Arithmetic of linear algebraic groups over two-dimensional geometric fields

by J.-L. Colliot-Thélène, P. Gille and R. Parimala

Dedicated to M. S. Raghunathan on his sixtieth birthday

§0. Introduction

Let K be the field of fractions of a two-dimensional, excellent, henselian, local domain with algebraically closed residue field of characteristic zero. In this introduction, we shall refer to such a field as a two-dimensional, strictly henselian field (indeed a henselian local ring with a separably closed residue field is called a strictly henselian local ring). An example of such a field is the field of fractions C((X, Y)) of the formal power series ring C[[X, Y]].

In the paper [CTOP], Ojanguren and two of the authors investigated quadratic forms over such a field K, and established properties analogous to those familiar over number fields. An analogue of the local-global principle for the Brauer group (going back to work of M. Artin) was also established.

In the present paper, we show that most well-known properties of linear algebraic groups and of their homogeneous spaces over totally imaginary number fields have counterparts for such groups over a two-dimensional strictly henselian field.

To prove this, we recall (§1) that both types of fields share a number of properties :

(i) their cohomological dimension is two ;

(ii) index and exponent of central simple algebras coincide ;

(iii) for any semisimple simply connected group G over K, we have H^{1}(K, G) = 1 ; (iv) on fields of either type, there is a natural set of rank one discrete valuations (i.e. with

values in Z), with respect to which one can take completions and then investigate such natural questions as the local-global principle and weak approximation for homogeneous spaces of linear algebraic groups.

We recall how properties (i) and (ii) are enough to deduce (iii) for groups without E_{8}-factors. Properties (i), (ii) and (iii) are satified by other fields of interest, such as the Laurent series

field l((t)) over a field l (of characteristic zero) of cohomological dimension one (such fields occur as completions of two-dimensional strictly henselian fields).

For function fields in two variables over an algebraically closed field of characteristic zero, (i) holds and (ii) is known for algebras of 2-primary or 3-primary index^{(1)}.

In §2, we show that if a field K satifies properties (i) and (ii), then for the standard isogeny

1 → μ → G^{sc} → G^{ad} → 1

between a simply connected algebraic group and its adjoint group, the associated boundary map H^{1}(K, G^{ad}) → H^{2}(K, μ) is onto. This is a generalization of a classical result over number fields. Using the results from §1, we simultaneously prove that if a semisimple group G over such a field K is not purely of type A, then it is isotropic. Again this is the generalization of a classical result over number fields.

In §3, for tori over two-dimensional strictly henselian fields, we establish the analogues of results of [CTS1] for tori over number fields : finiteness of R-equivalence on the set of rational points, computation of groups measuring the failure of weak approximation and of the Hasse

(1) This actually holds for any index, as recently proved by A. J. de Jong. For most problems studied in our paper, only the 2-primary and 3-primary parts of Brauer groups are relevant. One notable exception is Theorem 2.1, which by de Jong’s theorem now applies to function fields in two variables over an algebraically closed field of characteristic zero. principle for homogeneous spaces. Finiteness of R-equivalence holds for higher dimensional local henselian fields, and also for function fields.

In §4, we study the same problems for arbitrary connected linear algebraic groups. Here the restriction to fields of cohomological dimension two is crucial. We first discuss semisimple simply connected groups. Using a series of earlier works and the isotropy result of §2, we show that if the field K satisfies properties (i) and (ii) (and an additional property if factors of type E_{8} are allowed) then for any semisimple simply connected connected group G over K, the group G(K)/R is trivial. If K is a two-dimensional strictly henselian field, then weak approximation holds for such a group G. We then go on to study arbitrary linear algebraic groups over a twodimensional strictly henselian field. The analysis of [Gi1] and [Gi2] can be carried out : we thus show that for such a group G, the group G(K)/R is finite, and we give formulas for that group and for the group measuring the lack of weak approximation on G (with respect to a finite set of places). For G without E_{8}-factor, a number of properties, in particular finiteness of G(K)/R, also hold for K a two-dimensional function field.

In §5, we study the Hasse principle for homogeneous spaces. We prove that the Hasse principle holds for principal homogeneous spaces of a semisimple group G over a two-dimensional strictly henselian field, if G is adjoint or if the underlying K-variety of G is K-rational. The main result is a proof of the Hasse principle for projective homogeneous spaces of an arbitrary connected linear group over such a field. The proof uses the surjectivity statement established in §2, which enables us to adapt Borovoi’s proof in the number field case to the present context.

The main results of the paper were announced in the note [CTGP].

R-equivalence

Let F be a covariant functor from commutative k-algebras to sets. Let O denote the semilocal ring of the polynomial algebra k[t] at the points t = 0 and t = 1. Let us say that two elements a, b ∈ F (k) are elementarily related if there exists ξ ∈ F (O) such that ξ(0) = a and ξ(1) = b. By definition, R-equivalence on F (k) is the equivalence relation generated by the previous elementary relation. Thus two elements a, b ∈ F (k) are R-equivalent if and only if there exists a finite set of elements x_{0}, …, x_{n+1} ∈ F (k), with x_{0} = a and x_{n+1} = b, such that x_{i} is related to x_{i+1} for 0 ≤ i ≤ n. One lets F (k)/R denote the quotient set for this equivalence relation. For any field K containing k, one defines a similar equivalence relation on F (K) by using the semilocal ring of K[t] at the points t = 0 and t = 1. There is a natural, functorial map F (k)/R → F (K)/R.

If F = F_{X} is the functor associated to a k-variety X, namely F_{X}(A) = X(A), we get the R-equivalence on X(k), as defined by Manin. Suppose F = F_{G}, where G/k is a linear algebraic group (see [CTS1], [Gi1]). Then the set of k-points R-equivalent to 1 ∈ G(k) is a normal subgroup ; if k is infinite, then any k-point R-equivalent to 1 ∈ G(k) is elementarily related to 1 ([Gi1], Lemme II.1.1). The quotient G(k)/R is thus equipped with a natural group structure. For G linear connected, it is an open question whether this group is commutative (commutativity is known for simply connected groups of classical type).

In these notes, as in [Gi2], we shall also be concerned with the functors F defined by F (A) = H_{ét}^{1}(A, S), where S is a commutative k-group scheme.

Weak approximation

Let K be a field equipped with a set Ω of rank one discrete valuations (i.e. valuations the value group of which is Z). For each v ∈ Ω, let K_{v} denote the completion of K at v. For any K-variety X, and for any v ∈ Ω, the topology on K_{v} induces a topology on X(K_{v}). One says that weak approximation holds for X and a finite set S ⊂ Ω if the diagonal map X(K) → ∏_{v∈S} X(K_{v}) has dense image.

Acknowledgments We thank the Indo-French Centre for the Promotion of Advanced Research (IFCPAR) for continuous support of this project. Part of the paper grew out of lectures given at the Ecole de Printemps de l’Université d’Artois à Lens (June 2001). We thank the referees for their careful reading.

§1. The fields under consideration and their basic properties (a review)

Let k be an algebraically closed field of characteristic zero. We shall mainly be interested in fields of one of the following types :

(gl) a function field K in two variables over k, i.e. the function field of a smooth, projective, connected surface over k;

(ll) the field of fractions K of a henselian, excellent, two-dimensional local domain A with residue field k;

(sl) the Laurent series field l((t)) over a field l of characteristic zero and cohomological dimension 1.

For the definition of an excellent ring, see [EGA IV_{2}] 7.8.2 and 5.6.2 (and refer to [EGA IV_{1}] 16.1.4 and 14.3.2 for the definition of a “catenary” ring). See also [Ma], chap. 13.

Note that any finite field extension of a field of one of these types is of the same type. For type (sl), we shall mainly be interested in the case where l is either the function field of

a curve over k, or is itself a Laurent series field in one variable k((u)).

Let us now consider the following properties of a field K, which we assume to be of characteristic 0.

(a) The field K has the C_{2} property : that is, for any r ≥ 1 and any system of homogeneous forms f_{j}(X_{1}, …, X_{n}) (j = 1, …, r) with coefficients in K, if n > ∑_{j} d_{j}^{2}, where d_{j} is the degree of f_{j}, then there is a nontrivial common zero in K^{n} for these r forms. This property is preserved under finite field extensions ([Pf], Chap. 5, Thm. 1.3).

(b) The cohomological dimension cd(K) is at most 2. This property is preserved under algebraic field extensions ([Se2] §I.3.3, Prop. 14 p. 17).

(c) The reduced norm is surjective. That is, for any finite field extension L/K and any central simple algebra D/L, the reduced norm map Nrd : D^{∗} → L^{∗} is surjective.

(d) For a given prime p, index and exponent of p-primary algebras coincide : That is, for any finite field extension L/K, and any central simple p-primary division algebra D/L, the index of D, i.e. the square root of its dimension, and the exponent of D, i.e. the order of its class in the Brauer group of L, coincide. (An algebra is called p-primary if its order in the Brauer group is a power of p; equivalently, it is equivalent to a central simple algebra of index a power of p.)

(e) For a given prime p, over any finite field extension L of K, the tensor product of two central simple algebras of index p is of index at most p.

(f) Over any finite field extension L/K, any quadratic form in at least 5 variables has a nontrivial zero.

(g) The cohomological dimension cd(K^{ab}) of the maximal abelian extension of K is at most one (this property is stable under algebraic field extensions).

Here are the known relations between these various properties.

Theorem 1.1 Let K be a field of characteristic zero. The following implications hold :

(i) (a) implies (b) (ii) (b) is equivalent to (c). (iii) (a) implies (f). (iv) (d) is equivalent to (e).

(v) (f) implies (d) for p = 2.

(vi) (a) implies (d) for p = 2 and p = 3.

(vii) the combination of (b) and (e) for p = 2 implies (f).

Proof Statement (ii) is a consequence of the Merkurjev-Suslin results ([Su], Thm. 24.8). Statement (i) is a direct consequence of (ii). Statement (iii) is trivial. For statement (iv), which also relies on results of Merkurjev and Suslin, we refer to the proof of Prop. 7 in [CTG] and the references given there. Statement (v) follows from a well-known criterion of Albert (see [CTG]) together with Statement (iv) for p = 2. For the proof of (vi), see [CTG], Prop. 7. For the proof of (vii), see [CTOP], proof of Theorem 3.4.

Recall Serre’s conjecture II : for a perfect field K of cohomological dimension 2 and a simply connected group G/K, the set H^{1}(K, G) is reduced to one element. The following theorem gathers results known in that direction.

Theorem 1.2 Let K be a field of characteristic zero and of cohomological dimension at most 2. Let G/K be a simply connected group.

(i) If G is absolutely almost simple of classical type, or is of type G_{2} or F_{4}, then H^{1}(K, G) = 1. (ii) Suppose that index and exponent coincide for 2-primary algebras over finite field exten-

sions of K. Then H^{1}(K, G) = 1 for G of trialitarian type D_{4} and for G of type E_{7}.

(iii) Suppose that index and exponent coincide for 3-primary algebras over finite field extensions of K. Then H^{1}(K, G) = 1 for G of type E_{6}.

(iv) Suppose that the cohomological dimension of K^{ab} is at most one. If G is of type E_{8}, then H^{1}(K, G) = 1.

(v) Suppose that index and exponent coincide for 2-primary and 3-primary algebras over finite field extensions of K. If G has no factor of type E_{8}, then H^{1}(K, G) = 1, and the same holds in general if moreover the cohomological dimension of K^{ab} is at most one.

Proof For G of type ^{1}A_{n}, statement (i) is a consequence of the Merkurjev-Suslin theorems, as noticed by Suslin [Su]. For the other groups of classical type, the result is due to Bayer-Fluckiger and Parimala [BFP]. When exponent and index of algebras coincide, note that the proofs of [BFP] may be shortened. Statements (ii), (iii), (iv) are due to Gille. For trialitarian type D_{4}, this is [Gi2], Thm. 8 p. 313. For E_{6} (either ^{1}E_{6} or ^{2}E_{6}), this is [Gi2], Thm. 9. p. 314. For E_{7}, this is [Gi2], Thm. 10 p. 316. The argument for the E_{8} case (briefly sketched on p. 322 of [Gi2]) is inspired by the number theory case (Chernousov). Let G_{0} be the split group of type E_{8}. Over any field K, any element of H^{1}(K, G_{0}) is in the image of H^{1}(K, T) for the natural embedding T ⊂ G_{0} of a suitable maximal torus T of G (Steinberg, see [Se2], §III.2.3, Corollaire, p. 140). For T ⊂ G_{0} a maximal torus, the group H^{1}(K, T) is killed by a power of 2.3.5 (Harder, Tits, cf. [Se3], Thm. 3 p. 234). Under the assumption that cd(K^{ab}) ≤ 1, this implies that any element of H^{1}(K, G_{0}) belongs to the image of H^{1}(L/K, G_{0}) for some finite abelian extension of degree 2^{a}.3^{b}.5^{c}. Theorem 11, p. 317 of [Gi2], states that for G_{0} split of type E_{8} over a field K of characteristic zero and of cohomological dimension at most 2, and L/K cyclic of degree 2, 3 or 5, we have H^{1}(L/K, G_{0}) = 1. By induction, we conclude that H^{1}(K, G_{0}) = 1. Since the centre of G_{0} is trivial and there are no outer automorphisms, we conclude that the only form of G_{0} is the split form, which completes the proof of (iv).

The proof of (v) involves a well-known result, which will be used a number of times in this paper, and which we shall refer to as the standard reduction to the almost simple case. Here is the statement : If a semisimple group G/K is simply connected, resp. adjoint, then it is K-isomorphic to a product ∏_{i} R_{K_{i}/K}G_{i}, where the K_{i}/K are finite field extensions, R_{K_{i}/K} denotes the Weil restriction of scalars, and each G_{i} is an absolutely almost simple K_{i}-group which is simply connected, resp. adjoint. In the simply connected case, the centre of G is the product of the R_{K_{i}/K}C_{i}, where C_{i} denotes the centre of G_{i}.

Statement (v) now follows from the other statements thanks to the Shapiro type formula H^{1}(K, G) = ∏_{i} H^{1}(K_{i}, G_{i}).

Remark For the exceptional groups, an independent proof of Theorem 1.2 has been given by Chernousov [Ch].

We now specialize to the fields of “geometric type” of interest in this paper.

Theorem 1.3 (case (gl)) Let K be a function field in two variables over an algebraically closed field k of characteristic zero. The field K is a C_{2}-field, it has cd(K) = 2. Over K, index and exponent of a 2-primary or a 3-primary algebra coincide^{(2)}. If G/K is a semisimple simply connected group without E_{8}-factor, then H^{1}(K, G) = 1.

Proof That K is a C_{2}-field is a special case of the Tsen-Lang theorem (see [Pf], Chap. 5). That K has cohomological dimension 2 is a consequence of Theorem 1.1 (i) above, but it had been known from the very beginning of Galois cohomology ([Se2], II.4.2, Corollaire, p. 95). The rest of the theorem follows from Theorems 1.1 and 1.2.

Theorem 1.4 (case (ll)) Let K be the field of fractions of a two-dimensional, excellent, henselian local domain A with residue field k algebraically closed of characteristic zero. The field K has cd(K) = 2. All central simple algebras over K are cyclic, of exponent equal to the index. Quadratic forms in at least 5 variables have a nontrivial zero. The cohomological dimension cd(K^{ab}) of the maximal abelian extension of K is one. For any semisimple simply connected group G/K, we have H^{1}(K, G) = 1.

Proof That cd(K) = 2 is a special case of result of M. Artin ([SGA 4] XIX Cor. 6.3). That index and exponent coincide is also a result of M. Artin ([A], Theorem 1.1). That algebras are cyclic (of index equal to the exponent) is a result of T. Ford and D. Saltman in the geometric case, see Thm 2.1 of [CTOP] for the general statement. That cd(K^{ab}) ≤ 1 is Thm. 2.2 of [CTOP]. The rest of the theorem follows from Theorems 1.1 and 1.2.

Theorem 1.5 (case (sl)) Let K be the Laurent series field l((t)) over a field l of characteristic zero and cohomological dimension 1. Such a field is of cohomological dimension 2. Quadratic forms in at least 5 variables have a nontrivial zero. Index and exponent of algebras coincide. For any semisimple simply connected group G/K, we have H^{1}(K, G) = 1.

Proof For the property cd(K) = 2, see [Se2], II.4.3, Prop. 12 p. 95. The proof of the statement for quadratic forms is well-known ([Sc], §6.2, Corollary 2.6.(iv), p. 209). Any central simple algebra over K = k((t)) is similar to a cyclic algebra (K/k, σ, t), where K/k is a finite cyclic field extension and σ is a generator of the Galois group of K/k. Such an algebra being clearly a division algebra, its index is equal to its exponent. That Serre’s conjecture H^{1}(K, G) = 1 holds in this context is a theorem of Bruhat and Tits ([BT], Théorème 4.7).

Remark Fields of type (sl) were studied by J.-C. Douai in his thesis (Lille, 1976). Building upon the work of Bruhat and Tits, for such fields he established Theorem 2.1 (a), Prop. 5.3 and Prop. 5.4 of the present paper.

For fields of type (gl), there are various natural completions. More generally, let K be a function field of transcendence degree d over k (we shall mostly be interested in the case d = 2). For X/k a smooth projective model of K, let X^{1} denote the set of codimension 1 points on X, and let Ω_{X} denote the set of discrete valuations v associated to points x ∈ X^{1}. The completion K_{x} = K_{v} of K at such a discrete valuation ring is isomorphic to l((t)), where l = κ(x) is the

(2) This actually holds for arbitrary algebras, as recently proved by A. J. de Jong. residue field at v = v_{x}, which is a function field in d − 1 variables. One then denotes by Ω the union over all models X of the Ω_{X}.

For any X, the kernel of the diagonal restriction map on Brauer groups

Br(K) → ⊕_{v∈Ω_{X}} Br(K_{v})

is contained in the subgroup Br(X) ⊂ Br(K), and it is equal to that subgroup in the (gl) case under consideration above, namely if d = 2. The same two statements hold for the kernel of the diagonal restriction map

Br(K) → ∏_{v∈Ω} Br(K_{v}).

This torsion group is an extension of a finite group by a group of the shape (Q/Z)^{r}. For all this, see Grothendieck [Gr].

Similarly, for fields of type (ll), there are various natural completions. Let K be the field of fractions of a strictly henselian, two-dimensional, excellent, local domain A with residue field k algebraically closed of characteristic zero.

Let X be a regular, integral, two-dimensional scheme equipped with a proper birational morphism X → Spec(A). Such schemes exist (for references, see [CTOP]). The function field of X is K. Let Ω_{X} denote the set of discrete valuations v associated to points x ∈ X^{1}. If x lies above a codimension 1 point on Spec(A), then K_{v} is isomorphic to F_{v}((x)), where the field F_{v} is the field of fractions of a discrete, henselian valuation ring with residue field k. If x lies above the closed point of Spec(A), then K_{v} is isomorphic to F_{v}((x)), and F_{v} is the function field of a smooth, projective curve over k. In both cases, Theorem 1.5 ensures that K_{v} satisfies properties (a) to (g) listed before Theorem 1.1.

Given A ⊂ K as above, we let Ω = Ω_{A} be the union of all Ω_{X} for X varying among the regular integral schemes X equipped with a proper birational morphism X → Spec(A).

We refer to §1 of [CTOP] for the proof of the following theorem.

Theorem 1.6 Let K be the field of fractions of a strictly henselian two-dimensional local domain A with residue field k. Let X → Spec(A) be a regular proper model. With notation as above, the kernel of the diagonal restriction map on Brauer groups

Br(K) → ⊕_{v∈Ω_{X}} Br(K_{v})

is trivial. So is a fortiori the kernel of the diagonal restriction map on Brauer groups

Br(K) → ∏_{v∈Ω} Br(K_{v}).

§2. Surjectivity of the boundary map and isotropy

Part (a) of the following theorem was stated as Conjecture 5.4 in [CTOP]. It will play a key rôle in §5 (Hasse principle for projective homogeneous spaces over fields of type (ll)). Part (b) will play a key rôle in §4 (R-equivalence and weak approximation).

Theorem 2.1 Let k be a field of characteristic zero. Assume :

(i) the cohomological dimension cd(k) of k is at most 2 ;

(ii) for central simple algebras over any finite field extension of k, index coincides with exponent.

Let G be a semisimple, simply connected group over k. If G has some factor of type E_{8}, assume that the cohomological dimension of the maximal abelian extension k^{ab} of k is at most one. Let

1 → μ → G → G^{ad} → 1

be the central isogeny associated to the centre μ of G. Then :

(a) The boundary map δ : H^{1}(k, G^{ad}) → H^{2}(k, μ) is a bijection.

(b) If the group G is not purely of type A, then it is isotropic.

(c) Groups of type F_{4}, G_{2}, E_{8} are split, and groups of type ^{3,6}D_{4}, ^{1,2}E_{6}, E_{7} which are not split have the following Tits indices :

3,6D_{4}

1,2E_{6}

E_{7}

Proof The proof will occupy the whole section.

By the standard reduction recalled in the proof of Theorem 1.2, to prove the theorem one may assume that G is absolutely almost simple.

By Theorem 1.2, we have H^{1}(k, G) = 0 for any semisimple simply connected group G. This already implies that the connecting map H^{1}(k, G^{ad}) → H^{2}(k, μ) is one-to-one.

Let G_{0}, resp. G_{0}^{ad} be the unique, inner quasisplit form of G, resp. G^{ad}. We have the two exact sequences

1 → μ → G_{0} → G_{0}^{ad} → 1

and

1 → μ → G → G^{ad} → 1.

Let γ ∈ Z^{1}(k, G_{0}^{ad}) be a cocycle such that the twist G_{0,γ} is equal to G. Let ε ∈ H^{2}(k, μ) be the image of the class of γ under the connecting map δ_{0} : H^{1}(k, G_{0}^{ad}) → H^{2}(k, μ). We have a bijection θ_{γ} : H^{1}(k, G^{ad}) → H^{1}(k, G_{0}^{ad}) sending the trivial class to the class of γ. From the commutative diagram (28.12) p. 388 of [KMRT], we draw the following two conclusions :

(1) Surjectivity of the map δ_{0} : H^{1}(k, G_{0}^{ad}) → H^{2}(k, μ) is equivalent to surjectivity of the map δ : H^{1}(k, G^{ad}) → H^{2}(k, μ).

(2) The Tits class t_{G} of G (cf. [KMRT] p. 426) is the image under δ_{0} of the class of γ.

By statement (1), to prove the theorem we may restrict to the case where the absolutely almost simple simply connected group G is quasisplit, G = G_{0}.

For several types, the surjectivity statement in (a) and the isotropy statement in (b) (in the cases where (b) holds) will be proved simultaneously. What we shall prove is that for any class c ∈ H^{2}(k, μ) there exists an explicit isotropic internal form G of G_{0}, whose class in H^{1}(k, G_{0}^{ad}) has image c under the boundary map δ_{0}. That will clearly be enough to prove surjectivity and isotropy, because δ_{0} is one-to-one.

Groups of type ^{1}A_{n}

Let m = n + 1. The basic exact sequence here is

1 → μ_{m} → SL_{m} → P GL_{m} → 1

and the boundary map δ_{0} sends the isomorphy class of a central simple algebra A/k of degree m to its class in the m-torsion subgroup _{m}Br(k). Hypothesis (ii) of the theorem guarantees that δ is onto.

Groups of type B_{n} (n ≥ 2)

Here μ = μ_{2} and G_{0} is the split group. By hypothesis (ii), H^{2}(k, μ_{2}) = _{2}Br(k) consists of classes of quaternion algebras. Let D/k = (a, b) be a quaternion algebra. We have to show that the class [D] of D in Br(k) is equal to the Tits class t_{G} ([KMRT], p. 426) for some (automatically internal) form G of G_{0}.

By [KMRT] (31.9) p. 427, given q a quadratic form of dimension 2n + 1, the Tits class of Spin(q) is the class of the even Clifford algebra C_{0}(q). Let q be the (isotropic) quadratic form ⟨a, b, −ab⟩ ⊥ (n − 1)H, where H = ⟨1⟩ ⊥ ⟨−1⟩ and ⟨a, b, −ab⟩ is the standard notation for the quadratic form aX^{2} + bY^{2} − abZ^{2}. Since C_{0}(⟨a, b, −ab⟩) = (a, b), then [C_{0}(q)] = [(a, b)] (see [Sc], Chap. 9, §2). The associated group Spin(q) is isotropic, and its Tits class is [C_{0}(q)] ([KMRT], p. 378), hence equal to [D].

Groups of type C_{n} (n ≥ 2)

Here again μ = μ_{2}, and the group G_{0} is split (all forms are inner). By hypothesis, any class in _{2}Br(k) is represented by a quaternion algebra D. There is a unique symplectic involution γ on the quaternion algebra D/k ([KMRT], Prop. 2.21). Let P ∈ M_{n}(k) be the symmetric matrix with associated quadratic form 2x_{1}x_{2} + ∑_{i=3}^{n} x_{i}^{2}. The tensor product of the involution γ on D and the involution X ↦ −P^{t}XP on M_{n} is an isotropic symplectic involution on M_{n}(D), which we denote by γ̃. The group Sp(M_{n}(D), γ̃) is then an isotropic group of type C_{n} with Tits class [D] ([KMRT] (31.10) p. 427).

Groups of type ^{3}D_{4} or ^{6}D_{4}

Let G_{0} have type ^{3}D_{4} or ^{6}D_{4}, with associated cubic field extension L/k. The associated exact sequence of quasisplit groups

1 → μ → G_{0} → G_{0}^{ad} → 1

is sequence (44.11) of [KMRT], p. 563. The centre μ of G_{0} (which is denoted C_{δ} in [KMRT]) is the group μ = R_{L/k}^{1}(μ_{2}) ([KMRT], p. 564). Since the degree of L over k is 3, the composite of the diagonal embedding μ_{2} → R_{L/k}(μ_{2}) and of the norm map R_{L/k}(μ_{2}) → μ_{2} is identity. Thus the exact sequence

1 → R_{L/k}^{1}(μ_{2}) → R_{L/k}(μ_{2}) → μ_{2} → 1

is split, hence induces an isomorphism

H^{2}(k, μ) = ker[Cores : _{2}Br(L) → _{2}Br(k)].

By hypothesis (ii) of the Theorem, an element in H^{2}(k, μ) is represented by a quaternion algebra D over L whose corestriction in k is trivial in the Brauer group of k. The main Theorem of [Ga] states that there exists a unique (simply connected) trialitarian group G/k with Allen algebra M_{4}(D). This is an internal form of G_{0} and its Tits index of the type described in the Theorem. Let ξ ∈ H^{1}(k, G_{0}^{ad}) represent G/k. The map δ_{0} is the map denoted by Sn^{1} in sequence (44.12) of [KMRT]. The image of ξ under δ_{0} is called the Clifford invariant of G, it is the class of the Allen algebra, and is equal in that case to [D] ∈ Br(L).

Statements (b) and (c) now follow from the general argument mentioned before the discussion of case ^{1}A_{n}.

Groups of type E_{6}

Let G_{0} be the split simply connected group of type E_{6}. Its centre is the group μ_{3}. Fix a (split) Borel subgroup. Deleting the central root from the extended Dynkin diagram, we see that there is a natural k-homomorphism of split k-groups (SL_{2})^{3} → G_{0}^{ad} with finite kernel. Let H ⊂ G_{0} denote its image. One shows (see [Ti1], §6.4.4 and [Ti2], §1.2) that this homomorphism induces a commutative diagram

1 ⟶  μ_{3}  ⟶ G_{0} ⟶ G_{0}^{ad} ⟶ 1
↑  ↑  ↑
1 ⟶ ((μ_{3})^{3})^{0} ⟶ (SL_{3})^{3} ⟶ H ⟶ 1.

Here ((μ_{3})^{3})^{0} is the kernel of the product map (μ_{3})^{3} → μ_{3} sending (λ, μ, ν) to λ.μ.ν, and the map ((μ_{3})^{3})^{0} → μ_{3} in the diagram is the map which sends an element (λ, μ, ν) ∈ ((μ_{3})^{3})^{0} to λ/μ = μ/ν = ν/λ. Note that this map is split. Applying Galois cohomology to the above diagram, one sees that the image of δ_{0} in H^{2}(k, μ_{3}) contains the image of the composite map H^{1}(k, H) → H^{2}(k, ((μ_{3})^{3})^{0}) → H^{2}(k, μ_{3}). Because of the splitting mentioned above, the last map is surjective. Now the image of the first map consists exactly of the triples ([D_{1}], [D_{2}], [D_{3}]) ∈ H^{2}(k, ((μ_{3})^{3})^{0}) ⊂ _{3}Br(k) with sum zero, each D_{i} being a central simple algebra over k of degree 3. By our assumption on the field k, any class in H^{2}(k, ((μ_{3})^{3})^{0}) is of this shape. This concludes the proof of surjectivity in the ^{1}E_{6} case. For the isotropy, and the possible Tits indices, see [Gi2], Thm. 9. p. 314.

There is a natural action of the group Z/2 on the above diagram of morphisms of groups. On the upper row, it is given by the “opposition involution” associated to the nontrivial automorphism of the Dynkin diagram of E_{6} ([Ti1], 1.5.1) which induces the inverse map x → x^{−1} on the centre μ_{3} ([Ti1], 1.5.3 (c)). This involution respects the exended Dynkin diagram deprived of its central point. It thus gives rise to an automorphism of the group H which lifts to an automorphism of the group (SL_{3})^{3} which is given by (x, y, z) → (x, z, y). Let K/k be a quadratic field extension. One can twist the above diagram by means of the obvious nontrivial cocyle in H^{1}(Gal(K/k), Z/2). This gives rise to the commutative diagram

1 ⟶  R_{K/k}^{1}(μ_{3})  ⟶  G′_{0}  ⟶ G_{0}^{′ad} ⟶ 1
↑  ↑  ↑
1 ⟶ (μ_{3} × R_{K/k}μ_{3})^{0} ⟶ SL_{3} × R_{K/k}(SL_{3}) ⟶ H ⟶ 1,

where G′_{0} is the quasisplit form ^{2}E_{6} corresponding to the extension K/k and (μ_{3} × R_{K/k}μ_{3})^{0} denotes the kernel of the map μ_{3} × R_{K/k}μ_{3} → μ_{3} given by (λ, ξ) ↦ λ.N_{K/k}(ξ). The left vertical map sends (λ, ξ) to ξ/λ. This map is clearly split. To prove that δ_{0} is onto, it thus suffices to show that the boundary map H^{1}(k, H) → H^{2}(k, (μ_{3} × R_{K/k}μ_{3})^{0}) is onto. The target of this map consists of pairs (a, b) ∈ _{3}Br(k)⊕_{3}Br(K) such that a+N_{K/K}(b) = 0. The image of the map H^{1}(k, H) → H^{2}(k, (μ_{3} × R_{K/k}μ_{3})^{0}) consists of such pairs (a, b) having the additional property that a ∈ _{3}Br(k) is represented by an algebra over k of degree 3 and b ∈ _{3}Br(K) is represented by an algebra over K of degree 3. Under hypothesis (ii) of the Theorem, this last property is always satisfied.

For statements (b) and (c), see [Gi2], Thm. 9 p. 314.

Groups of type E_{7}

Let G_{0} be the simply connected split form of E_{7}. The centre of G_{0} is μ_{2}. Consideration of the extended Dynkin diagram and of [Ti2], §1.2 shows (see [MPW], §8.II, p. 156) that there exists a commutative diagram

1 ⟶ μ_{2} ⟶  G_{0}  ⟶ G_{0}^{ad} ⟶ 1
∥  ∪  ∪
1 ⟶ μ_{4}/μ_{2} ⟶ SL_{8}/μ_{2} ⟶ SL_{8}/μ_{4} ⟶ 1.

The diagonal map SL_{4} → SL_{4} × SL_{4} ⊂ SL_{8} induces the commutative diagram of exact sequences (where the last vertical arrow is an embedding)

1 ⟶ μ_{4}/μ_{2} ⟶ SL_{8}/μ_{2} ⟶ SL_{8}/μ_{4} ⟶ 1
↑  ↑  ↑
1 ⟶ μ_{4} ⟶ SL_{4} ⟶ P GL_{4} ⟶ 1.

Combining these two diagrams and taking the boundary map, we get the commutative diagram

H^{1}(k, G_{0}^{ad}) ⟶^{δ_{0}} H^{2}(k, μ_{2})
↑  ×2 ↑
H^{1}(k, P GL_{4}) ⟶ H^{2}(k, μ_{4}).

Since the cohomological dimension of k is at most 2, the induced map H^{2}(k, μ_{4}) → H^{2}(k, μ_{2}) is onto. The boundary map H^{1}(k, P GL_{4}) → H^{2}(k, μ_{4}) = _{4}Br(k) has image the set of classes of algebras of degree 4. Under the assumption that index coincides with exponent, that boundary map is surjective. Hence so is the map δ_{0} : H^{1}(k, G_{0}^{ad}) → H^{2}(k, μ_{2}).

For the isotropy statement and the more precise description of the possible Tits indices, we refer to [Gi2], Thm. 10 p. 316.

Groups of type G_{2}, F_{4}, E_{8}

In this case μ = 1 and forms are classified by H^{1}(k, G_{0}), where G_{0} is the simply connected split group. By Theorem 1.2, H^{1}(k, G_{0}) = 1, hence any group of this type is split and in particular isotropic.

Groups of type ^{2}A_{n}

Let K/k denote a quadratic field extension ; assume char(k) ≠ 2.

Lemma 2.2 Let k be a field such that quadratic forms of dimension 5 over k are isotropic. Let B be a quaternion algebra over K equipped with a K/k-involution τ. Given any λ ∈ k^{∗} there exists x ∈ B^{∗} such that τx = x and Nrdx = λ.

Proof Let τ_{0} denote the nontrivial automorphism of K/k. There exists a k-quaternion subalgebra B_{0} ⊂ B such that B = B_{0}.K and τ restricted to B_{0} is the canonical involution σ on B_{0}. (The algebra B_{0} is the fixed ring of the product στ, where σ denotes the canonical involution on the K-quaternion algebra B.) Let K = k(\sqrt{a}). A τ-symmetric element of B is of the form μ + y\sqrt{a} with μ ∈ k, y ∈ B_{0} satisfying σy = −y. We then have Nrd(μ + y\sqrt{a}) = μ^{2} − y^{2}a = μ^{2} + aN′(y), where N′ denotes the restriction of the reduced norm on B_{0} to trace zero elements. By the hypothesis on k, any 4-dimensional quadratic form over k represents any nonzero element in k.

Lemma 2.3 Let B/K be a central simple algebra with a K/k-involution τ. Let x ∈ B^{∗} be an element such that τx = −x and x^{2} = ν ∈ k^{∗}. Let ε ∈ k^{∗} be a norm from the quadratic

extension k(x)/k. Then the matrix ( \begin{matrix}0 & ε \\ 1 & 0 \end{matrix} ) in M_{2}(B) is conjugate to a τ-hermitian symmetric

matrix.

Proof Let ε = c^{2} − νd^{2}, c, d ∈ k. Then the product

( \begin{matrix}1 & c \\ 0 & xd \end{matrix} ) ( \begin{matrix}0 & ε \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & c \\ 0 & xd \end{matrix} )^{−1} = ( \begin{matrix}c & −xd \\ xd & −c \end{matrix} )

is hermitian symmetric with respect to τ.

Lemma 2.4 Let B be a central division algebra over K, of degree 2^{n}, n ≥ 1. Let ε ∈ k^{∗} be the reduced norm of a symmetric element with respect to some K/k-involution on B. Then there exist an odd degree field extension F/k, a KF/F -involution τ on B ⊗_{k} F and x ∈ B ⊗_{k} F such that τx = −x, x^{2} ∈ F^{∗} and ε is a norm from the quadratic extension F (x)/F.

Proof Let τ_{1} be a K/k-involution on B such that there exists u ∈ B^{∗} with τ_{1}u = u and Nrd(u) = ε. There exists a maximal subfield E of B/K such that u belongs to E and N_{E/K}(u) = ε. Since the degree [k(u) : k] is a power of 2, after replacing k by an odd degree extension, we may assume that k(u) contains a quadratic extension k(x)/k with x^{2} ∈ k. Let η be the automorphism of K(x)/k which restricts to the nontrivial automorphism of K/k and to the nontrivial automorphism of k(x)/k. Then η extends to a K/k-involution τ on B such that τ(x) = −x ([Kn1], Theorem on p. 37). Let w = N_{E/K(x)}(u). Since K(x) is stable under τ_{1}, we have τ_{1}w = w. Since k(x) is the field of τ_{1}-invariant elements in K(x), this implies w ∈ k(x) and N_{k(x)/k}(w) = ε.

To any central simple algebra B/K equipped with a K/k-involution τ there is an associated central simple algebra over k, called the discriminant algebra, denoted D(B, τ) ([KMRT], Definition 10.28 p. 128).

Lemma 2.5 Let B/K be a central division algebra with a K/k-involution τ. Let H_{2n}, n ≥ 1 be the standard hyperbolic form and τ_{H_{2n}} the involution on M_{2n}(B) adjoint to H_{2n}. Then the discriminant algebra of (M_{2n}(B), τ_{H_{2n}}) is a matrix algebra over k.

Proof Let X/K be the Severi-Brauer variety attached to B and Y = R_{K/k}(X) be its Weil restriction of scalars to k. The restriction map from Br(k) to Br(k(Y)) is injective ([MT]). To prove the lemma, we may replace k by k(Y) and thus assume that B/K is split. In this case the result follows from [KMRT] (10.35) p. 131.

Lemma 2.6 Let B/K be a central division algebra with a K/k-involution τ. Let τ̃ be the involution on M_{2}(B) given by x ↦ (τx)^{t}. If the cohomological dimension of k is at most 2, then the discriminant algebra of (M_{2}(B), τ̃) is a matrix algebra over k.

Proof The involution τ̃ is adjoint with respect to the hermitian form ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) over (B, τ)

([KMRT] (4.1) and (4.2) pp. 42, 43). Since this hermitian form has the same dimension and

discriminant as ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and cd(k) ≤ 2, the two forms ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) are isometric ([BFP]). Thus the adjoint involutions with respect to these two forms are isomorphic. The result now follows from Lemma 2.5.

We record the following well known result (cf. [CTG], Prop. 7).

Proposition 2.7 Let k be a field, char(k) ≠ 2, such that every 5-dimensional quadratic form over every finite extension of k is isotropic. Then over any finite extension of k, for an algebra of exponent a power of 2, the exponent is equal to the index.

Theorem 2.8 Let B/K be a central simple algebra with a K/k-involution τ. Suppose :

(a) cd(k) ≤ 2 ;

(b) every 5-dimensional quadratic form over every finite extension of k is isotropic.

Then every element of k^{∗}, up to a square, is the reduced norm of a τ-symmetric element. Proof Let λ ∈ k^{∗}. Suppose the theorem is true for division algebras. Write B = M_{n}(D), with D/K a central

division algebra equipped with a K/k-involution τ_{0} ([KMRT], (3.1)). There exists a τ_{0}-hermitian symmetric matrix H ∈ B^{∗} such that τ is the adjoint involution τ_{0,H} with respect to H. Since H is a hermitian symmetric matrix, Nrd_{B}(H) ∈ K^{∗} actually belongs to k^{∗}. By the assumption, there exists u ∈ D^{∗} such that τ_{0}(u) = u and λ.Nrd_{B}H = Nrd_{D}(u).μ^{2} ∈ k^{∗} with μ ∈ k^{∗}. Let U be the diagonal matrix < 1, ⋯, 1, u > and let X = U.H^{−1}. Then τ(X) = X and Nrd_{B}(X) = Nrd_{D}(u).Nrd_{B}(H^{−1}) = λ.μ^{−2}. We are reduced to the case where B is a central division algebra.

Suppose the theorem is true for division algebras with degree a power of 2. Write B = B_{1} ⊗_{K} B_{2} with deg_{K}(B_{1}) a power of 2 and deg_{K}(B_{2}) = m with m odd. By the characterisation of K/k-involutions (KMRT], (3.1)), Cores_{K/k}(B) = 0. The algebras B_{1} and B_{2} being of coprime order in the Brauer group, Cores_{K/k}(B_{i}) ∼ 0 for i = 1, 2. By the quoted characterisation, there exists a K/k-involution τ_{i} on B_{i} for i = 1, 2. By [KMRT] (2.7) there exists u ∈ B^{∗} with τu = u and τ = Int(u) ∘ (τ_{1} ⊗ τ_{2}). By the assumption, there exists v ∈ B_{1}^{∗} such that τ_{1}v = v and Nrd_{B_{1}}(v) = Nrd_{B}(u).ν^{2} for some ν ∈ k^{∗}. Let τ′_{1} = Int(v) ∘ τ_{1}. Then τ = Int(u) ∘ Int(v^{−1} ⊗ 1) ∘ (τ′_{1} ⊗ τ_{2}). By the assumption, there exists w ∈ B_{1} with τ′_{1}w = w and Nrd_{B_{1}}(w) = λδ^{2} for some δ ∈ k^{∗}. The element (w ⊗ 1)(u.(v^{−1} ⊗ 1))^{−1} is τ-invariant. We have

Nrd_{B}((w ⊗ 1)(u.(v^{−1} ⊗ 1))^{−1}) = Nrd_{B_{1}}(w)^{m}.Nrd_{B_{1}}(v)^{m}.Nrd_{B}(u)^{−1}

which is λ up to squares.

We are reduced to the case where B/K is a central division algebra of degree 2^{n}. In this case we prove more : every λ ∈ k^{∗} is the norm of a τ-symmetric element.

Note that, by Proposition 2.7, the exponent of B in Br(K) is 2^{n}. The proof is by induction on n. For n = 1, the theorem follows from Lemma 2.2. Suppose n > 1. Let D(B, τ) be the discriminant algebra of (B, τ). Its class is of order 2 in the Brauer group of k. Since every 5dimensional quadratic form is isotropic, the tensor product of two quaternion algebras is the class of a quaternion algebra. Combining this with Merkurjev’s theorem we conclude that the class of D(B, τ) in Br(k) is represented by a quaternion algebra A/k. The class of A_{K} in the Brauer group of K coincides with that of B^{⊗2^{n−1}} ([KMRT], Prop. 10.30 p. 129). By Proposition 2.7, the index of the latter is equal to its exponent, which is 2. Thus A_{K} and therefore A are division algebras.

Write K = k(\sqrt{a}) with a ∈ k^{∗}. Let N′_{A} denote the restriction of the reduced norm of A to trace zero elements of A. By hypothesis (b), the 5-dimensional quadratic form N′_{A} ⊥ a < 1, −λ > is isotropic over k. There exists ε = x^{2} − λy^{2} ∈ k^{∗}, with x, y ∈ k, such that −aε is a value of N′_{A}. Since A is a quaternion division algebra, aε is not a square and k(\sqrt{aε}) splits A. Since ε is a norm from k(\sqrt{λ}), λ is a norm from k(\sqrt{ε}). Let η ∈ k(\sqrt{ε}) be such that N_{k(\sqrt{ε})/k}(η) = λ. In the Brauer group of K(\sqrt{ε}) = k(\sqrt{a}, \sqrt{ε}), we have

B^{2^{n−1}} ⊗_{K} K(\sqrt{ε}) ∼ A ⊗_{k} K ⊗_{K} K(\sqrt{ε}) ∼ A ⊗_{k} k(\sqrt{aε}) ⊗_{k(\sqrt{aε})} K(\sqrt{ε}) ∼ 0.

Hence the exponent of B ⊗_{K} K(\sqrt{ε}) is at most 2^{n−1}. By Proposition 2.7 and hypothesis (b), the index of B ⊗_{K} K(\sqrt{ε}) is also at most 2^{n−1} ; it is in fact equal to 2^{n−1}, because the index can be divided at most by 2 under a quadratic extension. Hence K(\sqrt{ε}) embeds into B over K. The nontrivial automorphism of K(\sqrt{ε})/k(\sqrt{ε}) extends to a K/k-involution τ_{1} on B with τ_{1}(\sqrt{ε}) = \sqrt{ε} (Kneser, cf. [Sch], Chap. 8.10, Thm. 10.1 p. 311). Let C be the commutant of K(\sqrt{ε}) in B. Then τ_{1} restricts to a K(\sqrt{ε})/k(\sqrt{ε}) involution on C. The index of C/K(\sqrt{ε}) is 2^{n−1}. By the induction hypothesis applied to C/k(\sqrt{ε}) and τ_{1}, there exists x ∈ C with τ_{1}x = x and Nrd_{C}(x) = η. We have

Nrd_{B}(x) = N_{K(\sqrt{ε})/K}(Nrd_{C}(x)) = N_{k(\sqrt{ε})/k}(η) = λ.

There exists u ∈ B^{∗} such that τ_{1} = Int(u) ∘ τ with τ_{1}u = u. Let t = Nrd_{B}(u). Then τ(xu) = xu and Nrd_{B}(xu) = λt.

Our next step is to get rid of t by replacing k(\sqrt{ε}) by a conjugate field yk(\sqrt{ε})y^{−1} for a suitable y ∈ B^{∗}.

For y ∈ B^{∗} to be chosen later, let τ′_{1} = Int(y) ∘ τ_{1} ∘ Int(y)^{−1} and u′ = yuτ(y). Then τ′_{1} = Int(u′) ∘ τ. We look for y such that there exists x′ ∈ yCy^{−1} which is τ′_{1}-symmetric and Nrd_{B}(x′u′) = λ. This will complete the proof of the theorem because x′u′ is τ-symmetric.

Suppose we have y and x′ as above. Then

N_{yk(\sqrt{ε})y^{−1}/k}(Nrd_{yCy^{−1}}(x′)).N_{K/k}(Nrd_{B}(y)).t = λ.

As quadratic forms, we have N_{yk(\sqrt{ε})y^{−1}/k} ≃< 1, −ε > and N_{K/k} ≃< 1, −a > .

The existence of y and x′ guarantees the existence of a nontrivial zero of the quadratic form

q = t < 1, −ε >⊥ −λ < 1, −a >

over k.

Conversely, isotropy of q would give μ ∈ yk(\sqrt{ε})^{∗}y^{−1} and ν ∈ K^{∗} such that

t.N_{yk(\sqrt{ε})^{∗}y^{−1}/k}(μ).N_{K/k}(ν) = λ.

Since cd(k) ≤ 2, there exists y ∈ B^{∗} such that Nrd_{B}(y) = ν. Let τ′_{1} = Int(y) ∘ τ_{1} ∘ Int(y)^{−1} and u′ = yuτ(y). By the induction hypothesis, there exists x′ ∈ yCy^{−1} such that τ′_{1}(x′) = x′ and Nrd_{yCy^{−1}}(x′) = μ. Then τ(x′u′) = x′u′ and Nrd_{B}(x′u′) = λ.

Thus we need only to show that q is isotropic over k. It suffices to show that q is isotropic over the discriminant extension k(\sqrt{aε}).

By base change we replace k by k(\sqrt{aε}), K by K(\sqrt{aε}) = k(\sqrt{a}, \sqrt{ε}).

Over these new fields k and K, we have B = M_{2}(B_{1}) for B_{1}/K a central division algebra of degree 2^{n−1} which admits a K/k-involution. By the induction hypothesis, given any K/kinvolution on B_{1}, ε is the reduced norm of an element which is symmetric with respect to this involution.

To prove isotropy of q over k we may go over to an odd-degree extension of k. Using Lemma 2.4 we replace k by an odd-degree extension and assume that there exist a K/k-involution τ_{0} on B_{1} and x ∈ B_{1}^{∗} such that τ_{0}(x) = −x, x^{2} = d ∈ k^{∗} and ε = ξ_{1}^{2} − dξ_{2}^{2}, with ξ_{1}, ξ_{2} ∈ k. The involution τ on B is adjoint with respect to a τ_{0}-hermitian form W ∈ M_{2}(B_{1}), i.e. (τ_{0}W)^{t} = W.

Let τ̃_{0} be the involution on B adjoint to ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ).

By [KMRT] (10.36) p. 131, in the Brauer group of k, we have

D(B, τ) ∼ ((Nrd_{B}W) ∪ (a)) ⊗_{k} D(B, τ̃_{0}).

By Lemma 4, D(B, τ̃_{0}) ∼ 0. We note that D(B, τ) ∼ A is split over k(\sqrt{aε}) = k by our new choice of k. Hence Nrd_{B}W ∈ N_{K/k}K^{∗}. Let Nrd_{B}W = N_{K/k}(θ), θ ∈ K^{∗}.

Since q is isotropic if ε is a square in k, we may assume that ε is not a square. The element

\sqrt{ε} ∈ B is conjugate to ( \begin{matrix}0 & ε \\ 1 & 0 \end{matrix} ) in B (Skolem-Noether). By Lemma 2.3, ( \begin{matrix}0 & ε \\ 1 & 0 \end{matrix} ) is conjugate

to a τ_{0}-hermitian symmetric matrix in B = M_{2}(B_{1}). Hence there exists X ∈ GL_{2}(B_{1}) such that

τ_{0}(X\sqrt{ε}X^{−1})^{t} = X\sqrt{ε}X^{−1}

i.e.

W^{−1}τ(X\sqrt{ε}X^{−1})W = X\sqrt{ε}X^{−1}.

We have τ(\sqrt{ε}) = u^{−1}τ_{1}(\sqrt{ε})u = u^{−1}\sqrt{ε}u, so that

W^{−1}(τ(X))^{−1}u^{−1}\sqrt{ε}uτ(X)W = X\sqrt{ε}X^{−1}.

Thus the element b = uτ(X)W X commutes with k(\sqrt{ε}) hence belongs to C. From τ(X).W X = u^{−1}b we get

N_{K/k}(Nrd_{B}(X)).Nrd_{B}(W) = t^{−1}.N_{K(\sqrt{ε})/K}((Nrd_{C}(b)) ∈ K^{∗}.

Since W is τ-symmetric, and τ_{1} = Int(u) ∘ τ, the element b = uτ(X)W X is τ_{1}-symmetric. Thus Nrd_{C}(b) lies in k(\sqrt{ε}) and N_{K(\sqrt{ε})/K}((Nrd_{C}(b)) = N_{k(\sqrt{ε})/k}((Nrd_{C}(b)). We now have

N_{K/k}(Nrd_{B}(X)).N_{K/k}(θ) = t^{−1}.N_{k(\sqrt{ε})/k}(Nrd_{C}(b)) ∈ k^{∗}.

Thus q is isotropic over k.

Let n = 2m be an even integer and let k be a field, with char(k) prime to n. Let K be a separable field extension of k, let Γ = Gal(K/k) = {1, σ} and let μ be the kernel of the norm map R_{K/k}(μ_{n}) → μ_{n}; this is the group denoted μ_{n[K]} in the book [KMRT].

Lemma 2.9 For μ as above, there is an exact sequence of k-groups of multiplicative type

1 → μ → R_{K/k}G_{m} × G_{m} → R_{K/k}G_{m} × G_{m} → 1,

where the map R_{K/k}G_{m} × G_{m} → R_{K/k}G_{m} × G_{m} is given by

(α, b) ↦ (b.α^{m}, N_{K/k}(α)).

Proof Let μ̂ be the character group of μ. The dual of the sequence of k-groups

1 → μ → R_{K/k}(μ_{n}) → μ_{n} → 1

is the sequence

0 → Z/n → Z/n[Γ] → μ̂ → 0

where the map Z/n → Z/n[Γ] is given by 1 ↦ 1 + σ. As an abelian group, μ̂ is isomorphic to Z/n.

The map

M_{1} = Z[Γ] ⊕ Z → M_{2} = Z[Γ] ⊕ Z

given by (a + bσ, c) ↦ ((ma + c) + (mb + c)σ, a + b) is dual to the map R_{K/k}G_{m} × G_{m} → R_{K/k}G_{m} × G_{m} given in the proposition.

Consider the additive map Z[Γ]⊕Z → Z/n[Γ] given by (a+bσ, c) ↦ (a+bσ +cm) modulo n. Composition with the projection Z/n[Γ] → μ̂ produces a Γ-invariant, surjective homomorphism M_{2} → μ̂. One checks that the composite map M_{1} → M_{2} → μ̂ is zero and that the determinant of M_{1} → M_{2} has absolute value n. This gives the exact sequence of Γ-modules of finite type

0 → Z[Γ] ⊕ Z → Z[Γ] ⊕ Z → μ̂ → 0.

Dualizing gives the lemma.

Proposition 2.10 Let K/k and μ be as above. Then the group H^{2}(k, μ) is isomorphic to the subgroup of Br(K) ⊕ Br(k) consisting of pairs (x, y) satisfying Res_{k/K}(y) + mx = 0 ∈ Br(K) and Cores_{K/k}(x) = 0.

Proof This follows from the Galois cohomology of the exact sequence given in Lemma 2.9, noting that H^{1}(k, R_{K/k}(G_{m}) × G_{m}) = 0 by Hilbert’s Theorem 90.

Remark Let n > 1 be an odd integer. One easily checks that the sequence

1 → μ → R_{K/k}(μ_{n}) → μ_{n} → 1

is split. From this one identifies H^{2}(k, μ) with the kernel of the corestriction map _{n}Br(K) → nBr(k). One can use the same argument to compute H^{1}(k, μ) (compare [KMRT] (30.13) p. 418).

Theorem 2.11 Let n > 1 be an integer, let k be a field of characteristic prime to n, let G be a semisimple, simply connected, absolutely simple group of k of type ^{2}A_{n−1}. Let G → G^{ad} be the isogeny whose kernel is the center μ of G. Assume that k satisfies the two conditions :

(a) cd(k) ≤ 2 ;

(b) over any finite extension of k, the index of a central simple algebra coincides with its exponent.

Then the induced map H^{1}(k, G^{ad}) → H^{2}(k, μ) is surjective.

Remark Condition (b) implies that over any finite extension of k, 5-dimensional quadratic forms are isotropic.

Proof of Theorem 2.11 It is enough to prove the theorem when G = G_{0} is quasisplit.

The group G_{0} determines a quadratic extension K/k. To any ξ ∈ H^{1}(k, G_{0}^{ad}) there is an associated central simple algebra B/K of degree n, equipped with a K/k-involution τ. The group G_{0,ξ} is isomorphic to SU(B, τ).

Suppose that n is odd. Let x be an element of H^{2}(k, μ), identified with the kernel of the corestriction map _{n}Br(K) → _{n}Br(k). By assumption (b), there exists a central simple algebra B/K of degree n which in view of Cores_{K/k}(B) ∼ 0 admits a K/k-involution τ. Let G = SU(B, τ)^{ad}. Then δ_{0}(G) = t_{G} which is equal to the class of B ∈ H^{2}(k, μ) ([KMRT] (31.8) p. 427).

Suppose now that n = 2m. Under the identification of Proposition 2.10, the element δ_{0}(ξ) = t_{G} ∈ H^{2}(k, μ) is the pair ([B], D(B, τ)) ([KMRT] (31.8)).

By Proposition 2.10, any element of H^{2}(k, μ) is represented by a pair (x, y) ∈ Br(K) ⊕ Br(k) such that Res_{k/K}(y)+mx = 0 in Br(K) and Cores_{K/k}(x) = 0. Observe that 2y = 0 and 2mx = 0. By assumption (b), there exists a central simple algebra B/K of degree n which represents x. In view of Cores_{K/k}(B) ∼ 0, it admits a K/k-involution τ. The discriminant algebra D(B, τ) is of exponent 2 hence is represented by a quaternion algebra A/k. From D(B, τ) ⊗_{k} K ∼ B^{⊗m} we deduce that y − [A] ∈ Br(k) vanishes in Br(K). Let K = k(\sqrt{a}). There exists a λ ∈ k^{∗} such that y = [A]+(a, λ) ∈ Br(k). By Theorem 2.8, there exists u ∈ B^{∗} which is τ-symmetric and such that Nrd_{B}(u) = λ.μ^{2} for some μ ∈ k^{∗}. Let τ′ = Int(u) ∘ τ. Then D(B, τ′) ∼ (Nrd_{B}(u), a) ⊗ D(B, τ) by [KMRT] 10.36 p. 131, hence D(B, τ′) ∼ (Nrd_{B}(u), a) ⊗ A which coincides with the class of y in Br(k). We have now written (x, y) as δ_{0}(G), where is G is the group adjoint to SU(B, τ′).

Nontrialitarian groups of type D_{m} (m ≥ 2)

Let G_{0}/k be a quasisplit, semisimple, simply connected of type D_{m}, m ≥ 2. The group G_{0}/k is isomorphic to Spin(A_{0}, σ_{0}) where A_{0} = M_{2m}(k) is the split algebra and σ_{0} is an orthogonal involution corresponding to a quasisplit quadratic form q_{0} = (m − 1) < 1, −1 >⊥< 1, −d >. Let K be the centre of the Clifford algebra C(A_{0}, σ_{0}). This is a quadratic étale algebra over k isomorphic to k[x]/(x^{2} − d).

Let μ be the centre of G_{0}. If m is odd, μ = R_{K/k}^{1}(μ_{4}) ; if m is even, then μ = R_{K/k}(μ_{2}) ([KMRT], p. 371).

Let

1 → μ → G_{0} → G_{0}^{ad} → 1

be the standard isogeny.

For any ξ ∈ H^{1}(k, G_{0}^{ad}), there is an associated central simple algebra A/k of degree 2m and an orthogonal involution σ on A such that the twisted group G = G_{0,ξ} is isomorphic to Spin(A, σ). Let C(A, σ) denote the Clifford algebra of (A, σ) ([KMRT], p. 91). The cocycle ξ gives rise to an isomorphism η of its centre with K. Conversely, a triple (A, σ, η), where A is a central simple algebra over k of degree 2m with an orthogonal involution σ and an isomorphism η of the centre of C(A, σ) with K, determines a class ξ ∈ H^{1}(k, G_{0}^{ad}) (cf. [KMRT] p. 409).

Suppose m is odd. By Proposition 2.10, H^{2}(k, μ) is isomorphic to the group of pairs (x, y) ∈ Br(K) ⊕ Br(k) such that Res_{k/K}(y) + 2x = 0 ∈ Br(K) and Cores_{K/k}(x) = 0. The element δ_{0}(ξ) = t_{G} is the pair (η_{∗}[C(A, σ)], [A]), where η_{∗} is the induced map from the Brauer group of the centre of C(A, σ) to the Brauer group of K ([KMRT] (31.13) p. 428).

Suppose m is even. Then H^{2}(k, μ) = _{2}Br(K) and δ_{0}(ξ) = t_{G} is the class of η_{∗}([C(A, σ)]) in 2Br(K).

Proposition 2.12 Let m > 1 be an integer, k a field of characteristic prime to 2, and G a semisimple, quasisplit, simply connected group over k, of nontrialitarian type D_{m}. Let G → G^{ad} be the isogeny whose kernel is the center μ of G. Assume that k satisfies the two conditions :

(a) cd(k) ≤ 2 ;

(b) over any finite extension of k, every 5-dimensional quadratic form is isotropic.

Then the induced map H^{1}(k, G^{ad}) → H^{2}(k, μ) is surjective. Moreover, if m ≥ 4, then G is isotropic.

Proof

Suppose m = 2. Because D_{2} = A_{1}^{2} (for a rational description, see [KMRT] (15.B)), this case follows by the standard reduction from the case of A_{1} (over a field extension), handled earlier on. More explicitly, let x ∈ _{2}Br(K). By hypothesis (b) we may represent x as the class of a degree 2 central separable algebra B over K. The construction in [KMRT] (15.B) p. 210 yields a degree 4 algebra A over k and an orthogonal involution σ on A such that C(A, σ) is k-isomorphic to B. A suitable choice of an isomorphism of the centre of C(A, σ) with K defines a class ξ ∈ H^{1}(k, G^{ad}) with δ_{0}(ξ) = B.

Suppose m = 3. We have D_{3} = A_{3} (for a rational description, see [KMRT] (15.D) p. 220). This case has already been handled.

Suppose m ≥ 4 and for m = 4 exclude the trialitarian case.

Suppose m is odd. Let 2m = 6 + 4l for l ≥ 1. Let (x, y) ∈ Br(K) ⊕ Br(k) be an element representing a class in H^{2}(k, μ). By the case m = 3, there exists a degree 6 algebra D/k with an orthogonal involution σ such that the Tits class of Spin(A, σ) is (x, y). There exists a quaternion algebra A/k such that D = M_{3}(A). We fix an orthogonal involution τ on A. Then σ is the adjoint involution τ_{h} to a τ-hermitian form h on A^{3}. Let h_{l} be the hyperbolic hermitian form on A^{2l}. Let σ′ be the orthogonal involution on A′ = M_{3+2l}(A) which is adjoint to h ⊥ h_{l}. A suitable choice of an isomorphism of the centre of C(A′, σ′) with K defines a class ξ′ ∈ H^{1}(k, G_{0}^{ad}) with δ_{0}(ξ′) = (x, y) ([DLT], Proposition 3). By construction, the group Spin(A′, σ′) = SU(A, h ⊥ h_{l}) is isotropic.

Suppose m is even. Let 2m = 4 + 4l for l ≥ 1. Let x ∈ _{2}Br(K) be an element representing a class in H^{2}(k, μ). By the case m = 2, there exists a degree 4 algebra D/k with an orthogonal involution σ such that the Tits class of Spin(A, σ) is x. There exists a quaternion algebra A/k such that D = M_{2}(A). We fix an orthogonal involution τ on A. Then σ is the adjoint involution τ_{h} to a τ-hermitian form h on A^{2}. Let h_{l} be the hyperbolic hermitian form on A^{2l}. Let σ′ be the orthogonal involution on A′ = M_{2+2l}(A) which is adjoint to h ⊥ h_{l}. A suitable choice of an isomorphism of the centre of C(A′, σ′) with K defines a class ξ′ ∈ H^{1}(k, G_{0}^{ad}) with δ_{0}(ξ′) = x ([DLT], Proposition 3).

By the general argument given at the beginning of the proof, we conclude that any group of type ^{1,2}D_{m}, m ≥ 4 is isotropic.

§3 Groups of multiplicative type

§3.1 Tori and groups of multiplicative type : reminders.

We briefly recollect results from [CTS1] and [CTS2]. Given a finite group G, a G-lattice M is called coflasque if H^{1}(H, M) = 0 for all subgroups H ⊂ G. A G-lattice M is called flasque if the G-lattice M^{0} := Hom_{Z}(M, Z) is coflasque. If H is a normal subgroup of G, and if M is coflasque, then the G/H-lattice M^{H} is coflasque. If M is a coflasque G/H-module, it is a coflasque G-module. The notions thus extend to lattices equipped with a continuous action of a profinite group. Obvious examples of flasque and coflasque modules are the direct factors of permutation modules. If G is a metacyclic group, i.e. is a finite group all Sylow subgroups of which are cyclic, then a basic result of Endo and Miyata (cf. [CTS1], Prop. 2 p. 184) states that any flasque or coflasque G-lattice is a direct factor of a permutation module.

Given a G-module M of finite type over Z, there exists an exact sequence of G-modules

0 → M_{2} → M_{1} → M → 0,

where M_{1} is a G-permutation module and M_{2} is a coflasque G-lattice, and the isomorphism class of the G-lattice M_{2} is determined up to addition of a G-permutation module.

Given a G-module M of finite type over Z, there exists an exact sequence of G-modules

0 → M_{4} → M_{3} → M → 0,

with M_{4} a G-permutation module and M_{3} a flasque G-lattice, and the isomorphism class of M_{3} is determined up to addition of a G-permutation module.

Let now k be a field. A k-group of multiplicative type is an algebraic k-group which after a separable field extension of k becomes isomorphic to a closed subgroup of a product of copies of the multiplicative group G_{m}. The map which to a k-group of multiplicative type associates its character group (over a separable closure of k) defines an antiduality between k-groups of multiplicative type and finitely generated discrete Galois modules. In this antiduality, tori correspond to torsionfree Galois modules. If the character group of a k-torus is a permutation module, then the k-torus is quasitrivial : it is a product of Weil restrictions of scalars of the group G_{m}. A torus is called flasque, resp. coflasque, if its character group is flasque, resp. coflasque. Flasque k-tori F satisfy the following basic property : given any smooth connected k-variety X and any nonempty open set U ⊂ X, the restriction map H_{ét}^{1}(X, F) → H_{ét}^{1}(U, F) is onto ([CTS1]). One can actually define flasque tori over an arbitrary base scheme, and the same surjectivity property holds over regular schemes ([CTS2], Thm. 2.2).

Given any k-torus T, there exists an exact sequence of k-tori

1 → F → P → T → 1,

with P a quasitrivial k-torus and F a flasque k-torus. Such a sequence induces an isomorphism T (k)/R ≃ H^{1}(k, F) ([CTS1]).

Given any k-group of multiplicative type μ, there exists an exact sequence of k-groups of multiplicative type

1 → μ → F → P → 1,

with F a flasque k-torus and P a quasitrivial torus ([CTS2], Prop. 1.3). Such a sequence induces an isomorphism H^{1}(k, μ)/R ≃ H^{1}(k, F) (cf. [Gi2]).

The following lemma should have been recorded along with Lemme 2 of [CTS1]. Recall that for a finite group G, with group ring Z[G], one writes N_{G} = ∑_{g∈G} g ∈ Z[G] and one denotes by I_{G} the ideal of the group ring Z[G] generated by all g − 1 for g ∈ G.

Lemma 3.1 Let G be a finite group and H ⊂ G a normal subgroup. Let M be a flasque G-module. Then the G/H-module M/I_{H}M is naturally isomorphic to the Z-free G/H-module N_{H}M, and it is a flasque G/H-module.

Proof The G-module R = Hom_{Z}(M, Z) is coflasque. By [CTS1], Lemme 2, the G/H-module R^{H} is a coflasque G/H-module. Now R^{H} = Hom_{H}(M, Z) = Hom_{Z}(M/I_{H}M, Z). Multiplication by N_{H} on M induces a natural surjective map of G/H-modules from M/I_{H}M to the Ztorsionfree submodule N_{H}M of M. The kernel of this map is Ĥ^{−1}(H, M), the torsion subgroup of M/I_{H}M. The hypothesis that M is flasque implies that this map is an isomorphism. The G/Hmodule Hom_{Z}(M/I_{H}M, Z) is G/H-coflasque, hence the Z-free G/H-module M/I_{H}M = N_{H}M is flasque.

§3.2 Finiteness results

In [CTS1] and [CTS2], finiteness of H^{1}(K, F) for F a flasque torus over a function field K over k (i.e. a field K finitely generated over the ground field k), was proved when k is of one of the following types : a finite field, a number field, a p-adic field, a real closed field, a separably closed field, with immediate application to the finiteness of T (K)/R and of H^{1}(K, μ)/R for T an arbitrary K-torus and μ an arbitrary K-group of multiplicative type.

In this section, we establish the finiteness of H^{1}(K, F) for F a flasque torus over some more fields K, thereby proving the finiteness of T (K)/R for T a K-torus and of H^{1}(K, μ)/R for μ a K-group of multiplicative type over such fields.

Note the following easy remark. If H^{1}(K, F) is finite for all flasque K-tori F, then for any finite separable field extension L/K, and any flasque L-torus F, H^{1}(L, F) is finite. The point is that the K-torus R_{L/K}F is then a flasque K-torus and H^{1}(F, R_{L/K}T) ≃ H^{1}(L, T) for any L-torus T.

Theorem 3.2 Let k be a field of characteristic zero, and let K = k((t)) be the Laurent series field. If H^{1}(k, F) is finite for any flasque torus F over k, then H^{1}(K, F) is finite for any flasque torus F over K.

Proof Let F be a flasque k((t))-torus. There exists a finite Galois extension l/k and an integer n > 0 such that the field M = l((t^{1/n})) defines a Galois extension of K which splits the K-torus F. Let L = l((t)) ⊂ M. Let G = Gal(M/K). Let H = Gal(M/L) ≃ Z/n be the inertia group. This is a normal subgroup of G, and G/H = Gal(l/k). Let U = l[[t^{1/n}]]^{∗}. Valuation defines a G-equivariant exact sequence

0 → U → M^{∗} → Z → 0.

Evaluation at t = 0 defines a G-equivariant, G-split exact sequence

0 → U_{1} → U → l^{∗} → 1.

The groupe U_{1} of Einseinheiten is uniquely divisible.

The character group F̂ of the K-torus F is a finitely generated, Z-free G-module. By standard arguments, H^{1}(G, Hom_{Z}(F̂, M^{∗})) = H^{1}(K, F).

The first exact sequence is Z-split, hence we have the exact sequence

0 → Hom_{Z}(F̂, U) → Hom_{Z}(F̂, M^{∗}) → Hom_{Z}(F̂, Z) → 0.

The G-cohomology of this sequence yields the exact sequence

H^{1}(G, Hom_{Z}(F̂, U)) → H^{1}(G, Hom_{Z}(F̂, M^{∗})) → H^{1}(G, Hom_{Z}(F̂, Z)).

The finite group H^{1}(G, Hom_{Z}(F̂, Z)) is zero since the K-torus F is flasque.

Let us now study H^{1}(G, Hom_{Z}(F̂, U)). The second exact sequence is a split exact sequence of G-modules and the left term is uniquely divisible. We thus get an isomorphism

H^{1}(G, Hom_{Z}(F̂, U)) ⟶^{≃} H^{1}(G, Hom_{Z}(F̂, l^{∗})).

We have the restriction-inflation sequence associated to the normal subgroup H of G :

0 → H^{1}(G/H, Hom_{H}(F̂, l^{∗})) → H^{1}(G, Hom_{Z}(F̂, l^{∗})) → H^{1}(H, Hom_{Z}(F̂, l^{∗}))^{G/H}.

Since H acts trivially on l, we have Hom_{H}(F̂, l^{∗}) = Hom_{Z}(F̂/I_{H} F̂, l^{∗}). Since F̂ is G-flasque, Lemma 3.1 implies that F̂/I_{H} F̂ is a (torsionfree) flasque G/H-module. Let F_{0} be the flasque k-torus with character group F̂/I_{H} F̂. We have

H^{1}(G/H, Hom_{H}(F̂, l^{∗})) = H^{1}(G/H, Hom_{Z}(F̂/I_{H} F̂, l^{∗})) = H^{1}(k, F_{0}),

and by the hypothesis of the theorem this group is finite.

Let us now study the right hand side group H^{1}(H, Hom_{Z}(F̂, l^{∗}))^{G/H} in the restrictioninflation exact sequence. Since F̂ is G-flasque, hence H-flasque, and since H is cyclic, the theorem of Endo and Miyata implies that the H-module F̂ is a direct factor of a permutation H-module, i.e. of a direct sum of modules Z[H/H_{1}] for various subgroups H_{1} of H. Now

H^{1}(H, Hom_{Z}(Z[H/H_{1}], l^{∗})) ≃ H^{1}(H, Z[H/H_{1}] ⊗ l^{∗}) ≃ H^{1}(H_{1}, l^{∗}) = Hom(H_{1}, l^{∗})

and this last group is clearly finite. Hence H^{1}(H, Hom_{Z}(F̂, l^{∗})) is finite, which concludes the proof.

Remarks

1) The case where F has good reduction over K, i.e. comes from a k[[t]]-torus, hence from a k-torus, necessarily flasque, is clear, since in this case flasqueness of F implies H^{1}(k[[t]], F) = H^{1}(k((t)), F), and then H^{1}(k[[t]], F) = H^{1}(k, F).

2) Inspection of the proof shows that if one assumes finiteness of H^{1}(k, T) for arbitrary ktori, then one has finiteness of H^{1}(K, T) for any K-torus T which becomes flasque after an unramified extension of K.

3) Again, inspection of the proof shows that if one assumes H^{1}(k, T) finite for an arbitrary k-torus T, and one assumes that for each finite extension l of k, each quotient l^{∗}/l^{∗n} is finite (both assumptions being covered by the mere assumption H^{1}(k, M) finite for any k-group M of multiplicative type), then H^{1}(k((t)), T) is finite for any k((t))-torus T.

4) For the K-torus T = R_{L/K}^{1}G_{m}, with L = k((t^{1/2})), one easily computes H^{1}(K, T) = k^{∗}/k^{∗2}.

Let us say that a field k has finite cohomology if for any finite continuous Galois module M over the absolute Galois group of k, the Galois cohomology groups H^{n}(k, M) are finite for all n. Examples of such fields are the separably closed or real closed fields, the finite fields, finite extensions of a p-adic field Q_{p}. If k satisfies the property, then k((t)) also satisfies it.

Proposition 3.3 Let k be a field of characteristic zero with finite cohomology. Let A be an integral domain with fraction field K. Assume that A is of one of the following types

(a) a k-algebra of finite type over k;

(b) an excellent, henselian, local domain with residue field k.

Let U ⊂ Spec(A) be a nonempty open set and let T be a U-torus. Then the image of the restriction map H_{ét}^{1}(U, T) → H_{ét}^{1}(K, T) is finite.

Proof (suggested by O. Gabber) To prove the theorem, one may shrink U. We may thus assume that U is regular and of finite type over Spec(A). By considering character groups, one may easily produce an exact sequence

1 → F → P → T → 1

over U with P quasitrivial and F a U-torus (see [CTS2], Prop. 1.3), all being trivialized by a finite, connected, étale, Galois cover U′ → U with group G. Going over to character groups, and using the semisimplicity of Q[G], one sees that there exists a homomorphism of k-tori T → P such that the composite T → P → T (the last arrow being as in the exact sequence above) is multiplication by some positive integer n. The image of H_{ét}^{1}(U, P) → H_{ét}^{1}(K, P) is trivial, since H_{ét}^{1}(K, P) = 0 by Hilbert’s theorem 90. To prove the result it is thus enough to know the finiteness of H_{ét}^{2}(U,_{n}T), where _{n}T denotes the n-torsion subgroup of T. The finite group scheme nT/U is a twisted locally constant group, split by the Galois cover U′ → U. Here U′ is an open set of finite type of Spec(A′), where A′ is the integral closure of K in a finite Galois (field) extension K′ of K. Such a ring has the same property as A. Use of the Hochschild-Serre spectral sequence reduces the proof to that of the finiteness of the groups H_{ét}^{1}(U′, μ_{n}) and H_{ét}^{2}(U′, μ_{n}) for U′ regular open in Spec(A′). In case (a), we recognize a standard finiteness property of étale cohomology of k-varieties ([SGA 4], XVI, Thm. 5.1). In case (b), the finiteness follows from [SGA 4] XIX, Thm. 5.1.

Theorem 3.4 Let k be a field of characteristic zero with finite cohomology.

Let A be an integral domain with fraction field K. Assume that A is of one of the following types

(a) a k-algebra of finite type ;

(b) an excellent, henselian, local domain with residue field k.

Then

(i) For any flasque torus F over K, the group H^{1}(K, F) is finite.

(ii) For any K-torus, the groupe T (K)/R is finite.

(iii) For any K-group of multiplicative type μ, the quotient H^{1}(K, μ)/R is finite.

Proof According to the reminders in §3.1, it is enough to prove (i). In case (a), there exists an integral regular affine k-variety X, with fraction field K, over which the flasque torus F extends to a flasque torus F/X. In case (b), there exists a nonempty regular open set X ⊂ Spec(A) over which F extends to a flasque torus. Because F/X is flasque and X regular, the restriction map H_{ét}^{1}(X, F) → H_{ét}^{1}(K, F) is onto ([CTS2], Thm. 2.2). The result now follows from the previous proposition.

Remark If K is a function field over a number field, and F a flasque K-torus, then one may find a regular Z-algebra A of finite type over Z with fraction field K. Let us localize this algebra by inverting the primes dividing n. By a theorem of Deligne, the étale cohomology groups H_{ét}^{r}(A, μ_{n}) are finite. The same proof as above now gives the finiteness of H^{1}(K, F). The original proofs of that result ([CTS1], [CTS2]) used reduction to finite generation of the unit groups (Dirichlet) and finite generation of the Picard group, which if the transcendence degree of K/k is not zero involves the Mordell-Weil theorem.

§3.3. Weak approximation and Hasse principle for tori

In §4 and §5, weak approximation and the Hasse principle will be discussed for arbitrary linear algebraic groups over fields of type (gl) or (ll), i.e. in an essentially 2-dimensional situation. The case of tori is however particularly simple, and some of the results hold without the 2-dimensional restriction.

Let K be a field and S a finite set of distinct rank one discrete valuations on K. Let K_{v} denote the completion of K at v ∈ S. Let T be a K-torus and let

1 → F → P → T → 1

be a flasque resolution of T. We have the obvious diagram

P (K)  ⟶  T (K)  ⟶  H^{1}(K, F)  ⟶ 0
↓  ↓  ↓
∏_{v∈S} P (K_{v}) ⟶ ∏_{v∈S} T (K_{v}) ⟶ ∏_{v∈S} H^{1}(K_{v}, F) ⟶ 0,

The quasitrivial torus P is a Zariski open set of an affine space, hence it satisfies weak approximation. Since the morphism P → T is smooth, the induced maps on topological groups P (K_{v}) → T (K_{v}) are open ([Se1], Part II, Lie Groups, Chap. 3, §10.2). In particular, the image of P (K_{v}) in T (K_{v}) is open.

If we equip each H^{1}(K_{v}, F) with the discrete topology, each map T (K_{v}) → H^{1}(K_{v}, F) is continuous, and so is the induced map

φ : ∏_{v∈S} T (K_{v}) → ∏_{v∈S} H^{1}(K_{v}, F)/δ(H^{1}(K, F)),

where δ denotes the diagonal map H^{1}(K, F) → ∏_{v∈S} H^{1}(K_{v}, F), and the quotient group ∏_{v∈S} H^{1}(K_{v}, F)/δ(H^{1}(K, F)) is again equipped with the discrete topology. We have the straightforward :

Proposition 3.5 Let A_{S}(T) denote the quotient of the product ∏_{v∈S} T (K_{v}) by the closure of the image of T (K) under the diagonal map. The map φ induces an isomorphism between the (discrete) groups A_{S}(T) and ∏_{v∈S} H^{1}(K_{v}, F)/δ(H^{1}(K, F)).

Let K be a function field (of arbitrary transcendence degree) over the ground field k. For each smooth projective model X/k of K, let Ω_{X} be the set of discrete valuations on K associated to the codimension 1 points on X. Let Ω be the union of all Ω_{X}.

Theorem 3.6 Let k be a field of characteristic zero with finite cohomology. Let K/k be a function field. Let

1 → F → P → T → 1

be an exact sequence of K-tori, with P quasitrivial and F flasque. Then

(a) For any finite set S ⊂ Ω, the induced map A_{S}(T) → ∏_{v∈S} H^{1}(K_{v}, F)/δ(H^{1}(K, F)) is an isomorphism of finite groups.

(b) For any smooth model X/k of K/k, and any K-torus T, the kernel �_{X}^{1}(K, T) of the diagonal map H^{1}(K, T) → ∏_{v∈Ω_{X}} H^{1}(K_{v}, T) is a finite group. It embeds naturally into the kernel �_{X}^{2}(K, F) of the map H^{2}(K, F) → ∏_{v∈Ω_{X}} H^{2}(K_{v}, F).

(c) The kernel �^{1}(K, T) of the map H^{1}(K, T) → ∏_{v∈Ω} H^{1}(K_{v}, T) is finite. It embeds naturally into the kernel �^{2}(K, F) of the map H^{2}(K, F) → ∏_{v∈Ω} H^{2}(K_{v}, F).

Proof For any v ∈ S, the field K_{v} is isomorphic to the field k(Y)((t)) of Laurent series over the function field of a k-variety Y of dimension one less than that of X. Combining Theorems 3.2 and 3.4 (a), we see that for any flasque torus F over K, and any v, the group H^{1}(K_{v}, F) is finite (by Theorem 3.4, the group H^{1}(K, F) itself is also finite.) The rest of statement (a) is a special case of Proposition 3.5.

To prove (b), which clearly implies (c), we may restrict X to a smooth affine open set U over which the torus T extends to a U-torus. The kernel of the above map is contained in the image of H_{ét}^{1}(U, T) → H^{1}(K, T) (this well-known purity statement follows from the exact sequence at the bottom of page 163 of [CTS2]). Finiteness then follows from Proposition 3.3. The embedding statements in (b) and (c) are clear, since P being quasitrivial implies H^{1}(K, P) = 0 and H^{1}(K_{v}, P) = 0.

We now consider the local henselian case. Let k be a field of characteristic zero. Let A be an excellent, henselian, local domain with residue field k. Let K be its field of fractions. For a regular integral scheme X equipped with a proper birational map X → Spec(A), we let Ω_{X} be the set of discrete valuations on K associated to the codimension 1 points on X. We let Ω be the union of all Ω_{X}.

Theorem 3.7 Let k be a field of characteristic zero with finite cohomology. Let A be a two-dimensional, excellent, henselian, local domain with residue field k and let K be its fraction field. Let

1 → F → P → T → 1

be an exact sequence of K-tori, with P quasitrivial and F flasque. Then

(a) For any finite set S ⊂ Ω, the induced map A_{S}(T) → ∏_{v∈S} H^{1}(K_{v}, F)/δ(H^{1}(K, F)) is an isomorphism of finite groups.

(b) For any regular proper model X → Spec(A), and any K-torus T, the kernel �_{X}^{1}(K, T) of the diagonal map H^{1}(K, T) → ∏_{v∈Ω_{X}} H^{1}(K_{v}, T) is a finite group. It embeds naturally into the kernel �_{X}^{2}(K, F) of the map H^{2}(K, F) → ∏_{v∈Ω_{X}} H^{2}(K_{v}, F).

(c) The kernel �^{1}(K, T) of the map H^{1}(K, T) → ∏_{v∈Ω} H^{1}(K_{v}, T) is finite. It embeds naturally into the kernel �^{2}(K, F) of the map H^{2}(K, F) → ∏_{v∈Ω} H^{2}(K_{v}, F).

(d) Assume moreover that k is algebraically closed. Then there is an isomorphism of finite groups between the kernel �^{1}(K, T) of the map H^{1}(K, T) → ∏_{v∈Ω} H^{1}(K_{v}, T) and the kernel �^{2}(K, F) of the map H^{2}(K, F) → ∏_{v∈Ω} H^{2}(K_{v}, F).

(e) Under the same assumptions as in (d), if T/K is split by a metacyclic extension, more generally if T is a birational direct factor of a K-rational variety, then for any finite set S ⊂ Ω, we have A_{S}(T) = 0, and we have �^{1}(K, T) = 0.

Proof For v ∈ S, the field K_{v} is isomorphic to a Laurent series field L((t)) over a field L which is either a function field in one variable over k or is itself a Laurent series field k((u)). Combining Theorems 3.2 and 3.4, we see that for any flasque torus F over K, and any v, the group H^{1}(K_{v}, F) is finite. By Theorem 3.4, the group H^{1}(K, F) itself is also finite. The rest of statement (a) is a special case of Proposition 3.5. To prove (b), which clearly implies (c), we may restrict X to a smooth affine open set over which the torus T extends to a Utorus. By the purity statement referred to in the previous proof, the kernel of the above map is contained in the image of H_{ét}^{1}(U, T) → H^{1}(K, T). Finiteness then follows from Proposition 3.3. The embedding statements in (b) and (c) are clear, since P being quasitrivial implies H^{1}(K, P) = 0 and H^{1}(K_{v}, P) = 0. To prove (d), it is enough to know that �^{2}(K, P) = 0 and that is an immediate consequence of Corollary 1.10 (c) of [CTOP] (Theorem 1.6 of the present paper). The case of an arbitrary quasitrivial torus reduces immediately to that statement. If T is split by a metacyclic extension, then one may assume that the flasque resolution of T has the same property, hence by the Endo-Miyata theorem, there exists a K-torus F′ such that F ×_{K} F′ is a quasitrivial torus Q. This implies (Hilbert’s theorem 90) that the torsor P → T has a section over an open set, hence that T ×_{K} F is K-birational to P. Since weak approximation holds for P, it holds for T. We have �^{2}(K, F) ⊕ �^{2}(K, F′) = �^{2}(K, Q) and this last group is trivial by the argument given in the proof of (d). Statement (c) then yields �^{1}(K, T) = 0.

Remark Assume that k is algebraically closed, i.e. one is in the (ll) case. For X → Spec(A) a regular model of A, and F a flasque K-torus, for almost all points v ∈ Ω_{X}, one has H^{1}(K_{v}, F) = 0. Indeed F extends to a flasque torus over an open set U ⊂ X. For any v ∈ U^{(1)}, let O_{v} be the completion of the local ring of X at v and let H^{1}(O_{v}, F) = H_{ét}^{1}(Spec(O_{v}), F). Then the restriction map H^{1}(O_{v}, F) → H^{1}(K_{v}, F) is surjective. By Hensel’s lemma, H^{1}(O_{v}, F) ≃ H^{1}(k(v), F_{k(v)}), where k(v) denotes the residue field at v ([SGA3], Exp. XXIV, Prop. 8.2.(ii). But any such field k(v) is a field of cohomological dimension one, hence H^{1}(k(v), M) = 0 for any k(v)-torus M.

For any K-torus T over K, with flasque resolution 1 → F → P → T → 1 and any regular model X → Spec(A), this enables us to produce the finite group

A_{X}(T) = Coker[H^{1}(K, F) → ∏_{v∈Ω_{X}} H^{1}(K_{v}, F)]

which measures the lack of weak approximation with respect to all places in Ω_{X}. But if we consider the union Ω of all Ω_{X}, we have no such result. It seems hard to define an analogue of the Brauer-Manin obstruction to weak approximation in the present context.

§3.4. A counterexample to weak approximation

In the number field case, examples of K-tori with A_{S}(T) ≠ 0 or with �^{1}(K, T) ≠ 0 have been known and discussed for a long time. One may wonder whether such examples exist in our context. The next proposition handles weak approximation.

Proposition 3.8 Let A = C[[x, y]] ⊂ B = C[[x^{1/2}, y^{1/2}]], and let K ⊂ L be the inclusion of fraction fields. Let T be the K-torus R_{L/K}^{1}G_{m}. Let A_{x}, resp. A_{y}, be the discrete valuation ring which is the completion of A at the prime ideal x, resp. y; let K_{x}, resp. K_{y}, be the associated (topological) fraction field. The group T (K) of K-rational points is dense in each of T (K_{x}) and T (K_{y}), but its image under the diagonal embedding T (K) ⊂ T (K_{x}) × T (K_{y}) is not dense.

Proof Let G = Gal(L/K). Proposition 15 of [CTS1] states that for any Galois extension L/K with group G, for the K-torus R_{L/K}^{1}G_{m} there is a natural isomorphism T (K)/R ≃ Ĥ^{−1}(G, L^{∗}) = ^{N}L^{∗}/I_{G}L^{∗}, where ^{N}L^{∗} denotes the kernel of the norm map N : L^{×} → K^{×}. In the particular case under study, namely G ≃ Z/2 × Z/2, a transfer argument shows that the group Ĥ^{−1}(G, L^{∗}) is killed by 2.

The diagonal map T (K) → T (K_{x}) × T (K_{y}) induces the natural maps Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{x}^{∗}) and Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{y}^{∗}). We shall show that each of Ĥ^{−1}(G, L^{∗}), Ĥ^{−1}(G, L_{x}^{∗}) and Ĥ^{−1}(G, L_{y}^{∗}) is isomorphic to Z/2, each of the maps Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{x}^{∗}) and Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{y}^{∗}) being an isomorphism. The proposition will then follow from the quoted result of [CTS1].

The ring B is a unique factorization domain. Hence the map which to an element in L^{∗} associates its divisor defines a short exact sequence of G-modules

1 → B^{∗} → L^{∗} → Div(B) → 0.

Since Div(B) is a direct sum of G-permutation modules, one has Ĥ^{−1}(G, Div(B)) = 0, hence a surjection from Ĥ^{−1}(G, B^{∗}) to Ĥ^{−1}(G, L^{∗}). The G-module B^{∗} is the direct product of C^{∗} (with trivial action) and of the multiplicative group of series with value 1 at the origin, and the latter group is clearly uniquely divisible. Thus the inclusion C^{∗} ⊂ B^{∗} induces an isomorphism Ĥ^{−1}(G, C^{∗}) = Ĥ^{−1}(G, B^{∗}). One easily checks that Ĥ^{−1}(G, C^{∗}) = ^{N}C^{∗}/I_{G}C^{∗} = μ_{4}, the generator being the class of a primitive 4-th root of unity i. All in all, the inclusion C^{∗} ⊂ L^{∗} induces a surjection μ_{4} → Ĥ^{−1}(G, L^{∗}), the last group being of order at most 2, spanned by the class of i.

We have K_{x} = C((y))((x)), and L_{x} := L ⊗_{K} K_{x} = C((y^{1/2}))((x^{1/2})). This is a field extension of group G ≃ Z/2 × Z/2. Let G =< σ, τ >, with σ(x^{1/2}) = −x^{1/2}, σ(y^{1/2}) = y^{1/2}, τ(x^{1/2}) = x^{1/2}, τ(y^{1/2}) = −y^{1/2}. The multiplicative group of L_{x} = C((y^{1/2}))((x^{1/2})) may be written as a direct product C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z}.U with U a uniquely divisible G-module (namely the subgroup spanned by the Einseinheiten of C((y^{1/2}))[[x^{1/2}]]^{∗} and the Einseinheiten of C[[y^{1/2}]]^{∗}). As a G-module, L_{x}^{∗} is the direct product of its two subgroups C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z} and U.

One then has

Ĥ^{−1}(G, C((y^{1/2}))((x^{1/2}))^{∗}) = Ĥ^{−1}(G, C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z}).

One easily computes ^{N}(C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z}) = μ_{4} ⊂ C^{∗} and I_{G}(C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z}) =< −1 >⊂ C^{∗}, so that Ĥ^{−1}(G, C^{∗}.(x^{1/2})^{Z}.(y^{1/2})^{Z}) = Z/2 spanned by the class of i ∈ C^{∗}.

It is then clear that the map Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{x}^{∗}) is the identity on Z/2 induced by the identity on Z/2{i} = Ĥ^{−1}(G, C^{∗}). Permuting x and y yields the same result for Ĥ^{−1}(G, L^{∗}) → Ĥ^{−1}(G, L_{y}^{∗}).

The map T (K)/R → T (K_{x})/R × T (K_{y})/R therefore reads as the diagonal map Z/2 → Z/2⊕Z/2, and weak approximation fails for T. More precisely, the pair (i, 1) is not in the closure of T (K) (we use the fact, clear from a flasque resolution, that the set of points R-equivalent to 1 on T (K_{x}) is an open subgroup, and similarly for T (K_{y})).

There remains the following open question (to which we expect a negative answer) :

Question Let A be a two-dimensional, excellent, henselian, local domain with algebraically closed residue field k and let K be its fraction field. Let T be a K-torus. Is the group �^{1}(K, T) trivial ? That is, over such a field K, and with respect to the valuations in Ω, does the Hasse principle hold for principal homogeneous spaces under tori ?

Remarks

1) A closely related question is whether there exists a field K as above and a finite commutative K-group μ such that �^{2}(K, μ) ≠ 0.

2) Jaworski [Ja] has an interesting, explicit counterexample to the Hasse principle for H^{1}(K, Z/2) = K^{∗}/K^{∗2}. His (slightly generalized) example is as follows. Let

A = C[[x, y, z]]/(z^{2} − (x^{2} − y^{p})(y^{2} − x^{q})),

where the integers p, q satisfy pq ≥ 5. Let K be the field of fractions of the normal ring A. Then each of the classes (x^{2} − y^{p}), (y^{2} − x^{q}) is a square in any completion K_{v} of K at a discrete, rank one valuation. But one easily checks that neither of these is a square in K. In [CTOP], Remark 3.1.2, yet another (less explicit) example is given. O. Gabber points out that this example can be made explicit, for instance by taking A = C[[x, y, z]]/(xyz + x^{4} + y^{4} + z^{4}).

3) For the case considered in Proposition 3.8, using the triviality of the Picard group of B, one sees that �^{1}(K, T) = 0.

§4. R-equivalence and weak approximation on linear algebraic groups

In this section, unless otherwise mentioned, the ground field is assumed to be of characteristic 0. We first recall a series of basic results on simply connected semisimple groups.

Let K be a field and G/K be a simply connected group of type ^{1}A_{n}. Let B/K be a central simple algebra such that G = SL_{1,B}. Let B^{∗1} ⊂ B^{∗} be the group of elements of reduced norm 1. The commutator subgroup [B^{∗}, B^{∗}] ⊂ B^{∗1} is a normal subgroup of B^{∗1}, all elements of which are R-equivalent to 1 ∈ G(K). We therefore have a clearly surjective homomorphism φ : SK_{1}(B) := B^{∗1}/[B^{∗}, B^{∗}] → G(K)/R, and both groups are abelian. This homomorphism is functorial in the field K.

Theorem 4.1 Let G = SL_{1,B} be a simply connected K-group of type ^{1}A_{n}. With notation as above :

(i) (Voskresenskiǐ, [Vo1], [Vo2] p. 186) The homomorphism φ is an isomorphism.

(ii) (Yanchevskiǐ, [Ya2]) Suppose that over any finite field extension E of K, and any central simple algebra C/E, the reduced norm map Nrd : C^{∗} → E^{∗} is onto. Then B^{∗1}/[B^{∗}, B^{∗}] = 1, hence also G(K)/R = 1.

(iii) If cd(K) ≤ 2, then B^{∗1}/[B^{∗}, B^{∗}] = 1, hence also G(K)/R = 1.

Statement (iii) is a consequence of (ii) and the Merkurjev-Suslin theorem ([Su], Thm. 24.8).

Let K be a field and G/K be a group of type ^{2}A_{n}. Then G = SU(B, τ) where L/K is a quadratic, separable, field extension of K, and B/L is a central simple algebra equipped with an involution of the second kind τ. We have the natural exact sequence of algebraic groups over K :

1 → SU(B, τ) → U(B, τ) → R_{L/K}^{1}G_{m} → 1.

The homomorphism B^{∗} → L^{∗} defined by b ↦ Nrd(b)/τNrd(b) defines another exact sequence of algebraic groups over K :

1 → H → R_{L/K}(GL_{1,B}) → R_{L/K}^{1}G_{m} → 1.

The group H(K) of K-rational points of H is the preimage in B^{∗} of K^{∗} ⊂ L^{∗} under the reduced norm map B^{∗} → L^{∗}. Any τ-symmetric element in B^{∗} belongs to H(K), and it is R-equivalent to 1 on H since (B^{∗})^{τ} is the set of k–rational points of an open set of an affine space. Let Σ ⊂ H(K) be the (normal) subgroup spanned by the τ-symmetric elements. There is an obvious surjective homomorphism ψ : H(K)/Σ → H(K)/R, and that homomorphism is functorial in the field K.

The following theorem is the result of successive efforts by Yanchevskiǐ, Monastyrniǐ, Merkurjev, Chernousov (see [CM]).

Theorem 4.2 Let G = SU(B, τ) be a simply connected K-group of type ^{2}A_{n}. With notation as above :

(i) The underlying K-varieties of G and H are stably K-birationally equivalent.

(ii) There is a natural group isomorphism H(K)/R ≃ G(K)/R; that isomorphism is functorial in the field K.

(iii) The surjective homomorphism ψ : H(K)/Σ → H(K)/R is an isomorphism.

(iv) Suppose that over any finite field extension E of K, and any central simple algebra C/E, the reduced norm map Nrd : C^{∗} → E^{∗} is onto. Then H(K)/Σ = 1, hence also G(K)/R = 1.

(v) If cd(K) ≤ 2, then H(K)/Σ = 1, hence also G(K)/R = 1.

Proof Results (i) to (iii) may be read off from [CM]. A key property is that over any field extension F of K, the maps induced on F -points by U(B, τ) → R_{L/K}^{1}G_{m} and by R_{L/K}(GL_{1,B}) → R_{L/K}^{1}G_{m} have the same image (a result of Merkurjev, see [CM], Prop. 5.2). Denote by M the K-group which is the fibre product of U(B, τ) and R_{L/K}(GL_{1,B}) over R_{L/K}^{1}G_{m}. One then has two natural exact sequences of K-groups

1 → SU(B, τ) → M → R_{L/K}(GL_{1,B}) → 1

and

1 → H → M → U(B, τ) → 1,

and the above property implies that each of the morphisms M → R_{L/K}(GL_{1,B}) and M → U(B, τ) admits a section over an open set. Each of the K-groups R_{L/K}(GL_{1,B}) and U(B, τ) is a K-rational variety. This immediately gives statement (i). It also implies that each of the homomorphisms SU(B, τ) → M and H → M induces group isomorphisms SU(B, τ)(K)/R → M(K)/R and H(K)/R → M(K)/R. This is statement (ii) in the theorem. For (iii), see [CM], Lemma 5.1 (in the isotropic case, see [Ya3], Remark p. 537). Result (iv) is due to Yanchevskiǐ ([Ya1], Thm. 1). Result (v) follows from (iv) by the Merkurjev-Suslin results.

Theorem 4.3 (Chernousov-Platonov) Let K be a field of characteristic zero with cd(K) ≤ 2. Assume that over any finite field extension of K, index and exponent coincide for 2-primary and for 3-primary central simple algebras. Let G/K be a simply connected semisimple group without factor of type A_{n}. If G contains a factor of type E_{8}, assume cd(K^{ab}) ≤ 1. Then the K-variety G is K-rational, i.e. birational to affine space over K.

Proof In [CP], Chernousov and Platonov state the theorem for p-adic and totally imaginary number fields. Let us indicate how their arguments (easily) extend to the present situation. For L/K a finite field extension and X an L-rational L-variety, the K-variety R_{L/K}X is K-rational. By the standard reduction, we may thus assume that G is absolutely almost simple, simply connected, and not of type A_{n}.

Case B_{n} (n ≥ 2). Then G is the spinor group of a nondegenerate quadratic form q of rank 2n+1 ≥ 5 over K. Under our assumption on K, any quadratic form of rank at least 5 is isotropic (Theorem 1.1 (f)). The statement is now a special case of the general result : Over any field F of characteristic not 2, for any isotropic quadratic form q over F the group Spin(q) is an F -rational variety (Platonov). This result may be proved in a number of ways ([CP], Prop. 4 ; [Me2], Prop. 6.1). One may give a proof by induction. Let us write the underlying vector space V as an orthogonal sum W ⊥ K.v, in such a manner that a = q(v) ≠ 0 and the restriction q_{0} of q to W is still isotropic. Let X be the affine quadric defined by q(x) = a. The group Spin(q) acts on X, the isotropy group of v ∈ X being Spin(q_{0}). The isotropy assumption ensures that over any field extension F of K this action is transitive on F -points. This implies that the morphism g ↦ g.v admits a rational section, and Spin(q) is F -birational to the product of Spin(q_{0}) and the F -rational quadric X.

Case C_{n}. Over any field F, any simply connected group of type C_{n} is a (connected) unitary group, which is F -rational (Cayley transform) ([CP], Lemma 5).

Case of nontrialitarian D_{n} (n ≥ 4). Over any field F, any simply connected group of this type can be realized as the spinor group Spin(D, h) associated to a nondegenerate hermitian form over a central division algebra D of degree d with an orthogonal involution of the first kind. Chernousov and Platonov prove that if the index of D is at most 4 and G is isotropic, then G is F -rational ([CP], Cor. 5, Prop. 8 for inner forms, Prop. 10 and Prop. 11 for outer forms). Over the field K, the algebra D, which is of exponent 2, is of index at most 2, and the group G is isotropic by Theorem 2.1 (b) (see also Prop. 2.12). The result of Chernousov and Platonov therefore applies.

Case of trialitarian D_{n}. Over any field F, any isotropic group of that type is F -rational ([CP], Prop. 13). Over the field K, any such group is isotropic (Theorem 2.1).

Cases E_{6} and E_{7}. Over any field F, any F -isotropic group of type E_{6} or E_{7} with Tits index as described in Theorem 2.1 is F -rational ([CP], Prop. 14 for inner forms of E_{6}, Prop. 16 for outer forms of E_{6}, Prop. 17 for E_{7}). Over the field K, Theorem 2.1 says that either G is quasisplit, or it has the quoted Tits index, hence in particular is isotropic.

Cases F_{4}, G_{2}, E_{8}. In these cases, the centre is trivial, hence all forms of the split group G are classified by H^{1}(K, G), which by Theorem 1.2 is trivial. Thus any such group over K is K-split, hence K-rational.

Remark Chernousov and Platonov also investigate the rationality properties of arbitrary absolutely almost simple groups. In most cases, over p-adic and totally imaginary number fields, they establish the rationality when the group is not of type A_{n}. Presumably most of their results hold for our more general fields. Beware however of an inaccuracy in [CP] : In the main theorem, one must exclude adjoint outer forms of type D_{n} related to a quaternion division algebra A and a skew-hermitian form h over A whose anisotropic part has dimension 3 (this may occur over a totally imaginary number field).

Corollary 4.4 Let K be a field of type (ll), (sl) or (gl). Let G/K be a simply connected group without factor of type A_{n}. In the (gl) case, assume that G contains no E_{8}-factor. Then the underlying K-variety of the group G is K-rational, i.e. it is K-birational to an affine space over K.

Collecting the previous results, we obtain a general statement regarding R-equivalence on simply connected groups.

Theorem 4.5 Let G/K be a semisimple, simply connected group over a field K of characteristic zero. Assume cd(K) ≤ 2. Assume that over any finite field extension of K, index and exponent coincide for 2-primary and for 3-primary central simple algebras. If G/K contains a factor of type E_{8}, assume cd(K^{ab}) ≤ 1. Then G(K)/R = 1.

Proof The assumptions on the ground field k are stable by finite field extension. If L/K is a finite field extension and H/L is a linear algebraic group, then there is a natural isomorphism of groups R_{L/K}(K)/R ≃ H(L)/R. By the standard reduction, we may thus assume that G is an absolutely almost simple group. In the A_{n} case, G(K)/R = 1 by Theorems 4.1 and 4.2. In the other cases, G/K is a K-rational variety by Theorem 4.3, hence G(K)/R = 1.

Corollary 4.6 Let G/K be a semisimple, simply connected group over a field K of type (ll), (sl) or (gl). In the (gl) case, assume that G has no simple factor of type E_{8}. Then G(K)/R is trivial.

Remarks

(i) Note the big difference between simply connected groups of type A_{n} and other (absolutely almost simple) simply connected groups over fields of one of our three types (excluding E_{8} in case (gl)). For a group G not of type A_{n}, R-equivalence is universally trivial, i.e. G(F)/R = 1 for any field extension F/K. For a group of type A_{n}, we claim G(F)/R = 1 only for finite extensions of K. Merkurjev [Me1] has given examples with G(F)/R ≠ 1 for G/K of type ^{1}A_{n} and F/K a suitable function field.

(ii) For all we know, Theorem 4.5 might hold under the assumption cd(K) ≤ 3. Indeed the cohomological invariants on SK_{1}(B) = SL_{1,B}(K)/R studied by Suslin and later Merkurjev and Rost take their values in cohomology groups H^{i}(K, .) for i ≥ 4. The question whether SK_{1}(B) = 1 if cd(K) ≤ 3 was explicitly raised by Suslin.

(iii) Theorem 4.5 applies to p-adic fields : for such fields, the result had already been proved by Voskresenskiǐ. The theorem also applies to totally imaginary number fields. This algebraic proof of the triviality of G(K)/R for G simply connected should be compared with the available proof of finiteness of G(K)/R when K is an arbitrary number field (see [Gi1]) : that proof appeals to an ergodic result of Margulis.

We now consider the problem of weak approximation, and restrict attention to fields K of type (gl) or (ll). Recall from §1 that in each of these cases, there is an associated set Ω of places of K. For each place v ∈ Ω, the completion K_{v} is of the shape l((t)), where l is either a function field in one variable over the algebraically closed field k, or is a Laurent series field k((u)). For each v ∈ Ω, the topology on the local field K_{v} induces a topology on the group G(K_{v}).

Theorem 4.7 Let G/K be a simply connected group over a field K which is of type (gl) or (ll). In the (gl) case, assume that G has no E_{8}-factor. Then for any finite set of places S ⊂ Ω, the diagonal map G(K) → ∏_{v∈S} G(K_{v}) has dense image.

Proof The standard reduction allows us to assume that G/K is absolutely almost simple. If G is not of type A_{n}, then under our assumptions, the K-variety G is K-rational and weak approximation follows.

Suppose that G is of type ^{1}A_{n}, i.e. G = SL_{1,B} for B/K a central simple algebra. Let {g_{v}}_{v∈S} ∈ ∏_{v∈S} B_{K_{v}}^{∗1}. Each field K_{v} is a C_{2}-field ([Pf], Chap. 5, Thm. 2.32 p. 70). By Yanchevskiǐ’s results (Theorem 4.1.(ii)), each g_{v} is a product of commutators in B_{K_{v}}^{∗}. Since there are only finitely many places v ∈ S, there is an integer n > 0 such that each g_{v} for v ∈ S is a product of exactly n commutators in B_{K_{v}}^{∗}. Since B^{∗} is the set of K-points of an open subvariety of an affine space over K, the diagonal map B^{∗} → ∏_{v∈S} B_{K_{v}}^{∗} has dense image. Approximating all entries in the n commutators, we conclude that G(K) is dense in ∏_{v∈S} G(K_{v}).

Suppose that G is of type ^{2}A_{n}, i.e. G = SU(B, τ) where L/K is a quadratic, separable, field extension of K, and B/L is a central simple algebra equipped with an involution of the second kind τ. We refer to the discussion before Theorem 4.2 for the definition of the algebraic group H/K. Since G and H are stably K-birationally equivalent K-varieties, weak approximation for one of them is equivalent to weak approximation for the other. Let v be a place of K. Let {g_{v}}_{v∈S} ∈ ∏_{v∈S} H(K_{v}). By Theorem 4.2 (iv) applied to the completion K_{v}, any element of H(K_{v}) is a product of τ-symmetric elements in B_{K_{v}}^{∗}. Since S is finite, there is an integer n > 0 such that each g_{v} for v ∈ S is a product of exactly n τ-symmetric elements in B_{K_{v}}^{∗}. The set of τ-symmetric elements in B^{∗} is the set of K-points of a Zariski open set in an affine space over K. Thus weak approximation holds for such elements, and we conclude that H(K) is dense in ∏_{v∈S} H(K_{v}).

Remark In the ^{2}A_{n} case, the above proof essentially goes back to Yanchevskiǐ ([Ya3], Proposition following Thm. 5, p. 545). Use of [CM] enables one to streamline it.

We now discuss arbitrary connected linear algebraic groups. Recall the basic theorem ([Gi2], Thm. 6 p. 308).

Theorem 4.8 Let K be a field of characteristic zero such that cd(K) ≤ 2. Let

1 → μ → G′ → G → 1

be a K-isogeny of connected linear algebraic groups, where G′ is the product of a semisimple simply connected group and a quasitrivial torus. Let

1 → μ → F → P → 1

be a flasque resolution of μ (the torus F is flasque and the torus P is quasitrivial). The associated Galois cohomology sequences induce an exact sequence

G′(K)/R ⟶ G(K)/R ⟶ H^{1}(K, F) ⟶ 1.

We may now conclude :

Theorem 4.9 Let K be a field of characteristic zero such that cd(K) ≤ 2. Assume that over any finite field extension of K, index and exponent coincide for 2-primary and for 3-primary central simple algebras. Let

1 → μ → G′ → G → 1

be a K-isogeny of connected linear algebraic groups, where G′ is the product of a semisimple simply connected group and a quasitrivial torus. Let

1 → μ → F → P → 1

be a flasque resolution of μ (the torus F is flasque and the torus P is quasitrivial). If G′ contains a factor of type E_{8}, assume cd(K^{ab}) ≤ 1. Then the Galois cohomology sequences induce an isomorphism :

G(K)/R ⟶^{∼} H^{1}(K, F).

Proof A quasitrivial K-torus T is a K-rational variety, hence T (K)/R = 1. For G′/K as above, we have G′(K)/R = 1 by Theorem 4.5.

Corollary 4.10 Let K be a field of characteristic zero such that cd(K) ≤ 2. Assume that over any finite field extension of K, index and exponent coincide for 2-primary and for 3-primary central simple algebras. Let G/K be a connected linear algebraic group. If G contains a factor of type E_{8}, assume cd(K^{ab}) ≤ 1. Then the group G(K)/R is abelian.

Proof By a known reduction, one may assume that G is reductive. By Lemme 1.10 of [Sa], there then exists an isogeny 1 → μ → G′ → G^{m} × T → 1, with T a quasitrivial K-torus, m > 0 and G′ the product of a quasitrivial torus by a simply connected group with the same (absolute) simple factors as G. Let 1 → μ → F → P → 1 be a flasque resolution of μ. By Theorem 4.9, we have a group isomorphism (G(K)/R)^{m} = (G(K)/R)^{m} × T (K)/R ≃ H^{1}(K, F), hence G(K)/R is an abelian group.

Corollary 4.11 Let K be as in Theorem 4.9 and let G/K be a semisimple group. Under any of the following assumptions :

(i) G is simply connected,

(ii) G is adjoint,

(iii) G is absolutely almost simple,

(iv) G is an inner form of a group which is split by a metacyclic extension of K, we have G(K)/R = 1.

Proof Recall that a finite field extension L/K is called metacyclic if it is Galois and each Sylow subgroup of the Galois group is cyclic. We follow Sansuc’s argument [Sa, Cor. 5.4]. Case (i) has already been handled (Theorem 4.5). If G is adjoint, then it is a product of restrictions of scalars of groups of type (iii) over finite extensions of K. Thus (ii) reduces to (iii). An absolutely almost simple K-group is an inner form of an absolutely almost simple quasisplit K-group. The latter is split by a metacyclic extension, the automorphism group of the Dynkin diagram being either 0, Z/2 or 𝔖_{3}. One is thus reduced to case (iv). Let μ be the fundamental group of G. There exists a flasque resolution 1 → μ → F → P → 1 where F and P are split by the minimal splitting field of the multiplicative group μ (which by definition is the splitting field of the character group of μ). If a semisimple group is split, then its fundamental group μ is split (that is, the absolute Galois group acts trivally on the character group of μ). An inner twist does not affect the fundamental group. Thus F is split by a metacyclic extension, hence is a direct factor of a quasitrivial torus (theorem of Endo and Miyata), hence H^{1}(K, F) = 0. The triviality of G(K)/R now follows from Theorem 4.9.

Specializing to the fields of main interest in this paper, we have

Theorem 4.12 Let K be a field of type (ll), (sl) or (gl). Let G/K be a connected linear algebraic group. In the (gl) case, assume G has no E_{8} factor. Then

(i) The quotient G(K)/R is a finite abelian group.

(ii) Suppose the group G has a presentation

1 → μ → G′ → G → 1

where G′ is the product of a semisimple simply connected group and a quasitrivial torus, and G′ → G is an isogeny with kernel μ. Let

1 → μ → F → P → 1

be a flasque resolution of μ (the torus F is flasque and the torus P is quasitrivial). Then the Galois cohomology sequences induce an isomorphism of finite groups G(K)/R ≃ H^{1}(K, F).

Proof For part (ii), simply combine Theorems 4.6, 4.8 and the finiteness results 3.2 and 3.4. To prove the finiteness of G(K)/R in the general case, proceeding as in Corollary 4.10 above, one shows that for some m > 0, (G(K)/R)^{m} is finite.

We now consider weak approximation for fields of type (ll) or (gl), and the associated set Ω of discrete valuations.

Theorem 4.13 Let K be a field of type (ll) or (gl). Let S ⊂ Ω be a finite set. Let G/K be a connected linear group. In the case (gl), assume that G has no factor of type E_{8}.

(i) The closure \overline{G(K)} of the image G(K) under the diagonal map G(K) → ∏_{v∈S} G(K_{v}) is a normal subgroup, and the quotient A_{S}(G) = ∏_{v∈S} G(K_{v})/\overline{G(K)} is a finite abelian group.

(ii) Suppose the group G has a presentation 1 → μ → G′ → G → 1 where G′ is the product of a semisimple simply connected group and a quasitrivial torus, and G′ → G is an isogeny with kernel μ. Let 1 → μ → F → P → 1 be a flasque resolution of μ (the torus F is flasque and the torus P is quasitrivial). Then the composite maps G(K) → H^{1}(K, μ) → H^{1}(K, F) and G(K_{v}) → H^{1}(K_{v}, μ) → H^{1}(K_{v}, F) induce isomorphisms of finite abelian groups

A_{S}(G) ≃ Coker[H^{1}(K, F) → ∏_{v∈S} H^{1}(K_{v}, F)]

and

A_{S}(G) ≃ Coker[G(K)/R → ∏_{v∈S} G(K_{v})/R].

Proof The arguments due to Kneser [Kn2] and Sansuc ([Sa], §3) (for the last isomorphism, see also [N]) in the number field context may be adapted to our context, one key remark being that they do not involve any class field theory (for a minor difference between the two set-ups, see remark after the proof).

Using Proposition 3.2 and Lemme 1.10 of [Sa], one reduces the proof of (i) to that of (ii).

Let H be the group F ×^{μ} G′, i.e. the contracted product of H and G′ relatively to μ. There is a commutative diagram of exact sequences of algebraic groups

1  1
↓  ↓
1 ⟶ μ ⟶ G′ ⟶ G ⟶ 1
↓  ↓  ∥
1 ⟶ F ⟶ H ⟶ G ⟶ 1
↓  ↓
P = P
↓  ↓
1  1

The middle vertical sequence induces the following diagram, where horizontal maps are diagonal maps :

G′(K) ⟶ ∏_{v∈S} G′(K_{v})
↓  ↓
H(K) ⟶ ∏_{v∈S} H(K_{v})
↓  ↓
P (K)  ⟶ ∏_{v∈S} P (K_{v})
↓
H^{1}(K, G′)

By our assumptions, H^{1}(K, G′) = 1. Since P is a quasitrivial torus, the bottom horizontal map has dense image. By Theorem 4.7, the top horizontal map has dense image. The vertical maps on the right hand side are continuous. Since the K-morphism H → P is smooth, each projection map H(K_{v}) → P (K_{v}) is open ([Se1], Part II, Lie Groups, Chap. 3, §10.2). All these statements put together imply that the diagonal map H(K) → ∏_{v∈S} H(K_{v}) has dense image.

We have exact sequences of pointed sets

H^{1}(K, G′) → H^{1}(K, H) → H^{1}(K, P)

and

H^{1}(K_{v}, G′) → H^{1}(K_{v}, H) → H^{1}(K_{v}, P).

Since P is quasitrivial, H^{1}(K, P) = 0 and H^{1}(K_{v}, P) = 0. By our assumptions on G, hence G′, and on the field K, we have H^{1}(K, G′) = 1 and H^{1}(K_{v}, G′) = 0. Thus H^{1}(K, H) = 1 and H^{1}(K_{v}, H) = 1 for any v ∈ Ω.

From the initial diagram we deduce another commutative diagram, where the vertical maps are diagonal maps :

H(K)  ⟶  G(K)  ⟶  H^{1}(K, F)  ⟶ 0
↓  ↓  ↓
∏_{v∈S} H(K_{v}) ⟶ ∏_{v∈S} G(K_{v}) ⟶ ∏_{v∈S} H^{1}(K_{v}, F) ⟶ 0,

where the right hand side zeroes have just been explained.

By Theorems 3.4 and 3.2, the groups H^{1}(K, F) and ∏_{v∈S} H^{1}(K_{v}, F) are finite. Let A_{S}(μ) = Coker[H^{1}(K, F) → ∏_{v∈S} H^{1}(K_{v}, F)]. Let φ : ∏_{v∈S} G(K_{v}) → A_{S}(μ) be the obvious composite homomorphism. We have shown that the diagonal map H(K) → ∏_{v∈S} H(K_{v}) has dense image. An easy diagram chase now shows that the closure of G(K) in ∏_{v∈S} G(K_{v}) coincides with the subgroup φ^{−1}(1), which is clearly normal, and we have ∏_{v∈S} G(K_{v})/\overline{G(K)} ≃ A_{S}(μ).

Remark The above proof also works for a totally imaginary number field K and a finite set of completions K_{v}. In the number field case ([Sa], §3), one usually identifies A_{S}(G) with the cokernel of H^{1}(K, μ) → ∏_{v∈S} H^{1}(K_{v}, μ). This is all right because for a (usual) local field H^{1}(K_{v}, μ) is finite. But for a field K_{v} with residue field a function field in one variable over an algebraically closed field, a group H^{1}(K_{v}, μ) need not be finite. The replacement of H^{1}(K_{v}, μ) by H^{1}(K_{v}, F) does not however enable a total analogy. In the number field case, we have H^{1}(K_{v}, F) = 0 for almost all v. This enables one to define a finite abelian group A(G) which covers the defect of weak approximation at any finite set of places. In the (gl) and (ll) case, for a given model X, H^{1}(K_{v}, F) = 0 for almost all v ∈ Ω_{X} (remark after Theorem 3.7), hence a finite group measures the lack of weak approximation with respect to all places in Ω_{X}. As for the whole set Ω, we have the same comments as in the quoted remark.

The same arguments as in Corollary 4.11 above now yield :

Corollary 4.14 Let K be a field of type (ll) or (gl). Let G/K be a semisimple group. In the case (gl), assume that G has no factor of type E_{8}.

Under any of the following assumptions :

(i) G is simply connected,

(ii) G is adjoint,

(iii) G is absolutely almost simple,

(iv) G is an inner form of a group which is split by a metacyclic extension of K, for any finite S ⊂ Ω, we have A_{S}(G) = 0 : weak approximation holds for G.

Remark In the (ll) case, starting from the example in Proposition 3.8, one may easily produce a finite abelian K-group μ and a sequence 1 → μ → F → P → 1 such that for suitable S ⊂ Ω, the quotient A_{S}(μ) is not trivial. Starting from μ, one may then produce a semisimple group G such that A_{S}(G) ≠ 0.

§5. Hasse principle

The following theorem is an analogue of a result of Borel for number fields.

Theorem 5.1 Let K be a field of type (ll) or (gl). Let Ω be its associated set of places. Let G/K be a connected linear algebraic group. If K is of type (gl), assume that G has no E_{8}-factors. The set

�^{1}(K, G) = ker[H^{1}(K, G) → ∏_{v∈Ω} H^{1}(K_{v}, G)]

is a finite set.

Proof One reduces to the case where G is reductive. Using Lemme 1.10 of [Sa], one is reduced to the case of a group G equipped with an isogeny

1 → μ → G′ → G → 1

with G′ the product of a semisimple simply connected group (with the same factors as G) by a quasitrivial torus. By Theorems 1.3 and 1.4 we have H^{1}(K, G′) = 1, and this also holds true for all the twisted forms of G′, hence we have an injective map �^{1}(K, G) ↪ �^{2}(K, μ).

In the (gl) case, let X/k be an integral surface with function field K. In the (ll) case, let A be the defining strictly henselian local domain for K, and let X = Spec(A). In both cases, over a suitable nonempty regular open set U ⊂ X of finite type over X, there exists a commutative, finite étale group scheme μ_{U}/U whose generic fibre is our given μ. The elements of H^{2}(K, μ) with trivial restriction to H^{2}(K_{v}, μ) for each v ∈ U^{1} belong to the image of the restriction map H^{2}(U, μ_{U}) → H^{2}(K, μ) (purity, [SGA4] XVI 3.7 and XIX 3.2) and the group H^{2}(U, μ_{U}) is finite (see the proof of Prop. 3.3).

The following theorem is an analogue of results of Sansuc ([Sa], Cor. 5.4 and Cor. 5.9) for number fields.

Theorem 5.2 Let K be a field of type (ll) and Ω be its associated set of places. Let G/K be a connected semisimple group of fundamental group μ.

a) The boundary H^{1}(K, G) → H^{2}(K, μ) induces a bijection �^{1}(K, G) ⟶^{∼} �^{2}(K, μ).

b) In either of the following cases :

(i) G is adjoint or G is absolutely almost simple, (ii) G is semisimple and is a direct factor of a K-rational variety, i.e. there exists a

K-variety Y such that G ×_{K} Y is K-birational to affine space over K,

we have �^{1}(K, G) = 1.

Proof a) Since G is semisimple, we have the standard sequence

1 → μ → G′ → G → 1

with G′ simply connected. By Theorems 2.1 and 2.2, the map H^{1}(K, G) → H^{2}(K, μ) is bijective, and the same hold for the K_{v}. So �^{1}(K, G) → �^{2}(K, μ) is bijective.

b) Let us discuss case (i). By the standard reduction, we may assume that G is absolutely almost simple. As explained in the proof of Corollary 4.11, for such a G, the finite abelian K-group μ is then split by a metacyclic extension. The last property implies that we can find a sequence

1 → μ → F → P → 1

with P quasitrivial and F a direct factor of a quasitrivial torus : there exists a K-torus F′ such that F ×_{K} F′ ≃ Q with Q = ∏_{i} R_{K_{i}/K}G_{m}. We then have

�^{1}(K, G) ↪ �^{2}(K, μ) ⊂ �^{2}(K, F) ⊂ ∏_{i} �^{2}(K_{i}, G_{m}).

By Corollary 1.10 of [CTOP] (Theorem 1.6 in the present paper), the latter group vanishes. This completes the proof in case (i).

Let us consider case (ii). Let X be a smooth compactification of G. Let \overline{K} be an algebraic closure of K and Γ = Gal(\overline{K}/K). Because G is semisimple, the natural map \overline{K}^{∗} → \overline{K}[G]^{∗} is a bijection (the only units are the constants). On the other hand, for the semisimple group G, there is a natural Γ-isomorphism between the character group μ̂ and the Picard group of \overline{G} = G ×_{K} \overline{K}. The natural sequence of Γ-modules :

0 → \overline{K}[G]^{∗}/\overline{K}^{∗} → Div_{∞}(\overline{X}) → Pic(\overline{X}) → Pic(\overline{G}) → 0,

where Div_{∞}(\overline{X}) is the permutation module on the irreducible components of the complement of \overline{G} in \overline{X} therefore reads

0 → Div_{∞}(\overline{X}) → Pic(\overline{X}) → μ̂ → 0.

Under assumption (ii), there exists a Γ-module M, such that the Γ-module Pic(\overline{X}) ⊕ M is Γ-isomorphic to a permutation Γ-module ([CTS3], Prop. 2.A.1 p. 461). Dualizing the exact sequence above, we find an exact sequence

1 → μ → F → P → 1

with P a quasitrivial torus and F a direct factor (as a torus) of a quasitrivial torus. As above, this implies �^{2}(K, μ) = 0, hence �^{1}(K, G) = 1.

Remarks

(i) It is likely that �^{1}(K, G) = 1 holds for an arbitrary K-rational group. In the number field case, this is a result of Sansuc ([Sa], Cor. 9.7).

(ii) For an arbitrary connected linear algebraic group G over K of type (ll), one does not expect �^{1}(K, G) = 1 to always hold. But we have not been able to produce a counterexample (this is very closely connected with the Question and Remarks following Prop. 3.8).

We now follow Borovoi’s paper [Bo1] extremely closely. The aim is to show how Theorem 2.1 (conjecture 5.3 of [CTOP]) implies Conjecture 5.4 of [CTOP] (Corollary 5.7 below).

Borovoi’s paper elaborates on earlier work of Springer and of Douai. One preprint quoted in [Bo1] has since appeared ([Bo2]). A useful complement to Borovoi’s paper is the article [FSS].

Let k be a field of characteristic zero, and \overline{k} an algebraic closure of k. Let \overline{G} be a connected linear \overline{k}-group and L = (\overline{G}, κ) a k-kernel (otherwise known as k-band or k-lien). To \overline{G} one associates a \overline{k}-torus \overline{G}^{tor} ([Bo1], Notation p. 218).

To the k-kernel L one associates the second cohomology set H^{2}(k, L). This set may or may not be empty. Inside this set one defines special classes called neutral classes ([Bo1] 1.6). There may exist none, one, or more such classes. If \overline{G} is reductive, then H^{2}(k, L) contains a neutral element ([Bo1] Prop. 3.1). In general, if we let \overline{G}^{red} be the quotient of \overline{G} by its unipotent radical, then there are an induced k-kernel L^{red} = (\overline{G}^{red}, κ^{red}) and a canonical map of sets r : H^{2}(k, L) → H^{2}(k, L^{red}). An element η ∈ H^{2}(k, L) is neutral if and only r(η) ∈ H^{2}(k, L^{red}) is neutral ([Bo1], Prop. 4.1). When \overline{G} is reductive, one defines an abelian group H_{ab}^{2}(k, L) ([Bo1], 5.1) and an abelianization map ab^{2} : H^{2}(k, L) → H_{ab}^{2}(k, L) ([Bo1], 5.3). This map sends the neutral classes to zero. For an arbitrary connected linear \overline{k}-group, and a k-kernel L, one sets H_{ab}^{2}(k, L) = H_{ab}^{2}(k, L^{red}) and one defines the abelianization map to be the compositum

ab^{2} : H^{2}(k, L) → H^{2}(k, L^{red}) → H_{ab}^{2}(k, L^{red}) = H_{ab}^{2}(k, L).

Neutral elements are sent to zero.

Proposition 5.3 Let L = (\overline{G}, κ) be a connected k-kernel. Assume cd(k) ≤ 2, and assume that for central simple algebras over finite field extensions of k, index and exponent coincide. Then an element η ∈ H^{2}(k, L) is neutral if and only if ab^{2}(η) = 0.

Proof See the proof of Theorem 5.5 of [Bo1] (p. 229, Section 5.8). In our context, the rôle of [Bo1], Lemma 5.7 for number fields (surjectivity of the boundary map) is played by our Theorem 2.1.

Remark As in [Bo1], Cor. 5.6, when \overline{G} is semisimple simply connected, this implies that any element of H^{2}(k, L) is neutral.

There is a natural homomorphism of \overline{k}-algebraic groups t_{∗} : \overline{G} → \overline{G}^{tor}, where \overline{G}^{tor} is the biggest quotient torus of \overline{G}. The k-kernel L induces a k-kernel L^{tor} = (\overline{G}^{tor}, κ^{tor}) on the (commutative) \overline{k}-torus \overline{G}^{tor} : this defines a k-torus G^{tor}, k-form of \overline{G}^{tor} ([Bo1] 1.4, 1.7, 6.1). There is an identification H^{2}(k, L^{tor}) = H^{2}(k, G^{tor}), where the latter set is the usual second abelian cohomology group. There is a natural map

t_{∗} : H^{2}(k, L) → H^{2}(k, L^{tor}) = H^{2}(k, G^{tor}),

referred to as the canonical map.

As explained in [Bo1], 6.1, for L = (\overline{G}, κ) with \overline{G} connected, there is an exact sequence ([Bo1], (6.1.1))

H^{3}(k, μ) → H_{ab}^{2}(k, L) →^{t_{ab}} H^{2}(k, G^{tor}),

where μ is a finite abelian k-group (in the notation of [Bo1], section 5,1, μ is the kernel of the homomorphism of k-groups Z^{(sc)} → Z) and the composite map H^{2}(k, L) → H_{ab}^{2}(k, L) → H^{2}(k, G^{tor}) is the canonical map t_{∗}.

We now have the analogue of Proposition 6.2 of [Bo1].

Proposition 5.4 Assume cd(k) ≤ 2, and assume that for central simple algebras over finite field extensions of k, index and exponent coincide. Let L = (\overline{G}, κ) be a connected k-kernel. Then an element η ∈ H^{2}(k, L) is neutral if and only if t_{∗}(η) = 0, where t_{∗} is the canonical map.

Proof Since k has cohomological dimension 2, we have H^{3}(k, μ) = 0, hence the map H_{ab}^{2}(k, L) → H^{2}(k, G^{tor}) is injective. Thus t_{∗}(η) = 0 implies that the image of η under ab^{2} : H^{2}(k, L) → H_{ab}^{2}(k, L) is zero. By Proposition 5.3 this implies that η is neutral. The converse statement is obvious.

Let X be a smooth variety over k which is a right homogeneous space of a semisimple simply connected k-group H. Let \overline{G} be the isotropy group of a \overline{k}-point x of X(\overline{k}), and assume that this isotropy group is connected. Since the homogeneous space X is defined over k, there is an associated connected k-kernel L = (\overline{G}, κ) ([Bo1], 7.1), and an associated k-torus G^{tor}. These data are functorial in the ground field k. By cocycles computations, one may show that up to k-isomorphism the k-torus G^{tor} does not depend on the choice of the point x. If X has a k-point x, and G_{x} is its stablizer, then G^{tor} is the maximal torus quotient of G_{x}, i.e. the character group of G^{tor} coincides with the Galois module of characters of G_{x} ×_{k} \overline{k}.

Associated to the homogeneous space X there is a class η = η(X) ∈ H^{2}(k, L) ([Bo1], 7.7) which is neutral if and only if the homogeneous space X comes from a right principal homogeneous space under H, i.e. there exists such a space Y and a H-equivariant map Y → X.

We are now in a position to prove the analogue of a result of Borovoi ([Bo1], Thm. 7.3).

Theorem 5.5 Let A be an excellent henselian two-dimensional local domain, let K be its field of fractions and k its residue field. Assume that k is algebraically closed of characteristic zero. Let Ω be the set of all rank one discrete valuations on K. For v ∈ Ω, let K_{v} be the completion of K at v. Let X be a smooth variety over K which is a right homogeneous space of a semisimple simply connected group H over K. Assume that the geometric stabilizers are connected. Let G^{tor} be the associated K-torus. Assume that one of the following holds :

(i) �^{2}(K, G^{tor}) = 0 ;

(ii) the torus G^{tor} is the kernel of a surjective map of direct factors of quasitrivial tori ; (iii) the torus G^{tor} is split by a metacyclic extension of K; (iv) the torus G^{tor} has dimension at most one. Then the Hasse principle holds for X : if for each v ∈ Ω the set X(K_{v}) is not empty, then

X has a K-rational point.

Proof Let L = (\overline{G}, κ) be the K-kernel associated to the homogeneous space X. Consider the class η = η(X) ∈ H^{2}(K, L). By Proposition 5.4 and Theorem 2.1, η is neutral if and only if t_{∗}(η) ∈ H^{2}(K, G^{tor}) is trivial. For any place v, the assumption X(K_{v}) ≠ ∅ implies that η_{v} is neutral, hence t_{v}_{∗}(η) ∈ H^{2}(K_{v}, G^{tor}) is trivial. Thus t_{∗}(η) lies in �^{2}(K, G^{tor}) = Ker[H^{2}(K, G^{tor}) → ∏_{v∈Ω} H^{2}(K_{v}, G^{tor}).

Over any field, any 1-dimensional torus is split by a quadratic extension of the ground field. Over any field, for a K-torus T which is split by a metacyclic extension of K, there exists an exact sequence of K-tori

1 → T → P_{1} → P_{2} → 1,

where P_{1} and P_{2} are direct factors of quasitrivial tori (this uses the Endo-Miyata theorem, see §3). For the torus T = G^{tor} over the field K, this sequence induces an embedding �^{2}(K, G^{tor}) ↪ �^{2}(K, P_{1}) and the latter group vanishes by Theorem 1.6 (applied to finite field extensions of K).

Thus η is neutral, hence X comes from a right principal homogeneous space Y under H. But H^{1}(K, H) is trivial (Theorem 1.4), i.e. any principal homogeneous space under H has a K-point. Hence Y (K) ≠ ∅, hence also X(K) ≠ ∅.

The following lemma appears in the literature, but it is hard to locate a proof.

Lemma 5.6 Let k be a field and X a smooth projective k-variety which is a homogeneous space of a semisimple simply connected k-group H. With notation as above, the k-torus G^{tor} is a quasitrivial torus.

Proof Let F = k(X) be the function field of X. Since X is absolutely irreducible, the field k is algebraically closed in F. Let L denote the field F ⊗_{k} \overline{k}. The extension L/F is Galois, and the natural map Gal(L/F) → Gal(\overline{k}/k) = Γ is an isomorphism. Let G_{η} ⊂ H_{F} denote the stabilizer of the generic point of X, viewed as an F -point of X. By the functoriality mentioned before Theorem 5.5, the character group of the F -torus G^{tor} ×_{k} F, which splits over the field L, coincides, as a Γ-module, with the character group Ĝ_{η}(L) of the F -group G_{η}. The map h ↦ h.η makes H_{F} into a torsor over X_{F} under the group G_{F}. There is an associated exact sequence of Γ-modules ([Sa], Prop. 6.10)

0 → L[X]^{∗} → L[H]^{∗} → Ĝ_{η}(L) → Pic(X_{L}) → Pic(H_{L}).

Since H is semisimple, L^{∗} = L[H]^{∗}, and since H is simply connected, Pic(H_{L}) = 0. The exact sequence above reduces to an isomorphism Ĝ_{η}(L) ≃ Pic(X_{L}). The Galois module Pic(X ×_{k} \overline{k}) is a permutation module : it is the permutation module with basis the points of codimension 1 on X ×_{k} \overline{k} which are in the complement of the big cell, which is an affine space over \overline{k} (see [H2], proof of Satz 1.3.1 p. 118). This implies that the natural Γ-equivariant map Pic(X ×_{k} \overline{k}) → Pic(X_{K} ×_{K} L) is an isomorphism. We conclude that the character group of G^{tor} is a permutation module.

The following result is the analogue of a result of Harder over number fields ([H1] ; [Bo1], Cor. 7.5). It was put forward as a conjecture in §5 of [CTOP].

Corollary 5.7 Let K be as in Theorem 5.5 and let X be a smooth projective K-variety which is a homogeneous space of a connected linear algebraic K-group H. Then the Hasse principle holds for X.

Proof Let R ⊂ H be the radical of H, i.e. the maximal connected solvable normal subgroup of H. Let \overline{G} be the isotropy point of a \overline{K}-point of X. This is a parabolic group, hence it is connected. Its contains \overline{R}. It follows that the action of H on X factorizes through the semisimple group H^{ss} = H/R. Thus X is a homogeneous space under H^{ss}. Let λ : H^{sc} → H^{ss} be the universal covering of H. As λ is a surjective morphism, X is a homogeneous space under H^{sc}. We may thus assume that H is a semisimple simply connected group. The result now follows from Theorem 5.5 (ii) and Lemma 5.6.

We also have the analogue of Corollary 7.6 of [Bo1] (a generalization of a result of Rapinchuk) : Corollary 5.8 Let K be as above and let X be a symmetric homogeneous space of an

absolutely simple simply connected K-group H (i.e. the geometric isotropy group is the group of invariants of an involution of H ×_{K} \overline{K}). Then the Hasse principle holds for X.

Proof As Borovoi remarks, under the above assumption, the dimension of G^{tor} is at most one, hence the result by Theorem 5.5 (iv).

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J.-L. Colliot-Thélène,

C.N.R.S.,

UMR 8628, Mathématiques,

Bâtiment 425,

Université de Paris-Sud,

F-91405 Orsay

France

e-mail : colliot@math.u-psud.fr

Philippe Gille C.N.R.S., UMR 8628, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay France e-mail : Philippe.Gille@math.u-psud.fr

R. Parimala School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road 400005 Mumbai India e-mail : parimala@math.tifr.res.in