorem 3.1 and Theorem 1.9.Lemma A.7.