Norm tori of etale algebras and unramified Brauer groups

Let $k$ be a field, and let $L$ be an \'etale k-algebra of finite rank. If $a$ is a nonzero element in $k$, let $X_a$ be the affine variety defined by the norm equation $N_{L/k}(x) = a$. Assuming that $L$ has at least one factor that is a cyclic field extension of $k$, we give a combinatorial description of the unramified Brauer group of $X_a$.


Introduction
Let k be a field, let L be an étale k-algebra of finite rank, and let N L/k : L → k be the norm map.Let a ∈ k × , and let X a be the affine k-variety determined by N L/k (t) = a.
Let X c a be a smooth compactification of X a .The aim of this paper is to describe the group Br(X c a )/Im(Br(k)) under the hypothesis that L has at least one cyclic factor.We first give a combinatorial description of a group associated to the étale algebra L (see §3), and then give an explicit isomorphism between this group and Br(X c a )/Im(Br(k)) (see §4, in particular Theorem 4.3).Let us illustrate our results by a special case.Let p be a prime number and let n 1 be an integer; assume that char(k) = p and let F be a Galois extension of k with Galois group Z/p n Z × Z/p n Z. Suppose that L is a product of r linearly disjoint cyclic subfields of F of degree p n .Then we have (see Theorem 4.9) : Theorem.
Br(X c a )/Im(Br(k)) ≃ (Z/p n Z) r−2 .We also give explicit generators of this group, as follows.With the above notation, let K be one of the cyclic subfields of degree p n of F , and let χ be an injective morphism from Gal(K/k) to Q/Z.Let us write L = K × K ′ , with K ′ = i∈I K i , where K i is a cyclic subfield of F of degree p n of F for all i ∈ I, and assume that K and the fields K i are linearly disjoint in F .For all i ∈ I, set N i = N K i /k (y i ), considered as elements of k(X a ) × .Assume that the cardinal of I is r −1, so that L is a product of the r linearly independent cyclic subfields K and K i of F of degree p n .Let I ′ be a subset of cardinal r − 2 of I.
Let (N i , χ) denote the class of the cyclic algebra over k(X a ) associated to χ and the element N i ∈ k(X a ) × .
1 Theorem.The group Br(X c a )/Im(Br(k)) is generated by the elements (N i , χ) for i ∈ I ′ .This is also proved in Theorem 4.9.Note that the above results are generalizations of [BP 20], Theorems 11.1 and 11.2.
The paper is organized as follows.Throughout the paper, K is a finite cyclic extension of k, and L = K × K ′ , where K ′ is an étale k-algebra of finite rank.Sections 1 and 2 are preliminary : in particular, it is shown in §2 that we may assume K be cyclic of prime power degree.Sections 3 and 4 contain the description of the unramified Brauer group.When k is a global field, we obtain additional results concerning the "locally trivial" Brauer group (cf.§5).Finally, in §6 we apply Theorem 4.3 to give an alternative proof of [BLP 19] Theorem 7.1 for k a global field with char(k) = p; we show that the Brauer-Manin map of [BLP 19] is the Brauer-Manin pairing, and hence deduce the Hasse principle from results of [Sa 81] and [DH 17].

Generalities
Let k be a field, let k s be a separable closure of k and let G k = Gal(k s /k) be the absolute Galois group of k.We fix once and for all this separable closure k s , and all separable extensions of k that will appear in the paper will be contained in k s .We use standard notation in Galois cohomology; in particular, if M is a discrete G k -module and i is an integer ≥ 0, we set A G k -lattice will be a torsion free Z-module of finite rank on which G k acts continuously.For a k-torus T , we denote by T = Hom(T, G m ) its character group; it is a G k -lattice.
Let G be a finite group.A G-lattice is by definition a Z-torsion free Z[G]module of finite rank.If g ∈ G, we denote by g the cyclic subgroup of G generated by g.Let M be a G-lattice.Set We recall a result of Colliot-Thélène and Sansuc (cf.[CTS 87] Prop.9.5) Theorem 1.1.Let G be a finite group, let T be a k-torus, and assume that the character group of T is a G-lattice via a surjection G k → G. Let T c be a smooth compactification of T .We have

Norm equations
Let L be an étale k-algebra of finite rank; in other words, a product of a finite number of separable extensions of k.
Let a ∈ k × .Let X a be the affine k-variety associated to the norm equation The variety X a is a torsor under T L/k ; let X c a be a smooth compactification of X a .We have a natural map Br(k) → Br(X c a ); if a = 1 then X 1 = T L/k , and the map Br(k) → Br(T c L/k ) is injective, and moreover we have an injection Br(X c )/Im(Br(k)) → Br(T c )/Br(k) (see for instance [BP 20], §5).Recall a result from [BP 20], Theorem 6.1 :

Global fields
If k is a global field, we denote by V k be the set of all places of k; if v ∈ V k , we denote by k v the completion of k at v.
For any k-torus T , set

Norm equations and étale algebras
In the sequel, we consider norm equations of étale algebras having at least one cyclic factor.The aim of this section is to introduce some notation and prove some results that will be used throughout the paper.
Let K be a cyclic extension of k, and let K ′ be an étale k-algebra of finite rank; set L = K × K ′ .We first show that it is enough to consider the case when K/k is cyclic of prime power degree.

Reduction to the prime power degree case
Let P be the set of prime numbers dividing [K : k].For all p ∈ P, let K[p] be the largest subfield of K such that [K[p] : k] is a power of p, and set [BLP 19] the following result Proposition 2.1.Assume that k is a global field.We have Proof.This follows from [BLP 19], Lemma 3.1 and Proposition 5.16.
Let k ′ /k be a Galois extension of minimal degree splitting T L/k , and let Proof.Let us write K ′ = i∈I K i , where the K i are finite field extensions of k.
Let H be the subgroup of G such that K = (k ′ ) H , and for all i ∈ I, let H i be the subgroup of G such that For all p ∈ P, let H[p] be the subgroup of G such that Let ℓ ′ /ℓ be an unramified extension of number fields with Galois group G (cf. p] , and Proposition 2.3.Assume that k is a global field We have Proof.This follows from Proposition 2.2 and [BP 20], Corollary 3.4.

The prime power degree case
Let p be a prime number, and assume that K/k is cyclic of degree a power of p.Let us write K ′ = i∈I K i , where the K i are finite field extensions of k, and let [K : k] = p n .Notation 2.4.For all integers 1 ≤ m ≤ n, let K(m) be the unique subfield of K of degree p m over k.The K i -algebra K(m) ⊗ k K i is a product of cyclic extensions of K i ; let p e i (m) be the degree of these extensions, and set E(m) = {e i (m) | i ∈ I}.For all i ∈ I, let us chose one of the cyclic factors Let K be a Galois extension of k containing K and all the fields K i , and let For all integers 0 ≤ m ≤ n, let Γ m i be the set of conjugacy classes of elements Notation 2.5.Assume moreover that k is a global field.Let V m i be the set of places v of k such that there exists a place w of K i above v having the property that K ⊗ k (K i ) w is a product of fields extensions of degree at least p m of (K i ) w .
Proposition 2.6.Assume that k is a global field.Let V rm be the set of places of k which are ramified in K.For all integers 0 ≤ m ≤ n, sending a place In order to prove the Proposition, we need the following lemma.
Lemma 2.7.Let F be a field, and let E be a cyclic extension of F of prime degree.Let M be an extension of E, and assume that M is a Galois extension of Proof.Let G E/F be the Galois group of the extension E/F , and let D E/F be the decomposition group of v F ; note that v F is inert in E if and only if D E/F = G E/F .Since E/F is cyclic of prime degree, this amounts to saying that D E/F is not trivial.
We have the exact sequence Proposition 2.8.Let F , E and M be as in Lemma 2.7.Assume moreover that k is a subfield of F , and that M is a Galois extension of k.Let G = Gal(M/k).Let v k : k × → Z be a discrete valuation such that the extensions of v k to M are unramified; let D be the set of corresponding decomposition groups.The following are equivalent (a) There exists an extension of v k to F that is inert in E.
Proof.Let us prove that (a) implies (b).Let v F be an extension of v k to F that is inert in E, let v be an extension of v F to M, and let D be the decomposition group of v.By Lemma 2.7, we have (b).Conversely, assume that (b) holds.Let D ∈ D be as in (b), and let v be the corresponding valuation.Let v F be the restriction of v to F .By Lemma 2.7, we see that v F is inert in E, hence (a) holds.
, and hence also in E i (n − m + 1)/E i (n − m); therefore by Proposition 2.8 the conjugacy class of its Frobenius element f v belongs to Γ m i .Conversely, if the conjugacy class of g ∈ G belongs to Γ m i , then by Chebotarev's density theorem there exists an umramified place v such that its Frobenius element f v is the conjugacy class of g.Since the conjugacy class of g belongs to Γ m i , there is a place power degree and is unramified at w.This implies that v ∈ V m i \ V rm .We get immediately the following corollary.
Corollary 2.9.For i, j ∈ I, the map defined in Proposition 2.6 induces a surjection from Remark 2.10.Keep the notation in Proposition 2.6.For each conjugacy class in Γ m i , by Chebotarev's density theorem there are infinite many unramified places v ∈ V m i mapped to it.

Norm equations -unramified Brauer group
We keep the notation of the previous section.In particular, k is a field, K is a cyclic extension of k, and Let a ∈ k × , and let X a be the affine k-variety associated to the norm equation N L/k (t) = a.The variety X a is a torsor under T L/k .Let T c L/k be a smooth compactification of T L/k , and let X c a be a smooth compactification of X a .
The aim of this section is to describe the group Br(X c a )/Im(Br(k)).Using the results of §2, we can assume that K/k is of degree p n , where p is a prime number.
We use the notation of §2 (see Notation 2.4).In addition, we need the following Notation 3.1.For all integers n ≥ 1, we denote by C(I, Z/p n Z) the set of maps I → Z/p n Z.
If 1 ≤ m ≤ n, let π n,m be the projection C(I, Z/p n Z) → C(I, Z/p m Z).
For x ∈ Z/p m Z and y ∈ Z/p r Z, we denote by δ(x, y) the maximum integer d ≤ min{m, r} such that x = y (mod p d Z).
We start with some special cases, in which the results are especially simple.

K/k cyclic of degree p
Assume first that [K : k] = p, and that K is not contained in any of the fields K i .Then for all i ∈ I, E i is a cyclic field extension of degree p of and D be the subgroup of constant maps I → Z/pZ.
As a consequence of Theorem 3.8, we'll show the following Proposition 3.2.Assume that K/k is cyclic of degree p, and that K is not contained in any of the fields K i .Then we have . By Theorem 1.1, this implies the following Corollary 3.3.Assume that K/k is cyclic of degree p, and that K is not contained in any of the fields K i .Then we have K/k of degree p n and K linearly disjoint of all the K i For all integers m with 1 ≤ m ≤ n set and denote by D the subgroup of constant maps I → Z/p n Z.
Proposition 3.4.Assume that K/k is cyclic of degree p n , and that K is linearly disjoint of all the fields K i .Then we have As in the case where K/k is of degree p, this follows from Theorem 3.8, and has the immediate corollary Corollary 3.5.Assume that K/k is cyclic of degree p n , and that K is linearly disjoint of all the fields K i .Then we have

The general case
Recall that K/k is cyclic of degree p n , and that we use Notation 2.4.Recall that E = E(n).
Notation 3.6.For all e ∈ E, set I e = {i ∈ I | e i (n) = e}.Denote by ê the maximum element in E. Note that the index i belongs to I e if and only if K ∩ K i is an extension of degree p n−e of k.As K is a cyclic extension, this means that given 0 ≤ m ≤ n, the e i (m) are the same for all i ∈ I e and we denote it by e(m).
For all integers m with 1 ≤ m ≤ n set We still denote by π n,m the map from ⊕ Remark 3.7.
The main results of this section are Theorem 3.8.Assume that K/k is cyclic of degree p n .Then we have X 2 cycl (G, TL/k ) ≃ C(L)/D.By Theorem 1.1, this implies the following Corollary 3.9.Assume that K/k is cyclic of degree p n .then we have The proof of Theorem 3.8 will be given below, using some arithmetic results of [BLP 19].We start by recalling and developing some results concerning global fields.

Global fields
Assume that k is a global field.Recall that K/k is cyclic of degree p n , and that we use notations 2.4 as well as 3.6.In addition, for global fields, we also use notation 2.5.
We can continue the above process to get an infinite sequence of r l ∈ Z/p êZ such that δ(r l , r l+1 ) < δ(r l+1 , r l+2 ).It is a contradiction as δ(r l , r l+1 ) ranges from 0 to ê. Hence Corollary 3.11.Let k be a global field.Then X 2 ω (k, TL/k ) ≃ C(L)/D.Proof.By Corollary 2.9 and Remark 2.10, the two sets C m and C m ω are the same.Our claim then follows from Theorem 3.10.
Proof of Theorem 3.8.Recall that K is a Galois extension of k containing K and all the fields K i , and that G = Gal(K/k); if F is a subfield of K, we denote by G F the subgroup of G such that F = K G F .Note that k is not necessarily a global field here.However there is always an unramified extension ℓ ′ /ℓ with Galois group Gal(ℓ ′ /ℓ) ≃ G ([F 62]).Hence we can regard TL/K as a Gal(ℓ ′ /ℓ)-module.
To be precise, set construction, the G-lattices TE/ℓ and TL/k are isomorphic.Since the extension ℓ ′ /ℓ is unramified, we have

Unramified Brauer groups and generators
We keep the notation of the previous sections.Recall that p is a prime number, K/k a cyclic field extension of degree p n , and L = K × K ′ , where K ′ is an étale k-algebra of finite rank.In the previous section, we introduced a group C(L) and proved that Br(X c a )/Im(Br(k)) ≃ C(L)/D.The aim of this section is to give more precise information about the isomorphism C(L)/D → Br(X c a )/Im(Br(k)).
Let Br ur (k(X a )) be the subgroup of Br(k(X a )) consisting of all elements which are unramified at all discrete valuations of k(X a ) with residue fields containing k and with fields of fraction k(X a ); recall that Br ur (k(X a )) is isomorphic to Br(X c a ) (see Cesnavius [C 19], Theorem 1.2).As in the previous sections, let us write K ′ = i∈I K i , where the K i are finite separable field extensions of k.
Notation 4.1.We denote by G k the absolute Galois group of k, G k(Xa) the absolute Galois group of k(X a ).Let R be a discrete valuation ring of k(X a ) with residue field κ R containing k and with field of fractions k(X a ).We denote by G R the absolute Galois group of κ R .Notation 4.2.For all i ∈ I, let {β ij } be a basis of K i over k.Let where x ij are variables.Set N i = N K i ⊗k(Xa)/k(Xa) (y i ) considered as an element of k(X a ) × .We define N = N K⊗k(Xa)/k(Xa) (y) in a similar way.Fix an isomorphism χ : Gal(K/k) → Z/p n Z. Then χ gives rise to a morphism χ : G k(Xa) → Z/p n Z and a morphism χ R : G R → Z/p n Z.Let (N i , χ) denote the class of the cyclic algebra over k(X a ) associated to χ and the element The main result of this section is given by [BLP 19] Lemma 6.1).
We start with following lemmas.The following lemma can be found in [Lee 21] §3.Here we use the notation C(L) to simplify the proof.
(1) Let c ∈ C(L) \ D. Pick i 0 ∈ I ê.Let ĉ ∈ D be the image of the constant map from I to c(i 0 ).Set m to be the maximal integer such that π n,m (c) = π n,m (ĉ).Choose r ∈ Z/p êZ such that δ(r, c(i 0 )) = m.Consider the element c ′ ∈ ⊕ e∈E C(I e , Z/p e Z) defined as follows: Suppose l > m.If Γ l i ∩ Γ l j = ∅, then by Remark 3.7 e i (n) and e j (n) are at least l and c(i The proof of statement (2) is similar.
Lemma 4.7.Assume that char(k) = p.Let R be a discrete valuation ring as in Notation 4.1.Denote by ∂ R the residue map from Br(k(X a )) to H 1 (κ R , Q/Z).Suppose that the order of ∂ R (N i , χ) and the order of ∂ R (N j , χ) are both at least p m .Then (See [GS 06] 6.8.4 and 6.8.5.)Write ν R (N i ) as p m i q i where p ∤ q i .Let p n R be the order of χ R .As the order of ∂ Choose a factor M of K ⊗ k k(X a ) R and let M be its residue field.Let M i be the residue field of w i,R .Both fields are considered as subfields of a separable closure Consider the group action of g R on the set of left cosets of does not divide the order of the orbit of h i H i .Hence the stabilizer of h i H i is Let g and σ be the image of g R and σ R in G. Then σ fixes K i and σ is an element of order at least p m in Gal(K/k).Hence the conjuacy class of g belongs to Γ m i .The same argument proves that the conjuacy class of g belongs to Γ m j .Hence Γ m i ∩ Γ m j is nonempty.
Next we prove that for all c ∈ C(L), the element Proposition 4.8.Suppose that char(k) = p.The image of u is an unramified subgroup of Br(k(X a )).
Proof.By Lemma 4.5 we can assume that c(i) = 0 for some i ∈ I ê.
Let R be a discrete valuation ring of k with residue field κ R containing k and with field of fractions k(X a ).Let ν R be the discrete valuation associated to R. Denote by ∂ R the residue map from Br(k(X a )) to H 1 (κ, Q/Z).We claim that u(c) = Let M be the function field of A d .Denote by χ M the image of χ in H 1 (M, Q/Z).Suppose that u(c) = α ∈ Im(Br(k)).Then u(c) − α is in the kernel of Br(M) → Br(k(X a )),which is generated by (a ).Consider the discrete valuation v N i on M and let κ N i be its residue field.Denote by χ N i the image of χ in H 1 (κ N i , Q/Z).We claim that χ N i is of order p e i (n) .Let M i be the function field of the subvariety of A [K i :k] defined by N i .Then κ N i = M i (x jl ) where x jl are defined as in Notation 4.2 with which is a contradiction.Hence [ Mi : M i ] = p e i (n) and χ M i is of order p e i (n) .
As χ N i is of order p e i (n) , after taking residue of u(c) − α at v N i , we see that c(i) = −r (mod p e i (n) Z).Hence c ∈ D and u is injective.
Since u is injective, |C(L)/D| ≤ |Br(X c a )/Im(Br(k))|.By Theorem 3.8 and Theorem 1.1, the order of C(L)/D is equal to the order of Br(T c L/k )/Br(k).By Theorem 1.2, the map u is surjective and hence is an isomorphism.
We now prove the results announced in the introduction : Proposition 4.9.Let k be a field of char(k) = p.Let F be a bicyclic extension of k with Galois group Z/p n Z×Z/p n Z.Let K and K i be linearly disjoint cyclic subfields of F with degree p n for i = 1, ..., m.Then Br(X c a )/Im(Br(k)) ≃ (Z/p n Z) m−1 , and is generated by (N i , χ) for i = 1...m − 1.
Proof.There is a number field ℓ and an unramified Galois extension ℓ m−1 by Theorem 3.8.The assertion then follows from Theorem 4.3.

Global fields
We keep the notation of the previous section, and in addition we assume that k is a global field.Denote by B(X Proof.First note that [Sa 81] 6.1.4remains true over global fields.Hence B ω (X c a ) ≃ B ω (X a ) and B(X c a ) ≃ B(X a ).By [Sa 81] 6.8 and 6.9 (ii), we have B(X a ) ≃ X 2 (k, TL/k ) (resp.B ω (X a ) ≃ X 2 ω (k, TL/k )).By Theorem 3.10 and Corollary 3.11, we conclude that C(L)/D ≃ B ω (X c a ) and C arith (L)/D ≃ B(X c a ).As B ω (X c a ) is a subgroup of Br(X c a )/Im(Br(k)) with the same cardinality, the first statement follows from Theorem 4.3.
To see that u gives rise to the desired isomorphism in (2), it is sufficient to show that u By Lemma 4.5 we can assume that c(i 0 ) = 0 for some i Let |J(c)| = h + 1.Then c / ∈ D and J m (c) is nonempty.Pick j ∈ J m (c) and choose r ∈ Z/p êZ such that c(j) = r (mod p e j (n) Z).Let c ′ be defined as in Lemma 4.6. For By the definition of C arith (L), the set V m+1 More generally we have the following.
Proposition 5.3.Let k be a global field of char(k) = p.Let F be a bicyclic extension of k with Galois group Z/p n Z × Z/p n Z.Let K and K i be linearly disjoint cyclic subfields of F with degree p n for i = 1, ..., m.Assume moreover that F ⊗ k k v is a product of cyclic extensions for all v ∈ V k .Then (N i , χ) generates B(X c a ).

An application to Hasse principles
In this section we apply Theorem 4.3 to give an alternative proof of [BLP 19] Theorem 7.1 for k a global field with char(k) = p.Moreover we can assume that K/k is a cyclic extension of p-power degree.(See §2 and [BLP 19] 6.3.)We use the notation of the previous sections.In particular, X a is the affine variety defined in the introduction, K/k is a cyclic extension of p-power degree, and K i /k is a finite separable extension for all i ∈ I. Recall that χ an injective morphism from Gal(K/k) to Q/Z.
Let χ v be the image of χ in H 1 (k v , Q/Z).Let inv be the Hasse invariant map inv : Br(k v ) → Q/Z.
Denote by K v i the algebra K i ⊗ k k v .Suppose that there is a local point Theorem 6.1.Suppose that there is a local point Then the map α a is the Brauer-Manin pairing and X a has a k-point if and only if α a = 0.
Proof.First we consider the case where k is a number field.By Theorem 5.1 and [Sa 81] Lemma 6.2, the map α a is the Brauer-Manin pairing of X c a .Our claim then follows from Sansuc's result [Sa 81] Cor.8.7.
For k a global function field, we apply Theorem 5.1 and [DH 17] Theorem 2.5 to conclude.
e∈E C(I e , Z/p e Z) to ⊕ e∈E C(I e , Z/p e−e(n−m) Z) induced by the natural projection.Set C(L) = {c ∈ C n | π n,m (c) ∈ C m for all m ≤ n}, and denote by D the image of constant maps I → Z/p n Z in C n under the natural projection .

Lemma 4. 5 .
The group u(D) is contained in the image of Br(k) in Br(k(X c )). Proof.Since N • i∈I N i = c, we have i∈I (N i , χ) = (c/N, χ) = (c, χ), which is the image of (c, χ) in Br(k(X a )).Hence u(D) ⊆ Im(Br(k)).
c a ) the subgroup of Br(X c a )/Im(Br(k)) consisting of locally trivial elements; and by B ω (X c a ) the subgroup consisting of elements which are trivial at almost all places of k.Theorem 5.1.Suppose that k is a global field with char(k) = p.Then (1) u induces an isomorphism between C(L)/D and B ω (X c a ).(2) u induces an isomorphism between C arith (L)/D and B(X c a ).
We prove by induction on |J(c)|.If |J(c)| = 0, then c = 0 and our claim is trivial.Suppose that our claim is true for |J(c)| ≤ h.

..
any i ∈ J m (c) and for any j / ∈ J m (c).Let v ∈ V k .Suppose that v / Then for all i ∈ J m (c), χ i,w is of order at most m for all w | v.By the projection formula, (N i , χ) v has order at most p m .Hence c ′ (i)(N i , χ) v = 0 and u(c′ ) v = 0 in this case.Suppose that v ∈ V m+1 i for some i ∈ J m (c).Let d ∈ D be the image of the constant map from I to r. Set c = d − c ′ .As the set V m+1 i The same argument shows that u(c) v = 0. Hence u(c ′ ) v is in the image of Br(k v ).Since the cardinality of J(c − c ′ ) decreases by at least one, by induction hypothesis u(c− c ′ ) ∈ B(X c a ).In combination with u(c ′ ) ∈ B(X c a ), we see that u(c) is in B(X c a ).Example 5.2.Let k = Q(i).Let K = k( 4 √ 17), K 1 = k( 4 √ 17 × 13) and K 2 = k( 4 √13).By [Lee 21] Example 7.4, X 2 ω (k, TL/k ) ≃ Z/4Z and X 2 (k, TL/k ) ≃ Z/2Z.By Theorem 4.3 and Theorem 5.1, the element (N 2 , χ) generates the group Br(X c a )/Im(Br(k)) and 2(N 2 , χ) generates B(X c a ).