R-equivalence in adjoint classical groups over fields of virtual cohomological dimension 2

Amit Kulshrestha and R. Parimala

Abstract

Let F be a field of characteristic not 2 whose virtual cohomological dimension is at most 2. Let G be a semisimple group of adjoint type defined over F. Let RG(F) denote the normal subgroup of G(F) consisting of elements R-equivalent to identity. We show that if G is of classical type not containing a factor of type D_{n}, G(F)/RG(F) = 0. If G is a simple classical adjoint group of type D_{n}, we show that if F and its multi-quadratic extensions satisfy strong approximation property, then G(F)/RG(F) = 0. This leads to a new proof of the R-triviality of F -rational points of adjoint classical groups defined over number fields.

Mathematics Subject Classification 2000. 20G15, 14G05.

Key words. adjoint classical groups, R-equivalence, algebras with involutions, similitudes.

Introduction

In [Ma, Chapter II, §14] Manin introduced the notion of R-equivalence on a variety X over a field F as follows : two points x, y ∈ X(F) are R-equivalent if there exist x = x_{0}, x_{1}, x_{2}, ⋯, x_{n} = y ∈ X(F) and F -rational maps f_{i} : ℙ^{1} −−→ X; 1 ≤ i ≤ n, regular at 0 and 1 such that f_{i}(0) = x_{i−1} and f_{i}(1) = x_{i}. If X is the underlying variety of a connected algebraic group G, then the set of elements of G(F) which are R-equivalent to 1, is a normal subgroup RG(F) of G(F). We denote the quotient G(F)/RG(F) by G(F)/R. A connected algebraic group is called R-trivial, if for all field extensions E of F, we have G(E)/R = 0. Colliot-Thélène and Sansuc [CTS] proved that if the variety of a connected algebraic group G is stably rational then G is R-trivial. For example, if G is an adjoint classical group of type ^{1}A_{n}, ^{2}A_{2n} [VK, pp. 240] or B_{n}, then G is rational, and hence R-trivial.

Let G be a classical group of adjoint type defined over a number field. The group G(F)/R is then trivial. If G̃ is a simply connected cover of G, the triviality of G(F)/R can be deduced from the following results:

(i) [Y] If G̃ is of type ^{2}A_{n} then G̃(F)/R = 0.

(ii) [PR, Theorem 9.5] The group G̃(F) is projectively simple provided G̃ does not contain a factor of type A_{n}. In particular the non-central normal subgroup RG̃(F) coincides with G̃(F).

(iii) [G, pp. 222] and [CTGP, proof of Cor. 4.11] The natural map G̃(F)/R → G(F)/R is surjective.

Number fields are examples of fields of virtual cohomological dimension two. The aim of this paper is to study the group G(F)/R where G is a classical group of adjoint type defined over a field of virtual cohomological dimension two.

Let Γ_{F} be the Galois group Gal(F_{s}/F), where F_{s} is the separable closure of F. The cohomological dimension of F is the least positive integer n such that for all discrete torsion Γ_{F} -modules M, the Galois cohomology groups H^{i}(Γ_{F} , M) are zero for all i ≥ n + 1. A field F is said to have virtual cohomological dimension n if the cohomological dimension of F (\sqrt{−1}) is n. We write cd(F) to denote the cohomological dimension and vcd(F) to denote the virtual cohomological dimension of F. We prove that G(F)/R = 0 for adjoint groups G of type ^{2}A_{n} and C_{n} over a field F of virtual cohomological dimension at most 2. For classical groups of type D_{n}, we prove that if the cohomological dimension of F is at most 2 then G(F)/R = 0.

Further, if the virtual cohomological dimension of F is at most 2, then we show that G(F)/R = 0, provided F satisfies certain approximation properties. These results, in particular, lead to a new proof of the triviality of G(F)/R for adjoint classical groups over number fields.

The main ingredients in the proof of the our results are Merkurjev’s computation of G(F)/R for all adjoint groups of classical type [Me2, Th. 1], as well as results on the classification of hermitian forms over division algebras with involution over fields of virtual cohomological dimension two [BP2].

1 Some known results

In this section, we record some known results which are used in the paper. Let F be a field with char(F) ≠ 2. Let Z = F, or a quadratic extension of F. Let A be a central simple algebra over Z and σ be an involution on A of either kind. If σ is of second kind, then let Z^{σ} = F. An element a ∈ A^{∗} is said to be a similitude of (A, σ) if σ(a)a ∈ F^{∗}. The similitudes of (A, σ) form a group which we denote by Sim(A, σ). The map μ(a) = σ(a)a is a homomorphism μ : Sim(A, σ) → F^{∗} whose image is denoted by G(A, σ). Elements of G(A, σ) are called multipliers. Let σ be adjoint to a hermitian form h. Then λ ∈ G(A, σ) if and only if λh ≃ h [KMRT, Prop. 12.20]. Let Sim(A, σ) denote the algebraic group whose F rational points are given by Sim(A, σ). Let Sim_{+}(A, σ) be the connected component of identity of Sim(A, σ). Let Sim_{+}(A, σ) denote the F -rational points of Sim_{+}(A, σ). Elements of Sim_{+}(A, σ) are called proper similitudes. We denote the group μ(Sim_{+}(A, σ)) by G_{+}(A, σ). Let R_{Z/F} denote the Weil restriction to F. The group of projective similitudes is the quotient group

Sim(A, σ)/R_{Z/F} (G_{m})

which we denote by PSim(A, σ). The group of F -rational points of PSim(A, σ) is Sim(A, σ)/Z^{∗}. The connected component of the identity of the group PSim(A, σ) is denoted by PSim_{+}(A, σ)

Let N(Z) = F^{∗2} or N_{Z/F} (Z^{∗}) according as σ is of first kind or second kind, respectively. Let Hyp(A, σ) be the subgroup of F^{∗} generated by the norms from all those finite extensions of F, where the involution σ becomes hyperbolic. If A is split, the involution σ is adjoint to a quadratic form q over F. The group G_{+}(A, σ) is then denoted by G_{+}(q) and the group G(A, σ) is denoted by G(q). In fact G_{+}(q) = G(q), because of the existence of hyperplane reflections in the orthogonal group.

Theorem 1.1 ([Me2, Th. 1]) With the notation as above, N(Z). Hyp(A, σ) is a subgroup of G_{+}(A, σ) and further,

PSim_{+}(A, σ)(F)/R ≃ G_{+}(A, σ)/N(Z). Hyp(A, σ).
□

We now record a lemma due to Dieudonné.

Lemma 1.2 (Dieudonné, [KMRT, Lemma 13.22]) Let q be a quadratic form of even rank and d = disc(q). Let L = F (\sqrt{d}). Then G(q) ⊆ N_{L/F} (L^{∗}). □

The following result of Merkurjev-Tignol extends Dieudonné’s lemma.

Lemma 1.3 ([MT, Th. A]) Let A be a central simple algebra of even degree with an orthogonal involution σ. Let d = disc(σ) and let L = F (\sqrt{d}). Then G_{+}(A, σ) ⊆ N_{L/F} (L^{∗}). □

Let q be a non-degenerate quadratic form of rank r over F. Let τ_{q} be the adjoint involution on M_{r}(F). Then Hyp(M_{r}(F), τ_{q}) = Hyp(q), the subgroup of F^{∗} generated by N_{L/F} (L^{∗}); L varying over finite extensions of F where q becomes hyperbolic. If r is odd, then Hyp(q) = 1.

Theorem 1.4 ([Me2, pp. 200]) Let A be a central simple algebra of odd degree with an orthogonal involution σ. Let q be a quadratic form over F such that σ is adjoint to q. Then G_{+}(A, σ) = G(q) = Hyp(q).F^{∗2} = Hyp(A, σ).F^{∗2} = F^{∗2}. □

We now record a result due to Knebusch which describes the group of spinor norms of a quadratic form. Let q be a quadratic form over F and sn(q) denote the subgroup of F^{∗} generated by F^{∗2} and representatives of the square classes in the image of the spinor norm map sn : SO(q) → F^{∗}/F^{∗2}. For a central simple algebra A over F, let Nrd : A → F denote the reduced norm map. For S ⊆ F^{∗}, we denote by ⟨S⟩, the subgroup generated by S in F^{∗}.

Theorem 1.5 (Knebusch’s norm principle, [L, Theorem VII.5.1]) For a quadratic form q over F we have:

sn(q) = ⟨{N_{L/F} (L^{∗}) : L/F is a quadratic extension over F and q_{L} is isotropic}⟩.

□

The two results recorded below describe the group G(A, σ) in the case when σ is unitary or symplectic under further assumptions on the degree of A.

Theorem 1.6 ([Me2, §2]) Let F be a field with char(F) ≠ 2. Let A be a central simple algebra over Z of odd degree with an involution σ of second kind with Z^{σ} = F. Then G_{+}(A, σ) = G(A, σ) = Hyp(A, σ) = N(Z). □

Theorem 1.7 ([Me2, §2, Lemma 3]) Let F be a field with char(F) ≠ 2. Let A be a central simple algebra over F of degree 2n with n odd. Let σ be a symplectic involution on A. Then G_{+}(A, σ) = G(A, σ) = Hyp(A, σ) = Nrd(A). □

Next results we record are local criteria for elements to be reduced norms or spinor norms for formally real fields F with vcd(F) ≤ 2.

Theorem 1.8 ([BP2, Theorem 2.1]) Let F be a formally real field with vcd(F) ≤ 2. Let Ω denote the set of orderings on F. Let A be a central simple algebra over F and A_{v} = A ⊗_{F} F_{v}, F_{v} denoting the real closure of F at v. Let λ ∈ F^{∗} be such that λ >_{v} 0 at those orderings v ∈ Ω where A_{v} is non-split. Then λ ∈ Nrd(A^{∗}). □

Theorem 1.9 ([BP2, Cor. 7.10]) Let F be a formally real field with vcd(F) ≤ 2. Let q be a quadratic form over F. Then sn(q) consists of elements of F^{∗} which are positive at each v ∈ Ω such that q is definite at F_{v}. □

We say that a quadratic form q over F is locally isotropic if over each real closure F_{v}, v ∈ Ω, the form q is isotropic.

Corollary 1.10 With the notation as in 1.9, if q locally isotropic, then sn(q) = F^{∗}. □

Let Γ_{F} denote the Galois group Gal(F_{s}/F). For a discrete Γ_{F} -module M, let H^{n}(F, M) denote the Galois cohomology group H^{n}(Gal(F_{s}/F), M). We now record some results of Arason which we shall use in the paper.

Theorem 1.11 (Corollar 4.6, [A1]) Let Z = F (\sqrt{δ}) be a quadratic extension of F. Then we have a long exact sequence of abelian groups

⋯ → H^{n}(F, μ_{2}) ⟶^{res} H^{n}(Z, μ_{2}) ⟶^{cores} H^{n}(F, μ_{2}) \underrightarrow{⋃_{n,1}(δ)} H^{n+1}(F, μ_{2}) → ⋯

where res and cores denote the restriction and corestriction maps respectively. □

In view of 1.11 and the isomorphism H^{2}(F, μ_{2}) ≃ _{2} Br(F), we have the following

Proposition 1.12 Let Z = F (\sqrt{δ}) be a quadratic extension of F and let A be a central simple algebra over Z with exp(A) = 2 and cores_{Z/F} ([A]) = 0 ∈ H^{2}(F, μ_{2}). Then there exists a central simple algebra A_{0} over F such that A_{0}⊗_{F} Z is Brauer equivalent to A. □

We say that a field extension L/F is a quadratic tower over F if there exist fields F_{i} such that F = F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{r} = L and each F_{i}/F_{i−1} is a quadratic extension for 1 ≤ i ≤ r. We denote by F_{2}(F) the set of quadratic towers of F in an algebraic closure of F. Let I(F) denote the fundamental ideal of the Witt ring W (F) of F. For each n ≥ 1, we denote by I^{n}(F) to denote the ideal I(F)^{n}.

Lemma 1.13 ([A1, Satz 3.6]) Let I^{3}(F) = 0 and L/F be a quadratic tower. Then I^{3}(L) = 0. □

Theorem 1.14 ([A2, Prop. 2]) Let F be a field with cd(F) ≤ 2. Then I^{3}(F) = 0.

□

A non-trivial element χ ∈ H^{r}(F, μ_{2}) is called (−1)-torsion-free if for every s ≥ 1, the element χ ∪ (−1) ∪ (−1) ∪ ⋯ ∪ (−1) ∈ H^{r+s}(F, μ_{2}) is non-trivial. The following is a consequence of 1.11

Proposition 1.15 Let F be a field with vcd(F) ≤ n. Then H^{n+1}(F, μ_{2}) is (−1)- torsion-free. □

The following lemma relates the conditions vcd(F) ≤ 2 and I^{3}(F) being torsionfree.

Lemma 1.16 ([BP2, Lemma 2.4]) Let F be a field with virtual cohomological dimension at most two. Then I^{3}(F) is torsion-free.

Proof Since vcd(F) ≤ 2, by [AEJ] the invariants e_{r} : I^{r}(F) → H^{r}(F, μ_{2}) have kernel I^{r+1}(F) for each r ≥ 0 and H^{r}(F (\sqrt{−1}), μ_{2}) = 0 for r ≥ 3. Then it is evident from Arason exact sequence 1.11 for the quadratic extension F (\sqrt{−1})/F that H^{r}(F, μ_{2}) \underrightarrow{⋃(−1)} H^{r+1}(F, μ_{2}) is an isomorphism for r ≥ 3. Let q ∈ I^{3}(F) be a torsion-element. Then 2^{s}.q = 0 ∈ W (F) for some integer s ≥ 0. As a consequence e_{3}(q)∪(−1)∪(−1)∪⋯∪(−1) = 0 ∈ H^{3+s}(F, μ_{2}). Since H^{r}(F, μ_{2}) \underrightarrow{⋃(−1)} H^{r+1}(F, μ_{2}), r ≥ 3, are isomorphisms, we conclude that e_{3}(q) = 0; i.e. q ∈ ker(e_{3}) = I^{4}(F). By a similar argument q ∈ I^{r}(F) for each r ≥ 3 and hence q ∈ ⋂_{r} I^{r}(F). By a theorem of Arason-Pfister [L, Cor. X.3.2], q = 0 ∈ W (F) and hence I^{3}(F) is torsion-free. □

The following result is a weaker form of [Se, Prop. 10, §II.4.1].

Theorem 1.17 Let F be a field and cd(F) ≠ vcd(F). Then F has orderings. □

2 Some norm principles

Let F be a field with char(F) ≠ 2 and I^{3}(F) = 0. Let A be a central simple algebra with exp(A) = 2. Then by [Me1], there are quaternion algebras H_{i}; 1 ≤ i ≤ r, such that A ∼ H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{r}. We define an integer r(A) associated to A as follows:

r(A) := min{r : A ∼ H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{r}}.

If A is split, then we define r(A) = 0. Given a central simple algebra B over a field Z with [Z : F] ≤ 2 and a field extension L of F, we set: B_{L} = B ⊗_{F} L.

Proposition 2.1 Let I^{3}(F) = 0 and A be a central simple algebra over F. If exp(A) is a power of 2 then

F^{∗} = ⟨{N_{L/F} (L^{∗}) : L is a quadratic tower of F with A_{L} split}⟩ = Nrd(A^{∗})

In fact, for each λ ∈ F^{∗} there is a quadratic tower L/F and α ∈ L^{∗} such that λ = N_{L/F} (α).

Proof By the classical norm principle for reduced norms, over any field we have the inclusion

⟨{N_{L/F} (L^{∗}) : L is a quadratic tower of F with A_{L} split}⟩ ⊆ Nrd(A^{∗})

Thus to complete the proof, it suffices to show that under the assumption I^{3}(F) = 0,

F^{∗} ⊆ ⟨{N_{L/F} (L^{∗}) : L is a quadratic tower of F with A_{L} split}⟩.  (1)

Let exp(A) = 2^{m}. We prove the lemma by induction on m. Suppose m = 1. Then exp(A) = 2 and hence by Merkurjev’s Theorem [Me1], we write A ∼ H_{1} ⊗H_{2} ⊗⋯⊗ H_{r}, where r = r(A) and each H_{i} is a quaternion algebra over F. We proceed further by induction on r. If r = 1 the result holds by [BP2, Prop. 2.7]. Let r ≥ 2 and λ ∈ F^{∗}. By [BP2, Prop. 2.7] there exists a quadratic extension L of F which splits H_{1} and λ ∈ N_{L/F} (L^{∗}). Then r(A_{L}) < r and by 1.13 we have I^{3}(L) = 0. Induction on r leads to (1).

Suppose that m ≥ 2. Then exp(A ⊗_{F} A) = 2^{m−1}. Let λ ∈ F^{∗}. By induction, there exists a quadratic tower L over F and α ∈ L^{∗} such that λ = N_{L/F} (α) and (A ⊗_{F} A)_{L} is split. Then exp(A_{L}) = 2 and by 1.13, I^{3}(L) = 0. By the previous case, there exists a quadratic tower M of L with α ∈ N_{M/L}(M^{∗}) and A_{M} is split. Thus M is a quadratic tower of F such that λ ∈ N_{M/F} (M^{∗}) and A_{M} is split. This completes the proof. □

Proposition 2.2 Let I^{3}(F) = 0 and Z be a quadratic extension of F. Let A be a central simple algebra over Z such that cores_{Z/F} (A) = 0 and exp(A) = 2^{m}. Then for each λ ∈ F^{∗}, there exists a quadratic tower L/F such that λ ∈ N_{L/F} (L^{∗}) and A_{L} is split.

Proof We prove this by induction on m. Suppose m = 1. Since exp(A) = 2 and cores_{Z/F} (A) = 0, by 1.12 there exists a central simple algebra A_{0} of exponent 2 over F such that A ∼ A_{0} ⊗_{F} Z. Let λ ∈ F^{∗}. Since I^{3}(F) = 0, by 2.1, there exists a quadratic tower L/F such that (A_{0})_{L} is split and λ ∈ N_{L/F} (L^{∗}). Clearly the extension L splits A and the proposition follows.

Suppose m ≥ 2. Let λ ∈ F^{∗}. Since exp(A ⊗_{Z} A) = 2^{m−1}, by induction there exists a quadratic tower L/F such that λ = N_{L/F} (α) for some α ∈ L^{∗}, and (A⊗_{Z} A)_{L} splits. Clearly exp(A_{L}) = 2, and by the previous case we have a quadratic tower M/L such that A_{M} splits and α ∈ N_{M/L}(M^{∗}). Then M/F is a quadratic tower such that λ ∈ N_{M/F} (M^{∗}) and A_{M} is split. This completes the proof. □

We shall now describe norm principles for fields F with vcd(F) ≤ 2. If F has no orderings by 1.17, cd(F) ≤ 2, and the results follow from the previous discussion. We shall assume in the rest of the section that F has orderings. We denote by Ω, the set of orderings on F. If A is a central simple algebra over F then A is said to be locally split if A ⊗_{F} F_{v} = A_{v} is split for each v ∈ Ω.

Proposition 2.3 Let vcd(F) ≤ 2 and A be a central simple algebra over F with exp(A) = 2^{m}. Then

F^{∗} = ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and index(A_{M}) ≤ 2}⟩.

Proof We prove the proposition by induction on m. Let m = 1. Then exp(A) = 2 and we proceed by further induction on r(A). The statement is obvious if r(A) ≤ 1. Let r(A) ≥ 2 and A ∼ H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{r} with r = r(A) and H_{i}, 1 ≤ i ≤ r, quaternion algebras over F. Let H_{1} = (a, b) and H_{2} = (c, d). Then to the algebra H_{1} ⊗ H_{2} is associated the Albert form (cf. [KMRT, §16.A])

q = ⟨−a, −b, ab, c, d, −cd⟩.

Since disc(q) = 1 and dim(q) = 6, the form q is isotropic at F_{v} for each v ∈ Ω. Thus by 1.10, sn(q) = F^{∗} and by 1.5 we have

F^{∗} = sn(q) = ⟨{N_{L/F} (L^{∗}) : L is a quadratic extension of F and q_{L} is isotropic}⟩

Let L be a quadratic extension of F with q_{L} isotropic. By Albert’s Theorem [KMRT, Th. 16.5], we have r((H_{1} ⊗ H_{2})_{L}) ≤ 1. Thus r(A_{L}) < r(A) and by induction we have,

L^{∗} = ⟨{N_{M/L}(M^{∗}) : M ∈ F_{2}(L) and index(A_{M}) ≤ 2}⟩.

and therefore taking norms from L to F we have,

F^{∗} = sn(q) = ⟨{N_{L/F} (L^{∗}) : L is a quadratic extension of F and q_{L} is isotropic}⟩
⊆ ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and index(A_{M}) ≤ 2}⟩.

This completes the case m = 1. Now let m ≥ 2. Then exp(A ⊗_{F} A) = 2^{m−1} and by induction

F^{∗} = ⟨{N_{L/F} (L^{∗}) : L ∈ F_{2}(F) and index((A ⊗_{F} A)_{L}) ≤ 2}⟩.  (2)

Let L ∈ F_{2}(F) be such that index((A ⊗_{F} A)_{L}) ≤ 2. Since the Brauer group of a real-closed field is isomorphic to ℤ/2ℤ, it follows that (A ⊗_{F} A)_{L} is locally split. Thus by 1.8, Nrd((A ⊗_{F} A)_{L}) = L^{∗}. Since index((A ⊗_{F} A)_{L}) ≤ 2, we have

Nrd((A ⊗_{F} A)_{L})  (3)
⊆ ⟨{N_{N/L}(N^{∗}) : N is a quadratic extension of L and (A ⊗_{F} A)_{N} is split}⟩.

Let N be a quadratic extension of L such that (A⊗_{F} A)_{N} is split. Then exp(A_{N}) = 2 and by the case m = 1

N^{∗} = ⟨{N_{M/N}(M^{∗}) : M ∈ F_{2}(N) and index((A_{M}) ≤ 2}⟩.  (4)

Now it is clear from (2), (3) and (4) that

F^{∗} = ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and index(A_{M}) ≤ 2}⟩.
□

We refine 2.3 to the following:

Proposition 2.4 Let F be a field with vcd(F) ≤ 2. Let A be a central simple algebra over F with exp(A) = 2^{m} for some m ≥ 1. Then,

F^{∗} = ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and A_{M} ∼ (−1, −x) for some x ∈ M^{∗}}⟩.

Proof Let L be a quadratic tower over F such that index(A_{L}) ≤ 2. Then A_{L} ∼ (a, b), a, b ∈ L^{∗}. Let Ω_{L} denote the set of orderings on L. For each w ∈ Ω_{L}, the quadratic form q′ = ⟨ − 1, −a, −b, ab⟩ is isotropic over L_{w}, where L_{w} denotes the real closure of L at w. Therefore by [BP2, Prop. 7.7] we have sn(q′) = L^{∗}. Thus, in view of 1.5 we have:

L^{∗} = ⟨{N_{M/L}(M^{∗}) : M is a quadratic extension of L and q′_{M} is isotropic }⟩.

Let M be a quadratic extension of L such that q′_{M} is isotropic. Then the form ⟨ − a, −b, ab⟩_{M} represents 1, and we can write: ⟨ − a, −b, ab⟩_{M} ≃ ⟨1, x, y⟩_{M}; with x, y ∈ M^{∗}. Comparing the discriminants, we have ⟨ − a, −b, ab⟩_{M} ≃ ⟨1, x, x⟩_{M}. Thus ⟨1, −a, −b, ab⟩_{M} ≃ ⟨1, 1, x, x⟩_{M} and (a, b)_{M} ≃ (−1, −x). Thus,

L^{∗} = ⟨{N_{M/L}(M^{∗}) : M is a quadratic extension of F and q′_{M} is isotropic }⟩
⊆ ⟨{N_{M/L}(M^{∗}) : M ∈ F_{2}(L) and A_{M} ∼ (−1, −x) for some x ∈ M^{∗}}⟩

and

N_{L/F} (L^{∗}) ⊆ ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and A_{M} ∼ (−1, −x) for some x ∈ M^{∗}}⟩.

This together with 2.3 gives

F^{∗} ⊆ ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and A_{M} ∼ (−1, −x) for some x ∈ M^{∗}}⟩.

This completes the proof. □

Corollary 2.5 Let vcd(F) ≤ 2. Let A_{1} and A_{2} be central simple algebras over F with exp(A_{i}) a power of 2 for i = 1, 2. Then we have:

F^{∗} = ⟨{N_{M/F} (M^{∗}) : M ∈ F_{2}(F) and A_{1M} ∼ (−1, −x),
A_{2M} ∼ (−1, −y) for some x, y ∈ M^{∗}}⟩
□

The following is a refinement of the surjectivity of the reduced norm (Theorem 1.8) for locally split algebras with centre a quadratic extension of F.

Proposition 2.6 Let vcd(F) ≤ 2 and F has orderings. Let Ω denote the set of orderings on F. Let Z = F (\sqrt{δ}) be a quadratic extension of fields. Let A be a central simple Z-algebra which is split at each v ∈ Ω. Further assume that exp(A) = 2^{m} for some integer m and cores_{Z/F} (A) = 0. Then for each λ ∈ F^{∗}, there exist extensions E_{i} over F and λ_{i} ∈ E_{i}^{∗} such that each A ⊗_{F} E_{i} is split and λ = ∏_{i} N_{E_{i}/F} (λ_{i}).

Proof We proceed by induction on m. Let m = 1. Since cores_{Z/F} (A) = 0, by 1.12 there is a central simple algebra A_{0} over F with exp(A_{0}) = 2 and A ∼ A_{0} ⊗_{F} Z. By [Me1], there are quaternion algebras H_{i}; 1 ≤ i ≤ r = r(A_{0}) over F such that A_{0} ∼ H_{1} ⊗H_{2} ⊗⋯⊗H_{r}. Suppose r = 1. Then A_{0} ∼ H_{1} = (a, b) for some a, b ∈ F^{∗}. Let q denote the quadratic form ⟨1, −a, −b, abδ⟩ over F. Then by [CTSk, Prop. 2.3] we have sn(q) = Nrd((H_{1} ⊗_{F} F (\sqrt{δ}))^{∗}) ∩ F^{∗}. Since A is locally split, by 1.8 Nrd((H_{1} ⊗_{F} F (\sqrt{δ}))^{∗}) = Nrd(A^{∗}) = Z^{∗}. Therefore sn(q) = F^{∗}. Thus by 1.5, for each λ ∈ F^{∗}, there exist quadratic extensions E_{i}/F and λ_{i} ∈ E_{i}^{∗} such that each q_{E_{i}} is isotropic and λ = ∏_{i} N_{E_{i}/F} (λ_{i}). Further A ⊗_{F} E_{i} ∼ (a, b) ⊗_{E_{i}} E_{i}(\sqrt{δ}) and the norm form of (a, b) ⊗_{E_{i}} E_{i}(\sqrt{δ}) is isometric to q_{E_{i}(\sqrt{δ})}, which is isotropic. It follows therefore that each A ⊗_{F} E_{i} is split. Thus F^{∗} is generated by the norms from those extensions of F where the algebra A is split.

Now suppose r ≥ 2. Then by 2.3 we have

F^{∗} = ⟨{N_{L/F} (L^{∗}) : index((A_{0})_{L}) ≤ 2}⟩.

The proposition follows immediately from the case r = 1.

Let m ≥ 2. Then exp(A ⊗_{Z} A) = 2^{m−1} and at each v ∈ Ω the algebra (A ⊗_{Z} A) ⊗_{F} F_{v} is split since Br(F_{v}) = ℤ/2ℤ. Thus by induction, F^{∗} is generated by norms from extensions M_{i} over F such that the algebra (A ⊗_{Z} A) ⊗_{F} M_{i} splits. It is clear that exp(A ⊗_{F} M_{i}) = 2. Thus by the exponent 2 case, it follows that each M_{i}^{∗} is generated by norms from extensions E_{i} of M_{i} such that A ⊗_{F} E_{i} is split. We conclude therefore, that F^{∗} is generated by norms from those extensions of F where A splits. □

3 Fields with cd(F) ≤ 2

In this section, we prove that if cd(F) ≤ 2, then for adjoint classical groups G of type 2A_{n}, C_{n} and D_{n}, G(F)/R = 0. We begin with the result leading to the triviality of G(F)/R in the C_{n} case.

Theorem 3.1 Let F be a field with char(F) ≠ 2 and I^{3}(F) = 0. Let A be a central simple algebra of degree 2n over F and σ be a symplectic involution on A. Then Hyp(A, σ) = F^{∗}.

Proof Let λ ∈ F^{∗}. Since exponent of A is 2 and I^{3}(F) = 0, by 2.1, there exists a quadratic tower L/F such that L splits A and λ ∈ N_{L/F} (L^{∗}). The involution σ_{L} is adjoint to a skew-symmetric form h_{L} over L which is hyperbolic. Therefore λ ∈ Hyp(A, σ). □

Let q be a quadratic form over F of rank 2n. Let σ be the involution on M_{2n}(F) which is adjoint to q. We denote by C(q) the Clifford invariant of q.

Proposition 3.2 If I^{3}(F) = 0, then G(q) ⊆ Hyp(q).

Proof We first assume that the discriminant of q is trivial. Let λ ∈ F^{∗}. The algebra C(q) has exponent 2 and by 2.1, there exists a quadratic tower M of F such that C(q) ⊗_{F} M is split and λ ∈ N_{M/F} (M^{∗}). By 1.13, I^{3}(M) = 0. Since q_{M} is an even dimensional quadratic form with trivial discriminant and trivial Clifford invariant, in view of [EL, Th. 3] q_{M} is hyperbolic and hence λ ∈ Hyp(q). Thus Hyp(q) = F^{∗}.

Now suppose that disc(q) is non-trivial, d ∈ F^{∗} a representative of the square class of disc(q) in F^{∗}/F^{∗2} and L = F (\sqrt{d}). Let λ ∈ G(q). By 1.2, λ ∈ N_{L/F} (L^{∗}). Since disc(q_{L}) = 1, by the previous case L^{∗} = Hyp(q_{L}). Taking norms we get N_{L/F} (L^{∗}) ⊆ Hyp(q). Thus G(q) ⊆ Hyp(q). □

We prove a similar result when A is not split.

Theorem 3.3 Let I^{3}(F) = 0. Let A be a central simple algebra with an involution σ of orthogonal type. Let d be the discriminant of σ and L = F [X]/(X^{2} − d). Then we have G_{+}(A, σ) = Hyp(A, σ) = N_{L/F} (L^{∗}).

Proof Since A supports an involution of first kind, exp(A) ≤ 2. Suppose first that disc(σ) is trivial. Let M be a quadratic tower of F which splits A. By the proof of 3.2 we have M^{∗} = Hyp(A_{M}, σ_{M}). Thus N_{M/F} (M^{∗}) ⊆ Hyp(A, σ). This, together with 2.1 implies that F^{∗} = Nrd(A^{∗}) ⊆ Hyp(A, σ). Hence F^{∗} = Hyp(A, σ) = G_{+}(A, σ). Since L = F ×F, we have N_{L/F} (L^{∗}) = F^{∗}. Thus G_{+}(A, σ) = Hyp(A, σ) = N_{L/F} (L^{∗}).

Suppose that disc(σ) is not trivial. Let d ∈ F^{∗} represent the class of disc(σ) in F^{∗}/F^{∗2}. Let λ ∈ G_{+}(A, σ). Then by 1.3, we have λ ∈ N_{L/F} (L^{∗}) where L = F (\sqrt{d}). Clearly disc(σ_{L}) = 1 and by previous case L^{∗} = Hyp(A_{L}, σ_{L}). Thus λ ∈ N_{L/F} (L^{∗}) ⊆ Hyp(A, σ). Thus G_{+}(A, σ) ⊆ N_{L/F} (L^{∗}) ⊆ Hyp(A, σ). By 1.1, Hyp(A, σ).F^{∗2} ⊆ G_{+}(A, σ). Hence G_{+}(A, σ) = Hyp(A, σ) = N_{L/F} (L^{∗}). □

Let Z be a quadratic extension of F and A be a central simple algebra over Z with an involution σ of second kind such that Z^{σ} = F. In the next lemma, we consider the case where A splits and the involution σ is adjoint to a Z/F -hermitian form h. In view of 1.6, we further assume that h has even rank; i.e. deg(A) is even.

Lemma 3.4 Let I^{3}(F) = 0, let A be split and σ be an involution of second kind on A such that Z^{σ} = F. Then Hyp(A, σ) = F^{∗}.

Proof Let Z = F (\sqrt{δ}). Let q_{h} be the quadratic form over F defined by q_{h}(x) = h(x, x). Then q_{h} ≃ ⟨1, −δ⟩ ⊗ q [S, pp. 349, Remark 1.3] for some quadratic form q over F having the same rank as h, which is even. Therefore q_{h} ∈ I^{2}(F) and by a theorem of Jacobson [MH, pp. 114], the form h is hyperbolic over an extension M of F if and only if the quadratic form q_{h} is hyperbolic over M. Let C denote the Clifford invariant of q_{h}. Let λ ∈ F^{∗}. By 2.1, there exists a quadratic tower M over F such that such that C_{M} is split and λ ∈ N_{M/L}(M^{∗}). Since I^{3}(M) = 0, by [EL, Th. 3], (q_{h})_{M} is hyperbolic and hence the hermitian form h_{M} is hyperbolic. Therefore N_{M/F} (M^{∗}) ⊆ Hyp(A, σ). Thus Hyp(A, σ) = F^{∗}. □

Theorem 3.5 If I^{3}(F) = 0 and exp(A) = 2^{m} then Hyp(A, σ) = F^{∗}.

Proof Since A supports an involution σ of second kind, by [S, Th. 9.5] we have cores_{Z/F} (A) = 0. Therefore by 2.2, given λ ∈ F^{∗} there exists a quadratic tower L/F such that A_{L} splits and λ ∈ N_{L/F} (L^{∗}). Since A_{L} is split, by 3.4, L^{∗} = Hyp(A_{L}, σ_{L}). Taking norms we conclude that λ ∈ Hyp(A, σ). Therefore Hyp(A, σ) = F^{∗}. □

Theorem 3.6 Let cd(F) ≤ 2 and Z be a quadratic extension of F. Let A be a central simple algebra of even degree over Z with an involution σ of second kind such that Z^{σ} = F. Then Hyp(A, σ).F^{∗2} = F^{∗}.

Proof By [BP1, Lemma 3.3.1], there exists an odd degree extension L over F such that exp(A ⊗_{F} L) is a power of 2. Since the condition cd(F) ≤ 2 is preserved under finite extensions of fields [Ar, Th 2.1], we have cd(L) ≤ 2. By 1.14 I^{3}(L) = 0 and by 3.5, Hyp(A_{L}, σ_{L}) = L^{∗}. Hence N_{L/F} (L^{∗}) ⊆ Hyp(A, σ). Let λ ∈ F^{∗} and [L : F] = 2s + 1. Then λ^{2s+1} = N_{L/F} (λ) ∈ Hyp(A, σ) and we have λ ∈ Hyp(A, σ).F^{∗2}. This implies that Hyp(A, σ).F^{∗2} = F^{∗}. □

Theorem 3.7 If cd(F) ≤ 2 and G an adjoint group of classical type defined over F, then G(F)/R = 0.

Proof A classical adjoint group G is a direct product of groups R_{L_{i}/F} (G_{i}), where L_{i}/F are finite extensions and G_{i} are absolutely simple adjoint groups of classical type defined over L_{i} [T, 3.1.2]. Moreover, G_{i}(L_{i})/R = R_{L_{i}/F} (G_{i})(F)/R and Requivalence commutes with direct products [CTS, pp. 195]. In view of this, it suffices to prove the theorem for an absolutely simple classical adjoint group G defined over F. By [We] such an algebraic group is isomorphic to PSim_{+}(A, σ) for a central simple algebra A over a field Z, [Z : F] ≤ 2, with an involution σ. In view of 1.1 and 1.14, the result follows in ^{2}A_{n} case from 3.6 and 1.6, in B_{n} case from 1.4 in C_{n} case from 3.1 and 1.7, and in D_{n} case from 3.3. □

Remark Theorem 3.7 for groups of type A_{n} and C_{n} also follows from [CTGP, Cor. 4.11], using the fact that G(F)/R = 0 if G is simply connected of type A_{n} or C_{n}, and [G, pp. 222].

4 Fields with vcd(F) ≤ 2 : Symplectic groups

In this section F denotes a formally real field with vcd(F) ≤ 2, and Ω, the set of orderings on F. Let A be a central simple algebra over F of degree 2n and σ be an involution of symplectic type on A. In view of 1.7, we assume that n is even. We say that σ is locally hyperbolic if for each v ∈ Ω, the involution σ_{v} on A_{v} = A ⊗_{F} F_{v} is hyperbolic, F_{v} denoting the real closure of F at v.

Proposition 4.1 Let A be a central simple algebra over F of degree 2n, where n is an even integer. Let σ be a symplectic involution on A. If σ is locally hyperbolic then Hyp(A, σ) = F^{∗}.

Proof First assume that A = M_{n}(H), where H is a quaternion algebra over F. Let bar denote the canonical involution on H and h a hermitian form of rank n over (H, −) such that σ is adjoint to h. Since σ is locally hyperbolic, so is h and hence sgn(h) = 0. Thus h has even rank and trivial signature and by [BP2, Th. 6.2], the form h itself is hyperbolic. Thus Hyp(A, σ) = F^{∗}.

Suppose A is arbitrary. Since A supports an involution, exp(A) = 2 [S, Th. 8.4] and by 2.3, we have

F^{∗} = ⟨{N_{M/F} (M^{∗}) : index(A_{M}) ≤ 2}⟩ (∗)

Let M be a finite extension of F such that index(A_{M}) ≤ 2. Then A_{M} ≃ M_{n}(H) where H is a quaternion algebra over M. Since σ is locally hyperbolic, so is σ_{M} and by the previous case, M^{∗} = Hyp(A_{M}, σ_{M}). Therefore N_{M/F} (M^{∗}) ⊆ Hyp(A, σ) and in view of (∗) we get Hyp(A, σ) = F^{∗}. □

Theorem 4.2 Let F be a formally real field with vcd(F) ≤ 2. Let A be a central simple algebra over F of degree 2n and σ be a symplectic involution on A. Then G(A, σ) ⊆ Hyp(A, σ).

Proof In view of 1.7, we assume that n is even. Let λ ∈ G(A, σ) and K = F (\sqrt{−λ}). Let Ω_{K} denote the set of orderings on K. For each w ∈ Ω_{K}, λ ≡ −1 modulo K_{w}^{∗2} is a similarity factor for σ_{K} and hence sgn(σ_{K}) = 0. Further deg(A) is divisible by 4 and hence the involution σ_{K} is locally hyperbolic. Thus by 4.1, we have Hyp(A_{K}, σ_{K}) = K^{∗}. Therefore

λ = N_{K/F} (\sqrt{−λ}) ∈ N_{K/F} (K^{∗}) = N_{K/F} (Hyp(A_{K}, σ_{K})) ⊆ Hyp(A, σ).
□

5 Fields with vcd(F) ≤ 2 : Unitary groups

Let F be an arbitrary field with char(F) ≠ 2. Let Z = F (\sqrt{δ}) be a quadratic extension of F. Let A be a central simple algebra over Z and σ be an involution on A such that Z^{σ} = F. In view of 1.6, we assume throughout this section that A has even degree.

Let deg(A) = 2m and D = D(A, σ) denote the discriminant algebra of (A, σ) (cf. [KMRT, §10.E]). The algebra D is a central simple algebra over F and carries an involution \overline{σ} of first kind, which is of symplectic type if m is odd and of orthogonal type if m is even [KMRT, Prop. 10.30]. For 1 ≤ i ≤ 2m, let ∧^{i}A be the i^{th} ∧-power of A (cf. [KMRT, §10 (10.4)]). By [KMRT, Prop. 14.3], there is a homogeneous polynomial map ∧^{i} : A → ∧^{i}A of degree i, 1 ≤ i ≤ 2m. If A = End_{F} (V) then ∧^{i}A = End_{F} (Λ^{i}V) and ∧^{i}(f) = Λ^{i}(f), the i^{th} exterior power of the linear map f ∈ End_{F} (V).

Theorem 5.1 Let F be a field with char(F) ≠ 2. Let A be a central simple algebra of degree 2m over a field Z with m odd. Let σ be an involution of second kind on A such that Z^{σ} = F. Let D = D(A, σ) be the discriminant algebra of (A, σ). Then G(A, σ) ⊆ Nrd(D^{∗}).N_{Z/F} (Z^{∗}).

Proof Let x ∈ G(A, σ) and g ∈ Sim(A, σ) be such that μ(g) = σ(g)g = x. Then N_{Z/F} (Nrd(g)) = μ(g)^{2m} and by Hilbert Theorem-90, there exists α ∈ Z^{∗} such that μ(g)^{−m} Nrd(g) = α^{−1}\overline{α}, where bar denotes the non-trivial automorphism of Z over F. By [KMRT, Lemma 14.6], we have

\overline{σ}(α^{−1} ∧^{m} g)α^{−1} ∧^{m} g = N_{Z/F} (α)^{−1}μ(g)^{m}.

Since m is odd, x = μ(g) ∈ G(D, \overline{σ}).N_{Z/F} (Z^{∗}). Thus

G(A, σ) ⊆ G(D, \overline{σ}).N_{Z/F} (Z^{∗}) (∗)

Let y ∈ G(D, \overline{σ}) be arbitrary and h ∈ Sim(D, \overline{σ}) be such that μ(h) = \overline{σ}(h)h = y. Since m is odd, the involution \overline{σ} is of symplectic type and by [KMRT, Prop. 12.23] we have μ(h)^{m} = Nrd(h). Again, since m is odd, we have y = μ(h) ∈ Nrd(D^{∗}).F^{∗2}. Thus

G(D, \overline{σ}) ⊆ Nrd(D^{∗}).F^{∗2} (∗∗)

and combining the inclusions (∗) and (∗∗) above, we get

G(A, σ) ⊆ Nrd(D^{∗}).N_{Z/F} (Z^{∗}).

This completes the proof. □

In this section, from now onwards we assume that vcd(F) ≤ 2, and F has orderings and denote by Ω, the set of orderings on F. A quadratic form q over F is called locally hyperbolic if q is hyperbolic at every real closure F_{v}, v ∈ Ω.

Lemma 5.2 If q is a locally hyperbolic quadratic form of even rank and trivial discriminant over F, Hyp(q) = F^{∗}.

Proof Since q is locally hyperbolic the Clifford algebra C(q) of q is locally split. Thus by 1.8 we have Nrd(C(q)^{∗}) = F^{∗}. Let λ ∈ F^{∗} and let L/F be a finite extension such that λ ∈ N_{L/F} (L^{∗}) and C(q)_{L} is split. Then q_{L} has even dimension, trivial discriminant, trivial Clifford invariant and sgn(q_{L}) = 0. Therefore by [EL, Th. 3], the form q_{L} is hyperbolic and λ ∈ N_{L/F} (L^{∗}) ∈ Hyp(q). □

Proposition 5.3 Let Z = F (\sqrt{δ}) be a quadratic extension. Let A = M_{r}(Z), where r is an even positive integer, support a locally hyperbolic Z/F -involution σ. Then Hyp(A, σ) = F^{∗}.

Proof Let the involution σ be adjoint to a Z/F -hermitian form h. Then the rank of h is r. Let q_{h} be the quadratic form over F given by q_{h}(x) = h(x, x). Then q_{h} ≃ ⟨1, −δ⟩ ⊗ q [S, pp. 349, Remark 1.3], where q is a quadratic form over F of the same rank as that of h, which is even. Therefore q_{h} ∈ I^{2}(F). By Jacobson’s theorem [MH, p. 114], the form h_{M} is hyperbolic if and only if the quadratic form (q_{h})_{M} is hyperbolic. It follows that Hyp(A, σ) = Hyp(q_{h}). Since h is locally hyperbolic, the form q_{h} is locally hyperbolic as well. By 5.2, we have Hyp(q_{h}) = F^{∗}. Thus Hyp(A, σ) = Hyp(q_{h}) = F^{∗}. □

The following is a consequence of 5.3 and 2.6.

Proposition 5.4 Let A be a locally split central simple Z-algebra and σ be a locally hyperbolic Z/F -involution on A. Let exp(A) = 2^{m}. Then Hyp(A, σ) = F^{∗}.

Proof Since A supports an involution σ of second kind with Z^{σ} = F, by [S, Th. 9.5], cores_{Z/F} (A) = 0. Thus by 2.6 we have

F^{∗} = ⟨{N_{L/F} (L^{∗}) : A_{L} is split }⟩.

Let L/F be an extension which splits A. By 5.3, Hyp(A_{L}, σ_{L}) = L^{∗} and taking norm from L/F, we conclude that Hyp(A, σ) = F^{∗}. □

Proposition 5.5 Let A be a central simple algebra over Z of degree 2m, where m is odd. Let σ be a Z/F -involution on A with sgn(σ) = 0. Let D = D(A, σ) be the discriminant algebra of (A, σ). Then Nrd(D^{∗}) ⊆ Hyp(A, σ).F^{∗2}. Further

G(A, σ) = Nrd(D^{∗}).N_{Z/F} (Z^{∗}) = Hyp(A, σ).N_{Z/F} (Z^{∗}).

Proof We first show that Nrd(D^{∗}) ⊆ Hyp(A, σ).F^{∗2}. Assume first that exp(A) is a power of 2. Since deg(A) = 2m with m odd, index(A) = exp(A) = 2 and A = M_{m}(H) for some quaternion algebra H over Z. By [KMRT, §10.4], [KMRT, Prop. 10.30] and the hypothesis that m is odd, it follows that

D ⊗_{F} Z ≃ ∧^{m}(M_{m}(H)) ∼ H^{⊗m} ∼ H.

Thus if M is a finite extension of F such that D_{M} is split, then H_{M} is split and sgn(σ_{M}) = 0. Thus by 5.3, Hyp(A_{M}, σ_{M}) = M^{∗} and taking norms, N_{M/F} (M^{∗}) ⊆ Hyp(A, σ). In view of the classical norm principle for reduced norms, Nrd(D^{∗}) ⊆ Hyp(A, σ).

Now suppose that exp(A) is arbitrary. By [BP1, Lemma 3.3.1], there exists an odd degree extension L/F such that exp(A_{L}) is a power of 2. Let λ ∈ Nrd(D^{∗}). Then λ ∈ Nrd(D_{L}^{∗}). By the previous case, λ ∈ Hyp(A_{L}, σ_{L}). Taking norm from L/F and using the hypothesis that m is odd, we conclude that λ ∈ Hyp(A, σ).F^{∗2}. This proves the first assertion of 5.5. It follows immediately that

Nrd(D^{∗}).N_{Z/F} (Z^{∗}) ⊆ Hyp(A, σ).N_{Z/F} (Z^{∗}) (∗)

From 5.1, it is clear that

G(A, σ) ⊆ Nrd(D^{∗}).N_{Z/F} (Z^{∗}) (∗∗)

and further by 1.1, Hyp(A, σ).N_{Z/F} (Z^{∗}) ⊆ G(A, σ). In view of this and the inclusions (∗) and (∗∗) we conclude that

G(A, σ) = Nrd(D^{∗}).N_{Z/F} (Z^{∗}) = Hyp(A, σ).N_{Z/F} (Z^{∗}).
□

Theorem 5.6 Let F be a field with vcd(F) ≤ 2 and Z be a quadratic extension of F. Let A be a central simple algebra over Z of degree 2m, where m is odd. Let σ be a Z/F -involution on A. Then G(A, σ) = Hyp(A, σ).N_{Z/F} (Z^{∗}).

Proof Let λ ∈ G(A, σ). Let D = D(A, σ) be the discriminant algebra of (A, σ). By 5.1, λ ∈ Nrd(D^{∗}).N_{Z/F} (Z^{∗}). Let λ_{1} ∈ Nrd(D^{∗}) and α ∈ N_{Z/F} (Z^{∗}) be such that λ = λ_{1}α. Since N_{Z/F} (Z^{∗}) ⊆ Hyp(A, σ) ⊆ G(A, σ), it follows that λ_{1} ∈ G(A, σ). Let K = F (\sqrt{−λ_{1}}). Then sgn(σ_{K}) = 0 and by 5.5, Nrd(D_{K}^{∗}) ⊆ Hyp(A_{K}, σ_{K}).K^{∗2}. Further, since λ_{1} ∈ Nrd(D_{K}^{∗}) and λ_{1} ≡ −1 mod K^{∗2}, D_{K} is locally split and by 1.8, Nrd(D_{K}^{∗}) = K^{∗}. Thus Hyp(A_{K}, σ_{K}).K^{∗2} = K^{∗}. Taking norms, we get

λ_{1} ∈ N_{K/F} (K^{∗}) = N_{K/F} (Hyp(A_{K}, σ_{K})) ⊆ Hyp(A, σ).

Thus λ = λ_{1}α ∈ Hyp(A, σ).N_{Z/F} (Z^{∗}). This completes the proof. □

Let Σ(F) denote the set of elements of F which are positive at all orderings of F.

Lemma 5.7 Let α, δ ∈ F^{∗}. Then we have:

F^{∗} = ⟨{N_{L/F} (L^{∗}) :L/F is a quadratic extension such that there exists

u_{L} ∈ L(\sqrt{δ}) with N_{L(\sqrt{δ})/L}(u_{L}) = 1 and αu_{L} ∈ Σ(L(\sqrt{δ}))\}⟩.

Proof Since the quadratic form ϕ = ⟨1, δ, −α, δα⟩ is locally isotropic, by 1.5 and 1.10,

F^{∗} = sn(ϕ) = ⟨{N_{L/F} (L^{∗})⟩ : L is a quadratic extension of F such that ϕ_{L} is isotropic}⟩ (∗)

At an extension L/F where ϕ is isotropic, we choose a, b, c, d ∈ L^{∗} such that a^{2}+δb^{2}−αc^{2}+δαd^{2} = 0. If c^{2}+δd^{2} = 0 or a^{2}+δb^{2} = 0, clearly L(\sqrt{δ}) has no ordering and thus Σ(L(\sqrt{δ})) = L(\sqrt{δ})^{∗}. In this case, we may take u_{L} = 1. Otherwise, we let θ = c+d\sqrt{δ} and u_{L} = θ^{−1}\overline{θ}, where \overline{θ} = c−d\sqrt{δ}. It is immediate that Tr_{L(\sqrt{δ})/L}(u_{L}) = 2(c^{2} + δd^{2})(c^{2} − δd^{2})^{−1} and N_{L(\sqrt{δ})/L}(u_{L}) = 1. Since a^{2} + δb^{2} − αc^{2} + δαd^{2} = 0 and both c^{2} + δd^{2} and c^{2} − δd^{2} are units, it follows that

α = ((a^{2} + δb^{2})(c^{2} + δd^{2})^{−1}) ((c^{2} + δd^{2})(c^{2} − δd^{2})^{−1})

Thus

2αTr_{L(\sqrt{δ})/L}(u_{L}) = ((a^{2} + δb^{2})(c^{2} + δd^{2})^{−1}) (Tr_{L(\sqrt{δ})/L}(u_{L}))^{2} ∈ N_{L(\sqrt{−δ})/L}(L(\sqrt{−δ}))

and hence the quaternion algebra (2αTr_{L(\sqrt{δ})/L}(u_{L}), −δ) over L is split.

Let v be an ordering on L which extends to an ordering w on L(\sqrt{δ}). Then δ >_{v} 0 and hence 2αTr_{L(\sqrt{δ})/L}(u_{L}) >_{v} 0. Let bar denote the non-trivial automorphism of L(\sqrt{δ}) over L. Since αu_{L}\overline{αu_{L}} = α^{2} >_{v} 0, both αu_{L} and \overline{αu_{L}} have same sign at w. But αTr_{L(\sqrt{δ})/L}(u_{L}) = α(u_{L} + \overline{u_{L}}) >_{v} 0. Thus αu_{L} >_{w} 0. This is true for every ordering of L(\sqrt{δ}). Thus αu_{L} ∈ Σ(L(\sqrt{δ})) and N_{L(\sqrt{δ})/L}(u_{L}) = 1. This completes the proof of the lemma. □

Let D be a division algebra with centre Z and τ be an involution on D of the second kind. Let Z^{τ} = F. Let (V, h) be a non-degenerate hermitian space over (D, τ). Then the integer dim_{D}(V) is said to be the rank of h and is denoted by rank(h). Let rank(h) = n. For a choice {e_{1}, e_{2}, ⋯, e_{n}} of a D-basis of V, the form h determines a matrix M_{h} = (h(e_{i}, e_{j})) ∈ M_{n}(D). The matrix M_{h} is τhermitian symmetric. Let r = n deg(D). We define the discriminant of h to be (−1)^{r(r−1)/2} Nrd(M_{h}) ∈ F^{∗}/N_{Z/F} (F^{∗}) and denote it by disc(h).

We refine the notion of discriminant to the notion of Discriminant as follows:

Let M_{h} ∈ M_{n}(D) be a matrix as above, representing the hermitian form h. Let M′_{h} ∈ M_{n}(D) also represent h. Then there exists an invertible matrix T ∈ M_{n}(D) such that

Nrd(M′_{h}) = Nrd(M_{h}) Nrd(T)τ(Nrd(T))

Thus we have the following well defined notion of Discriminant:

Disc(h) = (−1)^{r(r−1)/2} Nrd(M_{h}) ∈ F^{∗}/N_{Z/F} (Nrd(D^{∗}))

where r = n deg(D).

We now quote a classification result for hermitian forms over division algebras with an involution of second kind over fields with vcd(F) ≤ 2.

Theorem 5.8 ([BP2, Theorem 4.8]) Let F be a field with vcd(F) ≤ 2 and D be a division algebra with an involution τ of second kind such that (centre(D))^{τ} = F.

Let h be a hermitian form over (D, τ), Then h is hyperbolic if and only if rank(h) is even, Disc(h) is trivial and h has trivial signature. □

Lemma 5.9 Let D be a central division algebra over Z, τ be a Z/F -involution over D and h be a hermitian of rank 2s over (D, τ). Let disc(h) = 1. Then

F^{∗} = ⟨{N_{M/F} (M^{∗}) : Disc(h_{M}) = 1}⟩.

Proof Let M_{h} ∈ M_{2s}(D) be a matrix representing h. Since disc(h) = 1 ∈ F^{∗}/N_{Z/F} (Z^{∗}), we have Nrd(M_{h}) = d ∈ N_{Z/F} (Z^{∗}). Let z ∈ Z be such that d = N_{Z/F} (z). Let β = Tr_{Z/F} (z) and γ = zβ^{−1}. Let w be an ordering on Z which extends an ordering v of F such that D_{w} is not split. Then Nrd(M_{h}) = d = N_{Z/F} (z) >_{w} 0. Thus Tr_{Z/F} (z) = β >_{w} 0 if and only if z >_{w} 0. This implies that γ = zβ^{−1} >_{w} 0 and thus by 1.8, γ ∈ Nrd(D^{∗}). Let x ∈ D^{∗} be such that Nrd(x) = γ. Let

M′_{h} = ( \begin{matrix}1 \\ & 1 \\ & & . \\ & & & x \end{matrix} ) M_{h} ( \begin{matrix}1 \\ & 1 \\ & & . \\ & & & τ(x) \end{matrix} )^{t}

Then Nrd(M′_{h}) = (dβ^{−1})^{2} and we conclude that for a suitable choice of a matrix M_{h} representing the hermitian form h, Nrd(M_{h}) = α^{2}, α ∈ F^{∗}. Let λ ∈ F^{∗}. By 5.7, there exist quadratic extensions L_{i}/F, λ_{i} ∈ L_{i}^{∗} and u_{i} ∈ L_{i}(\sqrt{δ}), 1 ≤ i ≤ r, such that λ = ∏_{i} N_{L_{i}/F} (λ_{i}), αu_{i} ∈ Σ(L(\sqrt{δ})) and N_{L_{i}(\sqrt{δ})/L}(u_{i}) = 1. Then

α^{2} = N_{L_{i}(\sqrt{δ})/L_{i}}(αu_{i}) ∈ N_{L_{i}(\sqrt{δ})/L_{i}}(Nrd(D_{L_{i}(\sqrt{δ})}))

and hence Disc(h_{L_{i}}) = 1 for 1 ≤ i ≤ r. Thus λ ∈ ⟨{N_{M/F} (M^{∗}) : Disc(h_{M}) = 1}⟩ and we conclude that F^{∗} = ⟨{N_{M/F} (M^{∗}) : Disc(h_{M}) = 1}⟩. □

The following propositions are used in the proof of 5.13, which is the main result of this section.

Proposition 5.10 Let A ≃ M_{r}(D) where D is a division algebra over Z and r is even. Let σ be a locally hyperbolic Z/F -involution on A. Then Hyp(A, σ) = F^{∗}.

Proof Let σ be adjoint to a hermitian form h of rank r. Let d ∈ F^{∗}/N_{Z/F} (Z^{∗}) denote the discriminant of h. Since σ is locally hyperbolic, for each v ∈ Ω, the quaternion algebra (δ, d) splits at F_{v}. Thus by 1.8, Nrd((δ, d)) = F^{∗}. Let λ ∈ F^{∗}. There exists a finite extension E/F such that λ ∈ N_{E/F} (E^{∗}) and (δ, d) splits over E. Then disc(h_{E}) is trivial. By 5.9 we have

E^{∗} = ⟨{N_{M/E}(M^{∗}) : Disc(h_{M}) = 1}⟩ (∗)

Let M/E be an extension such that and Disc(h_{M}) = 1. Since σ is locally hyperbolic, sgn(h_{M}) = 0. Thus by 5.8, the form h_{M} is hyperbolic and Hyp(h_{M}) = M^{∗}. Hence by (∗), Hyp(h_{E}) = E^{∗} and

λ ∈ N_{E/F} (E^{∗}) = N_{E/F} (Hyp(h_{E})) ⊆ Hyp(h) = Hyp(A, σ)

which implies that Hyp(A, σ) = F^{∗}. This completes the proof. □

Proposition 5.11 Let A be a central simple algebra over Z with deg(A) ≡ 0(4). Let exp(A) = 2^{m} for some positive integer m. Let σ be a locally hyperbolic Z/F - involution on A. Then Hyp(A, σ) = F^{∗}.

Proof Suppose m = 1. Since cores_{Z/F} (A) = 0, by 1.12 A ∼ A_{0} ⊗_{F} Z for some central simple F -algebra A_{0} with exp(A_{0}) = 2. Let M be a finite extension of F such that A_{0M} ∼ H for some quaternion algebra H over M. Then A_{M} = M_{r}(H ⊗_{F} Z). Since deg(A) ≡ 0(4), the integer r is even. Thus by 5.10, Hyp(A_{M}, σ_{M}) = M^{∗}. In view of 2.4 we have

F^{∗} = ⟨{N_{M/F} (M^{∗}) : A_{0M} ∼ H for some quaternion algebra H over M}⟩.

It follows that Hyp(A, σ) = F^{∗}.

Suppose m ≥ 2. Since Br(Z_{w}) = ℤ/2ℤ for each ordering w ∈ Ω_{Z}, the algebra A ⊗_{Z} A splits locally. Clearly exp(A ⊗_{Z} A) = 2^{m−1} and cores_{Z/F} (A ⊗_{Z} A) = 0. Let λ ∈ F^{∗}. By 2.6, there exist extensions L_{i}/F, 1 ≤ i ≤ s and λ_{i} ∈ L_{i}^{∗} such that each (A ⊗_{Z} A) ⊗_{F} L_{i} is split and λ = ∏_{i} N_{L_{i}/F} (λ_{i}). Then exp(A ⊗_{F} L_{i}) = 2 for each i and by the case m = 1, λ_{i} ∈ Hyp(A_{L_{i}}, σ_{L_{i}}). Hence

λ = ∏_{i} N_{L_{i}/F} (λ_{i}) ∈ ∏_{i} N_{L_{i}/F} (Hyp(A_{L_{i}}, σ_{L_{i}})) ⊆ Hyp(A, σ)

and it follows that Hyp(A, σ) = F^{∗}. □

Proposition 5.12 Let A be a central simple algebra over Z with deg(A) ≡ 0(4). Let σ be a locally hyperbolic Z/F -involution on A. Then we have Hyp(A, σ).F^{∗2} = F^{∗}.

Proof By [BP1, Lemma 3.3.1], there exists an odd degree extension M of F such that exp(A_{M}) is a power of 2 and by 5.11, Hyp(A_{M}, σ_{M}) = M^{∗}. Taking norm from M/F and using that [M : F] is odd, we conclude that Hyp(A, σ).F^{∗2} = F^{∗}. □

Theorem 5.13 Let F be a field with vcd(F) ≤ 2 and let Z be a quadratic extension over F. Let A be a central simple algebra over Z and σ be a Z/F -involution on A. Then, G(A, σ) ⊆ Hyp(A, σ).N_{Z/F} (Z^{∗}).

Proof The cases where deg(A) is odd or deg(A) ≡ 2(4) are covered by 1.6 and 5.6 respectively. We assume that deg(A) ≡ 0(4). Let λ ∈ G(A, σ). At each v ∈ Ω, the involution σ_{v} is adjoint to an even rank hermitian form which is hyperbolic if and only if sgn(σ_{v}) = 0. Therefore λ >_{v} 0 at those v ∈ Ω, where σ_{v} is not hyperbolic. Let K = F (\sqrt{−λ}). Then deg(A_{K}) ≡ 0(4) and σ_{K} is locally hyperbolic. Thus by 5.12, we have Hyp(A_{K}, σ_{K}).K^{∗2} = K^{∗}. Let \sqrt{−λ} = αβ^{2}, where α ∈ Hyp(A_{K}, σ_{K}) and β ∈ K^{∗}. Then λ = N_{K/F} (\sqrt{−λ}) = N_{K/F} (α)(N_{K/F} (β))^{2}. But N_{K/F} (α) ∈ N_{K/F} (Hyp(A_{K}, σ_{K})) ⊆ Hyp(A, σ). Thus λ ∈ Hyp(A, σ).F^{∗2}. This completes the proof. □

6 Fields with vcd(F) ≤ 2 : Orthogonal groups

Let F be an arbitrary field with char(F) ≠ 2. Let D be a central division algebra over F with an orthogonal involution τ. We first recall from [BP2], certain invariants associated to hermitian forms over (D, τ).

Discriminant: Let D and τ be as above and h be a hermitian form of even rank over (D, τ). Let rank(h) = 2m and let M_{h} ∈ M_{2m}(D) represent the hermitian form h. Let

Disc(h) = (−1)^{r(r−1)/2} Nrd(M_{h}) ∈ F^{∗}/(Nrd(D^{∗}))^{2},

where r = 2m deg(D). If M′_{h} ∈ M_{2m}(D) is another matrix representing h then there exists an invertible matrix T ∈ M_{2m}(D) such that M_{h} = T M_{h}(τ(T)^{t}). Thus Nrd(M′_{h}) = Nrd(M_{h}) Nrd(T)^{2} and Disc(h) is well defined. We call Disc(h) the Discriminant of h.

Clifford invariant: We recall from [KMRT, §8.B], the notion of the Clifford algebra C(A, σ) associated to a central simple algebra A over a field F with an involution σ of orthogonal type. If A is split and σ is adjoint to a quadratic form q then C(A, σ) is the even Clifford algebra C_{0}(q) of the quadratic form q. If disc(σ) is trivial, C(A, σ) decomposes into a product C_{+}(A, σ) × C_{−}(A, σ), each of the factors being a central simple algebra over F such that

[C_{+}(A, σ)] + [C_{−}(A, σ)] = [A] ∈ Br(F).

Let D, τ and h be as above. Let disc(h) be trivial and A = M_{2m}(D). Let τ_{h} be the orthogonal involution on A which is adjoint to h. We define the Clifford invariant of h as follows:

Cℓ(h) = [C_{+}(M_{2m}(D), τ_{h})] ∈ Br(F)/[D]

Let H_{2m} denote the matrix ( \begin{matrix}0 & I_{m} \\ I_{m} & 0 \end{matrix} ) ∈ M_{2m}(D) where I_{m} is the identity

matrix of size m. The matrix H_{2m} represents the hyperbolic form of rank 2m over (D, τ). Let U_{2m}(D, τ), SU_{2m}(D, τ) and Spin_{2m}(D, τ) denote respectively, the unitary, special unitary group and spin group with respect to the hyperbolic form H_{2m} over (D, τ). We have an exact sequence

1 → μ_{2} → Spin_{2m}(D, τ) → SU_{2m}(D, τ) → 1

from which one gets the exact sequence of pointed sets

→ H^{1}(F, Spin_{2m}(D, τ)) → H^{1}(F, SU_{2m}(D, τ)) →^{δ} H^{2}(F, μ_{2})

Let 𝔖 denote the set of ordered pairs (X, a), where X ∈ GL_{2m}(D) and a ∈ F^{∗} satisfy τ(X) = X^{t} and Nrd(X) = Nrd(H_{2m})a^{2}. The elements of H^{1}(F, SU_{2m}(D, τ)) are equivalence classes of 𝔖 under the following equivalence relation: (X, a) ∼ (X′, a′) if and only if there exists Y ∈ GL_{2m}(D) with X′ = Y X\overline{Y}^{t} and a′ = Nrd(Y)a.

Let h be a hermitian form over (D, τ) with rank(h) = 2m and disc(h) = 1. Let M_{h} be a matrix which represents h and Nrd(M_{h}) = a^{2}, a ∈ F^{∗}. The two elements ξ_{a} = (M_{h}, a) and ξ_{−a} = (M_{h}, −a) in H^{1}(F, SU_{2m}(D, τ)) map to [h] under H^{1}(F, SU_{2m}(D, τ)) → H^{1}(F, U_{2m}(D, τ)). Let C_{+}(h) = δ(ξ_{a}) and C_{−}(h) = δ(ξ_{−a}). We recall the following lemma from [BMPS, Lemma 3.1].

Lemma 6.1 If F is a formally real field and v is an ordering on F such that D_{v} is not split, then the algebra C_{+}(h) is split at v if and only if a >_{v} 0. □

Rost invariant: Let h be a hermitian form over (D, τ) with rank(h) = 2m, trivial discriminant and trivial Clifford invariant. Consider the exact sequence

1 → SU_{2m}(D, τ) → U_{2m}(D, τ) → μ_{2} → 1.

This gives rise to the following exact sequence of pointed sets

→ U_{2m}(D, τ)(F) → {±1} → H^{1}(F, SU_{2m}(D, τ)) → H^{1}(F, U_{2m}(D, τ)) →

Since Cℓ(h) = 0, there exists ξ ∈ H^{1}(F, SU_{2m}(D, τ)) mapping to [h] ∈ H^{1}(F, U_{2m}(D, τ)) such that δ(ξ) = 0. Let ξ̃ ∈ H^{1}(F, Spin_{2m}(D, τ)) be a preimage of ξ ∈ H^{1}(F, SU_{2m}(D, τ)). Let G = Spin_{2m}(D, τ) and R_{G} : H^{1}(F, G) → H^{3}(F, ℚ/ℤ(2)) denote the Rost invariant of G [Me3]. The Rost invariant of h is defined as follows: [BP2, pp. 664]

R(h) = R_{G}(ξ̃) ∈ \frac{H^{3}(F, ℚ/ℤ(2))}{F^{∗} ∪ [D]}

The element R_{G}(ξ̃) takes values in H^{3}(F, ℤ/4) [BP2, Remark 1], where ℤ/4 has the trivial Galois module structure. We now recall a proposition which we shall use often.

Proposition 6.2 ([BP2, Cor. 2.6]) Let F be a formally real field and let I^{3}(F) be torsion-free. Let Ω be the set of orderings on F. Then the natural map

H^{3}(F, ℤ/4) → ∏_{v∈Ω} H^{3}(F_{v}, ℤ/4)

is injective. □

We now record a classification result for hermitian forms over central division algebras with orthogonal involutions over fields with vcd(F) ≤ 2.

Theorem 6.3 ([BP2, Th. 7.3]) Let F be a field with vcd(F) ≤ 2 and D be a central division algebra over D with an orthogonal involution τ. Let h be a hermitian form over (D, τ). Then h is hyperbolic if and only if h has even rank, trivial Discriminant, trivial Clifford and Rost invariant and trivial signature. □

Let F be a field with vcd(F) ≤ 2 and (A, σ) be a central simple algebra over F with orthogonal involution. If A is split, deg(A) is even, σ is locally hyperbolic and disc(σ) = 1, so that by 5.2 we have Hyp(A, σ) = F^{∗}. We now consider the case where A is locally split.

Lemma 6.4 Let vcd(F) ≤ 2 and A be a central simple algebra of even degree over F. Let σ be an orthogonal involution on A. If A is locally split then

G_{+}(A, σ) = Hyp(A, σ).F^{∗2}.

Proof Let λ ∈ G_{+}(A, σ) and K = F (\sqrt{−λ}). Clearly λ ∈ N_{K/F} (K^{∗}). Let disc(σ) = d and L = F (\sqrt{d}). By 1.3, λ ∈ N_{L/F} (L^{∗}). Let M = F (\sqrt{−λ}, \sqrt{d}). Using [W, Lemma 2.14] for the biquadratic extension M/F, there exist x ∈ M^{∗} and y ∈ F^{∗} such that λ = N_{M/F} (x)y^{2}. Further A_{M} is locally split and by 1.8, Nrd(A_{M}) = M^{∗}. Let E/M be an extension such that x = N_{E/M}(α) for some α ∈ E^{∗} and A_{E} be split. Clearly disc(σ_{E}) = 1, σ_{E} is locally hyperbolic and A_{E} is split. Thus by 5.2, Hyp(A_{E}, σ_{E}) = E^{∗} and hence x = N_{E/M}(α) ∈ Hyp(A_{M}, σ_{M}). Thus λ = N_{M/F} (x)y^{2} ⊆ Hyp(A, σ).F^{∗2}. We conclude that G_{+}(A, σ) ⊆ Hyp(A, σ).F^{∗2}. In view of 1.1 we have G_{+}(A, σ) = Hyp(A, σ).F^{∗2}. □

We continue with some lemmas which will be used in the proofs of the main results of this section.

Lemma 6.5 Let vcd(F) ≤ 2 and χ ∈ H^{3}(F, μ_{2}). Then

F^{∗} = ⟨{N_{L/F} (L^{∗}) : L ∈ F_{2}(F) and χ_{L} = (−1) ∪ (−1) ∪ (−x) for some x ∈ L^{∗}}⟩.

Proof Since vcd(F) ≤ 2, H^{3}(F (\sqrt{−1}), μ_{2}) = 0 and in view of the Arason exact sequence 1.11, the map H^{2}(F, μ_{2}) ⟶^{∪(−1)} H^{3}(F, μ_{2}) is surjective. Let ξ ∈ H^{2}(F, μ_{2}) be such that (−1) ∪ ξ = χ. Let D_{ξ} be a central division algebra over F, whose Brauer class is represented by ξ. Then exp(D_{ξ}) = 2. Let L ∈ F_{2}(F) be such that (D_{ξ})_{L} ∼ (−1) ∪ (−x) for some x ∈ L. Then

χ_{L} = (−1) ∪ ξ_{L} = (−1) ∪ (D_{ξ})_{L} = (−1) ∪ (−1) ∪ (−x).

In view of this and 2.4, we have that

F^{∗} = ⟨{N_{L/F} (L^{∗}) : L ∈ F_{2}(F) and (D_{ξ})_{L} = (−1) ∪ (−x) for some x ∈ L^{∗}}⟩

⊆ ⟨{N_{L/F} (L^{∗}) : L ∈ F_{2}(F) and χ_{L} = (−1) ∪ (−1) ∪ (−x) for some x ∈ L^{∗}}⟩

□

For χ ∈ H^{r}(F, μ_{2}), we set N(χ) = ⟨{N_{L/F} (L^{∗}) : χ_{L} = 0}⟩.

Lemma 6.6 Let vcd(F) ≤ 2 and χ ∈ H^{r}(F, μ_{2}), r ≥ 2. Then following three groups coincide:

(i) N(χ)

(ii) {λ ∈ F^{∗} : λ >_{v} 0 at those v ∈ Ω where χ_{v} ≠ 0}.

(iii) {λ ∈ F^{∗} : (λ) ∪ χ = 0}.

Proof Since vcd(F) ≤ 2, in view of 1.15 the cohomology groups H^{r+1}(F, μ_{2}) are (−1)-torsion-free for r ≥ 2 and thus the groups (ii) and (iii) coincide. We show that N(χ) ⊆ {λ ∈ F^{∗} : (λ) ∪ χ = 0}. Let λ ∈ N(χ) be such that λ = N_{L/F} (μ) for an extension L/F with χ_{L} = 0. Then ((μ) ∪ χ)_{L} = 0 and thus we have

cores_{L/F} ((μ) ∪ χ)_{L} = (λ) ∪ χ = 0.

Hence N(χ) ⊆ {λ ∈ F^{∗} : (λ) ∪ χ = 0}. To complete the proof, we show that {λ ∈ F^{∗} : λ >_{v} 0 at those v ∈ Ω where χ_{v} ≠ 0} ⊆ N(χ). Let λ ∈ F^{∗} be such that λ >_{v} 0 at those v ∈ Ω where χ_{v} ≠ 0. Let L = F (\sqrt{−λ}). Then it follows that χ_{w} = 0 for each ordering w of L. It follows from [Ar, Th. 2.1] that vcd(L) ≤ 2 and thus by 1.15, H^{3}(L, μ_{2}) is (−1)-torsion-free. Therefore χ_{L} = 0. Thus λ = N_{L/F} (\sqrt{−λ}) ∈ N(χ). This completes the proof. □

In 6.7, 6.8 and 6.9 below, the only restriction on F is that char(F) ≠ 2. Let D be a central division algebra over F and τ be an orthogonal involution on D. Let h be a hermitian form of rank 2m and trivial discriminant over (D, τ). Let a ∈ F^{∗} be such that Nrd(M_{h}) = a^{2}, where M_{h} is a matrix representing the form h. Since disc(h) = 1, we recall from [MT, Prop. 1.12] that G_{+}(h) = G(h).

Lemma 6.7 Let D be a central division algebra over a field F of characteristic different from 2 with an orthogonal involution τ. Let h and h′ be two even rank hermitian forms of trivial discriminant over (D, τ). Then we have the following additive property for Clifford invariants:

Cℓ(h ⊥ h′) = Cℓ(h) + Cℓ(h′) ∈ H^{2}(F, μ_{2})/[D].

Proof We extend the scalars to the function field of the Brauer-Severi variety of D. Using the fact that the invariant e_{2} of quadratic forms is additive on forms of trivial discriminant and that the kernel of the scalar extension map H^{2}(F, μ_{2}) → H^{2}(F (X_{D}), μ_{2}) is generated by the class of D in H^{2}(F, μ_{2}) [MT, Cor. 2.7], the lemma follows. □

From this lemma and the fact that two similar hermitian forms with even rank and trivial discriminant have the same Clifford invariants [BP1, pp. 204], we immediately have

Corollary 6.8 Let D, τ and h be as in 6.7. Then for each λ ∈ F^{∗}, the Clifford invariant Cℓ(h ⊥ −λh) is trivial. □

In the following lemma, we compute the Rost invariant of the hermitian form h ⊥ −λh, where h is as in 6.7 and λ ∈ F^{∗} is an arbitrary scalar.

Lemma 6.9 Let D be a central division algebra of even degree over a field F of characteristic different from 2. Let τ be an orthogonal involution on D and h be a hermitian form over (D, τ) of even rank and trivial discriminant. Let λ ∈ F^{∗}. Then,

R(h ⊥ −λh) = (λ) ∪ [C_{+}(h)] ∈ H^{3}(F, ℚ/ℤ(2))/F^{∗} ∪ [D].

Proof Let rank(h) = 2m and A = M_{2m}(D). Let τ_{h} be the involution on A which is adjoint to h. We denote by PGO_{+}(h) the group PSim_{+}(A, τ_{h}) of similitudes. We have an exact sequence

1 → μ_{2} → SU(h) → PGO_{+}(h) → 1

which induces a map on the cohomology sets H^{1}(F, μ_{2}) → H^{1}(F, SU(h)). We claim that under this map (λ) ∈ H^{1}(F, μ_{2}) is mapped to an element ξ_{λ} ∈ H^{1}(F, SU(h)) which corresponds to the class of the hermitian form λh in H^{1}(F, U(h)). In fact, the cocycle (λ) ∈ Z^{1}(F, μ_{2}) given by s ↦ s(\sqrt{λ})(\sqrt{λ})^{−1} for s ∈ Gal(F_{s}/F), when treated as a cocycle with values in U(h), represents [λh] in H^{1}(F, U(h)).

Since deg(A) ≡ 0(4), the centre of Spin(h) is μ_{2} × μ_{2} and the kernel of the map Spin(h) → SU(h) is (ϵ, ϵ), where ϵ = ±1. The quotient of μ_{2} × μ_{2} by μ_{2} under the diagonal embedding maps isomorphically onto the centre of SU(h). By [MPT, Th. 1.14], the Rost invariant of the image ξ̃_{λ} of (1, λ) ∈ H^{1}(F, μ_{2} ×μ_{2}) in H^{1}(F, Spin(h)) is (λ) ∪ [C_{+}(h)]. Thus ξ̃_{λ} ∈ H^{1}(F, Spin(h)) maps to ξ_{λ} ∈ H^{1}(F, SU(h)), which in turn maps to the class of λh in H^{1}(F, U(h)) as is seen above. Thus we conclude that the hermitian form λh admits a lift ξ̃_{λ} such that R(ξ̃_{λ}) = (λ) ∪ [C_{+}(h)].

We now compute R(h ⊥ −λh). Let i : Spin(−h) → Spin(−h ⊥ h) be the natural map and ĩ: H^{1}(F, Spin(−h)) → H^{1}(F, Spin(−h ⊥ h)) the induced map on the cohomology sets. In view of [BP2, Lemma 3.6], R(ĩ(ξ)) = R(ξ) for every ξ ∈ H^{1}(F, Spin(−h)). The group Spin(−h ⊥ h) maps isomorphically onto Spin_{4m}(D, τ) preserving the Rost invariant. Further, the image of (1, λ) in the cohomology set H^{1}(F, Spin(−h)) maps to the isometry class of −λh in H^{1}(F, U(h)) and to the isometry class of −λh ⊥ h in H^{1}(F, U_{4n}(D, τ)). This implies that the Rost invariant R(h ⊥ −λh) is equal to (λ) ∪ [C_{+}(h)] ∈ H^{3}(F, ℚ/ℤ(2))/F^{∗} ∪ [D]. This completes the proof. □

From now on, we assume that vcd(F) ≤ 2. Let L/F be a formally real extension and Ω_{L} be the set of orderings on L. Let h be a hermitian form over (D, τ) - a central division algebra D over F with an orthogonal involution τ. We define S_{ℓ,L}(h) as follows:

S_{ℓ,L}(h) = {λ ∈ L^{∗} : h_{L_{w}} ≃ λh_{L_{w}} for all w ∈ Ω_{L}}.

If L = F then we simply write S_{ℓ}(h) to denote S_{ℓ,F} (h). In 6.10 - 6.14 below, h denotes an even rank hermitian form over (D, τ) with trivial discriminant and a ∈ F^{∗} denotes a scalar which satisfies Nrd(M_{h}) = a^{2} for a choice M_{h} of a matrix representing h. Further, for z ∈ F^{∗} and a central simple algebra B over F with exp(B) = 2, we denote by z ∪ B the element (z) ∪ [B] ∈ H^{3}(F, μ_{2}).

Proposition 6.10 We have (N(a ∪ D)N(−a ∪ D)) ∩ S_{ℓ}(h).

Proof We first prove that (N(a ∪ D)N(−a ∪ D)) ∩ S_{ℓ}(h) ⊆ G(h). Let λ ∈ (N(a ∪ D)N(−a ∪ D)) ∩ S_{ℓ}(h). We show that h ⊥ −λh is hyperbolic. It is clear that h ⊥ −λh has even rank. Further Nrd(M(h ⊥ −λh)) = a^{4}, and since a^{2} is totally positive, by 1.8 it belongs to Nrd(D^{∗}). Thus it follows that h ⊥ −λh has trivial Discriminant. Moreover since disc(h) is trivial, by 6.8, it follows that the Clifford invariant of h ⊥ −λh is trivial. Since λ ∈ S_{ℓ}(h), the form h ⊥ −λh has trivial signature as well.

By 6.9 we see that the Rost invariant R(h ⊥ −λh) = [(λ) ∪ C_{+}(h)]. We show that [(λ) ∪ C_{+}(h)] is trivial in H^{3}(F, ℤ/4)/F^{∗} ∪ [D]. Let x ∈ F^{∗} be such that x ∈ N(−a ∪ D) and λx^{−1} ∈ N(a ∪ D). We claim that (λ) ∪ [C_{+}(h)] = (x) ∪ [D]. In view of 6.2, it suffices to check that at each v ∈ Ω, we have (λ)∪[C_{+}(h)_{v}] = (x)∪[D_{v}].

Suppose v ∈ Ω is such that λ >_{v} 0 and x >_{v} 0. In this case (λ) ∪ [C_{+}(h)_{v}] and (x) ∪ [D_{v}] are both trivial.

Suppose λ >_{v} 0 and x <_{v} 0. Then λx^{−1} <_{v} 0. Since λx^{−1} ∈ N(a ∪ D) and x ∈ N(−a ∪ D), in view of 6.6 both (a) ∪ [D] and (−a) ∪ [D] are split at v. Thus −1 ∈ Nrd(D_{v}) and hence D is split at v. Thus both (λ) ∪ [C_{+}(h)_{v}] and (x) ∪ [D_{v}] are trivial in this case as well.

Now suppose that λ <_{v} 0 and D_{v} is split. Since λ ∈ S_{l}(h), we conclude that h_{v} is hyperbolic. Thus the Clifford invariant Cℓ(h)_{v} = 0. Further, since D_{v} is split and h_{v} is hyperbolic, we have C_{+}(h)_{v} = C_{−}(h)_{v} = 0. Thus we conclude that C_{+}(h)_{v} is split and thus (λ) ∪ [C_{+}(h)_{v}] and (x) ∪ [D_{v}] are both zero.

Next, suppose that λ <_{v} 0, D_{v} is not split and x <_{v} 0. Since x ∈ N(−a ∪ D), by 6.6 (−a) ∪ [D_{v}] = 0; i.e. −a ∈ Nrd(D_{v}). Hence a <_{v} 0. Since D_{v} is not split and a <_{v} 0, by 6.1 C_{+}(h)_{v} is not split. Thus we conclude in this case that both (λ) ∪ [C_{+}(h)_{v}] and (x) ∪ [D_{v}] are non-zero and hence equal.

Now the only remaining case is when λ <_{v} 0, D_{v} is not split and x >_{v} 0. In that case, λx^{−1} <_{v} 0 and since λx^{−1} ∈ N(a ∪ D), by 6.6 we have that (a) ∪ [D_{v}] = 0. Thus a ∈ Nrd(D_{v}) and hence a >_{v} 0. Since D_{v} is non-split, by 6.1 C_{+}(h)_{v} is split. Thus both (λ) ∪ [C_{+}(h)_{v}] and (x) ∪ [D_{v}] are zero in this case.

We conclude therefore that (λ) ∪ [C_{+}(h)_{v}] = (x) ∪ [D_{v}] for all v ∈ Ω. Thus by 6.2, we have (λ) ∪ [C_{+}(h)] = (x) ∪ [D] and

R(h ⊥ −λh) = (λ) ∪ [C_{+}(h)] = 0 ∈ H^{3}(F, ℤ/4)/F^{∗} ∪ [D].

Since vcd(F) ≤ 2, by 6.3 we have that h ⊥ −λh is hyperbolic. Thus λ ∈ G(h).

We now show the inclusion G(h) ⊆ (N(a ∪ D)N(−a ∪ D)) ∩ S_{ℓ}(h). It is clear that G(h) ⊆ S_{ℓ}(h). We thus show that G(h) ⊆ N(a ∪ D)N(−a ∪ D). Let λ ∈ G(h). Then the the form h ⊥ −λh is hyperbolic and hence its Rost invariant (λ) ∪ [C_{+}(h)] is trivial. Thus there exists x ∈ F^{∗} such that (λ) ∪ [C_{+}(h)] = (x) ∪ [D]. By reading this equality locally at each v ∈ Ω and observing the sign pattern, we conclude that x ∈ N(−a ∪ D) and λx ∈ N(a ∪ D). Therefore, λ ∈ N(a ∪ D)N(−a ∪ D). This completes the proof. □

The following lemma will be used in the proof of 6.12.

Lemma 6.11 Let D be a central division algebra over F and let τ be an orthogonal involution on D. Let h be an even rank locally hyperbolic hermitian form over (D, τ) with Disc(h) = 1 and Cℓ(h) = 0. Then Hyp(h) = F^{∗}.

Proof Since the hermitian form h has even rank, trivial discriminant and trivial Clifford invariant, there exists ξ̃ ∈ H^{1}(F, Spin_{2m}(D, τ)) which maps to [h] ∈ H^{1}(F, U_{2m}(D, τ)). Let R(ξ̃) ∈ H^{3}(F, μ_{2}) be the Rost invariant of ξ̃. Let L ∈ F_{2}(F) be such that R(ξ̃_{L}) = (−1) ∪ (−1) ∪ (−x) for some x ∈ L^{∗}. We claim that R(ξ̃_{L}) = (−x) ∪ D_{L}.

Let Ω_{L} be the set of orderings on L and w ∈ Ω_{L} be such that D_{Lw} is split. Then R(ξ̃_{Lw}) = e_{3}(h_{Lw}), where e_{3} is the Arason invariant of quadratic forms. Since h_{w} is hyperbolic by hypothesis, we have e_{3}(h_{Lw}) = 0. Thus R(ξ̃_{L}) and (−x) ∪ D_{L} are both zero at w.

Now suppose D_{Lw} is not split. Then D_{Lw} = (−1) ∪ (−1) and thus R(ξ̃_{Lw}) = (−1) ∪ (−1) ∪ (−x) = (−x) ∪ D_{Lw}. Thus R(ξ̃_{L}) = (−x) ∪ D_{L} at each w ∈ Ω_{L} and by 6.2, R(h_{L}) = 0. Therefore h_{L} is a locally hyperbolic form with even rank, trivial Discriminant, trivial Clifford invariant and trivial Rost invariant. By 6.3 the form h_{L} is hyperbolic. In view of this and 6.5 we conclude that Hyp(h) = F^{∗}. □

The following proposition gives an explicit description of the group Hyp(h).

Proposition 6.12 We have Hyp(h) = (N(a ∪ D) ∩ S_{ℓ}(h)).(N(−a ∪ D) ∩ S_{ℓ}(h)).

Proof We first prove that Hyp(h) ⊆ (N(a ∪ D) ∩ S_{ℓ}(h)).(N(−a ∪ D) ∩ S_{ℓ}(h)). Let L/F be a finite extension such that h_{L} is hyperbolic. Then Disc(h_{L}) is trivial in L^{∗}/ Nrd(D_{L}^{∗})^{2} and hence either a ∈ Nrd(D_{L}^{∗}) or −a ∈ Nrd(D_{L}^{∗}); i.e. either N(a ∪ D_{L}) = L^{∗} or N(−a ∪ D_{L}) = L^{∗}. We clearly have S_{ℓ,L}(h) = L^{∗}. Thus

(N(a ∪ D_{L}) ∩ S_{ℓ,L}(h)).(N(−a ∪ D_{L}) ∩ S_{ℓ,L}(h)) = L^{∗}.

Clearly N_{L/F} (N(a∪D_{L})) ⊆ N(a∪D) and N_{L/F} (N(−a∪D_{L})) ⊆ N(−a∪D). Further as in [KMRT, Prop. 12.21], N_{L/F} (S_{ℓ,L}(h)) ⊆ S_{ℓ}(h). Thus

N_{L/F} (L^{∗}) ⊆ N_{L/F} ((N(a ∪ D_{L}) ∩ S_{ℓ,L}(h)).(N(−a ∪ D_{L}) ∩ S_{ℓ,L}(h)))
⊆ (N(a ∪ D) ∩ S_{ℓ}(h)).(N(−a ∪ D) ∩ S_{ℓ}(h)).

Since N_{L/F} (L^{∗}) generate Hyp(h) as L runs over extensions where h is hyperbolic, it follows that Hyp(h) ⊆ (N(a ∪ D) ∩ S_{ℓ}(h)).(N(−a ∪ D) ∩ S_{ℓ}(h)).

To complete the proof, we show that N(a ∪ D) ∩ S_{ℓ}(h) ⊆ Hyp(h). The inclusion N(−a ∪ D) ∩ S_{ℓ}(h) ⊆ Hyp(h) follows in the similar manner. Let λ ∈ N(a ∪ D) ∩ S_{ℓ}(h). By 6.10, λ ∈ G(h). Let K = F (\sqrt{−λ}). Since λ ∈ N(a ∪ D), by 6.6 (λ) ∪ (a) ∪ [D] = 0 ∈ H^{4}(F, μ_{2}). Thus (−1) ∪ (a) ∪ [D_{K}] = 0 ∈ H^{4}(K, μ_{2}). By 1.15,

(a) ∪ [D_{K}] = 0 ∈ H^{3}(K, μ_{2}). Hence a ∈ Nrd(D_{K}) and Disc(h)_{K} = 1.

Let w be an ordering on K. Since λ <_{w} 0 and λ ∈ G(h_{K}), the form h_{K} is locally hyperbolic. Thus the Clifford invariant Cℓ(h)_{K} is trivial at w. Therefore, if D_{Kw} is split, then C_{+}(h)_{Kw} = C_{−}(h)_{Kw} = 0. If D_{Kw} is not split, then in view of 1.8, a >_{w} 0 as a ∈ Nrd(D_{K}). By 6.1, C_{+}(h)_{Kw} is split. We have thus shown that C_{+}(h_{K}) is locally split. By 1.8, it follows that Nrd(C_{+}(h_{K})) = K^{∗}. Let L/K be a finite extension and α ∈ L^{∗} be such that \sqrt{−λ} = N_{L/K}(α) and C_{+}(h_{L}) = 0. Then h_{L} is an even rank locally hyperbolic form with Disc(h_{L}) = 1 and C_{+}(h_{L}) = 0. By 6.11, Hyp(h_{L}) = L^{∗}. Thus

\sqrt{−λ} = N_{L/K}(α) ∈ N_{L/K}(Hyp(h_{L})) ⊆ Hyp(h_{K}).

Taking norm from K/F we have λ ∈ Hyp(h). Thus N(a ∪ D) ∩ S_{ℓ}(h) ⊆ Hyp(h). □

With the notation as above, we have following corollaries.

Corollary 6.13 If h is locally hyperbolic then

Hyp(h) = N(a ∪ D).N(−a ∪ D) = G(h).

Proof Since h is locally hyperbolic, S_{ℓ}(h) = F^{∗}. From 6.10 and 6.12, it is clear that Hyp(h) = N(a ∪ D).N(−a ∪ D) = G(h). □

Corollary 6.14 If h has trivial Discriminant, then Hyp(h) = S_{ℓ}(h) = G(h).

Proof Since Disc(h) = 1, it follows that either N(a∪D) = F^{∗} or N(−a∪D) = F^{∗}. In either case, it is immediate from 6.10 and 6.12, that Hyp(h) = S_{ℓ}(h) = G(h). □

Let A be a central simple algebra over F with an orthogonal involution σ. Suppose disc(σ) = 1 and C(A, σ) = C_{+}(A, σ) × C_{−}(A, σ). We have the following extension of 6.13 Proposition 6.15 Let F be a field with char(F) ≠ 2. Let (A, σ) be a central simple algebra of even degree over F with an orthogonal involution. Let deg(A) ≡ 0(4) and disc(σ) = 1. Then Hyp(A, σ) ⊆ Nrd(C_{+}(A, σ)) Nrd(C_{−}(A, σ)). Further if vcd(F) ≤ 2 and σ is locally hyperbolic then

Hyp(A, σ) = Nrd(C_{+}(A, σ)) Nrd(C_{−}(A, σ)) = G(A, σ).

Proof The first assertion follows from the fact that over any extension L/F where σ is hyperbolic, either C_{+}(A_{L}, σ_{L}) or C_{−}(A_{L}, σ_{L}) is split [KMRT, Prop. 12.21].

Suppose vcd(F) ≤ 2 and σ locally hyperbolic. Let λ ∈ Nrd(C_{+}(A, σ)). Let L/F be a finite extension such that λ ∈ N_{L/F} (L^{∗}) and C_{+}(A_{L}, σ_{L}) is split. We show that Hyp(A_{L}, σ_{L}) = L^{∗}. In view of 2.3, replacing L by a quadratic tower, we may assume that A_{L} ≃ M_{2r}(H) for some quaternion algebra H over L. Let τ be an orthogonal involution on H and h be a hermitian form over (H, τ) such that σ_{L} is adjoint to h. Let M_{h} ∈ M_{2r}(H) represent h and Nrd(M_{h}) = a^{2} for some a ∈ L^{∗}. Let w be an ordering on L such that H_{w} is not split. Since C_{+}(h) = C_{+}(A_{L}, σ_{L}) is split, by 6.1 a >_{w} 0. Thus (a) ∪ [H] = 0 ∈ H^{3}(L, μ_{2}) and N(a ∪ H) = L^{∗}. In view of 6.13, Hyp(A_{L}, σ_{L}) = Hyp(h) = N(a ∪ H).N(−a ∪ H) = L^{∗}. Thus Hyp(A_{L}, σ_{L}) = L^{∗}. Taking norms from L/F we have λ ∈ N_{L/F} (L^{∗}) = N_{L/F} (Hyp(A_{L}, σ_{L})) ⊆ Hyp(A, σ). Thus Nrd(C_{+}(A, σ)) ⊆ Hyp(A, σ).

The inclusion Nrd(C_{−}(A, σ)) ⊆ Hyp(A, σ) follows from a similar argument. We therefore conclude that Hyp(A, σ) = Nrd(C_{+}(A, σ)) Nrd(C_{−}(A, σ))

To complete the proof we show that G(A, σ) ⊆ Nrd(C_{+}(A, σ)) Nrd(C_{−}(A, σ)). Let λ ∈ G(A, σ). Then the hermitian form ⟨1, −λ⟩ is hyperbolic. Hence the Rost invariant R(⟨1, −λ⟩) is trivial. As in the proof of 6.9, R(⟨1, −λ⟩) = (λ) ∪ [C_{+}(A, σ)]. Since the Rost invariant is trivial, there exists x ∈ F^{∗} such that (λ) ∪ [C_{+}(A, σ)] = (x) ∪ [A]. If for an ordering v on F, the algebra A_{v} is split, then h_{v} being hyperbolic, C_{+}(A, σ)_{v} and C_{+}(A, σ)_{v} are both split. If A_{v} is not split and x <_{v} 0 then C_{+}(A, σ)_{v} is not split. Hence C_{−}(A, σ)_{v} is split. Thus x ∈ Nrd(C_{−}(A, σ)) and a similar argument gives λx ∈ Nrd(C_{+}(A, σ)). Hence λ = λx.x^{−1} ∈ Nrd(C_{+}(A, σ)) Nrd(C_{+}(A, σ)). We have thus shown that

G(A, σ) ⊆ Hyp(A, σ) = Nrd(C_{+}(A, σ)) Nrd(C_{−}(A, σ)).

The inclusion Hyp(A, σ) ⊆ G(A, σ) follows from 1.1 and this completes the proof. □

Theorem 6.16 Let vcd(F) ≤ 2 and let A be a central simple algebra over F with deg(A) even and an involution σ of orthogonal type. If disc(σ) = 1 and σ is locally hyperbolic then G(A, σ) = Hyp(A, σ).F^{∗2}.

Proof Let deg(A) = 2n. Suppose n is odd. Since σ is locally hyperbolic, the algebra A is locally split and by 6.4 the results holds. We can thus assume that n is even. In this case we are through by 6.15. □

7 Fields with vcd(F) ≤ 2 satisfying SAP

Let F be a field with orderings and let Ω denote the set of orderings on F. Given a ∈ F^{∗}, we define the corresponding Harrison set Ω_{a} as follows:

Ω_{a} := {v ∈ Ω : a >_{v} 0}.

The set Ω has Harrison topology for which {Ω_{a} : a ∈ F^{∗}} is a sub-basis. With this topology, Ω is a Hausdorff, compact and totally disconnected space. We say that F has strong approximation property (SAP), if every closed and open set of Ω is of the form Ω_{a} for some a ∈ F^{∗}. A quadratic form q is said to be weakly isotropic, if for some positive integer s, the s-fold orthogonal sum s.q =⊥_{s} q is isotropic. Combining [ELP, Th. C] and [P, Satz. 3.1] we have the following

Theorem 7.1 A field F with orderings has SAP if and only if for every a, b ∈ F^{∗}, the quadratic form ⟨1, a, b, −ab⟩ is weakly isotropic. □

In what follows, for a_{1}, a_{2}, ⋯, a_{r} ∈ F^{∗} the notation ⟨⟨a_{1}, a_{2}, ⋯, a_{r}⟩⟩ will denote the r-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⟨1, −a_{2}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{r}⟩. For a quadratic form q, we denote by D(q) the set of elements of F^{∗} represented by q. We remark that if q = ⟨1, −a, −b, ab⟩ then D(q) = Nrd(H^{∗}), where H is the quaternion algebra (a, b) over F. Set Ω(H) = {v ∈ Ω : H ⊗_{F} F_{v} is split }. The following lemma is recorded in [Ga].

Lemma 7.2 Let F be a field with orderings. Let a, b ∈ F^{∗}. Let q_{1} = ⟨⟨ − 1, −a⟩⟩, q_{2} = ⟨⟨ − 1, a⟩⟩ and H = (a, b). Suppose there does not exist c ∈ F^{∗} such that Ω_{c} = Ω(H), then −b ∉ D(q_{1})D(q_{2}).

Proof Suppose −b ∈ D(q_{1})D(q_{2}) and let x_{1} ∈ D(q_{1}) and x_{2} ∈ D(q_{2}) be such that −b = x_{1}x_{2}. Then q_{1} ⊥ bq_{2} is isotropic and hence 2⟨1, a, b, −ab⟩ ≃ q_{0} ⊥ ℍ for some Albert form q_{0} and ℍ ≃ ⟨1, −1⟩. We have

C(q_{0}) = C(q_{1} ⊥ bq_{2}) = (−1, −1) = C(4⟨1⟩ ⊥ ℍ) ∈ Br(F).

Therefore by [KMRT, Prop. 16.3], q_{0} ≃ 4⟨c⟩ ⊥ ℍ for some c ∈ F^{∗}. It is easy to see that Ω_{c} = Ω(H), which contradicts the hypothesis. Thus −b ∉ D(q_{1})D(q_{2}). □

Lemma 7.3 Let F be a field for which I^{3}(F) is torsion-free. Let a, b ∈ F^{∗}. Let q_{1} = ⟨⟨ − 1, −a⟩⟩, q_{2} = ⟨⟨ − 1, a⟩⟩ and H = (a, b). If −b ∉ D(q_{1})D(q_{2}) then there is no element c ∈ F^{∗} with Ω_{c} = Ω(H).

Proof Suppose there is an element c ∈ F^{∗} such that Ω_{c} = Ω(H). Let q′ = ⟨1, a, b, −ab, −c, −c⟩. For v ∈ Ω if c <_{v} 0, then by the choice of c we have a <_{v} 0 and b <_{v} 0. This implies that the form q′ is hyperbolic at v. If c >_{v} 0, then again by the choice of c, either a >_{v} 0 or b >_{v} 0 and in either case q′ is hyperbolic at v. We thus conclude that q′ is locally hyperbolic. Clearly q′ ∈ I^{2}(F), therefore 2q′ ∈ I^{3}(F). Since 2q′ is an even rank quadratic form with trivial signature, it is hyperbolic at each F_{v}, v ∈ Ω. Thus by Pfister’s local-global principle [L, Th. VIII.4.1], 2q′ is a torsion element in the Witt group W (F). By the hypothesis, I^{3}(F) is torsionfree. Thus 2q′ = 2⟨1, a, b, −ab, −c, −c⟩ is hyperbolic. Therefore the form q_{1} ⊥ bq_{2} is isotropic, which implies that −b ∈ D(q_{1})D(q_{2}). This is a contradiction to the hypothesis. □

Combining 7.2 and 7.3 above, we get the following

Corollary 7.4 Let I^{3}(F) be torsion-free. Let a, b ∈ F^{∗} and q_{1} = ⟨⟨ − 1, −a⟩⟩, q_{2} = ⟨⟨ − 1, a⟩⟩ and H = (a, b). Then −b ∈ D(q_{1})D(q_{2}) if and only if there exists c ∈ F^{∗} such that Ω_{c} = Ω(H). □

Using the results above, we have thus derived

Corollary 7.5 Let F be a field with I^{3}(F) torsion-free. Then the following statements are equivalent:

(i) For all a ∈ F^{∗} we have D (⟨⟨ − 1, −a⟩⟩) D (⟨⟨ − 1, a⟩⟩) = F^{∗}.

(ii) The field F has SAP.

(iii) Given a quaternion algebra H = (a, b) over F, there exists an element c ∈ F^{∗} such that Ω_{c} = Ω(H). □

From now on, in this section the field F satisfies vcd(F) ≤ 2. We say that a field F satisfies SAP for quadratic towers if each quadratic tower L ∈ F_{2}(F) has SAP.

Lemma 7.6 Let vcd(F) ≤ 2, F has orderings and has SAP for quadratic towers. Then for every a, b ∈ F^{∗} we have Nrd(a, b)^{∗}. Nrd(−a, b)^{∗} = F^{∗}.

Proof Let H_{1} = (a, b) and H_{2} = (−a, b). Let L ∈ F_{2}(F) be such that (a, b)_{L} = (−1, −x) and (−a, b)_{L} = (−1, −y) for some x, y ∈ L^{∗}. It is clear that at a given w ∈ Ω_{L} at least one of H_{1L_{w}} and H_{2L_{w}} is split, which means either x <_{w} 0 or y <_{w} 0. Thus if −x <_{w} 0 then y <_{w} 0. In other words, if (−1, −x)_{L_{w}} is non-split then so is (−1, y)_{L_{w}}. Thus by 1.8, we have Nrd(−1, y)^{∗} ⊆ Nrd(−1, −x)^{∗}. Thus Nrd(−1, y)^{∗} Nrd(−1, −y)^{∗} ⊆ Nrd(−1, −x)^{∗} Nrd(−1, −y)^{∗}. Since F has SAP for quadratic towers, L has SAP and thus by 7.5, we have Nrd(−1, −y)^{∗} Nrd(−1, y)^{∗} = L^{∗}. Therefore,

Nrd(H_{1L})^{∗} Nrd(H_{2L})^{∗} = Nrd(−1, −x)^{∗} Nrd(−1, −y)^{∗} = L^{∗}.

In view of this and 2.5 we have that Nrd(H_{1})^{∗}. Nrd(H_{2})^{∗} = F^{∗}. This completes the proof. □

Proposition 7.7 Let vcd(F) ≤ 2 and F has SAP for quadratic towers. Let h be a locally hyperbolic hermitian form of even rank over a central-division algebra D with an orthogonal involution τ. Let disc(h) = 1. Then we have Hyp(h) = F^{∗}.

Proof Let L ∈ F_{2}(F) be such that D_{L} ∼ (−1, −x) for some x ∈ L^{∗}. Since h_{L} is locally hyperbolic, by 6.13 we have Hyp(h_{L}) = N(a ∪ D_{L}).N(−a ∪ D_{L}). Clearly N(a ∪ D_{L}) = Nrd(a, −x)^{∗} and N(−a ∪ D_{L}) = Nrd(−a, −x)^{∗}. Since F has SAP for quadratic towers, so does L. Thus by 7.6, we have

Nrd(a, −x)^{∗}. Nrd(−a, −x)^{∗} = L^{∗}

and we conclude that Hyp(h_{L}) = L^{∗}. In view of this and 2.4 we have Hyp(h) = F^{∗}. □

Proposition 7.8 Let vcd(F) ≤ 2 and F has SAP with respect to quadratic towers. Let A be a central simple algebra of even degree over F and σ be a locally hyperbolic involution on A with disc(σ) = 1. Then Hyp(A, σ) = F^{∗}.

Proof Let L ∈ F_{2}(F) be such that A_{L} ∼ (−1, −x) for some x ∈ L^{∗}. Let H = (−1, −x). Then A_{L} = M_{r}(H) for some positive integer r. Let τ be an orthogonal involution on H and h be a hermitian form over (H, τ) such that σ_{L} is adjoint to h. Then Hyp(A_{L}, σ_{L}) = Hyp(h).

First assume that r is even. Then it follows from [KMRT, Prop. 7.3(1)] that disc(σ_{L}) = disc(h) = 1. Since σ_{L} is locally hyperbolic, the hermitian form h is locally hyperbolic. Thus h is a locally hyperbolic form of even rank and trivial discriminant over (H, τ). Therefore in view of 7.7 we have Hyp(A_{L}, σ_{L}) = Hyp(h) = L^{∗}.

Now suppose r is odd. Since σ_{L} is locally hyperbolic, the hermitian form h is locally hyperbolic. Thus the quaternion algebra H is locally split and by 1.8 we have Nrd(H)^{∗} = L^{∗}. Let λ ∈ L^{∗}. Let L_{i} ∈ F_{2}(L) be such that each H_{L_{i}} is split and λ = ∏_{i} N_{L_{i}/L}(λ_{i}), where λ_{i} ∈ L_{i}^{∗}. Then h_{L_{i}} is a locally hyperbolic quadratic form over L_{i} with even rank and trivial discriminant. Thus by 5.2, we have Hyp(h_{L_{i}}) = L_{i}^{∗}. Therefore

λ ∈ ∏_{i} N_{L_{i}/L}(Hyp(H_{L_{i}})) ⊆ Hyp(h)

and hence Hyp(h) = L^{∗}.

Thus it follows that if L ∈ F_{2}(F) is such that A_{L} ∼ (−1, −x) for some x ∈ L^{∗} then Hyp(A_{L}, σ_{L}) = Hyp(h) = L^{∗}. Therefore in view of 2.4, we have Hyp(A, σ) = F^{∗}. This completes the proof. □

Theorem 7.9 Let vcd(F) ≤ 2 and F has SAP for quadratic towers. Let A be a central simple algebra of degree 2n over F and σ be an orthogonal involution on A with disc(σ) = 1. Then Hyp(A, σ).F^{∗2} = G_{+}(A, σ).

Proof Let λ ∈ G_{+}(A, σ). Then λ = σ(a)a for some a ∈ A^{∗} with Nrd(a) = λ^{n}. Let K = F (\sqrt{−λ}). First suppose that n is odd. Then λ ∈ Nrd(A^{∗}) and thus λ >_{v} 0 at those v ∈ Ω where A_{v} is not split. If A_{v} is split then λ >_{v} 0 at those v ∈ Ω where sgn(σ_{v}) ≠ 0. Then A_{K} is locally split and σ_{K} is locally hyperbolic.

Now suppose that n is even let and v ∈ Ω is such that σ_{v} is not hyperbolic. Then by (Theorem 3.7, Chapter 10, [S]), A_{v} is split and σ_{v} is adjoint to a non-hyperbolic quadratic form q over F_{v} such that q ≃ λq. Thus we conclude that λ >_{v} 0 at those orderings v ∈ Ω where σ_{v} is not hyperbolic. Then σ_{K} is locally hyperbolic.

Thus in either case, by 7.8 we have that Hyp(A_{K}, σ_{K}) = K^{∗}. Taking norm of \sqrt{−λ} from K/F, we conclude that λ ∈ Hyp(A, σ) and therefore G_{+}(A, σ) ⊆ Hyp(A, σ). By 1.1 we have Hyp(A, σ).F^{∗2} ⊆ G_{+}(A, σ), and hence we conclude that G_{+}(A, σ) = Hyp(A, σ).F^{∗2}. □ Theorem 7.10 Let vcd(F) ≤ 2 and (A, σ) be an algebra of type ^{2}D_{n} over F. Let F have SAP for quadratic towers. Then we have G_{+}(A, σ) = Hyp(A, σ).F^{∗2}.

Proof Let disc(σ) = d. Let λ ∈ G_{+}(A, σ). By 1.3, we have G_{+}(A, σ) ⊆ N_{L/K}(L^{∗}), where L = F (\sqrt{d}). As in the proof of the 7.9, σ_{F (\sqrt{−λ})} is locally hyperbolic. Let M = L(\sqrt{−λ}). By the biquadratic lemma [W, Lemma 2.14], it follows that there exist x ∈ M^{∗} and y ∈ F^{∗} such that λ = N_{M/F} (x)y^{2}. It is clear that disc(σ_{M}) = 1 and σ_{M} is locally hyperbolic. Thus by 7.8 we have Hyp(A_{M}, σ_{M}) = M^{∗} and we easily see that λy^{−2} ∈ Hyp(A, σ). Thus G_{+}(A, σ) ⊆ Hyp(A, σ).F^{∗2}. Since by 1.1 Hyp(A, σ).F^{∗2} ⊆ G_{+}(A, σ), we conclude that G_{+}(A, σ) = Hyp(A, σ).F^{∗2}. □ Corollary 7.11 Let vcd(F) ≤ 2, F has SAP for quadratic towers and let PSim_{+}(A, σ) be of type ^{1}D_{n} or ^{2}D_{n}. Then PSim_{+}(A, σ)(F)/R = 0. □

Since number fields satisfy the conditions of 7.11, we have Corollary 7.12 Let F be a number field and let PSim_{+}(A, σ) be of type ^{1}D_{n} or ^{2}D_{n}. Then PSim_{+}(A, σ)(F)/R = 0. □

Remark It is a well known fact that SAP is not preserved under field extensions. As Detlev Hoffmann has pointed out to us, there are examples of fields F with vcd(F) = 2 and quadratic extensions E/F such that F satisfies SAP but not E. Thus the condition ‘SAP with respect to quadratic towers’ is not redundant in 7.10.

Hoffmann’s example is the following: One can construct a formally real field k with the following properties: (i) k has no extension of odd degree. (ii) There is only one ordering on k. (iii) The u-invariant of k is 2. Let α ∈ k^{∗}\k^{∗2} be a sum of two squares. Let F = k((X)) and E = F (\sqrt{α}). Then vcd(F) = 2 and F has SAP but not E.

Combining together the results of §4, §5 and §7, we have

Theorem 7.13 Let F be a field with vcd(F) ≤ 2. Let G is a classical group of adjoint type defined over F. Then,

(i) If G does not contain a factor of type D_{n} then G(F)/R = 0.

(ii) If F satisfies SAP for quadratic towers then G(F)/R = 0.

Proof If F does not have orderings, by 1.17 we have cd(F) ≤ 2 and we are through by 3.7. Thus we assume that F has orderings. As in the proof of 3.7, it suffices the prove the theorem for absolutely simple adjoint groups defined over F. In view of 1.1, the Assertion (i) follows from 1.4, 4.2 and 5.13. The Assertion (ii) of the theorem follows immediately from Assertion (i) and 7.11. □

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