An Alternative Proof of Scheiderer’s Theorem

on the Hasse Principle

for Principal Homogeneous Spaces

Vladimir Chernousov^{1}

Received: February 23, 1998 Revised: September 11, 1998

Communicated by Ulf Rehmann

Abstract. We give an alternative proof of the Hasse principle for principal homogeneous spaces defined over fields of virtual cohomological dimension at most one which is based on a special decomposition of elements in Chevalley groups.

1991 Mathematics Subject Classification: Primary 20G10; Secondary 11E72, 12D15.

1. Introduction

Let Y be a smooth irreducible projective curve defined over the real number field ℝ and k = ℝ ( Y) be the field of ℝ-rational functions on Y. For a point P ∈ Y (ℝ) we denote the completion of k at the point P by k_{P}. The present paper is devoted to the Hasse principle for the existence of a rational point on principal homogeneous spaces of a connected linear algebraic group G defined over k. It was Colliot-Thélène who conjectured ( [CT], Conjecture 2.9 ) that for any such space X the Hasse principle holds relative to all local fields k_{P}, P ∈ Y (ℝ), i.e. X(k) ≠ ∅ iff X(k_{P}) ≠ ∅ for each P ∈ Y (ℝ). Since principal homogeneous spaces of G are in natural one-to-one correspondence with elements of the set H^{1}(k, G) the latter statement is equivalent to the following: the natural map of pointed sets

H^{1}(k, G) ⟶ ∏_{P ∈Y (ℝ)} H^{1}(k_{P} , G) (1)

has trivial kernel ( [S] ).

In [CT] Colliot-Thélène proved the Hasse principle for algebraic k-tori and reduced the general case to that of a simple simply connected algebraic group G. The case of an arbitrary connected k-group G has been studied by Scheiderer ( [Sch1] ).

1The author gratefully acknowledges the support of the Alexander von Humboldt-Stiftung and SFB 343 “Diskrete Strukturen in der Mathematik” and the hospitality of the University of Bielefeld.

To prove the Hasse principle he first made an important observation ( which eventually turned out to be crucial ) that local objects k_{P} can be replaced by real closures k_{ξ} of k, ξ ∈ Ω_{k}, where Ω_{k} denotes the set of all orderings of k. Indeed, using the description of orderings of k and the so-called Artin-Lang homomorphism theorem ( [Srl], Theorem 3.1 ) it is easy to show that the condition X(k_{P}) ≠ ∅ for each real point P on Y implies X(k_{ξ}) ≠ ∅ for each ordering ξ of k and hence the triviality of the kernel of (1) follows immediately from the triviality of the kernel of

θ : H^{1}(k, G) ⟶ ∏_{ξ∈Ω_{k}} H^{1}(k_{ξ}, G) (2)

The question whether θ is injective makes sense not only for the function fields of curves but also for an arbitrary field k and it turned out that θ is indeed injective if k has virtual cohomological dimension ( vcd ) at most 1 (recall that function fields in one variable over ℝ are such). We have even more.

Theorem 1. ( Scheiderer, [Sch1] ) Let K be any field of virtual cohomological dimension ≤ 1. Then the Hasse principle holds for any homogeneous K-space X of a connected linear algebraic K-group G.

Scheiderer’s proof can be divided into two parts. In the first one it is proved that for X as in the theorem (here G may even be not connected) there exists a principal homogeneous space Z which is everywhere locally trivial and dominates X. The strategy of the proof in this part going back to Springer ( [S], [Sp] ) consists of replacing X by a homogeneous space which dominates X and has a smaller stabilizer. It is worth mentioning that in this part most arguments do not use specific properties of K and so most of them are valid over an arbitrary perfect field.

The second part of Scheiderer’s proof is devoted to the case of a principal homogeneous space. To treat such a space Scheiderer first constructs a locally constant sheaf of sets H^{1}(G) on Ω_{K} whose stalks are just the sets H^{1}(K_{ξ}, G). Then he shows that there exists a natural bijection between the set of global sections of H^{1}(G) and H^{1}(K, G). As a whole the proof in this part is quite complicated. It is based on using étale machinery and, in particular, strongly relies on results of the book [Sch2].

The aim of this paper is to provide a simpler and shorter self-contained proof which is based only on the Bruhat decomposition in semisimple algebraic groups and the so-called strong approximation property (SAP) of fields (see § 3). We show that in fact the Hasse principle follows immediately modulo two facts. Informally speaking one of them says that the kernel of the natural map H^{1}(K, T) → H^{1}(K, G), where G is an (absolutely) simple simply connected linear K-group and T is a K-torus splitting over K(\sqrt{−1}), can be parametrized by “good” rational functions (see § 2) and the other says that any field of virtual cohomological dimension ≤ 1 is an SAP field.

Acknowledgment. The author is grateful to J.-P. Serre and the referee for remarks that helped to improve the initial exposition.

2. Algebraic groups splitting over quadratic extensions

Throughout the section K denotes an arbitrary field of characteristic 0. Let G be an (absolutely) simple simply connected algebraic group of rank n defined over K and splitting over quadratic extension L = K(\sqrt{d}). Let

Θ = Gal (L/K) = ⟨τ | τ^{2} = 1⟩.

Consider a Borel L-subgroup B such that T = B ∩ τ(B) is a maximal torus which will be assumed for simplicity to be K-anisotropic. Since T is splitting over L, one has

T ≃ R_{L/K}^{(1)}(G_{m}) × … × R_{L/K}^{(1)}(G_{m}).

To prove the Hasse principle we need to describe Ker [H^{1}(Θ, T (L)) → H^{1}(Θ, G(L))]. This description can be easily extracted from [Ch]. However this paper is written in Russian and the translation made by the AMS is unreadable and contains a lot of misprints. So for the sake of expository completeness and the reader’s convenience we include here details.

First recall some basic facts about the structure of the group G(L) ( for details see [St1] ). Let Σ = R(T, G) be the root system of G relative to T. The Borel subgroup B determines an ordering on the set Σ and hence a system of simple roots Π = {α_{1}, …, α_{n}}. If α = ∑ n_{i}α_{i} ∈ Σ^{+}, then the number ht(α) = ∑ n_{i} is called the height of α. If {X_{α}, α ∈ Σ; H_{α_{1}}, …, H_{α_{n}}} is a Chevalley basis of the Lie algebra of G, then G(L) is generated by the corresponding root subgroups G_{α} = ⟨x_{±α}(t) | t ∈ L⟩, where

x_{α}(t) = ∑_{n=0}^{∞} t^{n}X_{α}^{n} /n! ,

and the torus T is generated by T_{α} = T ∩ G_{α} = ⟨h_{α}(t)⟩, where h_{α}(t) = w_{α}(t)w_{α}(1)^{−1} and w_{α}(t) = x_{α}(t)x_{−α}(−t^{−1})x_{α}(t).

Furthermore, since G is simply connected the following relations hold in G (cf. [St1], Lemma 28 b), Lemma 20 c), Lemma 15 ):

A) T = ⟨h_{α_{1}}(t_{1})⟩ × ⋯ × ⟨h_{α_{n}}(t_{n})⟩ and for α ∈ Σ we have

h_{α}(t) = ∏_{i=1}^{n} h_{α_{i}}(t)^{n_{i}}, where H_{α} = ∑_{i=1}^{n} n_{i}H_{α_{i}}; (3)

B) For α, β ∈ Σ let ⟨β, α⟩ = 2 (β, α)/(α, α). Then we have

h_{α}(t)x_{β}(u)h_{α}(t)^{−1} = x_{β}(t^{⟨β,α⟩}u) (4)

C) For all u, v ∈ L such that 1 + uv ≠ 0 we have

x_{−α}(u)x_{α}(v) = x_{α}(v(1 + uv)^{−1})h_{α}(1 + uv)^{−1}x_{−α}(u(1 + uv)^{−1}) (5)

D) For all α, β ∈ Σ, β ≠ −α, we have

x_{α}(v)x_{β}(u)x_{α}(v)^{−1}x_{β}(u)^{−1} = ∏_{i,j>0} x_{iα+jβ}(c_{i,j}v^{i}u^{j}) (6)

where the product on the right hand side is taken over all roots of the form iα + jβ and the c_{i,j} are integers which depend on α, β and on the chosen ordering of the roots but do not depend on v and u.

Since T is K-defined, τ acts on the root system Σ. More exactly, for any α ∈ Σ the character α + τ(α) is K-defined and hence is zero, i.e. τ(α) = −α, since, by assumption, T is K-anisotropic. It follows that there exists c_{α} ∈ L^{∗} such that τ(X_{α}) = c_{α}X_{−α}; in particular, the subgroup G_{α} is K-defined.

The constants c_{α} actually lie in K and c_{−α} = c_{α}^{−1}. Indeed, for rank one groups, i.e. of the form SL (1, D), where D is a quaternion K-algebra, this fact can be verified directly. The general case easily reduces to the rank one case since G_{α} is a simple simply connected K-group of rank 1. Thus, we have

Lemma 1. There exists constant c_{α} ∈ K^{∗} such that for any u ∈ L one has τ(x_{α}(u)) = x_{−α}(c_{α}τ(u)). Moreover, G_{α} ≃ SL(1, D), where D is a quaternion algebra over K of the form D = (d, c_{α}).

Proof: Straightforward computations.

Lemma 2. The positive roots Σ^{+} = {β_{1}, …, β_{m}} can be ordered in such a way that the following two properties hold:

1) for any pair of roots β_{i}, β_{j}, for which i < j and β_{i} + β_{j} = β_{k} ∈ Σ^{+}, the root β_{k} is between β_{i} and β_{j}, i.e. i < k < j;

2) if Σ is a root system of type either A_{2n−1} or D_{n} or E_{6} and σ is the outer automorphism of Σ induced by the non-trivial automorphism of order 2 (resp. 3) of the corresponding Dynkin diagram, then for any root β_{i} ∈ Σ^{+} the roots β_{i} and σ(β_{i}) (resp. β_{i}, σ(β_{i}), σ^{2}(β_{i})) are neighbours.

Proof. a) Let Σ = {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ 2n} be a root system of type A_{2n−1}. Let α_{1} = ε_{1}−ε_{2}, …, α_{2n−1} = ε_{2n−1}−ε_{2n} be a basis of Σ and Σ_{1} be the subsystem generated by the roots α_{2}, …, α_{2n−2}. By induction, we can pick an ordering Σ_{1}^{+} = {β_{1}, …, β_{k}} with the required properties. Let γ = α_{1}+⋯+α_{2n−1}. We number the remaining roots Σ^{+} \ {Σ_{1}^{+} ∪ γ} = {β_{k+1}, …, β_{m−1}} in the order of decreasing height. If β_{i} denotes the last root among {β_{k+1}, …, β_{m−1}} such that ht (β_{i}) ≥ n, then the ordering

Σ^{+} = {β_{1}, …, β_{k}, β_{k+1}, …, β_{i}, γ, β_{i+1}, …, β_{m−1}}

is as required.

b) Σ is a root system of type A_{2n}, B_{n}, C_{n}, D_{n}, E_{7}. It follows from the description of root systems of these types that there exists a subsystem Σ_{1} generated by n − 1 simple roots, say α_{1}, …, α_{n−1}, such that any root β ∈ Σ^{+} \ Σ_{1}^{+} can be written as a sum β = m_{1}α_{1} +⋯+m_{n−1}α_{n−1} +α_{n}. If Σ is of type D_{n} and |σ| = 2, we may assume in addition that the set {α_{1}, …, α_{n−1}} is stable under σ. The root system Σ_{1} has rank n − 1 and so by induction, there exists an ordering of the required type on the set Σ_{1}^{+} = {β_{1}, …, β_{k}}. We number the remaining roots Σ^{+} \ Σ_{1}^{+} = {β_{k+1}, …, β_{m}} in the order of decreasing height. Then the ordering {β_{1}, …, β_{m}} is as required.

c) Σ is a root system of type E_{6}, E_{8}, F_{4}, G_{2}. Here one can argue as in case a). Namely, there exists a subsystem Σ_{1} generated by simple roots α_{1}, …, α_{n−1} such that any root β ∈ Σ^{+} \ Σ_{1}^{+} is of the form β = m_{1}α_{1} + ⋯ + m_{n−1}α_{n−1} + α_{n} except for the maximal root α̃ and α̃ is of the form α̃ = m_{1}α_{1}+⋯+m_{n−1}α_{n−1}+2α_{n}. Let b = ht (α̃). Again, applying induction we can find an ordering Σ_{1}^{+} = {β_{1}, …, β_{k}} with the desired properties and then we number the roots Σ^{+} \ {Σ_{1}^{+} ∪ α̃} = {β_{k+1}, …, β_{m−1}} in the order of decreasing height. If Σ has type E_{6}, we may assume in addition that β and σ(β) are neighbours for all β ∈ Σ^{+}. Let β_{i} be the last root among {β_{k+1}, …, β_{m−1}} such that ht (β_{i}) ≥ b/2. We claim that the ordering

Σ^{+} = {β_{1}, …, β_{k}, β_{k+1}, …, β_{i}, α̃, β_{i+1}, …, β_{m−1}}

has the desired properties. Indeed, if β_{j} = β_{s}+β_{t}, where s < t and j ∈ {k+1, …, m− 1}, then clearly β_{s} belongs to Σ_{1}^{+}. It follows that β_{j} lies between β_{s} and β_{t}, since ht (β_{j}) ≥ ht (β_{s}), ht (β_{t}). Now let α̃ = β_{s} + β_{t}, s < t,. Then s, t ∈ {k + 1, …, m − 1} and ht (β_{s}) ≥ b/2, ht (β_{t}) < b/2 ( we use the fact that ht(α̃) is odd ), implying α̃ is also between β_{s} and β_{t}.

d) Σ has type D_{4} and |σ| = 3. Let α_{1}, …, α_{4} be simple roots such that σ permutes α_{1}, α_{3}, α_{4}. Then the required ordering is as follows: first we place α_{2}, then all roots of the height 2, then the maximal root and then the roots of heights 3, 4, 1 respectively.

Corollary 1. Let β_{i}, β_{j}, j < i, be any two positive roots. Then for any positive root β_{k} of the form β_{k} = rβ_{j} − lβ_{i}, r, l > 0, one has k < j. Analogously, for any negative root of the form −β_{k} = rβ_{j} − lβ_{i}, r, l > 0, one has k > i.

Proof. We distinguish three cases.

a) ⟨β_{i}, β_{j}⟩_{ℚ} ∩ Σ has type A_{2}. Then r = l = 1 and hence if β_{k} = β_{j} − β_{i} is a positive root then β_{k} + β_{i} = β_{j}, implying k < j < i. Analogously, if β_{j} − β_{i} = −β_{k} then we have j < i < k.

b) ⟨β_{i}, β_{j}⟩_{ℚ} ∩ Σ has type B_{2}. Then either r = l = 1 or r = 1 and l = 2 or r = 2 and l = 1. The case r = l = 1 was already handled in part a). Now let β_{k} = β_{j} − 2β_{i}. Then β_{j} − β_{i} = β_{s} is also a positive root implying s < j. Futhermore, β_{k} = β_{s} − β_{i} and s < j < i. So again we have k < s < j. The remaining cases can be handled in a similar way.

c) ⟨β_{i}, β_{j}⟩_{ℚ} ∩ Σ has type G_{2}. Here the proof is similar to that of case b) and we omit it.

Proposition 1. Fix an order in Σ^{+} as in Lemma 2. Then the regular map

ω : G_{m}^{n} × IA^{2m} ⟶ G , (t_{1}, …, t_{n}, u_{1}, …, u_{m}, v_{1}, …, v_{m}) ⟶

∏_{i=1}^{n} h_{α_{i}}(t_{i}) x_{−β_{1}}(u_{1})x_{β_{1}}(v_{1}) ⋯ x_{−β_{m}}(u_{m})x_{β_{m}}(v_{m})

is birational over L.

Remark 1. This statement is also true in positive characteristic. There is the only place which require additional work: one need additionally to check that ω is a separable map.

Proof. Both sides have the same dimension and hence it suffices to prove the injectivity of ω on some Zariski open subset, since char K = 0.

First we show that for any integer i and any parameters u_{1}, …, u_{i} and v_{1}, …, v_{i} from some Zariski open subset the element

A_{i} = x_{−β_{1}}(u_{1})x_{β_{1}}(v_{1}) ⋯ x_{−β_{i}}(u_{i})x_{β_{i}}(v_{i})

of the group G can be written in the form

A_{i} = ∏_{k=1}^{n} h_{α_{k}}(f_{k}) ∏_{j=1}^{m} x_{−β_{j}}(r_{j}) ∏_{j=1}^{i−1} x_{β_{j}}(s_{j}) x_{β_{i}}(v_{i}),

where f_{k}, r_{j}, s_{j} are rational functions depending on u_{1}, …, u_{i}, v_{1}, …, v_{i−1}. If i = 1 there is nothing to prove. By induction, we may write A_{i−1} in the form

∏_{k=1}^{n} h_{α_{k}}(f_{k}) ∏_{j=1}^{m} x_{−β_{j}}(r_{j}) ∏_{j=1}^{i−2} x_{β_{j}}(s_{j}) x_{β_{i−1}}(v_{i−1}).

To write A_{i} = A_{i−1} x_{−β_{i}}(u_{i}) x_{β_{i}}(v_{i}) in the same form we have to transpose x_{−β_{i}}(u_{i}) with each factor in the product ∏_{j=1}^{i−2} x_{β_{j}}(s_{j}) x_{β_{i−1}}(v_{i−1}). By (6) and by Corollary 1, every time doing so we obtain additional factors x_{β_{s}}( ) or x_{−β_{s}}( ), where s < i−1 in the first case and s > i in the second case. Collecting together all these factors corresponding to negative roots we can write the element ∏_{j=1}^{i−2} x_{β_{j}}(s_{j}) x_{β_{i−1}}(v_{i−1}) x_{−β_{i}}(u_{i}) in the form

∏_{k=1}^{n} h_{α_{i}}(f̃_{k}) ∏_{j=1}^{m} x_{−β_{j}}(r̃_{j}) ∏_{j=1}^{i−1} x_{β_{j}}(s̃_{j})

and so our claim follows.

Now we are ready to prove the injectivity of ω. Suppose that

ω(t_{1}, …, t_{n}, u_{1}, …, u_{m}, v_{1}, …, v_{m}) = ω(t̃_{1}, …, t̃_{n}, ũ_{1}, …, ṽ_{m}) (7)

From the above argument and the Bruhat decomposition we get immediately v_{m} = ṽ_{m}. To show that u_{m} = ũ_{m}, we use (4), (5). Namely, it follows from (4), (5) that the left hand side of (7) may be written in the form

∏_{i=1}^{n} h_{α_{i}}(f_{i}) [x_{β_{1}}(s_{1}) x_{−β_{1}}(r_{1})] ⋯ [x_{β_{m−1}}(s_{m−1}) x_{−β_{m−1}}(r_{m−1})]

x_{β_{m}}[v_{m}(1 + u_{m}v_{m})] x_{−β_{m}}[u_{m}(1 + u_{m}v_{m})^{−1}] ,

where f_{1}, …, f_{n}, s_{1}, …, s_{m−1}, r_{1}, …, r_{m−1} are rational functions. Rewriting the right hand side of (7) in the same form we conclude that

u_{m}(1 + u_{m}v_{m})^{−1} = ũ_{m}(1 + ũ_{m}ṽ_{m})^{−1},

hence u_{m} = ũ_{m}. After cancelling the factor x_{−β_{m}}(u_{m}) x_{β_{m}}(v_{m}) in (7) the same argument shows that v_{m−1} = ṽ_{m−1}, u_{m−1} = ũ_{m−1} and so on.

Now we are in position to formulate the main result of the section.

Theorem 2. Let g ∈ G(L) be such that g^{1−τ} ∈ T (L). Then there exist quaternion algebras D_{1}, …, D_{m} over K and elements w_{1}, …, w_{m} ∈ K which are reduced norm of D_{1}, …, D_{m} respectively and elements t_{1}, …, t_{n} ∈ L such that

g^{1−τ} = ∏_{i=1}^{n} h_{α_{i}}(t_{i}τ(t_{i})) ∏_{i=1}^{m} h_{β_{i}}(w_{i})

Proof. If g^{1−τ} ∈ T (L), then for any x ∈ G(K) one has g^{1−τ} = (gx)^{1−τ}. Since G(K) is Zariski dense in G, we may always assume that our element g is in “generic” position by which we mean point in some Zariski open subset U ⊂ G which can be easily specified from the argument. So let

g = ∏_{i=1}^{n} h_{α_{i}}(t_{i}) x_{−β_{1}}(u_{1})x_{β_{1}}(v_{1}) ⋯ x_{−β_{m}}(u_{m})x_{β_{m}}(v_{m})

where t_{i}, u_{i}, v_{i} ∈ L. Denote t = ∏_{i=1}^{n} h_{α_{i}}(t_{i}) and g_{i} = x_{−β_{i}}(u_{i})x_{β_{i}}(v_{i}), i = 1, …, m. Let also t′ = g^{1−τ}, so that

t · g_{1} ⋯ g_{m} = t′ · τ(t) · τ(g_{1}) ⋯ τ(g_{m}) (8)

By Lemma 1, we have τ(g_{i}) ∈ G_{β_{i}}. Then applying Proposition 1 we conclude that g_{m} and τ(g_{m}) coincide modulo T_{β_{m}}(L) = T (L) ∩ G_{β_{m}} and so the element g_{m}^{τ−1} is of the form h_{β_{m}}(w_{m}) for some parameter w_{m}. We claim that w_{m} ∈ K and it is a reduced norm of the quaternion K-algebra D_{m} = (d, d_{β_{m}}), where d_{β_{m}} = c_{β_{m}}. Indeed, by construction the cocycle (g_{m}^{τ−1}) ∈ Z^{1}(Θ, T_{β_{m}}(L)) is trivial in Z^{1}(Θ, G_{β_{m}}(L)) and by Lemma 1, G_{β_{m}} ≃ SL(1, D_{m}), hence our claim follows.

Substituting τ(g_{m}) = h_{β_{m}}(w_{m}) · g in (8) and cancelling g, we have then

t · g_{1} ⋯ g_{m−1} = t′ · τ(t) · h_{β_{m}}(w_{m}) · [h_{β_{m}}(w_{m})^{−1}τ(g_{1})h_{β_{m}}(w_{m})] ⋯
⋯ [h_{β_{m}}(w_{m})^{−1}τ(g_{m−1})h_{β_{m}}(w_{m})]

Applying again Proposition 1 and arguing analogously we have

[h_{β_{m}}(w_{m})^{−1}τ(g_{m−1})h_{β_{m}}(w_{m})] = h_{β_{m−1}}(w_{m−1}) · g_{m−1}

for some parameter w_{m−1}, which is again a reduced norm of the quaternion K-algebra D_{m−1} = (d, d_{β_{m−1}}), where

d_{β_{m−1}} = c_{β_{m−1}}w_{m}^{⟨β_{m−1},β_{m}⟩}.
To see it, let g̃_{m−1} = h_{β_{m}}(w_{m})^{−1}τ(g_{m−1})h_{β_{m}}(w_{m}). Using (4) we have
g̃_{m−1} = x_{β_{m−1}}(c_{β_{m−1}}^{−1}w_{m}^{−⟨β_{m−1},β_{m}⟩}τ(u_{m})) · x_{β_{m−1}}(c_{β_{m−1}}w_{m}^{⟨β_{m−1},β_{m}⟩}τ(v_{m})).

It follows that (h_{β_{m−1}}(w_{m−1})) = (g̃_{m−1} · g_{m−1}^{−1}) can be viewed as a trivial cocycle in an K-group of rank 1 whose K-structure, i.e. action of τ, is given by the constant d_{β_{m−1}}. This fact combined with Lemma 1 implies w_{m−1} is a reduced norm of D_{m−1}, as claimed, and so on. Theorem 2 is proved.

In § 4 we will also deal with a simple simply connected algebraic K-group G which is quasi-split over a quadratic extension L/K and for such a group we also need to describe elements of the form g^{1−τ} ∈ T (L), where g ∈ G(L).

Clearly, K-groups of type ^{2}A_{2n} split over a quadratic extension of K. Since this case has been already handled, we may assume that G is an outer form of type not A_{2n}. As above, let B be an L-Borel subgroup B of G such that T = B ∩ τ(B) is a maximal K-anisotropic torus.

Let F/K be the minimal extension over which G is an inner form and let E = F ·L. Let τ and σ be non-trivial automorphisms of E/K such that τ|_{F} = 1 and σ|_{L} = 1 respectively. In the case ^{3,6}D_{4} by σ we denote any automorphism of order 3.

Clearly, σ induces an outer automorphism of the root system Σ = R (T, G) which will be denoted by the same letter. Let Λ = {γ_{1}, …, γ_{s}} ⊂ Σ^{+} ( resp. Λ′) be a set of representatives of all orbits of σ in Σ^{+} ( resp. in Π ). We divide Λ into two parts: Λ_{1} = {γ_{i} ∈ Λ | σ(γ_{i}) = γ_{i}} and Λ_{2} = Λ \ Λ_{1}. Let also Λ′_{i} = Λ′ ∩ Λ_{i}, i = 1, 2. For γ_{i} ∈ Λ_{1} ( resp. Λ_{2} ) we denote by H_{i} the subgroup in G generated by G_{γ_{i}} ( resp. G_{γ_{i}}, G_{σ(γ_{i})} and G_{σ^{2}(γ_{i})}, if |σ| = 3 ).

Lemma 3. H_{i} is a simple simply connected K-group of type A_{1} ( resp. A_{1} × A_{1} or A_{1} × A_{1} × A_{1} ) if γ_{i} ∈ Λ_{1} ( resp. γ_{i} ∈ Λ_{2} and |σ| = 2 or |σ| = 3 ).

Proof. It suffices to note that τ acts on Σ as either −1, if Σ has type D_{2n}, or −σ otherwise, since it permutes positive and negative roots. Moreover, the combination β_{i} ± σ(β_{i}) is not a root, hence G_{γ_{i}} and G_{σ(γ_{i})} commute.

Theorem 3. Let g ∈ G(L) be such that g^{1−τ} ∈ T (L). Then there exist quaternion algebras D_{1}, …, D_{s} and elements w_{1}, …, w_{s} which are reduced norm of D_{1}, …, D_{s} respectively and elements t_{1}, …, t_{p} such that:

1) If Σ is not of type ^{3,6}D_{4}, then

g^{1−τ} = ∏_{α_{i}∈Λ′_{1}} h_{α_{i}}(t_{i}τ(t_{i})) ∏_{α_{i}∈Λ′_{2}} h_{α_{i}}(t_{i}τ(t_{i}))h_{σ(α_{i})}[σ(t_{i})(τ ∘ σ)(t_{i})]·

∏_{γ_{i}∈Λ_{1}} h_{γ_{i}}(w_{i}) ∏_{γ_{i}∈Λ_{2}} h_{γ_{i}}(w_{i})h_{σ(γ_{i})}(σ(w_{i}))

2) If Σ is of type ^{3,6}D_{4}, then

g^{1−τ} = ∏_{α_{i}∈Λ′_{2}} h_{α_{i}}(t_{i}τ(t_{i}))h_{σ(α_{i})}[σ(t_{i})(τ ∘ σ)(t_{i})]h_{σ^{2}(α_{i})}[σ^{2}(t_{i})(τ ∘ σ^{2})(t_{i})]·

∏_{α_{i}∈Λ′_{1}} h_{α_{i}}(t_{i}τ(t_{i})) ∏_{γ_{i}∈Λ_{1}} h_{γ_{i}}(w_{i}) ∏_{γ_{i}∈Λ_{2}} h_{γ_{i}}(w_{i})h_{σ(γ_{i})}(σ(w_{i}))h_{σ^{2}(γ_{i})}(σ^{2}(w_{i}))

Here D_{i} is over K ( resp. over F ) and w_{i} ∈ K (resp. F), if γ_{i} ∈ Λ_{1} ( resp. γ_{i} ∈ Λ_{2} ), and t_{i} ∈ L (resp. E), if α_{i} ∈ Λ′_{1} (resp. α_{i} ∈ Λ′_{2}).

Proof. As in the L-split case first we may assume that g is in “generic” position and so by property 2 in Lemma 2 and by Proposition 1, it can be written in the form g = t g_{1} ⋯ g_{s}, where t ∈ T, g_{i} ∈ H_{i}, i = 1, …, s. Then the rest of the proof works exactly as in the L-split case, since by Lemma 3 all subgroups H_{i} are of the form R_{K′/K}(SL(1, D)), where D is a quaternion algebra over K′ and K′ is either F or K.

3. Some cohomological computations

From now on we assume that vcd (K) ≤ 1 and we let L = K(\sqrt{−1} ). We also assume that the set Ω_{K} of all orderings on K is non-empty; this means, in particular, that char K = 0. Recall ( [Srl] ) that there is a canonical topology on Ω_{K} under which Ω_{K} is compact and totally disconnected.

Remark 2. If Ω_{K} = ∅, then −1 is a sum of squares in K and so cd (K) = cd (K(\sqrt{−1})) ≤ 1 ( [S], Ch. 2, Prop. 10′). Therefore, if Ω_{K} = ∅, then by Steinberg’s theorem ( [St2] ) one has H^{1}(K, G) = 1 for any connected linear algebraic K-group G.

To reduce the proof of the Hasse principle to the case of simply connected semisimple groups we need two auxiliary cohomological statements (Propositions 2 and 4 below) which are very particular cases of the general Theorem 12.13 in [Sch2]. Since we do not need to consider such a generality as in [Sch2] we include here the straightforward proofs of these statements.

Let A be a discrete Γ-module, where Γ = Gal (\overline{K}/K), and let

φ_{i} : H^{i}(K, A) → ∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, A)

be the canonical map induced by res_{K_{ξ}}. We want to describe Ker φ_{i}, i ≥ 2, and Im φ_{1}. To do so first remind that there is not a canonical way of choosing a real closure of K at ξ ∈ Ω_{K}. If K_{ξ} and K′_{ξ} are two real closures of K at ξ, then by the theorem of Artin-Schreier ( [Srl] Ch. 3, Theorem 2.1 ) there is a unique K-isomorphism K_{ξ} ≃ K′_{ξ}, hence there is an element g ∈ Γ such that gτ_{ξ}g^{−1} = τ′_{ξ}, where τ_{ξ} (resp. τ′_{ξ}) is the involution (= element of order 2) in Γ corresponding to K_{ξ} (resp. K′_{ξ}) (in other words, there is a natural one-to-one correspondence between points of the set Ω_{K} and conjugacy classes of involutions in Γ).

The element g induces a natural map λ_{i,g} : H^{i}(K_{ξ}, A) → H^{i}(K′_{ξ}, A) and obviously we have res_{K′_{ξ}} = λ_{i,g} ∘ res_{K_{ξ}}. It follows that the question on whether φ_{i} is injective does not depend on a choice of real closures K_{ξ}, ξ ∈ Ω_{K}.

Clearly, any cocycle from Z^{1}(K_{ξ}, A) is determined by the single element a ∈ A such that aτ_{ξ}(a) = 1. We will say that an element {a_{ξ}}_{ξ∈Ω_{K}} ∈ ∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, A) is locally constant if there are a decomposition Ω_{K} = U_{1} ∪ … ∪ U_{l} into disjoint clopen ( = open and closed ) sets and elements {a_{1}, …, a_{l}} of A for which the following condition holds: for any ξ ∈ U_{i} there are a cocycle c_{ξ} representing a_{ξ} and g_{ξ} ∈ Γ such that the cocycle λ_{1,g_{ξ}}(c_{ξ}) is determined by a_{i}. Analogously, for any i ≥ 1 one defines the subset of elements in ∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, A) which are locally constant. We denote

this subset by (∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, A))^{lc}. Since for any ζ ∈ H^{i}(K, A) the element φ_{i}(ζ) is locally constant we denote by the same letter the canonical map

φ_{i} : H^{i}(K, A) ⟶ ( ∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, A))^{lc} ⊂ ∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, A)

Proposition 2. If A is a finite discrete Γ-module, then the maps φ_{i} are injective for all integers i ≥ 2.

Proof. Since H^{i}(L, A) = 1, i ≥ 2, the “res-cores” argument shows that H^{i}(K, A) has exponent 2. So replacing A, if necessary, by its 2-Sylow subgroup we may assume that A is a 2-group. First examine the case A = ℤ/2ℤ.

Lemma 4. Let A = ℤ/2ℤ. Then φ_{i} is surjective if i ≥ 1 and injective if i ≥ 2.

Proof. Recall ( [L], §17 ) that a field F is said to be an SAP field (strong approximation property) if for any two disjoint closed subsets A, B ⊂ Ω_{F} there exists an element f ∈ F such that f is positive at all orderings in A, but negative at all orderings in B. We need Proposition 3. ( [L], Theorem 17.9 ) If vcd (K) ≤ 1, then K is a SAP field.

Surjectivity of φ_{i}, i ≥ 1. In view of the periodicity of H^{i}(K_{ξ}, ℤ/2ℤ) it suffices to consider the cases i = 1, 2. If i = 1 then H^{1}(K, ℤ/2ℤ) = K^{∗}/K^{∗2}, hence the surjectivity of φ_{1} follows immediately from Proposition 3. Furthermore, any element from H^{2}(K, ℤ/2ℤ) splits over L and so can be represented by a quaternion algebra having L as a maximal subfield. Then clearly, the surjectivity of φ_{2} again follows from Proposition 3. Injectivity of φ_{i}, i ≥ 2. The proof is similar to that of [B-P], Lemma 2.3. Namely, by Arason’s theorem ( [A1], Satz 3 ), local triviality of ζ ∈ H^{i}(K, ℤ/2ℤ) implies that ζ ∪ (−1)^{r} = 0 for some integer r, where ∪ denotes the cup product. On the other hand from the exact sequence

H^{i}(L, ℤ/2ℤ) ⟶^{cor} H^{i}(K, ℤ/2ℤ) ⟶^{∪(−1)} H^{i+1}(K, ℤ/2ℤ) ⟶^{res} H^{i+1}(L, ℤ/2ℤ)

( [A2], Corollary 4.6 ) and from the equalities

H^{i}(L, ℤ/2ℤ) = H^{i+1}(L, ℤ/2ℤ) = 1, i ≥ 2

we conclude that the product ∪(−1) is an isomorphism. Therefore, ζ = 1, as required. Lemma 4 is proved.

We come back to an arbitrary finite 2-primary module A. Let Γ_{2} be a Sylow 2-subgroup of Γ. Since the restriction map H^{i}(K, A) → H^{i}(Γ_{2}, A) is injective, after replacing Γ by Γ_{2} we may assume that Γ is a pro-2-group. But for such a group any irreducible module is isomorphic to ℤ/2ℤ ( [S], §4, Proposition 20 ). Therefore there exists a submodule A′ ⊂ A such that A/A′ = ℤ/2ℤ. It induces the commutative diagram

H^{i}(K, ℤ/2ℤ)  ⟶  H^{i+1}(K, A′)  ⟶
↓θ_{1}  ↓θ_{2}
( ∏_{ξ∈Ω_{K}} H^{i}(K_{ξ}, ℤ/2ℤ))^{lc} ⟶ ( ∏_{ξ∈Ω_{K}} H^{i+1}(K_{ξ}, A′))^{lc} ⟶

H^{i+1}(K, A)  ⟶  H^{i+1}(K, ℤ/2ℤ)
↓θ_{3}  ↓θ_{4}
( ∏_{ξ∈Ω_{K}} H^{i+1}(K_{ξ}, A))^{lc} ⟶ ( ∏_{ξ∈Ω_{K}} H^{i+1}(K_{ξ}, ℤ/2ℤ))^{lc}

By what has been proved above, θ_{1} (resp. θ_{4}) is surjective (resp. injective) and by induction, θ_{2} is injective. It follows that θ_{3} is injective as well. Proposition 2 is proved.

Proposition 4. If A is a finite discrete Γ-module, then φ_{1} is surjective.

Proof. Since φ_{i}, i ≥ 2, are injective, one can easily verify that if the statement holds both for a submodule A′ ⊂ A and the quotient A/A′, then it also holds for A. So we may assume, if necessary, that A is irreducible. It suffices to prove that for a given ξ ∈ Ω_{K} and an element a ∈ A for which aτ_{ξ}(a) = 1 there exist a small clopen neighbourhood U ⊂ Ω_{K} of ξ and a cocycle ζ ∈ Z^{1}(K, A) such that for a proper real closure K_{ξ′} of K at ξ′ the cocycle res_{K_{ξ′}} (ζ) is determined by the element a if ξ′ ∈ U, and is trivial otherwise.

We need the following simple property of orderings of K ( see [Srl] ):

if F/K is an extension of odd degree then for any ordering ξ ∈ Ω_{K} there is an extension of ξ to F; moreover, the restriction map ϕ : Ω_{F} → Ω_{K} is a local homeomorphism.

Let E be a finite Galois extension of K over which A is a trivial module and let F ⊂ E be the subfield corresponding to a Sylow 2-subgroup of Gal (E/K). Denote Δ = Gal (\overline{K}/F). Let ϕ^{−1}(ξ) = \{ξ_{1}, …, ξ_{t}\} ⊂ Ω_{F}, where, as above, ϕ : Ω_{F} → Ω_{K} is the restriction map.

By construction, ϕ(ξ_{i}) = ξ. So we can pick a small clopen neighbourhood U ⊂ Ω_{K} of ξ and disjoint small clopen neighbourhoods U_{i} ⊂ Ω_{F} of ξ_{i}, i = 1, …, t, such that the restriction map ϕ|_{U_{i}} : U_{i} → U is a homeomorphism and ϕ^{−1}(U) = U_{1} ∪ … ∪ U_{t}. Taking smaller neighbourhoods, if necessary, one can additionally assume that for any ξ′ ∈ U_{1} there is an involution τ_{ξ′} ∈ Δ corresponding to ξ′ for which the following property holds:

if g ∈ Γ\Δ be such that τ̃_{ξ′} = g τ_{ξ′} g^{−1} ∈ Δ then the point of Ω_{F}

(9) corresponding to the involution τ̃_{ξ′} does not lie in U_{1}.

Indeed, let I_{Δ} ⊂ Δ be a subset of involutions and τ ∈ I_{Δ} be an involution which corresponds to ξ_{1}. Assume the contrary. Since I_{Δ}, Γ are compact and totally disconnected there exist then in Δ a sequence of involutions (τ_{1}, τ_{2}, …) converging to τ and a converging sequence of elements (g_{1}, g_{2}, …) in Γ\Δ such that g_{i} τ_{i} g_{i}^{−1} ∈ Δ. Letting g = lim g_{i}, one has g ∈ Γ\Δ and τ′ = g τ g^{−1} ∈ Δ. But by assumption, the point ξ′ of Ω_{F} corresponding to τ′ lies in U_{1} and ϕ(ξ′) = ξ. This means that ξ′ = ξ_{1}, hence there is δ ∈ Δ such that τ′ = δ τ δ^{−1}, implying g^{−1} δ lies in the centralizer C_{Γ}(τ). But every involution in Γ is self-centralizing, i.e. C_{Γ}(τ) = ⟨τ⟩,– a contradiction.

The map φ_{1} is clearly surjective for the field F, since A can be viewed as Gal (E/F)-module and Gal (E/F) is a 2-group, implying that any irreducible Gal (E/F)-module is of the form ℤ/2ℤ. Therefore, we can pick ζ′ ∈ Z^{1}(F, A) such that for proper real closures the cocycle res_{F_{ξ′}} (ζ′) is determined by the element a if ξ′ ∈ U_{1} and is trivial otherwise. We claim that the cocycle ζ = cor _{K}^{F} (ζ′) has the same property. To verify it we need

Proposition 5. ( [Br], Ch. III, Proposition 9.5 ) Let A be a Γ-module and Θ ⊂ Δ ⊂ Γ be subgroups. If [Γ : Δ] < ∞ and z ∈ H^{∗}(Δ, A) then we have

res _{Θ}^{Γ} ∘ cor _{Δ}^{Γ} (z) = ∑_{g∈Λ} cor _{Θ∩gΔg^{−1}}^{Θ} ∘ res _{Θ∩gΔg^{−1}}^{gΔg^{−1}} (ĝ(z)),

where Λ is a set of representatives of double cosets Θ g Δ and

ĝ : H^{∗}(Δ, A) → H^{∗}(g Δ g^{−1}, A)

is the natural map induced by pair ( int(g^{−1}), g).

To prove our claim first take η ∈ U. Let ξ′ = ϕ^{−1}(η) ∩ U_{1} and let τ_{ξ′} ∈ Δ be an involution corresponding to ξ′ and satisfying (9). Then applying Proposition 5 we have

res_{K_{ξ′}} (ζ) = ∑ res _{Θ_{ξ′} ∩gΔg^{−1}}^{gΔg^{−1}} (ĝ(ζ′)) = ∑ res _{g^{−1}Θ_{ξ′} g∩Δ}^{Δ}(ζ′) = res _{Θ_{ξ′}}^{Δ} (ζ′)

where Θ_{ξ′} = ⟨τ_{ξ′}⟩, hence res_{K_{ξ′}} (ζ) is defined by a. Analogously, one shows that res_{K_{η}} (ζ) is trivial if η ∉ U. Proposition 4 is proved.

Corollary 2. Let A be a commutative connected linear algebraic K-group. Then φ_{2} is injective.

Proof. One has H^{i}(L, A) = 1, i ≥ 1. So H^{i}(K, A) has exponent 2 and hence the map H^{i}(K, _{2}A) → H^{i}(K, A) is surjective, where _{2}A consists of all elements of A killed by 2. By Proposition 4, it gives the surjectivity of φ_{1} for A. Then the result follows from the injectivity of φ_{2} for _{2}A.

Corollary 3. The Hasse principle holds for algebraic K-tori.

Proof. Let T be a K-torus. There exists K-quasi-split torus S and its connected K-subtorus H such that T = S/H. Then the commutative diagram

H^{1}(K, S) = 1  ⟶  H^{1}(K, T)  ⟶  H^{2}(K, H)
↓θ_{1}  ↓θ_{2}  ↓θ_{3}
∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, S) = 1 ⟶ ∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, T) ⟶ ∏_{ξ∈Ω_{K}} H^{2}(K_{ξ}, H)

shows that the injectivity of θ_{2} follows from that of θ_{3}.

4. The Hasse principle for principal homogeneous spaces

Let us keep the notations of § 3. In particular, we assume that K is a field with vcd (K) ≤ 1, L = K(\sqrt{−1} ) and Ω_{K} ≠ ∅. Let also τ be the non-trivial element of Gal (L/K). Using the results of the previous sections we may produce a simple proof of the triviality of the kernel of (2).

a) Let G′ be a connected linear algebraic K-group, Z ≤ G′ be a finite central K-subgroup and let G = G′/Z.

Lemma 5. If the Hasse principle holds for G′ then it also holds for G.

Proof. Consider the commutative diagram

H^{1}(K, Z)  ⟶  H^{1}(K, G′)  ⟶  H^{1}(K, G)  ⟶^{λ_{1}}
↓θ_{1}  ↓θ_{2}  ↓θ_{3}
∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, Z) ⟶ ∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, G′) ⟶ ∏_{ξ∈Ω_{K}} H^{1}(K_{ξ}, G) ⟶^{λ_{2}}
⟶^{λ_{1}}  H^{2}(K, Z)
↓θ_{4}
⟶^{λ_{2}}  ∏_{ξ∈Ω_{K}} H^{2}(K_{ξ}, Z)

By assumption and by Proposition 2, the maps θ_{2}, θ_{4} are injective. Then from the above diagram and from Proposition 4 we have Ker θ_{3} = 1.

b) Reduction to semisimple groups. Since unipotent K-groups have trivial cohomology we may assume without loss of generality that G is reductive. Then G = T · H is an almost direct product of the central torus T and the semisimple group H = [G, G]. Let G′ = T × H. Clearly, the kernel of the natural morphism G′ → G is finite and by induction and by Corollary 3, the Hasse principle holds for H and T. So by Lemma 5, it holds for G as well.

c) Reduction to simple simply connected groups. One can again apply Lemma 5 to a simply connected covering G′ of G.

d) Let G be an (absolutely) simple simply connected K-group. By Steinberg’s theorem ( [St2] ), G has a Borel subgroup B over L. We may assume that T = B ∩ τ(B) is a maximal K-torus of G. Since H^{1}(L, G) = 1, the map H^{1}(L/K, G(L)) → H^{1}(K, G) is surjective. By Lemma 6.28 [Pl-R], the map H^{1}(L/K, T (L)) → H^{1}(L/K, G(L)) is surjective as well, hence any class [ζ] ∈ H^{1}(K, G) can be represented by a cocycle ζ′ ∈ Z^{1}(L/K, T (L)). Let S be a maximal K-split subtorus of T.

First let S ≠ 1. Then C_{G}(S) is a proper connected subgroup of G. Since C_{G}(S) is a reductive part of some parabolic K-subgroup, one has Ker (H^{1}(E, C_{G}(S)) → H^{1}(E, G)) = 1 for any extension E/K ( [Pr-R], Lemma 5.1 ). So if in addition ζ ∈ Ker θ, then for each ξ ∈ Ω_{K} the element res_{K_{ξ}}(ζ′) is trivial as an element of H^{1}(K_{ξ}, C_{G}(S)), hence the claim follows by induction.

e) S = 1, i.e. T is a K-anisotropic torus. By Steinberg’s theorem, G is either split or quasi-split over L. We examine the L-splitting case only, since the L-quasi-splitting case can be handled analogously. Identify Z^{1}(Θ, T (L)) with (K^{∗})^{n}. Arguing as in d) we get that any element from Ker θ can be represented by a cocycle ζ ∈ Z^{1}(Θ, T (L)). We claim that there exist a maximal K-torus T′ ⊂ G isomorphic to T over K and a cocycle ζ′ ∈ Z^{1}(Θ, T′(L)) equivalent to ζ in Z^{1}(Θ, G(L)) such that ζ′ is everywhere locally positive. By Corollary 3, the last would mean that ζ′ is trivial as an element of H^{1}(Θ, T′(L)), hence ζ is trivial in H^{1}(Θ, G(L)) as well.

To show it, we proceed as in Theorem 2. Namely, we construct inductively quaternion algebras D_{1}, …, D_{m} over K and elements g_{i} ∈ G_{β_{i}}(L) such that for g = g_{1} ⋯ g_{m} the element g^{1−τ} ∈ T (L) and the components of the cocycles (g^{1−τ}) and ζ everywhere locally have the same signs.

As in Theorem 2, we begin with D_{m} = (−1, d_{β_{m}}), where d_{β_{m}} = c_{β_{m}}. For ξ ∈ Ω_{K} let g_{ξ} ∈ G(\overline{K}_{ξ}) be such that ζ = (g_{ξ}^{1−τ}) ( note that T is still anisotropic over K_{ξ}). We may assume that g_{ξ} is in “generic” position and so we may write g_{ξ} as a product g_{ξ} = t_{ξ} g_{ξ,1} ⋯ g_{ξ,m}, where t_{ξ} ∈ T, g_{ξ,i} ∈ G_{β_{i}}, i = 1, …, m.

We have already known that τ(g_{ξ,m}) = h_{β_{m}}(w_{ξ,m}) g_{ξ,m} for some parameter w_{ξ,m} ∈ K_{ξ}. By virtue of the facts that our field K has the property SAP and the Hasse principle holds for groups of type A_{1} ( [B-P], [Sch1] ) we can pick w_{m} ∈ K, which has everywhere locally the same sign as w_{ξ,m}, and g_{m} ∈ G_{β_{m}}(L) such that h_{β_{m}}(w_{m}) = g_{m}^{1−τ}.

Next consider the quaternion K-algebra D_{m−1} = (−1, d_{β_{m−1}}), where

d_{β_{m−1}} = c_{β_{m−1}}w_{m}^{⟨β_{m−1},β_{m}⟩}.

Let w_{ξ,m−1} ∈ K_{ξ} be such that h_{β_{m−1}}(w_{ξ,m−1}) h_{β_{m}}(w_{ξ,m}) = (g_{ξ,m−1} g_{ξ,m})^{1−τ} Again we can pick w_{m−1} ∈ K such that for all ξ ∈ Ω_{K} the elements w_{m−1} and w_{ξ,m−1} have the same sign. By construction, the equation h_{β_{m−1}}(w_{m−1}) h_{β_{m}}(w_{m}) = (x g_{m})^{1−τ}, where x ∈ G_{β_{m−1}}(L), has solution everywhere locally, so it has solution g_{m−1} globally, and so on.

Thus, there exists g ∈ G(L) such that the components of both cocycles (gτ(g^{−1})) and ζ have the same signs in K_{ξ} for each ξ ∈ Ω_{K}. To complete the proof of the theorem it remains to notice that the cocycle ζ′ = τ(g)^{−1}ζ g is equivalent to ζ in Z^{1}(Θ, G(L)), takes values in the K-defined and L-splitting torus T′ = τ(g)^{−1}T τ(g) and ζ′ is everywhere locally positive.

Remark 3. The same argument shows that θ is still injective if we replace Ω_{K} by a dense set of orderings.

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Vladimir Chernousov Fakultät für Mathematik Universität Bielefeld D 33501 Bielefeld, Germany chernous@mathematik.uni-bielefeld.de