Reciprocity obstruction to strong approximation over p-adic function fields

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SL_n with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.


Introduction
In recent years, there has been growing interest in studying the arithmetic of algebraic groups and their homogeneous spaces defined over fields of cohomological dimension strictly greater than 2, and in particular, function fields K of p-adic curves.For example, in [HS16] and [HSS15], Harari, Szamuely and Scheiderer studied local-global principle and weak approximation questions for tori over such fields K, with respect to the valuations coming from the codimension 1 points of the p-adic curve.For both questions, they showed that the reciprocity obstruction is the only one.Tian generalized their results on weak approximation to quasi-split reductive groups in his work [Tia21].Very recently in [Lin22], Linh studied homogeneous spaces of SL n with toric stabilizers, and more generally, stabilizers in the form of an extension of a group of multiplicative type by a unipotent group.He showed that the reciprocity obstruction is the only one to both the local-global principle and weak approximation.
Being a generalized form of the Chinese Remainder Theorem, strong approximation is a natural question in the study of the arithmetic of algebraic groups and their homogeneous spaces.For tori over p-adic function fields, Harari and Izquierdo described the defect of strong approximation in [HI19], based on a Poitou-Tate type exact sequence obtained in [HSS15].
We study strong approximation questions for homogeneous spaces over p-adic function fields.Let K be the function field of a smooth proper geometrically integral curve X defined over a p-adic field k.We first show that we have very different behavior of strong approximation over K compared to number fields: Theorem 1.1.There are homogeneous spaces of the form Z = SL n /H with H semisimple simple connected such that strong approximation away from a non-empty set S ⊆ X (1) does not hold for Z.
Our constructions include H of inner type A (Theorem 2.6, Corollary 2.7 and Example 2.8), and outer type A (Corollary 2.11 and Example 2.13).
For H of inner type A, we have H = SL 1 (A) for a simple central K-algebra A and Z = SL n /H is a stably rational K-variety.We show that the maps H 1 (K v , H) → H3 (K v , Q/Z(2)) given by the Rost invariant composed with the sum in the complex of the generalized Weil reciprocity law can be calculated in terms of the Lichtenbaum duality pairing Br X × Pic X → Q/Z induced by evaluation on closed points.As a consequence, we give explicit descriptions of H 1 (K v , H), which are finite cyclic groups, and conditions under which the map H 1 (K, H) → v∈X (1) \S H 1 (K v , H) is not surjective, yielding counter-examples to strong approximation for SL n /H.
For H of outer type A, we have H = SU(A, τ ) for a central simple L-algebra A with L/K-unitary involution τ , where L/K is a separable quadratic extension.We describe the images of H 1 (K v , H) under the Rost invariant with respect to different ramification types, and we give counter-examples to strong approximation for SL n /H based on calculations in the case of inner type A.
To account for such failure, we define a reciprocity obstruction.We use the group H 3 nr (Z, Q/Z(2)), which is the subgroup of elements in H 3 (K(Z), Q/Z(2)) that are unramified with respect to all codimension 1 points of Z.We define the subgroup of "trivial on S" elements H 3 nr,S (Z, Q/Z(2)) := Ker(H 3 nr (Z, Q/Z(2)) and we denote by H 3 nr,S (Z, Q/Z(2)) its quotient by constant elements.We define a pairing The subset of elements in Z(A S K ) that are orthogonal to H 3 nr,S (Z, Q/Z(2)) contains Z(K) in virtue of the generalized Weil reciprocity law, and also its closure Z(K) S by continuity of the pairing, giving rise to a reciprocity obstruction to strong approximation away from S.
When Z is of the form SL n /H, we show that we can also define a pairing using Inv 3 (H, Q/Z(2)) norm , the normalized degree 3 cohomological invariants of H with coefficients in Q/Z(2), and we have the following compatibility given be the commutative diagram As a consequence, there is a reciprocity obstruction to strong approximation for the examples we constructed.Then we show that for certain Z (e.g. the one in Example 2.8, or SL n /T for T a torus), this obstruction is the only one.
Theorem 1.2 (Theorem 3.10).Let Z = E/ SL 1 (A) with [A] ∈ Br X of exponent m, and the special rational group E is split semisimple simply connected.For the p-adic curve X, suppose Pic 0 (X)/m = 0, or equivalently m H 1 (k, Pic X) = 0 (for example X = P 1 k the projective line satisfies this condition).Then there is an exact sequence of pointed sets for S ⊆ X (1) a non-empty finite set of places.In particular, the reciprocity obstruction to strong approximation away from S is the only one for Z.
The group H 3 nr,S (Z, Q/Z(2)) measures the defect of strong approximation away from S for Z, and is finite cyclic of order gcd( I(S) I(X) , m) where I(X) (resp.I(S)) is the index of X (resp.S).In particular, strong approximation away from S holds for Z if and only of I(S)/I(X) is coprime to m, and such an S always exists, for example S such that I(S) = I(X).
The following result is obtained by exploiting the Poitou-Tate type exact sequence in [HSS15] along with the description of Inv 3 (T, Q/Z(2)) norm given by Blinstein and Merkurjev in [BM13].
Theorem 1.3 (Theorem 3.11).Let Z = E/T be a classifying variety of a torus T over K a p-adic function field, where the special rational group E is split semisimple simply connected.Then there is an exact sequence of pointed sets for S ⊆ X (1) a non-empty finite set of places.In other words, the reciprocity obstruction to strong approximation away from S is the only one for Z.If we suppose furthermore that CH 2 (Z) → H 0 (K, CH 2 (Z)) is surjective, then there is an exact sequence of pointed sets Then we show that our reciprocity obstruction to strong approximation not only works for classifying varieties, but also applies to tori, compatible with the results in [HI19] obtained by Harari and Izquierdo: measures the defect of strong approximation, and A(T )/ Div denotes the quotient of A(T ) by its maximal divisible subgroup.
Finally, we explain that for classifying varieties, our comparison between the two pairings of reciprocity obstruction and cohomological invariants has a parallel form for weak approximation problems too, given by the commutative diagram where As a quick and direct application, we obtain the following theorem giving answers to weak approximation problems for classifying varieties of tori over K, combining the Poitou-Tate type exact sequence in [HSS15] and the description of Inv 3 nr (T, Q/Z(2)) norm in [BM13].This result was also obtained by Linh in a different way in his very recent work (cf.Theorem B of [Lin22]).
Theorem 1.5 (Theorem 4.2).Let Z = E/T be a classifying variety of a torus T over K. Then the reciprocity obstruction to weak approximation is the only one for Z.In fact, there is a morphism with respect to the pairing (1) already equals the closure

Motivic complexes
Let X be a smooth scheme.Lichtenbaum (cf.[Lic87][Lic07]) defined the motivic complexes Z(i) for i = 0, 1, 2, of étale sheaves on X.We write H * (X, Z(i)) for the étale (hyper)cohomology groups of X with values in Z(i).The complex Z(0) equals the constant sheaf Z and In particular, H 3 (X, Z(1)) = Br X, the cohomological Brauer group of X.The complex Z(2) is concentrated in degrees 1 and 2 and there is a product map Z(1) ⊗ L Z(1) → Z(2) (cf.Proposition 2.5 of [Lic87]).Over the small Zariski site X Zar , we have the complex Z(2) Zar concentrated in degree ≤ 2 defined in a similar way to Z(2).We denote by A(i) the complex A ⊗ Z(i) for an abelian group A (similarly for Z(i) Zar ).If X is defined over a field whose characteristic does not divide m, we have a quasi-isomorphism of complexes of étale sheaves Z/mZ ⊗ L Z(i) ≃ µ ⊗i m , where µ m is the étale sheaf of mth roots of unity, and we set µ 0 m = Z/mZ.Thus we shall also use the notation Q/Z(i) for the direct limit of the sheaves µ ⊗i m for all m > 0 when the base field has characteristic 0. The exact triangle in the derived category of étale sheaves yields the connecting morphism which is an isomorphism if X = Spec F for a field F and n > i (cf.Lemma 1.1 of [Kah93]).For every field extension F/K, the set H 1 (F, H) classifying H-torsors over F can be identified with Z(F )/E(F ), the orbit space of the action of E(F ) on Z(F ).For a point x ∈ Z(F ), we write x * E for the fiber of the H-torsor E → Z above the point x and it is an H-torsor over F , and its class in

Classifying varieties and cohomological invariants
In particular, we have the generic H-torsor ξ * E over K(Z) corresponding to the generic point ξ ∈ Z.
Let H be an algebraic group over a field K.The group Inv d (H, Q/Z(d − 1)) of degree d cohomological invariants of H consists of morphisms of functors from the category of field extensions of K to the category of sets.An invariant is called normalized if it takes the trivial H-torsor to zero.The normalized invariants form a subgroup Inv d (H, Q/Z(d − 1)) norm of Inv d (H, Q/Z(d − 1)) and there is a natural isomorphism Rost proved that if H is absolutely simple simply connected, then Inv 3 (H, Q/Z(2)) norm is a finite cyclic group generated by the Rost invariant R H (cf. Theorem 9.11 of [GMS03]).

p-adic function fields
Let k be a finite extension of Q p , and let X be a smooth proper geometrically integral curve over k.Let K be the function field of X.There is a natural set of places: the set of all closed points of X that we denote by X (1) .For a closed point v ∈ X (1) , we denote by κ(v) its residue field, and K v the completion of K for the discrete valuation induced by v. Then κ(v) is also the residue field of the ring of integers O v and is a finite extension of k.Therefore K v is a 2-dimensional local fields, i.e. a field complete with respect to a discrete valuation whose residue field is a classical local field.
We have a complex in virtue of the generalized Weil reciprocity law (cf.Theorem 6.4.3 of [GS17]): where the map Σ is given by the composition The above notation of our fields will be fixed for the rest of the article.

Strong approximation
Given a smooth geometrically integral K-scheme Z, for a non-empty open subscheme U ⊆ X sufficiently small, we can find a smooth geometrically integral U -scheme Z such that Z × U K ≃ Z.We can thus define the adelic space (which is actually independent of the choice of the model Z): The problem of strong approximation studies the closure Z(K) inside the adelic space Z(A K ) with the restricted product topology, possibly away from a finite set S of places (in this case, we ∈S Z(K v ) the restricted product with respect to integral points excluding places in S).If Z(K) is dense in Z(A S K ), we say that Z satisfies strong approximation away from S: we can simultaneously approximate finitely many K v -points with v / ∈ S by a single K-rational point with the condition that this point is integral at all other places.For S ⊆ S ′ , strong approximation away from S implies strong approximation away from S ′ .
Strong approximation for semisimple groups has been studied in the general setting.A Kgroup G is said to be quasi-split if it has a Borel subgroup defined over K.A K-group G is called absolutely almost simple if over the algebraic closure it becomes an extension of a simple group by a finite normal (hence central) subgroup.
Theorem 1.6 (Gille, Corollaire 5.1 of [Gil09]).Let R be a Dedekind domain with function field K and let G be a semisimple simply connected K-group which is absolutely almost simple and isotropic.Suppose the K-variety G is retract rational.Then strong approximation holds for G over Spec R.

The above theorem implies an earlier result:
Theorem 1.7 (Harder, Satz 2.2.1 of [Har67]).Let R be a Dedekind domain with function field K. Let G be a quasi-split semisimple simply connected K-group.Then strong approximation holds for G over Spec R.
Corollary 1.8.Let X be a smooth proper geometrically integral curve over a p-adic field k, with function field K = k(X).Let G be a quasi-split semisimple simply connected K-group.Then strong approximation away from a non-empty finite set of places S ⊆ X (1) holds for G.
Proof.The open subset V := X\S is affine and the ring of regular functions k[V ] is a Dedekind domain.
When studying strong approximation for classifying varieties Z = E/H, we usually make use of the following commutative diagram denotes the restricted topological product of the H i (K v , H) for all v ∈ X (1) \S with respect to the images of the maps

Notation
Unless otherwise stated, all (hyper)cohomology groups are taken with respect to the étale cohomology.
Given an abelian group A, we denote by m A the m-torsion subgroup of A. The notation A ∧ is the inverse limit of the quotients A/nA for all n > 0. For A a topological abelian group, denote by A D := Hom cont (A, Q/Z) the group of continuous homomorphisms from A to Q/Z.When there is no other topology defined on A, we equip A with the discrete topology.The functor A → A D is an anti-equivalence of categories between torsion abelian groups and profinite groups.

Failure of strong approximation
Over p-adic function fields, we construct varieties of the form SL n /H with H semisimple simply connected and we show that strong approximation away from a finite set of places fails for such varieties.

Inner type
For a smooth proper geometrically integral curve X over a p-adic field k, we have the following dualities due to Lichtenbaum: Theorem 2.1 (Lichtenbaum, cf.[Lic69] Theorem 4).There are pairings where ψ is induced by evaluation on closed points.These pairings induce dualities Construction 2.3.Let H be an absolutely simple simply connected group over K of inner type A. Then H = SL 1 (A) for A a simple central K-algebra.Suppose the class [A] ∈ Br K lies in Br nr (K/k) = Br X. Embed H into a special rational group E which is split semisimple simply connected (for example, E = SL n or Sp 2n ).Consider the classifying variety Z := E/H.Since H also embeds into the special rational group GL 1 (A), we have that Z is stably birational to the classifying variety GL 1 (A)/ SL 1 (A) = G m which is rational.Therefore, the variety Z is stably rational.In particular, weak approximation holds for Z.
The Rost invariant R H has an explicit formula in this situation (cf. and m is the exponent of A. Moreover, the order of R H ∈ Inv 3 (H, Q/Z(2)) norm equals m (cf.Theorem 11.5 of [GMS03]).
We have the following commutative diagram where the first row is the complex (2) by the generalized Weil reciprocity: Now we show that the composed map f H defined by this diagram is related to the pairing ψ in (4): . By linearity, it suffices to show Choose a uniformizer π v of the place v, and write In fact with respect to the uniformizer −π v (cf.Construction 6.8.6 of [GS17]).Actually giving the desired result.
The above calculations also give the following result, which will be used later.
Proposition 2.5.There is a bijection Proof.The map given by the Rost invariant is injective in this case (cf.Theorem 5.3 of [CTPS12]).Therefore, we can identify with the calculations in Proposition 2.4, where v * ([A]) denotes the evaluation of [A] at v. By Proposition 6.8.7 and Corollary 6.8.8 of [GS17], the class under the injection Br(κ(v)) ֒→ Br K v , and thus they have the same order.
In virtue of the Lichtenbaum duality (5), the non-vanishing of the pairing ψ gives non-vanishing f H , which can then be used to fabricate examples of non-surjective Moreover, this map can be non-surjective even away from a finite set S of places: Theorem 2.6.If the set of values taken by ψ([A], −) at the divisors supported in S ⊆ X (1) do not cover all the values taken at the divisors supported in X (1) \ S, i.e.

{ψ([
is not surjective.As a result, strong approximation away from S does not hold for Z.
Proof.Take a divisor and thus cannot cancel out f H (a) = ψ([A], z 0 ).Then chasing the diagram (7) shows that a is not in the image of H 1 (K, H).Now consider the following commutative diagram with exact rows.We identify ⊕ v∈X (1) \S H 1 (K v , H) with the topological restricted product for example the a we just constructed.There exists an open subset V of z ∈ Z(A S K ) which maps to a.A diagram chasing shows that V doesn't meet Z(K).
For S ⊆ X (1) , the index I(S) of S is defined to be the greatest common divisor of the degrees of all v ∈ S. The index I(X) of X is I(S) for S = X (1) .
Corollary 2.7.Let A be a simple central k-algebra of exponent m and consider not meeting the support of z 0 such that deg(z 0 ) is not contained in the subgroup generated by I(S) in Z/mZ.Then strong approximation away from S does not hold for Z.

Proof. By Remark 2.2, we have ψ(
When z is supported in S, the value ψ([A], z) is always a multiple of I(S)/m, which cannot be equal to deg(z 0 )/m.The condition in Proposition 2.6 is thus satisfied.
Example 2.8.Let X = P 1 k be the projective line over k = Q p .Then the function field K = Q p (t).Consider the quaternion algebra A = (p, u) over K with u a non-square unit in Z × p .Then [A] ∈ 2 BrK comes from the generator of 2 BrX = 2 Brk.Let H be the semisimple simply connected group SL 1 (A), and consider the stably rational homogeneous space Z = SL n /H.If S ⊆ X (1) is a subset containing only places of even degrees, for example (t 2 − p), (t 4 − p), (t 6 − p) etc, then strong approximation away from S does not hold for Z.In fact, we will see later that strong approximation away from S holds for Z if and if only the index of S is odd, and the reciprocity obstruction is the only one (cf.Theorem 3.10).
Remark 2.9.Our example shows that we have a different behavior over p-adic function fields compared to the classical situations over number fields.For H a semisimple and connected algebraic group defined over a number field F , the canonical map ).The group SL n satisfies strong approximation away from a non-empty set S ⊆ Ω F of places.A diagram chasing argument shows that strong approximation away from a non-empty S ⊆ Ω F always holds for SL n /H.In our case over a p-adic function field K, we still have strong approximation away from a non-empty S ⊆ X (1) for SL n (cf.Corollary 1.8), but not for SL n /H where H is an absolutely simple simply connected algebraic group.We will give another example with H of outer type A (cf. Example 2.13).

Outer type
Now consider H a simply connected group of outer type A n−1 , i.e.H = SU(A, τ ), where A is a central simple algebra of degree n ≥ 3 over a (separable) quadratic extension L of K with a unitary involution τ which leaves K element-wise invariant.The existence of such a L/K-unitary involution on A is equivalent to which is a homogeneous space for GL 1 (A) under the operation given by x : (s, z) → (xsτ (x), z Nrd A (x)).
There is an exact sequence for any field extensions F/K.In other words, there is a canonical bijection between H 1 (F, H) and SSym(A F ⊗L , τ ) × / ≈ where the equivalence relation ≈ is defined by (s, z) ≈ (s ′ , z ′ ) if and only of s ′ = xsτ (x) and z ′ = z Nrd A F ⊗L (x) for some x ∈ A × F ⊗L (cf.(29.18) of [Knu98] and §5.5 of [KJ69]).Over the quadratic extension L/K, the group H is isomorphic to SL 1 (A).Under the field extension map, the Rost invariant R H maps to the Rost invariant R H L .The corestriction map for the field extension L/K takes R H L to 2R H . Using the formula of the Rost invariant for SL 1 (A), we have over a field extension F/K, where (s, z) ∈ SSym(A F ⊗L , τ ) × / ≈ .We investigate different places v ∈ X (1) in terms of their ramification types.If L v := K v ⊗ L is a field, then the valuation of v on K v extends uniquely to L v (which we denote by ṽ); moreover, if the residue field κ(ṽ) of L v is a field extension of degree 2 (resp.1) of the residue field κ(v) of K v , we call such a place inert (resp.ramified).For an inert place v, the field extension and such a place is called totally split.
If v is totally split, then L embeds into K v .The group H Kv is of inner type, and equals SL 1 (A Kv ).We can thus apply our results in §2.1.In particular, the order of the Rost invariant R H Kv equals the exponent m v of A Kv .If [A Kv ] is unramified at v, we can show that there is a bijection H 1 (K v , H) ≃ Z/m v Z as in Proposition 2.5.
Proposition 2.10.For an inert place v such that [A Lv ] is unramified at ṽ, the image of 2R H : Proof.We use the diagram in Proposition 8.6 of [GMS03] which is commutative with exact rows: where the ramification index e = 1 in our case.Since [A Lv ] is unramified at ṽ, it comes from a certain ᾱ ∈ Br(κ(ṽ)) in the top row of the diagram.The condition Cor Lv/Kv ([A Lv ]) = 0 then gives Cor κ(ṽ)/κ(v) (ᾱ) = 0 by taking i = 2 and j = 1 in the left square of the diagram.Now we compose by taking i = 3 and j = 2 in the right square of the diagram, where we denote by ṽ(z) the valuation of z with respect to ṽ.
Corollary 2.11.Suppose there is a totally split place v 0 such that [A Kv 0 ] is unramified at v 0 with exponent ≥ 3. Let S ⊆ X (1) be a finite set containing inert places v such that [A Lv ] is unramified at ṽ. Then is not surjective.As a result, strong approximation away from S does not hold for Z = E/H which is a classifying variety of H.
Proof.Proposition 2.5 applies to and we know that it is in fact a cyclic group of order equal to the exponent of ) for a uniformizer π v 0 of K v 0 and a v = 0 elsewhere.Then apply Proposition 2.4 and we get f H (a) is of order ≥ 3.For v ∈ S, the images of H 1 (K v , H) under f H only give elements of order at most 2, and thus cannot cancel out f H (a). The same diagram chasing argument as in Theorem 2.6 then gives the result.
There are only finitely many places left: those v such that the field extension L v /K v is ramified, or [A Lv ] is ramified at ṽ.In fact, these two conditions cannot hold at the same time. If The compatibility between residue maps and restriction maps (cf.Proposition 1.4.6 of [CTS21]) shows that [A Lv ] is unramified at ṽ.Moreover, the exponent of Lemma 2.12.For v ∈ X (1) , we have where π is a uniformizer for K v , the element δ ∈ K v is a unit whose image δ ∈ κ(v) is a uniformizer for κ(v), and u is a unit in If K v contains a primitive mth root of unity ω (e.g. when m|(p − 1)), then we have where (a, b) ω is a cyclic algebra of degree m.
Proof.The exact rows in (8) are in fact split (cf.Corollary 6.8.8 of [GS17]).Taking i = 1, j = 1 gives the first statement, noting that which is obtained by applying the same exact sequence to the complete discretely valued field κ(v).
A primitive mth root of unity ω induces an isomorphism µ m ≃ Z/mZ.The group H 2 (κ(v), µ m ) ≃ m Br(κ(v)) ≃ Z/mZ is then generated by the cyclic algebra ( δ, ū) ω, where ω denotes the image of ω in the residue field κ(v).Take i = 2, j = 1 in the split exact sequence (8).A retract of the injection m Br κ(v) ֒→ m Br K v is given by the specialization map s 2 π : a → ∂ v ((−π) ∪ a).The calculations give the second statement.
If the field extension L v /K v is unramified and [A Lv ] has index n ≥ 3, then by Proposition 6.6 of [PS22], there is a primitive nth root of unity ρ in L v such that N Lv/Kv (ρ) = 1 and [A Lv ] = (π, δ) ρ , where π and δ are as in Lemma 2.12.In particular, [A Lv ] is ramified at ṽ.There exists an L v /K vunitary involution τ on (π, δ) ρ such that τ leaves i and j invariant, where i and j are generators of (π, δ) ρ such that i n = π, j n = δ, ij = ρji.
Example 2.13.Let X = P 1 k be the projective line over k with function field K = k(t).Let k ′ = k[T ]/g(T ) be an unramified quadratic extension of k, such that there is a primitive mth root of unity ρ with N k ′ /k (ρ) = 1 and m ≥ 3. Examples include k ′ = k[T ]/(T 2 + T + 1) over k = Q 5 with ρ a primitive 3rd root of unity, and Let δ be a uniformizer of k, which stays a uniformizer of k ′ since the extension k ′ /k is supposed to be unramified.Let u ∈ k ′ be a unit with order m in k ′× /k ′×m .Consider the cyclic L-algebra A = (δ, u) ρ .The non-trivial L/K-automorphism extends to an L/K-unitary involution τ on A. Let H = SU(A, τ ) and Z = E/H be a classifying variety.The place v 0 = (g(t)) is totally split, and H 1 (K v 0 , H) is cyclic of order m.There are no ramified places or places v where [A L k ] is ramified.Let S be a subset of places v where L v /K v are (unramified) field extensions.Then strong approximation away from S does not hold for Z.

Reciprocity obstruction to strong approximation
We establish a reciprocity obstruction to strong approximation.This obstruction for classifying varieties is related to the degree 3 cohomological invariants.As a result, the failure of strong approximation constructed in Section 2 is explained by this reciprocity obstruction.We prove that this reciprocity obstruction to strong approximation is the only one in certain situations, including the case of Example 2.8, and classifying varieties of tori.We also explain that for K-tori, this reciprocity obstruction to strong approximation is compatible with the known results in [HI19].

Defining the reciprocity obstruction
Let Z be a smooth geometrically integral K-variety.As a consequence of Gersten's conjecture for étale cohomology proved by Bloch and Ogus (cf.[BO74]), we have: Theorem-Definition 3.1 (cf.Theorem 4.1.1 of [CT95], see also (1.1) of [CT15]).The following subgroups of H 3 (K(Z), Q/Z(2)) coincide, that we define to be H 3 nr (Z, Q/Z(2)): (1) the group H 0 Zar (Z, H 3 (Q/Z(2))) of global sections of the Zariski sheaf H 3 (Z, Q/Z(2)) which is the sheaf associated to the Zariski presheaf U → H 3 ét (U, Q/Z(2)); (2) the group of elements α ∈ H 3 (K(Z), Q/Z(2)) that are unramified with respect to any codimension 1 point P of Z, i.e. we have (3) the group of elements in H 3 (K(Z), Q/Z(2)) which at any point P ∈ Z come from a class in Remark 3.2.Since we do not suppose Z to be proper, the group H 3 nr (Z, Q/Z(2)) can be strictly bigger than its subgroup H 3 nr (K(Z)/K, Q/Z(2)), which is the group of elements in H 3 (K(Z), Q/Z(2)) that are unramified with respect to all discrete valuations of K(Z) trivial on K. Let Z c be a smooth compactification of Z, then we have ).We will see later in Theorem 3.10 that for Z in our Construction 2.3, the group )) is of order m which accounts for the defect of strong approximation for Z, but since Z is stably birational.

Now we define a pairing
in a similar way as in §2.2 of [CTPS16], where their original definition applies to For any field extension F/K and any point x ∈ Z(F ) with image P ∈ Z, we can lift α to a (unique) element of H 3 (O Z,P , Q/Z(2)) in virtue of 3.1(3), and define α(x) to be its image under the pullback ).This defines a pairing By shrinking the open subset U ⊆ X over which we have the model ) for all but finitely many codimension 1 points Q of the k-variety Z.By the assumption on α, the exceptional Q lie in finitely many closed fibers ) for all points P ∈ Z v .Therefore, for x coming from has cohomological dimension 2. We can thus sum up these maps to get a well-defined pairing (9) for the adelic space.
Remark 3.3.In the particular case when Z is proper (so strong approximation is equivalent to weak approximation), the above pairing (9) becomes which is exactly the pairing in §4 of [HSS15] giving the reciprocity obstruction to weak approximation.
For a subset S ⊆ X (1) , we define the subgroup of "trivial on S" elements: In virtue of the complex (2) of the generalized Weil reciprocity law, the subset Z(K) ⊆ Z(A S K ) is orthogonal to H 3 nr,S (Z, Q/Z(2)) with respect to the pairing The same holds for the closure Z(K) S of Z(K) in Z(A S K ) by continuity of the pairing, giving rise to a reciprocity obstruction to strong approximation away from S. Modulo constant elements, we can also replace H 3 nr,S (Z, Q/Z(2)) by which is a subgroup of ).We say that the reciprocity obstruction to strong approximation away from S is the only one if the subset of elements in Z(A S K ) orthogonal to H 3 nr,S (Z, Q/Z(2)) (or equivalently Remark 3.4.In the study of local-global principal for torsors under tori over K, Harari and Szamuely (cf.[HS16]) used the reciprocity obstruction given by a different pairing This pairing is compatible with our pairing (9) via a map H 3 (Z, Q/Z(2)) → H 3 nr (Z, Q/Z(2)) that we describe now.By results of Bloch and Ogus ([BO74]), the spectral sequence gives rise to an exact sequence which for j = 2 can be rewritten as where CH 2 (Z) denotes the second Chow group of codimension 2 cycles modulo rational equivalence.
When CH 2 (Z)/m vanishes, the obstructions given by H 3 (Z, µ ⊗2 m ) and H 3 nr (Z, µ ⊗2 m ) are the same.In general, the obstruction given by H 3 nr (Z, Q/Z(2)) is potentially more restrictive than the obstruction given by H 3 nr (Z, Q/Z(2)).
Remark 3.5.In the literature of strong approximation problems away from S over a number field F , the Brauer-Manin obstruction used to be defined using the projection π S : Z(A F ) → Z(A S F ), saying that the Brauer-Manin obstruction to strong approximation away from S is the only one if Z(F ) is dense in π S (Z(A) Br ).Demeio introduced the modified Brauer group of "trivial on S" elements (cf.§6 of [Dem22]) and the corresponding Brauer set Z(A S F ) Br S (for which we can also replace Br S Z by Br S Z/ Im(Br S F )).This gives a less restrictive obstruction compared to the one defined using projection, since π S (Z(A F ) Br ) ⊆ Z(A S F ) Br S ; moreover, we have Z(F ) S ⊆ Z(A S F ) Br S while π S (Z(A F ) Br ) can be strictly smaller than Z(F ) S (cf.Proposition 4.8 of [Dem22]).We adopt this point of view and we will show later (cf. Theorem 3.10 and 3.11) that in our case, the group H 3 nr,S (Z, Q/Z(2)) of "trivial on S" elements cuts out exactly the closure Z(K) S with respect to the reciprocity obstruction.

Application to classifying varieties
Now we calculate this reciprocity obstruction we just defined for classifying varieties.Let H be an algebraic group over K. Let Z = E/H be a classifying variety where the special rational group E is split semisimple simply connected (for example, E = SL n or Sp 2n ).
The following morphism is defined by evaluating an invariant I ∈ Inv 3 (H, Q/Z(2)) at the generic H-torsor ξ * E (the fiber of the H-torsor E → Z above the generic point ξ ∈ Z): Rost proved that this map is injective, and more precisely, there is an isomorphism (cf.Part 2, Theorem 3.3 and Part 1, Appendix C of [GMS03]).The decomposition then induces an isomorphism and modulo the constant elements, we get Proposition 3.6.There is a well-defined pairing induced by evaluating an invariant I at the H-torsors x * v E over K v , where x * v E denotes the fiber of the H-torsor E → Z above the point x v ∈ Z(K v ).This pairing ( 13) is compatible with (9) in the sense of the commutative diagram: ) and x ∈ Z(F ) be a point over any field extension F/K.We use the description of the isomorphism θ in Appendix A.2, p.461 of [Mer02].We define the class b(x) ∈ H 3 (F, Q/Z(2)) as the image of b under the pull-back morphism with respect to x : Spec F → Z. Thus we get a map which is exactly how we defined our pairing (10).This ÎK defines an invariant I ∈ Inv 3 (H, Q/Z(2)) with θ(I) = b since the map b is constant on the orbits of the E(K)-action on Z(E).Therefore, we have a commutative diagram Now we sum up these maps and quotient by H 3 (K, Q/Z(2)) and we get the desired result.
As a result, there is an reciprocity approximation to strong approximation which accounts for our constructions in Section 2. Now we study more precise questions of how this reciprocity obstruction controls strong approximation, for example, is this obstruction the only one?For this end, we first need to take a closer took at our p-adic function field K. Proposition 3.7.Let X be a smooth proper geometrically integral curve over a p-adic field k, with index I(X) and function field K. gcd(deg v,m 0 ) , which is also the exponent of A Kv because of the injection Br κ(v) ֒→ Br K v .Each exponent m v of A Kv divides the exponent m of A, and thus their least common multiple lcm(m v ) v∈X (1) divides m.On the other hand lcm(m v ) v∈X (1) annihilates all the v * [A Kv ], and thus lcm(m v ) v∈X (1) [A] ∈ Br vanishes at all the closed points v ∈ X (1) .By Lichtenbaum's duality (5), this implies that lcm(m v ) v∈X (1) [A] = 0 ∈ Br X and hence m| lcm(m v ) v∈X (1) .Therefore, we have m = lcm(m v ) v∈X (1) .But

and we can always choose
giving the first statement.
In particular, taking m = 1 shows that Ker(Br k → Br K) is the cyclic subgroup of order I(X) and Remark 3.8.In fact, the particular case m = 1 showing that Ker(Br k → Br K) has order I(X) gives exactly Theorem 1 of [Roq66] and Theorem 3 of [Lic69], where in the latter paper Lichtenbaum reduced the calculations to the period of X and used the duality (6).
Remark 3.9.In the proof of Proposition 3.7, we see that Lichtenbaum's duality (5) can actually give a local-global principle for Br K, i.e. the natural map Br K v is injective.If α ∈ Br K vanishes in Br K v , then α is unramified because the residue map factors through the completion.Hence α lies in Br X with evaluation v * (α) = 0 ∈ Br κ(v) for all places v, and the duality (5) shows that α = 0 ∈ Br K. See also Corollary 10.5.5 and Remark 10.5.7(2) of [CTS21].
Theorem 3.10.Let Z = E/ SL 1 (A) as in Construction 2.3, with A of exponent m.For the p-adic curve X, suppose Pic 0 (X)/m = 0, or equivalently m H 1 (k, Pic X) = 0 by the Lichtenbaum duality (6) (for example X = P 1 k the projective line satisfies this condition).Then there is an exact sequence of pointed sets for S ⊆ X (1) a non-empty finite set of places.In particular, the reciprocity obstruction to strong approximation away from S is the only one for Z.
The group H 3 nr,S (Z, Q/Z(2)) measures the defect of strong approximation away from S for Z, and is finite cyclic of order gcd( I(S) I(X) , m) where I(X) (resp.I(S)) is the index of X (resp.S).In particular, strong approximation away from S holds for Z if and only of I(S)/I(X) is coprime to m, and such an S always exists, for example S such that I(S) = I(X).
Proof.We claim that there is a commutative diagram as follows, and the right column is an exact Given any l ∈ Z/mZ, there exists v∈X (1) by the exact sequence (15).Then (l v ) v ∈ v∈X (1) Z/m v Z has image l = Z/mZ under the map f H , proving the exactness at the third term.Definition and commutativity of the bottom square.By Proposition 2.4, we can identify H 3 nr (Z, Q/Z(2)) D with (Inv 3 (H, Q/Z(2)) norm ) D in the sequence.Since Inv 3 (H, Q/Z(2)) norm is a finite cyclic group of order m generated by the Rost invariant R H , we have an isomorphism (Inv 3 (H, Q/Z(2)) norm ) D → Z/mZ by evaluating an element at R H . Then the formula (13), the definition of f H , and the identification of Z/mZ as the cyclic subgroup of Z/mI(X)Z generated by I(X) altogether give the commutativity.Now consider S = ∅.We use the identification (12).Under the field extension map, the Rost invariant R H maps to the Rost invariant R H Kv of order m v .Therefore, an element n 0 R H ∈ is a generator of this cyclic group of order gcd( I(S) I(X) , m), and evaluation at this generator gives an isomorphism H 3 nr,S (Z, Q/Z(2)) D → Z/ gcd( I(S) I(X) , m)Z.Any (n v ) v ∈ v∈X (1) \S Z/m v Z can be completed by elements from v∈S Z/m v Z, whose images under f H give exactly the subgroup generated by I(S)/I(X) in Z/mZ.Therefore, the exactness of the right column and also the commutativity follow from the previous case S = ∅.
Since the special rational group E is split semisimple simply connected, it satisfies strong approximation away from S (cf.Corollary 1.8).The rows in the diagram are exact sequences, then a diagram chasing gives the desired result.
As another application of this reciprocity obstruction to strong approximation, we study classifying varieties of tori.
Theorem 3.11.Let Z = E/T be a classifying variety of a torus T over K a p-adic function field, where the special rational group E is split semisimple simply connected.Then there is an exact sequence of pointed sets for S ⊆ X (1) a non-empty finite set of places.In other words, the reciprocity obstruction to strong approximation away from S is the only one for Z.If we suppose furthermore that CH 2 (Z) → H 0 (K, CH 2 (Z)) is surjective, then there is an exact sequence of pointed sets which are defined exactly by using (20).Taking T to be T ′ in (23), and noting that any elements of P 1 S (T ′ ) can be completed by 0's into an element of P 1 (T ′ ), we get an exact sequence which can be completed into the exact sequence by definition.The pairing (24) is actually a perfect duality, and the subgroups ) are exact annihilators of each other (Proposition 1.1 and 1.3 of [HSS15]).Therefore, by dualizing (25), we get the exactness of the right column in (18).The commutative diagram Now applying Proposition 3.6 and the compatibility between the morphism α (20) and the pairing (24), we get the commutativity of the two squares at the bottom of (18).Since E satisfies strong approximation away from S (cf.Corollary 1.8), chasing the diagram (18) gives the desired exactness of (16).
Under the further assumption of the surjectivity of CH 2 (Z) → H 0 (K, CH 2 (Z)), the morphism α is thus surjective by (19), and so is the induced morphism (26).Hence the two bottom rows in (18) are injective.Now a diagram chasing gives the exact sequence (17).
Example 3.12.We consider the torus T ′ = R L/K (G m,L )/G m in Example 4.14 of [BM13], where L/K is a degree n field extension induced by a continuous surjective group morphism from the absolute Galois group of K to the symmetric group S n .For example, the generic maximal torus of the group PGL n is of this form.The dual torus T of T ′ is the norm one torus R (1) L/K (G m,L ).Let Z = E/T be a classifying variety.Then H 0 (K, CH 2 (Z))/ Im(CH(Z)) is trivial, and we have H 3 nr (Z, Q/Z(2)) ≃ Br(L/K).Since local-global principle holds for the Brauer group of K (cf.Remark 3.9), the group H 3 nr,X (1) (Z, Q/Z(2)) is trivial.We apply (17) and get the exact sequence of pointed sets Remark 3.13.The exact sequence in Theorem 3.10 actually also takes the form (17), because for some finite set S ′ such that I(S ′ ) = I(X).Therefore, we may hope this exact sequence to hold in other more general situations.
In fact, the commutative diagram in the proof of Proposition 3.6 showing local compatibility is generally true for Inv d (H, Q/Z(d−1)) and H d nr (Z, Q/Z(d−1)) over any field of characteristic 0 (while in characteristic p, we can identify Inv d (H, Q/Z(d − 1)) with the subgroup of "balanced elements" in )) under certain conditions, cf.Theorem A of [BM13]).In particular, taking d = 2, we get the group Inv(H, Br) of invariants with values in the Brauer group, and This enables us to relate Inv(H, Br) with the Brauer-Manin pairing, which is used for studying the arithmetic of varieties over a number field F , for which we get the following commutative diagram Now we explain how this is compatible with known results of strong approximation over F a number field.
For H a connected linear group over a field K of characteristic 0, Blinstein and Merkurjev (cf.Theorem 2.4 of [BM13]) showed that with the same proof as in Theorem A of [BM13].Now for F a number field, the above diagram (27) along with the identification (29) gives rise to the following commutative diagram, where the right column is an exact sequence of pointed sets (cf.Corollary 2.5 and Proposition 2.6 of [Kot86]): ≃ For S ⊆ Ω F such that E satisfies strong approximation away from S (e.g.E = SL n with any nonempty finite set S), a diagram chasing argument shows that the Brauer-Manin obstruction to strong approximation away from S (defined using projection and the whole Brauer group, cf.Remark 3.5) is the only one for the classifying variety Z. See Theorem 3.7 of [CTX09] where these results were obtained for the first time.Now we apply Demeio's version of obstruction given by Br S Z (see Remark 3.5).Let S be a set containing only non-archimedean places and denote by Z(A F ) • the adelic points where each Now we explain that this is compatible with our reciprocity obstruction given by H 3 nr (T, Q/Z(2)).
Proposition 3.15.There is a morphism H 2 (K, T ′ ) → H 3 nr (T, Q/Z(2)) which fits into the following commutative diagram Proof.Given an element a : Spec K v → T in H 0 (K v , T ), the functoriality of the cup-products induced by T ⊗ T ′ → Z(2)[2] gives the following commutative diagram: where a * denotes the pullback maps induced by a.The identity Id ∈ H 0 (T, T ) is mapped to a under a * , and thus we get the commutativity of the upper part of the following diagram: where the morphism H 2 (K, T ′ ) → H 4 (T, Z(2)) is given by taking the cup-product with Id ∈ H 0 (T, T ).Now we construct and prove the commutativity of the lower part of this diagram.Recall that (cf.Proposition 2.9 of [Kah12]) for a smooth K-variety Z, we have a natural map H 4 (Z, Z(2)) → H 3 nr (Z, Q/Z(2)) defined as follows.Let α : Z ét → Z Zar be the change-of-sites map.The natural map Q(2) Zar → Rα * Q(2) is an isomorphism in the derived category of Zariski sheaves (cf.Lemma 2.5 and Theorem 2.6 of [Kah12]).Since Q(2) Zar is concentrated in degree ≤ 2, we have R 4 α * Q(2) = R 3 α * Q(2) = 0, hence R 4 α * Z(2) ≃ R 3 α * Q/Z(2).Then the Leray spectral sequence for α yields an edge map The construction of the Gersten resolution implies we have the commutative diagram H 4 (Z, Z(2)) H 4 (K(Z), Z(2)) where H 4 (Z, Z(2)) → H 4 (K(Z), Z(2)) is the restriction map.This gives the commutativity of the lower part of (33).Proof.Proposition 3.15 along with (32) gives the exact sequence Now apply Lemme 2.7 and Théorème 6.6 of [HI19], and we get the first injection.The morphism constructed in Proposition 3.15 gives H 2 (K, T ′ )/m → H 3 nr (T, Q/Z(2))/m for all m > 0, and thus a morphism H 2 (K, T ′ ) ∧ → H 3 nr (T, Q/Z(2)) ∧ , compatible with the pairing in Theorem 3.14, yielding the second injection.

Weak approximation for classifying varieties
In the study of weak approximation problems over K a p-adic function field, we can define a pairing (cf.[HSS15]) whose left kernel contains the closure Z(K) inside v∈X (1) Z(K v ) with respect to the product topology, defining the reciprocity obstruction to weak approximation.For Z = E/H a classifying variety where the special rational group E is split semisimple simply connected, we can establish the same compatibility between the reciprocity obstruction and the cohomological invariants.
An invariant I ∈ Inv 3 (H, Q/Z(2)) is called unramified if for every field extension F/K and every element H 1 (F, H), we have I(a) in H 3 nr (F/K, Q/Z(2)) the group of elements unramified with respect to all discrete valuations of F trivial on K.Under the identification (11), the subgroup H 3 nr (K(Z)/K, Q/Z(2)) of H 3 nr (Z, Q/Z(2)) is identified with the group Inv 3 nr (H, Q/Z(2)) of unramified invariants (cf.Proposition 4.1 of [Mer16]).We denote by Inv 3 nr (H, Q/Z(2)) norm the group of unramified normalized invariants.induced by evaluating an invariant I at the H-torsors x * v E over K v , where x * v E denotes the fiber of the H-torsor E → Z above the point x v ∈ Z(K v ).This pairing is compatible with (35) in the sense of the following commutative diagram Proof.Same as that of Proposition 3.6.
As an application, we show how this method gives answers to weak approximation problems for classifying varieties of tori over K.The following result was also obtained by Linh in a different way in his very recent work (cf.Theorem B of [Lin22]).Now we give our proof using cohomological invariants.
Theorem 4.2.Let Z = E/T be a classifying variety of a torus T over K. Then the reciprocity obstruction to weak approximation is the only one for Z.In fact, there is a morphism X 1 S (T ′ ) → H 3 nr (K(Z)/K, Q/Z(2))/ Im(H 3 (K, Q/Z(2))) It is still an open question whether weak approximation holds for Z = SL n /H where H is a semisimple simply connected group over K a p-adic function field.In case of a negative answer, we might hope to find a reciprocity obstruction using H 3 nr (K(Z)/K, Q/Z(2)) ≃ Inv 3 nr (H, Q/Z(2)).Merkurjev studied Inv 3 nr (H, Q/Z(2)) for all classical semisimple simply connected groups G, and he showed in [Mer02] that Inv 3 nr (H, Q/Z(2)) norm is trivial except for H of type 2 A n−1 , 2 D 3 or 1 D 4 under certain conditions, where Inv 3 nr (H, Q/Z(2)) norm ≃ Z/2Z.Garibaldi computed the remaining cases when H is exceptional, i.e. when H is of type G 2 , 3 D 4 , 6 D 4 , F 4 , E 6 , E 7 or E 8 , and he showed in [Gar06] that for H a simple simply connected exceptional algebraic group, the group Inv 3 nr (G, Q/Z(2)) norm is Z/2Z if H is of type 3 D 4 with a nontrivial Tits algebra; otherwise Inv 3 nr (H, Q/Z(2)) norm is trivial.It would be interesting to see if weak approximation could fail for SL n /H in the case of a non-trivial Inv 3 nr (H, Q/Z(2)) norm .

Proposition 4. 1 .
There is a well-defined pairingv∈X (1) Z(K v ) × Inv 3 nr (H, Q/Z(2)) → Q/Z ((x v ) v , I) → v∈X (1) Cor κ(v)/k (∂ v (I Kv (x * v E))) 1 for all field extensions F/K.Let H be a smooth algebraic group over K. Choose an embedding ρ : H ֒→ E into a special rational group E. Examples of special rational groups E include SL n , Sp 2n , GL 1 (A) for a central simple K-algebra A. The variety Z = E/ρ(H) is called a classifying variety of H. Different choices of ρ give stably birational classifying varieties.(cf.§2 of [Mer02].) Br) ≃ Br Z whose construction and proof are based on the exact sequence Pic W → Pic H → Br Y → Br W (28) associated to a smooth K-variety Y and a Y -torsor W under H, as defined in Proposition 6.10 of [San81].In fact, if we use the exact sequence in Theorem 2.8 of [BD13] which is of the same form as (28) but with Pic H → Br Y constructed in a different way via the abelian group Ext c F (H, G m ) of isomorphism classes of central extensions of K-algebraic groups of H by G m , we can still get an isomorphism Pic H ⊕ Br K ≃Inv(H, Br) ≃ Br Z