Multipliers of Improper Similitudes

R. Preeti^{1} and J.-P. Tignol^{2}

Abstract. For a central simple algebra with an orthogonal involution (A, σ) over a field k of characteristic different from 2, we relate the multipliers of similitudes of (A, σ) with the Clifford algebra C(A, σ). We also give a complete description of the group of multipliers of similitudes when deg A ≤ 6 or when the virtual cohomological dimension of k is at most 2.

2000 Mathematics Subject Classification: 11E72.

Keywords and Phrases: Central simple algebra with involution, hermitian form, Clifford algebra, similitude.

Introduction

A. Weil has shown in [22] how to obtain all the simple linear algebraic groups of adjoint type D_{n} over an arbitrary field k of characteristic different from 2: every such group is the connected component of the identity in the group of automorphisms of a pair (A, σ) where A is a central simple k-algebra of degree 2n and σ: A → A is an involution of orthogonal type, i.e., a map which over a splitting field of A is the adjoint involution of a symmetric bilinear form. (See [7] for background material on involutions on central simple algebras and classical groups.) Every automorphism of (A, σ) is inner, and induced by an element g ∈ A^{×} which satisfies σ(g)g ∈ k^{×}. The group of similitudes of (A, σ) is defined by that condition,

GO(A, σ) = {g ∈ A^{×} | σ(g)g ∈ k^{×}}.

The map which carries g ∈ GO(A, σ) to σ(g)g ∈ k^{×} is a homomorphism

μ: GO(A, σ) → k^{×}

1The first author gratefully acknowledges the generous support of the Université catholique de Louvain, Belgium and the ETH-Z, Switzerland.

2Work supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The second author is supported in part by the National Fund for Scientific Research (Belgium).

called the multiplier map. Taking the reduced norm of each side of the equation σ(g)g = μ(g), we obtain

Nrd_{A}(g)^{2} = μ(g)^{2n},

hence Nrd_{A}(g) = ±μ(g)^{n}. The similitude g is called proper if Nrd_{A}(g) = μ(g)^{n}, and improper if Nrd_{A}(g) = −μ(g)^{n}. The proper similitudes form a subgroup GO_{+}(A, σ) ⊂ GO(A, σ). (As an algebraic group, GO_{+}(A, σ) is the connected component of the identity in GO(A, σ).) Our purpose in this work is to study the multipliers of similitudes of a central simple k-algebra with orthogonal involution (A, σ). We denote by G(A, σ) (resp. G_{+}(A, σ), resp. G_{−}(A, σ)) the group of multipliers of similitudes of (A, σ) (resp. the group of multipliers of proper similitudes, resp. the coset of multipliers of improper similitudes),

G(A, σ) = {μ(g) | g ∈ GO(A, σ)},
G_{+}(A, σ) = {μ(g) | g ∈ GO_{+}(A, σ)},
G_{−}(A, σ) = {μ(g) | g ∈ GO(A, σ) \ GO_{+}(A, σ)}.

When A is split (A = End_{k} V for some k-vector space V), hyperplane reflections are improper similitudes with multiplier 1, hence

G(A, σ) = G_{+}(A, σ) = G_{−}(A, σ).

When A is not split however, we may have G(A, σ) ≠ G_{+}(A, σ). Multipliers of similitudes were investigated in relation with the discriminant disc σ by Merkurjev–Tignol [14]. Our goal is to obtain similar results relating multipliers of similitudes to the next invariant of σ, which is the Clifford algebra C(A, σ) (see [7, §8]). As an application, we obtain a complete description of G(A, σ) when deg A ≤ 6 or when the virtual cohomological dimension of k is at most 2.

To give a more precise description of our results, we introduce some more notation. Throughout the paper, k denotes a field of characteristic different from 2. For any integers n, d ≥ 1, let μ_{2^{n}} be the group of 2^{n}-th roots of unity in a separable closure of k and let H^{d}(k, μ_{2^{n}}^{⊗(d−1)}) be the d-th cohomology group of the absolute Galois group with coefficients in μ_{2^{n}}^{⊗(d−1)} (= Z/2^{n}Z if d = 1). Denote simply

H^{d}k = \underrightarrow{lim}_{n} H^{d}(k, μ_{2^{n}}^{⊗(d−1)}),

so H^{1}k and H^{2}k may be identified with the 2-primary part of the character group of the absolute Galois group and with the 2-primary part of the Brauer group of k, respectively,

H^{1}k = X_{2}(k), H^{2}k = Br_{2}(k).

In particular, the isomorphism k^{×}/k^{×2} ≃ H^{1}(k, Z/2Z) derived from the Kummer sequence (see for instance [7, (30.1)]) yields a canonical embedding

k^{×}/k^{×2} ↪ H^{1}k.  (1)

The Brauer class (or the corresponding element in H^{2}k) of a central simple k-algebra E of 2-primary exponent is denoted by [E]. If K/k is a finite separable field extension, we denote by N_{K/k}: H^{d}K → H^{d}k the norm (or corestriction) map. We extend the notation above to the case where K ≃ k × k by letting H^{d}(k × k) = H^{d}k × H^{d}k and

N_{(k×k)/k}(ξ_{1}, ξ_{2}) = ξ_{1} + ξ_{2} for (ξ_{1}, ξ_{2}) ∈ H^{d}(k × k).

Our results use the product

·: k^{×} × H^{d}k → H^{d+1}k for d = 1 or 2

induced as follows by the cup-product: for x ∈ k^{×} and ξ ∈ H^{d}k, choose n such that ξ ∈ H^{d}(k, μ_{2^{n}}^{⊗(d−1)}) and consider the cohomology class (x)_{n} ∈ H^{1}(k, μ_{2^{n}}) corresponding to the 2^{n}-th power class of x under the isomorphism H^{1}(k, μ_{2^{n}}) = k^{×}/k^{×2^{n}} induced by the Kummer sequence; let then

x · ξ = (x)_{n} ∪ ξ ∈ H^{d+1}(k, μ_{2^{n}}^{⊗d}) ⊂ H^{d+1}k.

In particular, if d = 1 and ξ is the square class of y ∈ k^{×} under the embedding (1), then x · ξ is the Brauer class of the quaternion algebra (x, y)_{k}. Throughout the paper, we denote by A a central simple k-algebra of even degree 2n, and by σ an orthogonal involution of A. Recall from [7, (7.2)] that disc σ ∈ k^{×}/k^{×2} ⊂ H^{1}k is the square class of (−1)^{n} Nrd_{A}(a) where a ∈ A^{×} is an arbitrary skew-symmetric element. Let Z be the center of the Clifford algebra C(A, σ); thus, Z is a quadratic étale k-algebra, Z = k[\sqrt{disc σ}], see [7, (8.10)]. The following relation between similitudes and the discriminant is proved in [14, Theorem A] (see also [7, (13.38)]):

Theorem 1. Let (A, σ) be a central simple k-algebra with orthogonal involution of even degree. For λ ∈ G(A, σ),

λ · disc σ = \begin{cases}0 & if λ ∈ G_{+}(A, σ), \\ [A] & if λ ∈ G_{−}(A, σ). \end{cases}

For d = 2 (resp. 3), let (H^{d}k)/A be the factor group of H^{d}k by the subgroup {0, [A]} (resp. by the subgroup k^{×} · [A]). Theorem 1 thus shows that for λ ∈ G(A, σ)

λ · disc σ = 0 in (H^{2}k)/A.

Our main results are Theorems 2, 3, 4, and 5 below.

Theorem 2. Suppose A is split by Z. There exists an element γ(σ) ∈ H^{2}k such that γ(σ)_{Z} = [C(A, σ)] in H^{2}Z. For λ ∈ G(A, σ),

λ · γ(σ) = 0 in (H^{3}k)/A.

Remark 1. In the conditions of the theorem, the element γ(σ) ∈ H^{2}k is not uniquely determined if Z ≄ k × k. Nevertheless, if λ · disc σ = 0 in (H^{2}k)/A, then λ · γ(σ) ∈ (H^{3}k)/A is uniquely determined. Indeed, if γ, γ′ ∈ H^{2}k are such that γ_{Z} = γ′_{Z}, then there exists u ∈ k^{×} such that γ′ = γ + u · disc σ, hence

λ · γ′ = λ · γ + λ · u · disc σ.

The last term vanishes in (H^{3}k)/A since λ · disc σ = 0 in (H^{2}k)/A. The proof of Theorem 2 is given in Section 1. It shows that in the split case, where A = End_{k} V and σ is adjoint to some quadratic form q on V, we may take for γ(σ) the Brauer class of the full Clifford algebra C(V, q). Note that the statement of Theorem 2 does not discriminate between multipliers of proper and improper similitudes, but Theorem 1 may be used to distinguish between them. Slight variations of the arguments in the proof of Theorem 2 also yield the following result on multipliers of proper similitudes:

Theorem 3. Suppose the Schur index of A is at most 4. If λ ∈ G_{+}(A, σ), then there exists z ∈ Z^{×} such that λ = N_{Z/k}(z) and

N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A.

The proof is given in Section 1. Note however that the theorem holds without the hypothesis that ind A ≤ 4, as follows from Corollaries 1.20 and 1.21 in [12]. Using the Rost invariant of Spin groups, these corollaries actually yield an explicit element z as in Theorem 3 from any proper similitude with multiplier λ. Remark 2. The element N_{Z/k}(z · [C(A, σ)]) ∈ (H^{3}k)/A depends only on N_{Z/k}(z) and not on the specific choice of z ∈ Z. Indeed, if z, z′ ∈ Z^{×} are such that N_{Z/k}(z) = N_{Z/k}(z′), then Hilbert’s Theorem 90 yields an element u ∈ Z^{×} such that, denoting by ι the nontrivial automorphism of Z/k,

z′ = zuι(u)^{−1},

hence

N_{Z/k}(z′ · [C(A, σ)]) =

N_{Z/k}(z · [C(A, σ)]) + N_{Z/k}(u · [C(A, σ)]) − N_{Z/k}(ι(u) · [C(A, σ)]).

Since N_{Z/k} ∘ ι = N_{Z/k} and since the properties of the Clifford algebra (see [7, (9.12)]) yield

[C(A, σ)] − ι[C(A, σ)] = [A]_{Z},

it follows that

N_{Z/k}(u · [C(A, σ)]) − N_{Z/k}(ι(u) · [C(A, σ)]) = N_{Z/k}(u · [A]_{Z}).

By the projection formula, the right side is equal to N_{Z/k}(u) · [A]. The claim follows.

Remark 3. Theorems 2 and 3 coincide when they both apply, i.e., if A is split by Z (hence ind A = 1 or 2), and λ ∈ G_{+}(A, σ). Indeed, if λ = N_{Z/k}(z) and γ(σ)_{Z} = [C(A, σ)] then the projection formula yields

N_{Z/k}(z · [C(A, σ)]) = λ · γ(σ).

Remarkably, the conditions in Theorems 1 and 2 turn out to be sufficient for λ to be the multiplier of a similitude when deg A ≤ 6 or when the virtual cohomological 2-dimension^{3} of k is at most 2.

Theorem 4. Suppose n ≤ 3, i.e., deg A ≤ 6.

• If A is not split by Z , then every similitude is proper,

G(A, σ) = G_{+}(A, σ), G_{−}(A, σ) = ∅.

Moreover, for λ ∈ k^{×}, we have λ ∈ G(A, σ) if and only if there exists z ∈ Z^{×} such that λ = N_{Z/k}(z) and

N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A.

• If A is split by Z, let γ(σ) ∈ H^{2}k be as in Theorem 2. For λ ∈ k^{×}, we have λ ∈ G(A, σ) if and only if

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A.

The proof is given in Section 2.

Note that if deg A = 2, then A is necessarily split by Z and we may choose γ(σ) = 0, hence Theorem 4 simplifies to

λ ∈ G(A, σ) if and only if λ · disc σ = 0 in (H^{2}k)/A,

a statement which is easily proved directly. (See [14, p. 15] or [7, (12.25)].)

If deg A = 4, multipliers of similitudes can also be described up to squares as reduced norms from a central simple algebra E of degree 4 such that [E] = γ(σ) if A is split by Z (see Corollary 4.5) or as norms of reduced norms of C(A, σ) if A is not split by Z (see Corollary 2.1).

For the next statement, recall that the virtual cohomological 2-dimension of k (denoted vcd_{2} k) is the cohomological 2-dimension of k(\sqrt{−1}). If v is an ordering of k, we let k_{v} be a real closure of k for v and denote simply by (A, σ)_{v} the algebra with involution (A ⊗_{k} k_{v}, σ ⊗ Id_{k_{v}}).

Theorem 5. Suppose vcd_{2} k ≤ 2, and A is split by Z. For λ ∈ k^{×}, we have λ ∈ G(A, σ) if and only if

λ > 0 at every ordering v of k such that (A, σ)_{v} is not hyperbolic,

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A.

The proof is given in Section 3.

3The authors are grateful to Parimala for her suggestion to investigate the case of low cohomological dimension.

1 Proofs of Theorems 2 and 3

Theorems 2 and 3 are proved by reduction to the split case, which we consider first. We thus assume A = End_{k} V for some k-vector space V of dimension 2n, and σ is adjoint to a quadratic form q on V. Then disc σ = disc q and C(A, σ) is the even Clifford algebra C(A, σ) = C_{0}(V, q). We denote by C(V, q) the full Clifford algebra of q, which is a central simple k-algebra, and by I^{m}k the m-th power of the fundamental ideal Ik of the Witt ring W k.

Lemma 1.1. For λ ∈ k^{×}, the following conditions are equivalent:

(a) λ · disc q = 0 in H^{2}k and λ · [C(V, q)] = 0 in H^{3}k;

(b) ⟨λ⟩ · q ≡ q mod I^{4}k.

Proof. For α_{1}, …, α_{m} ∈ k^{×}, let

⟨⟨α_{1}, …, α_{m}⟩⟩ = ⟨1, −α_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −α_{m}⟩.

Let e_{2} : I^{2}k → H^{2}k be the Witt invariant and e_{3} : I^{3}k → H^{3}k be the Arason invariant. By a theorem of Merkurjev [9] (resp. of Merkurjev–Suslin [13] and Rost [17]), we have ker e_{2} = I^{3}k and ker e_{3} = I^{4}k. Therefore, the lemma follows if we prove

λ · disc q = 0 if and only if ⟨⟨λ⟩⟩ · q ∈ I^{3}k,  (2)

and that, assuming that condition holds,

e_{3}(⟨⟨λ⟩⟩ · q) = λ · [C(V, q)].  (3)

Let δ ∈ k^{×} be such that disc q = (δ)_{1} ∈ H^{1}(k, Z/2Z) ⊂ H^{1}k. Then

q ≡ ⟨⟨δ⟩⟩ mod I^{2}k,  (4)

hence

e_{2}(⟨⟨λ⟩⟩ · q) = e_{2}(⟨⟨λ, δ⟩⟩) = λ · disc q,

proving (2). Now, assuming λ · disc q = 0, we have ⟨⟨λ, δ⟩⟩ = 0 in W k, hence

⟨⟨λ⟩⟩ · q = ⟨⟨λ⟩⟩ · (q ⊥ ⟨⟨δ⟩⟩).

By (4), we have q ⊥ ⟨⟨δ⟩⟩ ∈ I^{2}k, hence

e_{3}(⟨⟨λ⟩⟩ · q) = λ · e_{2}(q ⊥ ⟨⟨δ⟩⟩).  (5)

The computation of Witt invariants in [8, Chapter 5] yields

e_{2}(q ⊥ ⟨⟨δ⟩⟩) = [C(V, q)] + (−1) · disc q.  (6)

Since λ · disc q = 0 by hypothesis, (3) follows from (5) and (6).

Proof of Theorem 2. If A is split, then using the same notation as in Lemma 1.1 we may take γ(σ) = [C(V, q)], and Theorem 2 readily follows from Lemma 1.1. For the rest of the proof, we may thus assume A is not split, hence disc σ ≠ 0 since Z is assumed to split A. Let G = {Id, ι} be the Galois group of Z/k. The properties of the Clifford algebra (see for instance [7, (9.12)]) yield

[C(A, σ)] − ι[C(A, σ)] = [A]_{Z} = 0.

Therefore, [C(A, σ)] lies in the subgroup (Br Z)^{G} of Br Z fixed under the action of G. The “Teichmüller cocycle” theory [6] (or the spectral sequence of group extensions, see [19, Remarque, p. 126]) yields an exact sequence

Br k → (Br Z)^{G} → H^{3}(G, Z^{×}).

Since G is cyclic, H^{3}(G, Z^{×}) = H^{1}(G, Z^{×}). By Hilbert’s Theorem 90, H^{1}(G, Z^{×}) = 1, hence (Br Z)^{G} is the image of the scalar extension map Br k → Br Z, and there exists γ(σ) ∈ Br k such that γ(σ)_{Z} = [C(A, σ)]. Then, by [7, (9.12)],

2γ(σ) = N_{Z/k}([C(A, σ)]) = \begin{cases}0 & if n is odd, \\ [A] & if n is even, \end{cases}  (7)

hence 4γ(σ) = 0. Therefore, γ(σ) ∈ Br_{2}(k) = H^{2}k. Note that ind A = 2, since A is split by the quadratic extension Z/k, hence A is Brauer-equivalent to a quaternion algebra Q. Let X be the conic associated with Q; the function field k(X) splits A. Since Theorem 2 holds in the split case, we have

λ · γ(σ) ∈ ker(H^{3}k → H^{3}k(X)).

By a theorem of (Arason–) Peyre [16, Proposition 4.4], the kernel on the right side is the subgroup k^{×} · [A] ⊂ H^{3}k, hence

λ · γ(σ) = 0 in (H^{3}k)/A.

Proof of Theorem 3. Suppose first A is split, and use the same notation as in Lemma 1.1. If λ ∈ G(A, σ), then ⟨λ⟩ · q ≃ q and Lemma 1.1 yields

λ · disc q = 0 in H^{2}k  and  λ · [C(V, q)] = 0 in H^{3}k.

The first equation implies that λ = N_{Z/k}(z) for some z ∈ Z^{×}. Since

[C(A, σ)] = [C_{0}(V, q)] = [C(V, q)]_{Z},

the projection formula yields

N_{Z/k}(z · [C(A, σ)]) = N_{Z/k}(z) · [C(V, q)] = λ · [C(V, q)] = 0,

proving the theorem if A is split. If A is not split, we extend scalars to the function field k(X) of the Severi– Brauer variety of A. For λ ∈ G_{+}(A, σ), there still exists z ∈ Z^{×} such that λ = N_{Z/k}(z), by Theorem 1. Since Theorem 3 holds in the split case, we have

N_{Z/k}(z · [C(A, σ)]) ∈ ker(H^{3}k → H^{3}k(X)),

and Peyre’s theorem concludes the proof. (Note that applying Peyre’s theorem requires the hypothesis that ind A ≤ 4.)

2 Algebras of low degree

We prove Theorem 4 by considering separately the cases ind A = 1, 2, and 4.

2.1 Case 1: A is split

Let A = End_{k} V, dim V ≤ 6, and let σ be adjoint to a quadratic form q on V. Since C(A, σ) = C_{0}(V, q), we may choose γ(σ) = [C(V, q)]. The equations

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A

are then equivalent to

λ · disc q = 0 in H^{2}k  and  λ · [C(V, q)] = 0 in H^{3}k,

hence, by Lemma 1.1, to ⟨⟨λ⟩⟩ · q ∈ I^{4}k. Since dim q = 6, the Arason–Pfister Hauptsatz [8, Chapter 10, Theorem 3.1] shows that this relation holds if and only if ⟨⟨λ⟩⟩ · q = 0, i.e., λ ∈ G(V, q) = G(A, σ), and the proof is complete.

2.2 Case 2: ind A = 2

Let Q be a quaternion (division) algebra Brauer-equivalent to A. We represent A as A = End_{Q} U for some 3-dimensional (right) Q-vector space. The involution σ is then adjoint to a skew-hermitian form h on U (with respect to the conjugation involution on Q), which defines an element in the Witt group W^{−1}(Q). Let X be the conic associated with Q. The function field k(X) splits Q, hence Morita-equivalence yields an isomorphism

W^{−1}(Q ⊗ k(X)) ≃ W k(X).

Moreover, Dejaiffe [4] and Parimala–Sridharan–Suresh [15] have shown that the scalar extension map

W^{−1}(Q) → W^{−1}(Q ⊗ k(X)) ≃ W k(X)  (8)

is injective. Let (V, q) be a quadratic space over k(X) representing the image of (U, h) under (8). We may assume dim V = deg A ≤ 6 and σ is adjoint to q after scalar extension to k(X). An element λ ∈ k^{×} lies in G(V, q) if and only if ⟨⟨λ⟩⟩ · q = 0; by the injectivity of (8), this condition is also equivalent to ⟨⟨λ⟩⟩ · h = 0 in W^{−1}(Q), i.e., to λ ∈ G(A, σ). Therefore,

G(V, q) ∩ k^{×} = G(A, σ).  (9)

Suppose first A is not split by Z. Theorem 1 then shows that every similitude of (A, σ) is proper, and it only remains to show that if λ = N_{Z/k}(z) for some z ∈ Z^{×} such that

N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A,

then λ ∈ G(A, σ). Extending scalars to k(X), we derive from the last equation by the projection formula

N_{Z(X)/k(X)}(z) · [C(V, q)] = 0 in H^{3}k(X).

Therefore, by Lemma 1.1, ⟨λ⟩ · q ≡ q mod I^{4}k(X), i.e.,

⟨⟨λ⟩⟩ · q ∈ I^{4}k(X).

Since dim q ≤ 6, the Arason–Pfister Hauptsatz implies ⟨⟨λ⟩⟩ · q = 0, hence λ ∈ G(V, q) and therefore λ ∈ G(A, σ) by (9). Theorem 4 is thus proved when ind A = 2 and A is not split by Z. Suppose next A is split by Z. In view of Theorems 1 and 2, it suffices to show that if λ ∈ k^{×} satisfies

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A,

then λ ∈ G(A, σ). Again, extending scalars to k(X), the conditions become

λ · disc q = 0 in H^{2}k(X)  and  λ · [C(V, q)] = 0 in H^{3}k(X).

By Lemma 1.1, these equations imply ⟨⟨λ⟩⟩ · q ∈ I^{4}k(X), hence ⟨⟨λ⟩⟩ · q = 0 by the Arason–Pfister Hauptsatz since dim q ≤ 6. It follows that λ ∈ G(V, q), hence λ ∈ G(A, σ) by (9).

2.3 Case 3: ind A = 4

Since deg A ≤ 6, this case arises only if deg A = 4, i.e., A is a division algebra. This division algebra cannot be split by the quadratic k-algebra Z, hence all the similitudes are proper, by Theorem 1. Theorem 3 shows that if λ ∈ G(A, σ), then there exists z ∈ Z^{×} such that λ = N_{Z/k}(z) and N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A, and it only remains to prove the converse. Let z ∈ Z^{×} be such that N_{Z/k}(z · [C(A, σ)]) = u · [A] for some u ∈ k^{×}. Since by [7, (9.12)], N_{Z/k}([C(A, σ)]) = [A], it follows that

N_{Z/k}(u^{−1}z · [C(A, σ)]) = 0 in H^{3}k.  (10)

Since deg A = 4, the Clifford algebra C(A, σ) is a quaternion algebra over Z. Let

C(A, σ) = (z_{1}, z_{2})_{Z}.

Suppose first disc σ ≠ 0, i.e., Z is a field. Let s: Z → k be a k-linear map such that s(1) = 0, and let s_{∗}: W Z → W k be the corresponding (Scharlau) transfer map. By [2, Satz 3.3, Satz 4.18], Equation (10) yields

s_{∗}(⟨⟨u^{−1}z, z_{1}, z_{2}⟩⟩) ∈ I^{4}k.

However, the form s_{∗}(⟨⟨u^{−1}z, z_{1}, z_{2}⟩⟩) is isotropic since ⟨⟨u^{−1}z, z_{1}, z_{2}⟩⟩ represents 1 and s(1) = 0. Moreover, its dimension is 2^{4}, hence the Arason–Pfister Hauptsatz implies

s_{∗}(⟨⟨u^{−1}z, z_{1}, z_{2}⟩⟩) = 0 in W k.

It follows that

s_{∗}(⟨u^{−1}z⟩ · ⟨⟨z_{1}, z_{2}⟩⟩) = s_{∗}(⟨⟨z_{1}, z_{2}⟩⟩),

hence the form on the left side is isotropic. Therefore, the form ⟨u^{−1}z⟩·⟨⟨z_{1}, z_{2}⟩⟩ represents an element v ∈ k^{×}. Then v^{−1}u^{−1}z is represented by ⟨⟨z_{1}, z_{2}⟩⟩, which is the reduced norm form of C(A, σ), hence z ∈ k^{×} Nrd(C(A, σ)^{×}), and

N_{Z/k}(z) ∈ k^{×2}N_{Z/k}(Nrd(C(A, σ)^{×})).

By [7, (15.11)], the group on the right is G_{+}(A, σ). We have thus proved N_{Z/k}(z) ∈ G(A, σ), and the proof is complete when Z is a field. Suppose finally disc σ = 0, i.e., Z ≃ k × k. Then C(A, σ) ≃ C′ × C′′ for some quaternion k-algebras C′ = (z′_{1}, z′_{2})_{k} and C′′ = (z′′_{1}, z′′_{2})_{k}, and [7, (15.13)] shows

G(A, σ) = Nrd(C′^{×}) Nrd(C′′^{×}).

We also have z = (z′, z′′) for some z′, z′′ ∈ k^{×}, and (10) becomes

u^{−1}z′ · [C′] + u^{−1}z′′ · [C′′] = 0 in H^{3}k.

It follows that

⟨⟨u^{−1}z′, z′_{1}, z′_{2}⟩⟩ ≃ ⟨⟨u^{−1}z′′, z′′_{1}, z′′_{2}⟩⟩.

By [2, Lemma 1.7], there exists v ∈ k^{×} such that

⟨⟨u^{−1}z′, z′_{1}, z′_{2}⟩⟩ ≃ ⟨⟨v, z′_{1}, z′_{2}⟩⟩ ≃ ⟨⟨v, z′′_{1}, z′′_{2}⟩⟩ ≃ ⟨⟨u^{−1}z′′, z′′_{1}, z′′_{2}⟩⟩,

hence v^{−1}u^{−1}z′ ∈ Nrd(C′) and v^{−1}u^{−1}z′′ ∈ Nrd(C′′). Therefore,

N_{Z/k}(z) = z′z′′ ∈ Nrd(C′^{×}) Nrd(C′′^{×}),

and the proof of Theorem 4 is complete. To finish this section, we compare the descriptions of G_{+}(A, σ) for deg A = 4 or 6 in [7] with those which follow from Theorem 4 (and Remark 3).

Corollary 2.1. Suppose deg A = 4. If disc σ ≠ 0, then

G_{+}(A, σ) = k^{×2}N_{Z/k}(Nrd(C(A, σ)^{×}))
= {N_{Z/k}(z) | N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A}.

If disc σ = 0, then C(A, σ) ≃ C′ × C′′ for some quaternion k-algebras C′, C′′, and

G_{+}(A, σ) = Nrd(C′^{×}) Nrd(C′′^{×})
= {z′z′′ | z′ · [C′] + z′′ · [C′′] = 0 in (H^{3}k)/A}.

Proof. See [7, (15.11)] for the case disc σ ≠ 0 and [7, (15.13)] for the case disc σ = 0.

Corollary 2.2. Suppose deg A = 6. If disc σ ≠ 0, let ι be the nontrivial automorphism of the field extension Z/k and let \underline{σ} be the canonical (unitary) involution of C(A, σ). Let also

GU(C(A, σ), \underline{σ}) = \{g ∈ C(A, σ) | \underline{σ}(g)g ∈ k^{×}\}.

Then

G_{+}(A, σ) =

\{N_{Z/k}(z) | zι(z)^{−1} = (\underline{σ}(g)g)^{−2} Nrd(g) for some g ∈ GU(C(A, σ), \underline{σ})\}

= {N_{Z/k}(z) | N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A}.

If disc σ = 0, then C(A, σ) ≃ C × C^{op} for some central simple k-algebra C of degree 4, and

G_{+}(A, σ) = k^{×2} Nrd(C^{×})
= {z ∈ k^{×} | z · [C] = 0 in (H^{3}k)/A}.

Proof. See [7, (15.31)] for the case disc σ ≠ 0 and [7, (15.34)] for the case disc σ = 0. In the latter case, Theorem 3 shows that G_{+}(A, σ) consists of products z′z′′ where z′, z′′ ∈ k^{×} are such that

z′ · [C] + z′′ · [C^{op}] = 0 in (H^{3}k)/A.

However, [C^{op}] = −[C], and 2[C] = [A] by [7, (9.15)], hence

z′ · [C] + z′′ · [C^{op}] = z′z′′ · [C] in (H^{3}k)/A.

Note that the equation

k^{×2} Nrd(C^{×}) = {z ∈ k^{×} | z · [C] = 0 in (H^{3}k)/A}

can also be proved directly by a theorem of Merkurjev [11, Proposition 1.15].

3 Fields of low virtual cohomological dimension

Our goal in this section is to prove Theorem 5. Together with Theorem 2, the following lemma completes the proof of the “only if” part:

Lemma 3.1. If λ ∈ G(A, σ), then λ > 0 at every ordering v such that (A, σ)_{v} is not hyperbolic.

Proof. If (A, σ)_{v} is not hyperbolic, then A_{v} is split, by [18, Chapter 10, Theorem 3.7]. We may thus represent A_{v} = End_{k_{v}} V for some k_{v}-vector space V, and σ ⊗ Id_{k_{v}} is adjoint to a non-hyperbolic quadratic form q. If λ ∈ G(A, σ), then λ ∈ G(V, q), hence

⟨λ⟩ · q ≃ q.

Comparing the signatures of each side, we obtain λ > 0.

For the “if” part, we use the following lemma:

Lemma 3.2. Let F be an arbitrary field of characteristic different from 2. If vcd_{2} F ≤ 3, then the torsion part of the 4-th power of IF is trivial,

I_{t}^{4}F = 0.

Proof. Our proof uses the existence of the cohomological invariants e_{n}: I^{n}F → H^{n}(F, μ_{2}), and the fact that ker e_{n} = I^{n+1}F, proved for fields of virtual cohomological 2-dimension at most 3 by Arason–Elman–Jacob [3]. Suppose first −1 ∉ F^{×2}. From vcd_{2} F ≤ 3, it follows that H^{n}(F (\sqrt{−1}), μ_{2}) = 0 for n ≥ 4, hence the Arason exact sequence

H^{n}(F (\sqrt{−1}), μ_{2}) ⟶^{N} H^{n}(F, μ_{2}) ⟶^{(−1)_{1}∪} H^{n+1}(F, μ_{2}) → H^{n+1}(F (\sqrt{−1}), μ_{2})

(see [2, Corollar 4.6] or [7, (30.12)]) shows that the cup-product with (−1)_{1} is an isomorphism H^{n}(F, μ_{2}) ≃ H^{n+1}(F, μ_{2}) for n ≥ 4. If q ∈ I_{t}^{4}F, there is an integer ℓ such that 2^{ℓ}q = 0, hence the 4-th invariant e_{4}(q) ∈ H^{4}(F, μ_{2}) satisfies

\underbrace{(−1)_{1} ∪ ⋯ ∪ (−1)_{1}}_{ℓ} ∪e_{4}(q) = 0 in H^{ℓ+4}(F, μ_{2}).

Since (−1)_{1}∪ is an isomorphism, it follows that e_{4}(q) = 0, hence q ∈ I_{t}^{5}F. Repeating the argument with e_{5}, e_{6}, …, we obtain q ∈ ⋂_{n} I^{n}F, hence q = 0 by the Arason–Pfister Hauptsatz [8, p. 290]. If −1 ∈ F^{×2}, then the hypothesis implies that H^{n}(F, μ_{2}) = 0 for n ≥ 4, hence for q ∈ I^{4}F we get successively e_{4}(q) = 0, e_{5}(q) = 0, etc., and we conclude as before.

Proof of Theorem 5. As observed above, the “only if” part follows from Theorem 2 and Lemma 3.1. The proof of the “if” part uses the same arguments as the proof of Theorem 2 in the case where ind A = 2.

We first consider the split case. If A = End_{k} V and σ is adjoint to a quadratic form q on V, then we may choose γ(σ) = C(V, q), and the conditions

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A

imply, by Lemma 1.1, that ⟨⟨λ⟩⟩ · q ∈ I^{4}k. Moreover, for every ordering v on k, the signature sgn_{v}(⟨⟨λ⟩⟩·q) vanishes, since λ > 0 at every v such that sgn_{v}(q) ≠ 0. Therefore, by Pfister’s local-global principle [8, Chapter 8, Theorem 4.1], ⟨⟨λ⟩⟩ · q is torsion. Since the hypothesis on k implies, by Lemma 3.2, that I_{t}^{4}k = 0, we have ⟨⟨λ⟩⟩ · q = 0, hence λ ∈ G(V, q) = G(A, σ). Note that Lemma 3.2 yields I_{t}^{4}k = 0 under the weaker hypothesis vcd_{2} k ≤ 3. Therefore, the split case of Theorem 5 holds when vcd_{2} k ≤ 3. Now, suppose A is not split. Since A is split by Z, it is Brauer-equivalent to a quaternion algebra Q. Let k(X) be the function field of the conic X associated with Q. This field splits A, hence there is a quadratic space (V, q) over k(X) such that A ⊗ k(X) may be identified with End_{k(X)} V and σ ⊗ Id_{k(X)} with the adjoint involution with respect to q. As in Section 2 (see Equation (9)), we have

G(V, q) ∩ k^{×} = G(A, σ).

Therefore, it suffices to show that the conditions on λ imply λ ∈ G(V, q). If v is an ordering of k such that (A, σ)_{v} is hyperbolic, then q_{w} is hyperbolic for any ordering w of k(X) extending v, since hyperbolic involutions remain hyperbolic over scalar extensions. Therefore, λ > 0 at every ordering w of k(X) such that q_{w} is not hyperbolic. Moreover, the conditions

λ · disc σ = 0 in (H^{2}k)/A  and  λ · γ(σ) = 0 in (H^{3}k)/A

imply

λ · disc q = 0 in H^{2}k(X)  and  λ · [C(V, q)] = 0 in H^{3}k(X).

Since X is a conic, Proposition 11, p. 93 of [20] implies

vcd_{2} k(X) = 1 + vcd_{2} k ≤ 3.

As Theorem 5 holds in the split case over fields of virtual cohomological 2dimension at most 3, it follows that λ ∈ G(V, q).

Remark. The same arguments show that if vcd_{2} k ≤ 2 and ind A = 2, then G_{+}(A, σ) consists of the elements N_{Z/k}(z) where z ∈ Z^{×} is such that

N_{Z/k}(z · [C(A, σ)]) = 0 in (H^{3}k)/A.

4 Examples

In this section, we give an explicit description of the element γ(σ) of Theorem 2 in some special cases. Throughout this section, we assume the algebra A is not split, and is split by Z (hence Z is a field and disc σ ≠ 0). Our first result is easy:

Proposition 4.1. If A is split by Z and σ becomes hyperbolic after scalar extension to Z, then we may choose γ(σ) = 0.

Proof. Let ι be the nontrivial automorphism of Z/k. Since Z is the center of C(A, σ),

C(A, σ) ⊗_{k} Z ≃ C(A, σ) ×^{ι}C(A, σ).  (11)

On the other hand, C(A, σ) ⊗_{k} Z ≃ C(A ⊗_{k} Z, σ ⊗ Id_{Z}), and since σ becomes hyperbolic over Z, one of the components of C(A ⊗_{k} Z, σ ⊗ Id_{Z}) is split, by [7, (8.31)]. Therefore,

[C(A, σ)] = [^{ι}C(A, σ)] = 0 in Br Z.

Corollary 4.2. In the conditions of Proposition 4.1, if deg A ≤ 6 or vcd_{2} k ≤ 2, then

G_{+}(A, σ) = {λ ∈ k^{×} | λ · disc σ = 0 in H^{2}k}

and

G_{−}(A, σ) = {λ ∈ k^{×} | λ · disc σ = [A] in H^{2}k}.

Proof. This readily follows from Proposition 4.1 and Theorem 2 or 5.

To give further examples where γ(σ) can be computed, we fix a particular representation of A as follows. Since A is assumed to be split by Z, it is Brauer-equivalent to a quaternion k-algebra Q containing Z. We choose a quaternion basis 1, i, j, ij of Q such that Z = k(i). Let A = End_{Q} U for some right Q-vector space U, and let σ be the adjoint involution of a skew-hermitian form h on U with respect to the conjugation involution on Q. For x, y ∈ U, we decompose

h(x, y) = f(x, y) + jg(x, y) with f(x, y), g(x, y) ∈ Z.

It is easily verified that f (resp. g) is a skew-hermitian (resp. symmetric bilinear) form on U viewed as a Z-vector space. (See [18, Chapter 10, Lemma 3.1].) We have

A ⊗_{k} Z = (End_{Q} U) ⊗_{k} Z = End_{Z} U.

Moreover, for x, y ∈ U and φ ∈ End_{Q} U, the equation

h(x, φ(y)) = h(σ(φ)(x), y)

implies

g(x, φ(y)) = g(σ(φ)(x), y),

hence σ ⊗_{k} Id_{Z} is adjoint to g.

Proposition 4.3. With the notation above,

[C(A, σ)] = [C(U, g)] in Br Z.

Proof. Since σ ⊗ Id_{Z} is the adjoint involution of g,

C(A ⊗_{k} Z, σ ⊗ Id_{Z}) ≃ C_{0}(U, g).  (12)

Now, disc σ is a square in Z, hence C_{0}(U, g) decomposes into a direct product

C_{0}(U, g) ≃ C′ × C′′  (13)

where C′, C′′ are central simple Z-algebras Brauer-equivalent to C(U, g). The proposition follows from (11), (12), and (13).

To give an explicit description of g, consider an h-orthogonal basis (e_{1}, …, e_{n}) of U. In the corresponding diagonalization of h,

h ≃ ⟨u_{1}, …, u_{n}⟩,

each u_{ℓ} ∈ Q is a pure quaternion, since h is skew-hermitian. Let u_{ℓ}^{2} = a_{ℓ} ∈ k^{×} for ℓ = 1, …, n. Then

disc σ = (−1)^{n} Nrd(u_{1}) … Nrd(u_{n}) = a_{1} … a_{n},

so we may assume i^{2} = a_{1} … a_{n}. Write

u_{ℓ} = μ_{ℓ}i + jv_{ℓ}  where μ_{ℓ} ∈ k and v_{ℓ} ∈ Z.  (14)

Each e_{ℓ}Q is a 2-dimensional Z-vector space, and we have a g-orthogonal decomposition

U = e_{1}Q ⊕ ⋯ ⊕ e_{n}Q.

If v_{ℓ} = 0, then g(e_{ℓ}, e_{ℓ}) = 0, hence e_{ℓ}Q is hyperbolic. If v_{ℓ} ≠ 0, then (e_{ℓ}, e_{ℓ}u_{ℓ}) is a g-orthogonal basis of e_{ℓ}Q, which yields the following diagonalization of the restriction of g:

⟨v_{ℓ}, −a_{ℓ}v_{ℓ}⟩.

Therefore,

g = g_{1} + ⋯ + g_{n}  (15)

where

g_{ℓ} = \begin{cases}0 & if v_{ℓ} = 0, \\ ⟨v_{ℓ}⟩⟨1, −a_{ℓ}⟩ & if v_{ℓ} ≠ 0. \end{cases}  (16)

We now consider in more detail the cases n = 2 and n = 3.

4.1 Algebras of degree 4

Suppose deg A = 4, i.e., n = 2, and use the same notation as above. If v_{1} = 0, then squaring each side of (14) yields a_{1} = μ_{1}^{2}a_{1}a_{2}, hence a_{2} ∈ k^{×2}, a contradiction since Q is assumed to be a division algebra. The case v_{2} = 0 leads to the same contradiction. Therefore, we necessarily have v_{1} ≠ 0 and v_{2} ≠ 0. By (15) and (16),

g = ⟨v_{1}⟩⟨1, −a_{1}⟩ + ⟨v_{2}⟩⟨1, −a_{2}⟩,

hence by [8, p. 121],

[C(A, σ)] = (a_{1}, v_{1})_{Z} + (a_{2}, v_{2})_{Z} + (a_{1}, a_{2})_{Z}
= (a_{1}, −v_{1}v_{2})_{Z}.  (17)

Since the division algebra Q contains the pure quaternions u_{1}, u_{2} and i with u_{1}^{2} = a_{1}, u_{2}^{2} = a_{2} and i^{2} = a_{1}a_{2}, we have a_{1}, a_{2}, a_{1}a_{2} ∉ k^{×2} and we may consider the field extension

L = k(\sqrt{a_{1}}, \sqrt{a_{2}}).

We identify Z with a subfield of L by choosing in L a square root of a_{1}a_{2}, and denote by ρ_{1}, ρ_{2} the automorphisms of L/k defined by

ρ_{1}(\sqrt{a_{1}}) = −\sqrt{a_{1}},  ρ_{2}(\sqrt{a_{1}}) = \sqrt{a_{1}},
ρ_{1}(\sqrt{a_{2}}) = \sqrt{a_{2}},  ρ_{2}(\sqrt{a_{2}}) = −\sqrt{a_{2}}.

Thus, Z ⊂ L is the subfield of ρ_{1} ∘ ρ_{2}-invariant elements. Let j^{2} = b. Then (14) yields

a_{1} = μ_{1}^{2}a_{1}a_{2} + bN_{Z/k}(v_{1}), a_{2} = μ_{2}^{2}a_{1}a_{2} + bN_{Z/k}(v_{2}),

hence N_{Z/k}(−v_{1}v_{2}) = a_{1}a_{2}b^{−2}(1 − μ_{1}^{2}a_{2})(1 − μ_{2}^{2}a_{1}) and

\frac{−v_{1}v_{2}}{ρ_{1}(−v_{1}v_{2})} = \frac{−v_{1}v_{2}}{ρ_{2}(−v_{1}v_{2})} = \frac{a_{1}a_{2}}{b^{2}ρ_{1}(−v_{1}v_{2})^{2}} (1 − μ_{1}^{2}a_{2})(1 − μ_{2}^{2}a_{1}).

Since L = Z(\sqrt{a_{1}}) = Z(\sqrt{a_{2}}), it follows that 1 − μ_{1}^{2}a_{2} and 1 − μ_{2}^{2}a_{1} are norms from L/Z. Therefore, the preceding equation yields

\frac{−v_{1}v_{2}}{ρ_{1}(−v_{1}v_{2})} = \frac{−v_{1}v_{2}}{ρ_{2}(−v_{1}v_{2})} = N_{L/Z}(ℓ) for some ℓ ∈ L^{×}.

Since N_{Z/k}(−v_{1}v_{2}ρ_{1}(−v_{1}v_{2})^{−1}) = 1, we have N_{L/k}(ℓ) = 1. By Hilbert’s Theorem 90, there exists b_{1} ∈ L^{×} such that

ρ_{1}(b_{1}) = b_{1}  and  b_{1}ρ_{2}(b_{1})^{−1} = ℓρ_{1}(ℓ).  (18)

Set b_{2} = −v_{1}v_{2}ρ_{1}(ℓ)b_{1}^{−1}. Computation yields

ρ_{2}(b_{2}) = b_{2}  and  ρ_{1}(b_{2})b_{2}^{−1} = ℓρ_{2}(ℓ).  (19)

Define an algebra E over k by

E = L ⊕ Lr_{1} ⊕ Lr_{2} ⊕ Lr_{1}r_{2}

where the multiplication is defined by

r_{1}x = ρ_{1}(x)r_{1},  r_{2}x = ρ_{2}(x)r_{2}  for x ∈ L,
r_{1}^{2} = b_{1},  r_{2}^{2} = b_{2},  and  r_{1}r_{2} = ℓr_{2}r_{1}.

Since b_{1}, b_{2} and ℓ satisfy (18) and (19), the algebra E is a crossed product, see [1]. It is thus a central simple k-algebra of degree 4.

Proposition 4.4. With the notation above, we may choose γ(σ) = [E] ∈ Br k.

Proof. The centralizer C_{E}Z of Z in E is L ⊕ Lr_{1}r_{2}. Computation shows that

(r_{1}r_{2})^{2} = −v_{1}v_{2}.

Since conjugation by r_{1}r_{2} maps \sqrt{a_{1}} ∈ L to its opposite, it follows that

C_{E}Z = (a_{1}, −v_{1}v_{2})_{Z}.

Since [C_{E}Z] = [E]_{Z}, the proposition follows from (17).

Corollary 4.5. Let

E_{+} = C_{E}Z = {x ∈ E^{×} | xz = zx for all z ∈ Z}

and

E_{−} = {x ∈ E^{×} | xz = ρ_{1}(z)x for all z ∈ Z}.

Then

G_{+}(A, σ) = k^{×2} Nrd_{E}(E_{+})  and  G_{−}(A, σ) = k^{×2} Nrd_{E}(E_{−}).

Proof. As observed in the proof of Proposition 4.4, C_{E}Z ≃ C(A, σ). Since, by [5, Corollary 5, p. 150],

Nrd_{E}(x) = N_{Z/k}(Nrd_{C_{E}Z} x) for x ∈ C_{E}Z,

the description of G_{+}(A, σ) above follows from [7, (15.11)] (see also Corollary 2.1). To prove k^{×2} Nrd_{E}(E_{−}) ⊂ G_{−}(A, σ), it obviously suffices to prove Nrd_{E}(E_{−}) ⊂ G_{−}(A, σ). From the definition of E, it follows that r_{1} ∈ E_{−}. By [10, p. 80],

Nrd_{E}(r_{1}) · [E] = 0  in H^{3}k.  (20)

Let L_{1} ⊂ L be the subfield fixed under ρ_{1}. We have r_{1}^{2} = b_{1} ∈ L_{1}, hence

Nrd_{E}(r_{1}) = N_{L_{1}/k}(b_{1}).

On the other hand, the centralizer of L_{1} is

C_{E}L_{1} = L ⊕ Lr_{1} ≃ (a_{1}a_{2}, b_{1})_{L_{1}},

hence

[N_{L_{1}/k}(C_{E}L_{1})] = (a_{1}a_{2}, N_{L_{1}/k}(b_{1}))_{k} = Nrd_{E}(r_{1}) · disc σ in H^{2}k. (21)

Since [C_{E}L_{1}] = [E_{L_{1}}], we have [N_{L_{1}/k}(C_{E}L_{1})] = 2[E]. But 2[E] = 2γ(σ) = [A] by (7), hence (21) yields

Nrd_{E}(r_{1}) · disc σ = [A]  in H^{2}k.  (22)

From (20), (22) and Theorems 1, 2 it follows that Nrd_{E}(r_{1}) ∈ G_{−}(A, σ). Now, suppose x ∈ E_{−}. Then r_{1}x ∈ E_{+}, hence Nrd_{E}(r_{1}x) ∈ G_{+}(A, σ) by the first part of the corollary. Since

G_{+}(A, σ)G_{−}(A, σ) = G_{−}(A, σ)

it follows that

Nrd_{E}(x) ∈ Nrd_{E}(r_{1})G_{+}(A, σ) = G_{−}(A, σ).

We have thus proved k^{×2} Nrd_{E}(E_{−}) ⊂ G_{−}(A, σ). To prove the reverse inclusion, consider λ ∈ G_{−}(A, σ). Since

G_{−}(A, σ)G_{−}(A, σ) = G_{+}(A, σ),

we have λ Nrd_{E}(r_{1}) ∈ G_{+}(A, σ), hence by the first part of the corollary,

λ Nrd_{E}(r_{1}) ∈ k^{×2} Nrd_{E}(E_{+}).

It follows that

λ ∈ k^{×2} Nrd_{E}(r_{1}E_{+}) = k^{×2} Nrd_{E}(E_{−}).

4.2 Algebras of degree 6

Suppose deg A = 6, i.e., n = 3, and use the same notation as in the beginning of this section. If σ (i.e., h) is isotropic, then h is Witt-equivalent to a rank 1 skew-hermitian form, say ⟨u⟩. Hence i^{2} = disc σ = u^{2} ∈ k^{×}. Hence we may assume that h is Witt-equivalent to the rank 1 skew-hermitian form ⟨μi⟩ for some μ ∈ k^{×}. This implies that the form g is hyperbolic and C(U, g) is split. Hence we may choose γ(σ) = 0. By Theorem 4, we then have λ ∈ G(A, σ) if and only if λ. disc σ = 0 in (H^{2}k)/A. If σ becomes isotropic over Z, the form g is isotropic, hence we may choose a diagonalization of h

h ≃ ⟨u_{1}, u_{2}, u_{3}⟩

such that g(u_{3}, u_{3}) = 0, i.e., in the notation of (14), u_{3} = μ_{3}i. Raising each side to the square, we obtain

a_{3} = μ_{3}^{2}a_{1}a_{2}a_{3},

hence a_{1} ≡ a_{2} mod k^{×2}. It follows that u_{2} is conjugate to a scalar multiple of u_{1}, i.e., there exists x ∈ Q^{×} and θ ∈ k^{×} such that

u_{2} = θxu_{1}x^{−1} = θ Nrd_{Q}(x)^{−1}xu_{1}\overline{x}.

Since ⟨u_{1}⟩ ≃ ⟨xu_{1}\overline{x}⟩, we may let ν = −θ Nrd(x)^{−1} ∈ k^{×} to obtain

h ≃ ⟨u_{1}, −νu_{1}, μ_{3}i⟩.

If v_{1} = 0, then g is hyperbolic, hence we may choose γ(σ) = 0 by Proposition 4.1. If v_{1} ≠ 0, then (15) and (16) yield

g = ⟨v_{1}⟩⟨1, −a_{1}⟩ + ⟨−νv_{1}⟩⟨1, −a_{1}⟩ = ⟨v_{1}⟩⟨⟨a_{1}, ν⟩⟩.

The Clifford algebra of g is the quaternion algebra (a_{1}, ν)_{Z}, hence we may choose

γ(σ) = (a_{1}, ν)_{k}.

Suppose finally that σ does not become isotropic over Z, hence v_{1}, v_{2}, v_{3} ≠ 0. Then

g = ⟨v_{1}⟩⟨1, −a_{1}⟩ + ⟨v_{2}⟩⟨1, −a_{2}⟩ + ⟨v_{3}⟩⟨1, −a_{3}⟩

and, by Proposition 4.3,

[C(A, σ)] = (a_{1}, v_{1})_{Z} + (a_{2}, v_{2})_{Z} + (a_{3}, v_{3})_{Z} + (a_{1}, a_{2})_{Z} + (a_{1}, a_{3})_{Z} + (a_{2}, a_{3})_{Z}.

Since Z = k(\sqrt{a_{1}a_{2}a_{3}}), the right side simplifies to

[C(A, σ)] = (a_{1}, v_{1}v_{3})_{Z} + (a_{2}, v_{2}v_{3})_{Z} + (a_{1}, a_{2})_{Z} + (a_{1}a_{2}, −1)_{Z}.  (23)

By [7, (9.16)], N_{Z/k}C(A, σ) is split, hence

(a_{1}, N_{Z/k}(v_{1}v_{3}))_{k} = (a_{2}, N_{Z/k}(v_{2}v_{3}))_{k} in Br k.

By the “common slot lemma” (see for instance [2, Lemma 1.7]), there exists α ∈ k^{×} such that

(a_{1}, N_{Z/k}(v_{1}v_{3}))_{k} = (α, N_{Z/k}(v_{1}v_{3}))_{k} =

(α, N_{Z/k}(v_{2}v_{3}))_{k} = (a_{2}, N_{Z/k}(v_{2}v_{3}))_{k}.

Then

(αa_{1}, N_{Z/k}(v_{1}v_{3}))_{k} = (αa_{2}, N_{Z/k}(v_{2}v_{3}))_{k} = (α, N_{Z/k}(v_{1}v_{2}))_{k} = 0.

By [21, (2.6)], there exist β_{1}, β_{2}, β_{3} ∈ k^{×} such that

(αa_{1}, v_{1}v_{3})_{Z} = (αa_{1}, β_{1})_{Z}, (αa_{2}, v_{2}v_{3})_{Z} = (αa_{2}, β_{2})_{Z},
(α, v_{1}v_{2})_{Z} = (α, β_{3})_{Z}.

Since

(a_{1}, v_{1}v_{3})_{Z} + (a_{2}, v_{2}v_{3})_{Z} = (αa_{1}, v_{1}v_{3})_{Z} + (αa_{2}, v_{2}v_{3})_{Z} + (α, v_{1}v_{2})_{Z},

it follows from (23) that

[C(A, σ)] = (αa_{1}, β_{1})_{Z} + (αa_{2}, β_{2})_{Z} + (α, β_{3})_{Z} + (a_{1}, a_{2})_{Z} + (a_{1}a_{2}, −1)_{Z}.

We may thus take

γ(σ) = (a_{1}, β_{1})_{k} + (a_{2}, β_{2})_{k} + (α, β_{1}β_{2}β_{3})_{k} + (a_{1}, a_{2})_{k} + (a_{1}a_{2}, −1)_{k}
= (a_{1}, −a_{2}β_{1})_{k} + (a_{2}, −β_{2})_{k} + (α, β_{1}β_{2}β_{3})_{k}.

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R. Preeti Université de Rennes 1 F-35042 Rennes, France preeti.raman@math.univrennes1.fr J.-P. Tignol Université catholique de Louvain B-1348 Louvain-la-Neuve, Belgium tignol@math.ucl.ac.be