The Arason Invariant

of Orthogonal Involutions of Degree 12 and 8,

and Quaternionic Subgroups of the Brauer Group

To Sasha Merkurjev on his 60th birthday

Anne Quéguiner-Mathieu^{1} and Jean-Pierre Tignol^{2}

Received: June 30, 2014

Abstract. Using the Rost invariant for torsors under Spin groups one may define an analogue of the Arason invariant for certain hermitian forms and orthogonal involutions. We calculate this invariant explicitly in various cases, and use it to associate to every orthogonal involution σ with trivial discriminant and trivial Clifford invariant over a central simple algebra A of even co-index an element f_{3}(σ) in the subgroup F^{×} · [A] of H^{3}(F, ℚ/ℤ(2)). This invariant f_{3}(σ) is the double of any representative of the Arason invariant e_{3}(σ) ∈ H^{3}(F, ℚ/ℤ(2))/F^{×}·[A]; it vanishes when deg A ≤ 10 and also when there is a quadratic extension of F that simultaneously splits A and makes σ hyperbolic. The paper provides a detailed study of both invariants, with particular attention to the degree 12 case, and to the relation with the existence of a quadratic splitting field.

As a main tool we establish, when deg(A) = 12, an additive decomposition of (A, σ) into three summands that are central simple algebras of degree 4 with orthogonal involutions with trivial discriminant, extending a well-known result of Pfister on quadratic forms of dimension 12 in I^{3}F. The Clifford components of the summands generate a subgroup U of the Brauer group of F, in which every element is represented by a quaternion algebra, except possibly the class of A. We show that the Arason invariant e_{3}(σ), when defined, generates the homology of a complex of degree 3 Galois cohomology groups, attached to the subgroup U, which was introduced and studied by Peyre. In the final section, we use the results on degree 12 algebras to extend the definition of the Arason invariant to trialitarian triples in which all three algebras have index at most 2.

2010 Mathematics Subject Classification: 11E72, 11E81, 16W10. Keywords and Phrases: Cohomological invariant, orthogonal group, algebra with involution, Clifford algebra.

1. Introduction

In quadratic form theory, the Arason invariant is a degree 3 Galois cohomology class with μ_{2} coefficients attached to an even-dimensional quadratic form with trivial discriminant and trivial Clifford invariant. Originally defined by Arason in [1], it can also be described in terms of the Rost invariant of a split Spin group, as explained in [26, § 31.B]. It is not always possible to extend this invariant to the more general setting of orthogonal involutions, see [7, §3.4]. Nevertheless, one may use the Rost invariant of some possibly non-split Spin groups to define relative and absolute Arason invariants for some orthogonal involutions (see [41] or section 2 below for precise definitions). This was first noticed by Bayer-Fluckiger and Parimala in [6], where they use the Rost invariant to prove classification theorems for hermitian or skew-hermitian forms, leading to a proof of the so-called Hasse Principle conjecture II. Below, we will focus on the absolute Arason invariant, which we will refer to simply as the Arason invariant. For orthogonal involutions, it was considered by Garibaldi, who uses the notation e_{3}^{hyp}, in [15], and by Berhuy in the index 2 case in [8]. In particular, the latter covers the case of central simple algebras of degree 2m with m odd, since such an algebra has index 1 or 2 when it is endowed with an orthogonal involution. Based on the Rost invariant for the exceptional group E_{8}, Garibaldi also defined, for orthogonal involutions on degree 16 central simple algebras, another invariant related to the Arason invariant of quadratic forms, denoted by e_{3}^{16}. Bermudez and Ruozzi [9] extended this definition to all degrees divisible by 16. It follows from the proof of Corollary 10.11 in [15], and Remark 4.10 in Barry’s paper [2], that these invariants do not coincide with what we call Arason invariant in this paper. A systematic study of the relative and absolute Arason invariants for orthogonal involutions was recently initiated in [35], where the degree 8 case is studied in detail. In this paper, we continue with an investigation of absolute invariants in degree 12. Let (A, σ) be a central simple algebra with orthogonal involution over a field F of characteristic different from 2. The Arason invariant e_{3}(σ), when defined, belongs to the quotient

H^{3}(F, ℚ/ℤ(2))/F^{×} · [A],

where F^{×} · [A] denotes the subgroup consisting of cup products (λ) · [A], for λ ∈ F^{×}, [A] the Brauer class of A. In § 2 below, we give a general formula for computing the Arason invariant of an algebra with involution admitting a rank 2 factor. It follows from this formula that the Arason invariant is not always represented by a cohomology class of order 2. This reflects the fact that the Dynkin index of a non-split Spin group, in large enough degree, is equal to 4. We define a new invariant f_{3}(σ) ∈ H^{3}(F, μ_{2}), attached to any orthogonal involution for which the Arason invariant is defined, and which vanishes if and only if the Arason invariant is represented by a cohomology class of order 2. This invariant is zero if the algebra is split, or of degree ≤ 10; starting in degree 12, we produce explicit examples where it is non-zero. This is an important motivation for studying the degree 12 case in detail. The main results of the paper are given in sections 3 to 5. First, we prove that a degree 12 algebra with orthogonal involution (A, σ), having trivial discriminant and trivial Clifford invariant, admits a non-unique decomposition as a sum—in the sense of algebras with involution—of three degree 4 algebras with orthogonal involution of trivial discriminant. This can be seen as a refinement of the main result of [17], even though our proof in index 4 relies on the openorbit argument of [17], see Remark 3.5. Using this additive decomposition, we associate to (A, σ) in a non-canonical way some subgroups of the Brauer group of F, which we call decomposition groups of (A, σ), see Definition 3.6. Such subgroups U ⊂ Br(F) are generated by (at most) three quaternion algebras; they were considered by Peyre in [32], where the homology of the following complex is studied:

1The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005.

2The second author is grateful to the first author and the Université Paris 13 for their hospitality while the work for this paper was carried out. He acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grants n^{∘} 1.5009.11 and 1.5054.12.

Documenta Mathematica · Extra Volume Merkurjev (2015) 529–576

F^{×} · U → H^{3}(F, ℚ/ℤ(2)) → H^{3}(F_{U}, ℚ/ℤ(2)),

where F^{×} · U denotes the subgroup generated by cup products (λ) · [B], for λ ∈ F^{×}, [B] ∈ U, and F_{U} is the function field of the product of the Severi-Brauer varieties associated to the elements of U. Peyre’s results are recalled in § 3.3. In § 4, we restrict to those algebras with involution of degree 12 for which the Arason invariant is defined, and we prove e_{3}(σ) detects isotropy of σ, and vanishes if and only if σ is hyperbolic. We then explore the relations between the decomposition groups and the values of the Arason invariant. Reversing the viewpoint, we also prove that the Arason invariant e_{3}(σ) provides a generator of the homology of Peyre’s complex, where (A, σ) is any algebra with involution admitting U as a decomposition group. In § 5, we give a necessary and sufficient condition for the vanishing of f_{3}(σ) in degree 12, in terms of decomposition groups of (A, σ). When there is a quadratic extension that splits A and makes σ hyperbolic, an easy corestriction argument shows that f_{3}(σ) = 0, see Proposition 2.5. We give in § 5.3 an explicit example to show that the converse does not hold. This also provides new examples of subgroups U for which the homology of Peyre’s complex is nontrivial, which differ from Peyre’s example in that the homology is generated by a Brauer class of order 2. In the last section, we extend the definition of the Arason invariant in degree 8 to index 2 algebras with involution of trivial discriminant, and such that the two components of the Clifford algebra have index 2. In this case also, the algebra with involution has an additive decomposition, and the Arason invariant detects isotropy.

Acknowledgements. Both authors thank A. Sivatski for pointing out a mistake in a preliminary version of the paper, and S. Garibaldi for useful comments.

Notation. Throughout this paper, we work over a base field F of characteristic different from 2. We use the notation H^{n}(F, M) = H^{n}(Gal(F_{sep}/F), M) for any discrete torsion Galois module M. For every integer n ≥ 0 we let

H^{n}(F) = H^{n}(F, ℚ/ℤ(n − 1)),

(see [16, Appendix A, p. 151] for a precise definition). The cohomology classes we consider actually are in the 2-primary part of these groups, hence we shall not need the modified definition for the p-primary part when char(F) = p ≠ 0. For each integer m ≥ 0 we let _{m}H^{n}(F) denote the m-torsion subgroup of H^{n}(F). Using the norm-residue isomorphism, one may check that

2H^{n}(F) = H^{n}(F, μ_{2})  and  _{4}H^{3}(F) = H^{3}(F, μ_{4}^{⊗2}),

(see for instance [32, Remark 4.1]). In particular, _{2}H^{1}(F) = F^{×}/F^{×2}. For every a ∈ F^{×} we let (a) ∈ _{2}H^{1}(F) be the square class of a. For a_{1}, …, a_{n} ∈ F^{×} we let (a_{1}, …, a_{n}) ∈ _{2}H^{n}(F) be the cup-product

(a_{1}, …, a_{n}) = (a_{1}) · · · · · (a_{n}).

We refer to [26] and to [28] for background information on central simple algebras with involution and on quadratic forms. However, we depart from the notation in [28] by letting ⟨⟨a_{1}, …, a_{n}⟩⟩ denote the n-fold Pfister form

⟨⟨a_{1}, …, a_{n}⟩⟩ = ⟨1, −a_{1}⟩ · · · · · ⟨1, −a_{n}⟩ for a_{1}, …, a_{n} ∈ F^{×}.

Thus, the discriminant, the Clifford invariant and the Arason invariant, viewed as cohomological invariants e_{1}, e_{2} and e_{3}, satisfy:

e_{1}(⟨⟨a_{1}⟩⟩) = (a_{1}), e_{2}(⟨⟨a_{1}, a_{2}⟩⟩) = (a_{1}, a_{2}), e_{3}(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩) = (a_{1}, a_{2}, a_{3}).

For every central simple F -algebra A, we let [A] be the Brauer class of A, which we identify to an element in H^{2}(F). If L is a field extension of F, we let A_{L} = A ⊗_{F} L be the L-algebra obtained from A by extending scalars. Recall that the object function from the category Fields_{F} of field extensions of F to abelian groups defined by

L ↦ ∐_{n≥0} H^{n}(L)

is a cycle module over SpecF (see [37, Rem.1.11]). In particular, each group H^{n}(L) is a module over the Milnor K-ring K_{∗}L. The Brauer class [A] of the algebra A generates a cycle submodule; we let M_{A} denote the quotient cycle module. Thus, for every field L ⊇ F, we have

M_{A}^{n}(L) = \begin{cases}H^{n}(L) & if n = 0 or 1; \\ H^{n}(L)/(K_{n−2}L · [A_{L}]) & if n ≥ 2. \end{cases}

In particular, M_{A}^{2}(L) = Br(L)/{0, [A_{L}]} and M_{A}^{3}(L) = H^{3}(L)/(L^{×} · [A_{L}]).

Let F_{A} denote the function field of the Severi–Brauer variety of A, which is a generic splitting field of A. Scalar extension from F to F_{A} yields group homomorphisms

M_{A}^{2}(F) → M_{A}^{2}(F_{A}) = Br(F_{A})  and  M_{A}^{3}(F) → M_{A}^{3}(F_{A}) = H^{3}(F_{A}).

The first map is injective by Amitsur’s theorem, see [18, Th. 5.4.1]; the second one is injective if the Schur index of A divides 4 or if A is a division algebra that decomposes into a tensor product of three quaternion algebras, but it is not always injective (see [32], [23] and [24]).

2. Cohomological invariants of orthogonal forms and involutions

Most of this section recalls well-known facts on absolute and relative Arason invariants that will be used in the sequel of the paper. Since we will consider additive decompositions of algebras with involution, we need to state the results both for hermitian forms and for involutions. Some new results are also included. In Proposition 2.6 and Corollary 2.13, we give a general formula for the Arason invariant of an algebra with involution which has a rank 2 factor. In Definitions 2.4 and 2.15, we introduce a new invariant, called the f_{3}-invariant, which detects whether the Arason invariant is represented by a cohomology class of order 2. Finally, we state and prove in Proposition 2.7 a general formula for computing the f_{3} invariant of a sum of hermitian forms, which is used in the proof of the main results of the paper. Throughout this section, D is a central division algebra over an arbitrary field F of characteristic different from 2, and θ is an F -linear involution on D (i.e., an involution of the first kind). To any nondegenerate hermitian or skew-hermitian module (V, h) over (D, θ) we may associate the corresponding adjoint algebra with involution Ad_{h} = (End_{D} V, ad_{h}). Conversely, any central simple algebra A over F Brauer-equivalent to D and endowed with an F -linear involution σ can be represented as (A, σ) ≃ Ad_{h} for some nondegenerate hermitian or skew-hermitian module (V, h) over (D, θ). The hermitian or skew-hermitian module (V, h) is said to be a hermitian module of orthogonal type if the adjoint involution ad_{h} on End_{D} V is of orthogonal type. This occurs if and only if either h is hermitian and θ is of orthogonal type, or h is skew-hermitian and θ is of symplectic type, see [26, (4.2)]. Abusing terminology, we also say that h is a hermitian form of orthogonal type when (V, h) is a hermitian module of orthogonal type (even though h may actually be skew-hermitian if θ is symplectic).

2.1. Invariants of hermitian forms of orthogonal type. Let (V, h) be a hermitian module of orthogonal type over (D, θ); we call r = dim_{D} V the relative rank of h and n = deg End_{D} V the absolute rank of h. These invariants are related by n = r deg D. Cohomological invariants of h are defined in terms of invariants of the adjoint involution ad_{h}. Namely, if n is even, the discriminant of h, denoted e_{1}(h) ∈ H^{1}(F, μ_{2}), is the discriminant of ad_{h}; the corresponding quadratic étale extension K/F is called the discriminant extension. If n is even and e_{1}(h) is trivial, the Clifford invariant of h, denoted e_{2}(h), is the class in M_{D}^{2}(F) of any component of the Clifford algebra of ad_{h}.

Remark 2.1. It follows from the relations between the components of the Clifford algebra (see [26, (9.12)]) that the Clifford invariant is well-defined. However, since we do not assume n is divisible by 4, this invariant need not be represented by a cohomology class of order 2 in general.

Our definitions of rank and discriminant differ slightly from the definitions used by Bayer and Parimala in [5, §2], who call “rank” what we call the relative rank of h. The discriminant d(h) of h in the sense of [5, §2.1] is related to e_{1}(h) by

e_{1}(h) = d(h) disc(θ)^{r},

where disc(θ) is the discriminant of θ as defined in [26, §7], and H^{1}(F, μ_{2}) is identified with the group of square classes F^{×}/F^{×2}. In particular, e_{1}(h) = d(h) when h has even relative rank r. By [5, 2.1.3], the Clifford invariant Cℓ(h) used by Bayer and Parimala coincides with our e_{2}(h) when they are both defined, i.e., when h has even relative rank and trivial discriminant. Assume now that the hermitian form h has even relative rank, i.e., dim_{D} V is even. The vector space V then carries a hyperbolic hermitian form h_{0} of orthogonal type, and the standard nonabelian Galois cohomology technique yields a canonical bijection between H^{1}(F, O(h_{0})) and the set of isomorphism classes of nondegenerate hermitian forms of orthogonal type on V, under which the trivial torsor corresponds to the isomorphism class of h_{0}, see [26, §29.D]. If e_{1}(h) and e_{2}(h) are trivial, the torsor corresponding to the isomorphism class of h has two different lifts to H^{1}(F, O^{+}(h_{0})), and one of these lifts can be further lifted to a torsor ξ in H^{1}(F, Spin(h_{0})). Bayer and Parimala consider the Rost invariant R(ξ) ∈ H^{3}(F) and define in [6, §3.4, p. 664] an Arason invariant of h by the formula

e_{3}(h) = R(ξ) + F^{×} · [D] ∈ M_{D}^{3}(F).

This invariant satisfies the following properties:

Lemma 2.2 (Bayer–Parimala [6, Lemma 3.7, Corollary 3.9]). Let h and h′ be two hermitian forms of orthogonal type over (D, θ) with even relative rank, trivial discriminant, and trivial Clifford invariant.

(i) If h is hyperbolic, then e_{3}(h) = 0; (ii) e_{3}(h ⊥ h′) = e_{3}(h) + e_{3}(h′); (iii) e_{3}(λh) = e_{3}(h) for any λ ∈ F^{×}.

In particular, it follows immediately that e_{3}(h) is a well-defined invariant of the Witt class of h. Moreover, we have:

Corollary 2.3. The Arason invariant e_{3} has order 2.

Proof. Indeed, for any h as above, we have 2e_{3}(h) = e_{3}(h) + e_{3}(h) = e_{3}(h) + e_{3}(−h) = e_{3}(h ⊥ (−h)) = 0, since h ⊥ (−h) is hyperbolic. □

Using the properties of the Arason invariant, we may define a new invariant as follows. Assume h is as above, a hermitian form of orthogonal type with even relative rank, trivial discriminant, and trivial Clifford invariant. Let c, c′ ∈ H^{3}(F) be two representatives of the Arason invariant e_{3}(h). Since c − c′ ∈ F^{×} · [D], we have 2c = 2c′ ∈ H^{3}(F), hence 2c depends only on h and not on the choice of the representative c of e_{3}(h). Because of Corollary 2.3, the image of 2c in M_{D}^{3}(F) vanishes, hence 2c ∈ F^{×} · [D]. These observations lead to the following definition:

Definition 2.4. Given an arbitrary representative c ∈ H^{3}(F) of the Arason invariant e_{3}(h) ∈ M_{D}^{3}(F), we let f_{3}(h) = 2c ∈ F^{×} · [D] ⊂ _{2}H^{3}(F).

Thus, the invariant f_{3}(h) is well-defined; it vanishes if and only if the Arason invariant e_{3}(h) is represented by a class of order at most 2, or equivalently, if every representative of e_{3}(h) is a cohomology class of order at most 2. It is clear from the definition that the f_{3} invariant is trivial when D is split. Another case where the f_{3} invariant vanishes is the following:

Proposition 2.5. If there exists a quadratic extension K/F such that D_{K} is split and h_{K} is hyperbolic, then f_{3}(h) = 0.

Proof. Assume such a field K exists, and let c ∈ H^{3}(F) be any representative of e_{3}(h) ∈ M_{D}^{3}(F). Since h_{K} is hyperbolic, we have e_{3}(h_{K}) = c_{K} = 0 ∈ M_{D}^{3}(K) = H^{3}(K). Hence, cor_{K/F} (c_{K}) = 2c = 0, that is f_{3}(h) = 0. □

In §5, we will see that the f_{3} invariant is trivial if the absolute rank is ≤ 10. In addition, we will prove that the converse of Proposition 2.5 does not hold, even in absolute rank 12. Since the Dynkin index of the group Spin(Ad_{h_{0}}) divides 4, the Arason invariant e_{3}(h) is represented by a cohomology class of order dividing 4. Moreover, there are examples where it is represented by a cohomology class of order equal to 4. Therefore, f_{3}(h) is nonzero in general. Explicit examples can be constructed by means of Proposition 2.6 below, which yields the e_{3} and f_{3} invariants of hermitian forms with a rank 2 factor. (See also Corollary 2.19 for examples in the lowest possible degree, which is 12.)

2.2. Hermitian forms with a rank 2 factor. Consider a hermitian form which admits a decomposition as ⟨1, −λ⟩ ⊗ h for some λ ∈ F^{×} and some hermitian form h. In this case, we have the following explicit formulae for the Arason and the f_{3}-invariant, when they are defined:

Proposition 2.6. Let h be a hermitian form of orthogonal type with even absolute rank n, and let K/F be the discriminant quadratic extension. For any μ ∈ K^{×}, the hermitian form ⟨1, −N_{K/F} (μ)⟩h has even relative rank, trivial discriminant and trivial Clifford invariant. Moreover,

e_{3}(⟨1, −N_{K/F} (μ)⟩h) = cor_{K/F} (μ · e_{2}(h_{K}))

and

f_{3}(⟨1, −N_{K/F} (μ)⟩h) = \begin{cases}0 & if n ≡ 0 mod 4, \\ N_{K/F} (μ) · [D] & if n ≡ 2 mod 4. \end{cases}

In particular, if h has trivial discriminant, then for λ ∈ F^{×} we have

e_{3}(⟨1, −λ⟩h)= λ · e_{2}(h)

and

f_{3}(⟨1, −λ⟩h)= \begin{cases}0 & if n ≡ 0 mod 4, \\ λ · [D] & if n ≡ 2 mod 4. \end{cases}

Proof. We first need to prove that the hermitian form ⟨1, −N_{K/F} (μ)⟩h has trivial discriminant and trivial Clifford invariant. This can be checked after scalar extension to a generic splitting field of D, since the corresponding restriction maps _{2}H^{1}(F) → _{2}H^{1}(F_{D}) and M_{D}^{2}(F) → H^{2}(F_{D}) are injective. In the split case, the result follows from an easy computation for the discriminant, and from [28, Ch. V, §3] for the Clifford invariant. Alternatively, one may observe that the algebra with involution Ad_{⟨1,−N_{K/F} (μ)⟩h} decomposes as Ad_{⟨1,−N_{K/F} (μ)⟩} ⊗ Ad_{h}, and apply [26, (7.3)(4)] and [39]. This computation also applies to the trivial discriminant case, where λ = N_{F ×F/F} (λ, 1). With this in hand, we may compute the Arason invariant by using the description of cohomological invariants of quasi-trivial tori given in [30]. Let us first assume h has trivial discriminant. Consider the multiplicative group scheme 𝔾_{m} as a functor from the category Fields_{F} to the category of abelian groups. For any field L containing F, consider the map

φ_{L}: 𝔾_{m}(L) → M_{D}^{3}(L) defined by λ ↦ e_{3}(⟨1, −λ⟩h_{L}).

To see that φ_{L} is a group homomorphism, observe that in the Witt group of D_{L} we have for λ_{1}, λ_{2} ∈ L^{×}

⟨1, −λ_{1}λ_{2}⟩h_{L} = ⟨1, −λ_{1}⟩h_{L} + ⟨λ_{1}⟩⟨1, −λ_{2}⟩h_{L}.

Therefore, Lemma 2.2 yields

e_{3}(⟨1, −λ_{1}λ_{2}⟩h_{L}) = e_{3}(⟨1, −λ_{1}⟩h_{L}) + e_{3}(⟨1, −λ_{2}⟩h_{L}).

The collection of maps φ_{L} defines a natural transformation of functors 𝔾_{m} → M_{D}^{3}, i.e., a degree 3 invariant of 𝔾_{m} with values in the cycle module M_{D}. By [29, Prop. 2.5], there is an element u ∈ M_{D}^{2}(F) such that for any L and any λ ∈ L^{×}

φ_{L}(λ) = λ · u_{L} in M_{D}^{3}(L).

To complete the computation of e_{3}(⟨1, −λ⟩h), it only remains to show that u = e_{2}(h). Since the restriction map M_{D}^{2}(F) → M_{D}^{2}(F_{D}) = H^{2}(F_{D}) is injective, it suffices to show that u_{F_{D}} = e_{2}(h)_{F_{D}}. Now, since F_{D} is a splitting field for D, there exists a quadratic form q over F_{D}, with trivial discriminant, such that (Ad_{h})_{F_{D}} ≃ Ad_{q}. Let t be an indeterminate over F_{D}. We have

Ad_{⟨1,−t⟩} ⊗(Ad_{h})_{F_{D}(t)} ≃ Ad_{⟨1,−t⟩} ⊗(Ad_{q})_{F_{D}(t)}

hence e_{3}(⟨1, −t⟩h_{F_{D}(t)}) is the Arason invariant of the quadratic form ⟨1, −t⟩q_{F_{D}(t)}, which is t · e_{2}(q) = t · e_{2}(h)_{F_{D}(t)}. Therefore, we have

t · u_{F_{D}(t)} = t · e_{2}(h)_{F_{D}(t)}.

Taking the residue ∂: H^{3}(F_{D}(t)) → H^{2}(F_{D}) for the t-adic valuation, we obtain u_{F_{D}} = e_{2}(h)_{F_{D}}, which completes the proof of the formula for e_{3}(⟨1, −λ⟩h). To compute f_{3}(⟨1, −λ⟩h), recall that e_{2}(h) is represented by any of the two components C_{+}, C_{−} of the Clifford algebra of Ad_{h}. Therefore, e_{3}(⟨1, −λ⟩h) is represented by λ · [C_{+}] or λ · [C_{−}], and

f_{3}(⟨1, −λ⟩h) = 2(λ · [C_{+}]) = 2(λ · [C_{−}]).

By [26, (9.12)] we have

2[C_{+}] = 2[C_{−}] = \begin{cases}0 & if n ≡ 0 mod 4, \\ [D] & if n ≡ 2 mod 4. \end{cases}

The formula for f_{3}(⟨1, −λ⟩h) follows. Assume now h has nontrivial discriminant. The proof in this case follows the same pattern. Let K/F be the discriminant field extension. We consider the group scheme R_{K/F} (𝔾_{m}), which is the Weil transfer of the multiplicative group. For every field L containing F, the map

μ ∈ R_{K/F} (𝔾_{m})(L) = (L ⊗_{F} K)^{×} ↦ e_{3}(⟨1, −N_{L⊗K/L}(μ)⟩h_{L}) ∈ M_{D}^{3}(L)

defines a degree 3 invariant of the quasi-trivial torus R_{K/F} (𝔾_{m}) with values in the cycle module M_{D}. By [30, Th. 1.1], there is an element u ∈ M_{D}^{2}(K) such that for any field L containing F and any μ ∈ (L ⊗ K)^{×},

e_{3}(⟨1, −N_{L⊗K/L}(μ)⟩h_{L}) = cor_{L⊗K/L}(μ · u_{L⊗K}) in M_{D}^{3}(L).

It remains to show that u = e_{2}(h_{K}). To prove this, we consider the field L = K(t), where t is an indeterminate. Since e_{1}(h_{K(t)}) = 0, the previous case applies. We thus get for any μ ∈ (K(t) ⊗_{F} K)^{×}

N_{K(t)⊗K/K(t)}(μ) · e_{2}(h_{K(t)}) = cor_{K(t)⊗K/K(t)}(μ · u_{K(t)⊗K}) in M_{D}^{3}(K(t)).

Let ι be the nontrivial F -automorphism of K. The K(t)-algebra isomorphism K(t) ⊗_{F} K ≃ K(t) × K(t) mapping α ⊗ β to (αβ, αι(β)) yields an isomorphism M_{D}^{2}(K(t) ⊗ K) ≃ M_{D}^{2}(K(t)) × M_{D}^{2}(K(t)) that carries u_{K(t)⊗K} to (u_{K(t)}, ι(u)_{K(t)}). Thus, for every (μ_{1}, μ_{2}) ∈ K(t)^{×} × K(t)^{×},

μ_{1}μ_{2} · e_{2}(h_{K(t)}) = μ_{1} · u_{K(t)} + μ_{2} · ι(u)_{K(t)} in M_{D}^{3}(K(t)).

In particular, if μ_{1} = t and μ_{2} = 1 we get t · e_{2}(h_{K(t)}) = t · u_{K(t)}, hence taking the residue for the t-adic valuation yields e_{2}(h_{K}) = u, proving the formula for e_{3}(⟨1, −N_{K/F} (μ)⟩h). To complete the proof, we compute f_{3}(⟨1, −N_{K/F} (μ)⟩h). Let C be the Clifford algebra of Ad_{h}, so [C] represents e_{2}(h_{K}) and

f_{3}(⟨1, −N_{K/F} (μ)⟩h) = 2 cor_{K/F} (μ · [C]).

By [26, (9.12)] we have

2[C] = \begin{cases}0 & if n ≡ 0 mod 4, \\ [D_{K}] & if n ≡ 2 mod 4. \end{cases}

The formula for f_{3}(⟨1, −N_{K/F} (μ)⟩h) follows by the projection formula. □

2.3. Hermitian forms with an additive decomposition. We now present another approach for computing the f_{3}-invariant, which does not rely on the computation of the Arason invariant. This leads to an explicit formula in a more general situation, which will be used in the proof of Theorem 5.4:

Proposition 2.7. Let (V_{1}, h_{1}), …, (V_{m}, h_{m}) be hermitian modules of orthogonal type and even absolute rank n_{1}, …, n_{m} over (D, θ), and let λ_{1}, …, λ_{m} ∈ F^{×}. Let also h = ⟨1, −λ_{1}⟩h_{1} ⊥ … ⊥ ⟨1, −λ_{m}⟩h_{m}. If ∑_{i=1}^{m} λ_{i}·e_{1}(h_{i}) = 0, then h has trivial Clifford invariant, and

f_{3}(h) = λ_{1}^{n_{1}/2} … λ_{m}^{n_{m}/2} · [D].

To prove this proposition, we need some preliminary results. Let (V, h_{0}) be a hyperbolic module of orthogonal type over (D, θ). Recall from [26, (13.31)] the canonical map (“vector representation”)

χ: Spin(h_{0}) → O^{+}(h_{0}).

Since proper isometries have reduced norm 1, we also have the inclusion

i: O^{+}(h_{0}) → SL(V).

Lemma 2.8. The following diagram, where R is the Rost invariant, is commutative:

\xymatrix{H^{1}(F, Spin(h_{0})) \ar[r]^{(i∘χ)_{∗}} \ar[d]_{R} & H^{1}(F, SL(V)) \ar[d]^{R} \\ H^{3}(F) \ar[r]^{2} & H^{3}(F)}

Proof. This lemma is just a restatement of the property that the Rost multiplier of the map i ∘ χ is 2, see [16, Ex. 7.15, p. 124]. □

We next recall from [26, (29.27)] (see also [13]) the canonical description of the pointed set H^{1}(F, O^{+}(h_{0})). Define a functor SSym(h_{0}) from Fields_{F} to the category of pointed sets as follows: for any field L containing F, set

SSym(h_{0})(L) = {(s, λ) ∈ GL(V_{L}) × L^{×} | ad_{h_{0}}(s) = s and Nrd(s) = λ^{2}},

where the distinguished element is (1, 1). Let F_{s} be a separable closure of F and let Γ = Gal(F_{s}/F) be the Galois group. We may identify SSym(h_{0})(F_{s}) with the quotient GL(V_{F_{s}})/ O^{+}((h_{0})_{F_{s}}) by mapping a class a · O^{+}((h_{0})_{F_{s}}) to (a ad_{h_{0}}(a), Nrd(a)) for a ∈ GL(V_{F_{s}}). Therefore, we have an exact sequence of pointed Γ-sets

1 → O^{+}((h_{0})_{F_{s}}) → GL(V_{F_{s}}) → SSym(h_{0})(F_{s}) → 1.

Since H^{1}(F, GL(V_{F_{s}})) = 1 by Hilbert’s Theorem 90, the induced exact sequence in Galois cohomology yields a canonical bijection between H^{1}(F, O^{+}(h_{0})) and the orbit set of GL(V) on SSym(h_{0})(F). Abusing notation, we write simply SSym(h_{0}) for SSym(h_{0})(F). The orbits of GL(V) on SSym(h_{0}) are the equivalence classes under the following relation:

(s, λ) ∼ (s′, λ′) if s′ = as ad_{h_{0}}(a) and λ′ = λ Nrd(a) for some a ∈ GL(V).

Therefore, we may identify

H^{1}(F, O^{+}(h_{0})) = SSym(h_{0})/∼.

Lemma 2.9. The composition H^{1}(F, O^{+}(h_{0})) ⟶^{i_{∗}} H^{1}(F, SL(V)) ⟶^{R} H^{3}(F) maps the equivalence class of (s, λ) to λ · [A].

Proof. Let π: SSym(h_{0})(F_{s}) → F_{s}^{×} be the projection (s, λ) ↦ λ. We have a commutative diagram of pointed Γ-sets with exact rows:

[figure: 1; O^{+}((h_{0})_{F_{s}}); i; GL(V_{F_{s}}); SSym(h_{0})(F_{s}); π; 1; 1; SL(V_{F_{s}}); GL(V_{F_{s}}); Nrd; F_{s}^{×}; 1]

This diagram yields the following commutative square in cohomology:

\xymatrix{SSym(h_{0}) \ar[r] \ar[d]_{π} & H^{1}(F, O^{+}(h_{0})) \ar[d]^{i_{∗}} \\ F^{×} \ar[r] & H^{1}(F, SL(V))}

On the other hand, the Rost invariant and the map F^{×} → H^{3}(F) carrying λ to λ · [A] fit in the following commutative diagram (see [26, p. 437]):

\xymatrix{F^{×} \ar[rr] \ar[dr] &  & H^{1}(F, SL(V)) \ar[dl]^{R} \\  & H^{3}(F)}

The lemma follows. □

For the next statement, let ∂: H^{1}(F, O^{+}(h_{0})) → _{2}H^{2}(F) be the connecting map in the cohomology exact sequence associated to

1 → μ_{2} → Spin(h_{0}) ⟶^{χ} O^{+}(h_{0}) → 1.

For any hermitian form h of orthogonal type on V, there exists a unique linear transformation s ∈ GL(V) such that h(x, y) = h_{0}(x, s^{−1}(y)) for all x, y ∈ V, hence ad_{h} = Int(s) ∘ ad_{h_{0}} and ad_{h_{0}}(s) = s. If the discriminant of h is trivial we have Nrd(s) ∈ F^{×2}, hence there exists λ ∈ F^{×} such that λ^{2} = Nrd(s), and we may consider (s, λ) and (s, −λ) ∈ SSym(h_{0}). By the main theorem of [13], ∂(s, λ) and ∂(s, −λ) are the Brauer classes of the two components of the Clifford algebra of Ad_{h_{0}⊥−h}, so if the Clifford invariant of h is trivial we have

{∂(s, λ), ∂(s, −λ)} = {0, [D]}.

Lemma 2.10. With the notation above, we have f_{3}(h) = λ · [D] if ∂(s, λ) = 0.

Proof. By definition of s, the torsor in H^{1}(F, O(h_{0})) corresponding to h lifts to (s, λ) ∈ H^{1}(F, O^{+}(h_{0})). If ∂(s, λ) = 0, then (s, λ) lifts to some ξ ∈ H^{1}(F, Spin(h_{0})), and by definition of the invariants e_{3} and f_{3} we have

e_{3}(h) = R(ξ) + F^{×} · [D] ∈ M_{D}^{3}(F)  and  f_{3}(h) = 2R(ξ) ∈ H^{3}(F).

Lemma 2.8 then yields f_{3}(h) = R ∘ (i ∘ χ)_{∗}(ξ) = R ∘ i_{∗}(s, λ), and by Lemma 2.9 we have R ∘ i_{∗}(s, λ) = λ · [D]. □

In order to check the condition ∂(s, λ) = 0 in Lemma 2.10, the following observation is useful: Suppose (V_{1}, h_{1}) and (V_{2}, h_{2}) are hermitian modules of orthogonal type over (D, θ). The inclusions V_{i} ↪ V_{1} ⊥ V_{2} yield an F -algebra homomorphism C(Ad_{h_{1}}) ⊗_{F} C(Ad_{h_{2}}) → C(Ad_{h_{1}⊥h_{2}}), which induces a group homomorphism Spin(h_{1}) × Spin(h_{2}) → Spin(h_{1} ⊥ h_{2}). This homomorphism fits into the following commutative diagram with exact rows

\xymatrix{1 \ar[r] & μ_{2} × μ_{2} \ar[r] \ar[d]_{∏} & Spin(h_{1}) × Spin(h_{2}) \ar[r]^{χ_{1}×χ_{2}} \ar[d] & O^{+}(h_{1}) × O^{+}(h_{2}) \ar[r] \ar[d]^{⊕} & 1 \\ 1 \ar[r] & μ_{2} \ar[r] & Spin(h_{1} ⊥ h_{2}) \ar[r]^{χ} & O^{+}(h_{1} ⊥ h_{2}) \ar[r] & 1}

The left vertical map is the product, and the right vertical map carries (g_{1}, g_{2}) to g_{1} ⊕ g_{2}. The induced diagram in cohomology yields the commutative square

\xymatrix{H^{1}(F, O^{+}(h_{1})) × H^{1}(F, O^{+}(h_{2})) \ar[r]^{∂_{1}×∂_{2}} \ar[d]_{⊕} & _{2}H^{2}(F) ×_{2}H^{2}(F) \ar[d]^{∏} \\ H^{1}(F, O^{+}(h_{1} ⊥ h_{2})) \ar[r]^{∂} & _{2}H^{2}(F)}

The following additivity property of the connecting maps ∂ follows: for (s_{1}, λ_{1}) ∈ H^{1}(F, O^{+}(h_{1})) and (s_{2}, λ_{2}) ∈ H^{1}(F, O^{+}(h_{2})),

(1)  ∂_{1}(s_{1}, λ_{1}) + ∂_{2}(s_{2}, λ_{2}) = ∂(s_{1} ⊕ s_{2}, λ_{1}λ_{2}).

Proof of Proposition 2.7. Let h_{0} = ⟨1, −1⟩h_{1} ⊥ … ⊥ ⟨1, −1⟩h_{m}, which is a hyperbolic form, and let V = V_{1}^{⊕2}⊕⋯⊕V_{m}^{⊕2} be the underlying vector space of h and h_{0}. The linear transformation s ∈ GL(V) such that h(x, y) = h_{0}(x, s^{−1}(y)) for all x, y ∈ V is

s = 1 ⊕ λ_{1}^{−1} ⊕ 1 ⊕ λ_{2}^{−1} ⊕ ⋯ ⊕ 1 ⊕ λ_{m}^{−1}.

By the additivity property (1), the connecting map

∂: H^{1}(F, O^{+}(h_{0})) → _{2}H^{2}(F)

satisfies

∂(s, λ_{1}^{−n_{1}/2} … λ_{m}^{−n_{m}/2}) = ∂_{1}(λ_{1}^{−1}, λ_{1}^{−n_{1}/2}) + ⋯ + ∂_{m}(λ_{m}^{−1}, λ_{m}^{−n_{m}/2}).

A theorem of Bartels [4, p. 283] (see also [13]) yields ∂_{i}(λ_{i}^{−1}, λ_{i}^{−n_{i}/2}) = λ_{i}^{−1} · e_{1}(h_{i}) for all i. Therefore, if ∑_{i=1}^{m} λ_{i} · e_{1}(h_{i}) = 0 we have f_{3}(h) = λ_{1}^{n_{1}/2} … λ_{m}^{n_{m}/2} · [D] by Lemma 2.10. □

2.4. Relative Arason invariant of hermitian forms of orthogonal type. By using the Rost invariant, one may also define a relative Arason invariant, in a broader context:

Definition 2.11. Let h_{1} and h_{2} be two hermitian forms of orthogonal type over (D, θ) such that their difference h_{1} + (−h_{2}) has even relative rank, trivial discriminant, and trivial Clifford invariant. Their relative Rost invariant is defined by

e_{3}(h_{1}/h_{2}) = e_{3}(h_{1} ⊥ (−h_{2}))∈ M_{D}^{3}(F).

In particular, if both h_{1} and h_{2} have even relative rank, trivial discriminant, and trivial Clifford invariant, then e_{3}(h_{1}/h_{2}) = e_{3}(h_{1}) + e_{3}(h_{2}) = e_{3}(h_{1}) − e_{3}(h_{2}).

Remark 2.12. Under the conditions of this definition, one may check that the involution ad_{h_{2}} corresponds to a torsor which can be lifted to a Spin(Ad_{h_{1}}) torsor (see [41, §3.5]). As explained in [6, Lemma 3.6], the relative Arason invariant e_{3}(h_{1}/h_{2}) coincides with the class in M_{D}^{3}(F) of the image of this torsor under the Rost invariant of Spin(Ad_{h_{1}}).

Combining the properties of the Arason invariant recalled in Lemma 2.2 and the computation of Proposition 2.6, we obtain:

Corollary 2.13. (i) Let h be a hermitian form of orthogonal type with even absolute rank, and let K/F be the discriminant quadratic extension. For any μ ∈ K^{×}, the relative Arason invariant e_{3}(⟨N_{K/F} (μ)⟩h/h) is well-defined, and

e_{3}(⟨N_{K/F} (μ)⟩h/h) = cor_{K/F} (μ · e_{2}(h_{K})).

(ii) Let h_{1} and h_{2} be two hermitian forms of orthogonal type with even absolute rank and trivial discriminant. We have

e_{3}(h_{1} ⊥ ⟨λ⟩h_{2}/h_{1} ⊥ h_{2}) = e_{3}(⟨λ⟩h_{2}/h_{2}) = λ · e_{2}(h_{2}).

2.5. Arason and f_{3} invariants of orthogonal involutions. Let (A, σ) be an algebra with orthogonal involution, Brauer-equivalent to the division algebra D over F. We pick an involution θ on D, so that (A, σ) can be represented as the adjoint (A, σ) ≃ Ad_{h} of some hermitian module (V, h) over (D, θ). The co-index of A, which is the dimension over D of the module V, is equal to the relative rank of h. If the form h has even relative rank, trivial discriminant, and trivial Clifford invariant, then its Arason invariant is well-defined. Moreover, by Lemma 2.2, we have e_{3}(h) = e_{3}(λh) for any λ ∈ F^{×}, and, as explained in [6, Prop 3.8], e_{3}(h) does not depend on the choice of θ. Therefore, we get a well-defined Arason invariant for the involution σ, provided the algebra A has even co-index, i.e. deg(A)/ ind(A) = deg(A)/ deg(D) ∈ 2ℤ, and the involution σ has trivial discriminant and trivial Clifford invariant:

e_{3}(σ) = e_{3}(h) ∈ M_{A}^{3}(F) = M_{D}^{3}(F).

Remarks 2.14. (1) Under the assumptions above on (A, σ), one may also check that the algebra A carries a hyperbolic orthogonal involution σ_{0}, and the Arason invariant e_{3}(σ) can be defined directly in terms of the Rost invariant of the group Spin(A, σ_{0}), see [41, §3.5].

(2) Similarly, we may also define a relative Arason invariant e_{3}(σ_{1}/σ_{2}) if the involutions σ_{1} and σ_{2} both have trivial discriminant and trivial Clifford invariant. But we cannot relax those assumptions, as we did for hermitian forms. Indeed, if e_{2}(h_{2}) = e_{2}(ad_{h_{2}}) is not trivial, then e_{3}(⟨λ⟩h_{1}/h_{2}) and e_{3}(h_{1}/h_{2}) are generally different, as Corollary 2.13 shows.

In the setting above, we may also define an f_{3}-invariant by f_{3}(σ) = f_{3}(h), or equivalently:

Definition 2.15. Let (A, σ) be an algebra with orthogonal involution. We assume A has even co-index, and σ has trivial discriminant and trivial Clifford invariant. We define f_{3}(σ) ∈ F^{×} · [A] ⊂ _{2}H^{3}(F) by f_{3}(σ) = 2c, where c is any representative of the Arason invariant e_{3}(σ) ∈ M_{A}^{3}(F).

Remark 2.16. (i) If A is split, then F^{×} · [A] = {0}, and f_{3}(ad_{φ}) = 0 for all quadratic forms φ ∈ I^{3}(F). This also follows from the fact that e_{3}(φ) ∈ 2H^{3}(F). (ii) Using the same process, one may define an invariant f_{3}^{16} from Garibaldi’s invariant e_{3}^{16}, and from Bermudez-Ruozzi’s generalization (see [15], [9]). This invariant has values in _{2}H^{3}(F), but need not have values in F^{×} · [A] in general.

Example 2.17. Let Q be a quaternion algebra over F, and consider the algebra with involution (A, σ) = (Q, ρ) ⊗ Ad_{φ}, where ρ is an orthogonal involution with discriminant δ · F^{×2} ∈ F^{×}/F^{×2}, and φ is a even-dimensional quadratic form with trivial discriminant. We have e_{3}(σ) = δ · e_{2}(φ) mod F^{×} · [Q], and f_{3}(σ) = 0. Indeed, since the restriction map M_{Q}^{3}(F) → M_{Q}^{3}(F_{Q}) = H^{3}(F_{Q}) is injective, it is enough to check the formula in the split case, where it follows from a direct computation.

The computation in Proposition 2.6 can be again rephrased as follows:

Corollary 2.18. Let (A, σ) be a central simple F -algebra of even degree n with orthogonal involution, and let K/F be the discriminant quadratic extension. For any μ ∈ K^{×}, the algebra with involution Ad_{⟨1,−N_{K/F} (μ)⟩} ⊗(A, σ) has even co-index, trivial discriminant and trivial Clifford invariant. Its Arason invariant is given by

e_{3}(ad_{⟨1,−N_{K/F} (μ)⟩} ⊗σ) = cor_{K/F} (μ · e_{2}(σ_{K})),

and

f_{3}(ad_{⟨1,−N_{K/F} (μ)⟩} ⊗σ) = \begin{cases}0 & if n ≡ 0 mod 4, \\ N_{K/F} (μ) · [A] & if n ≡ 2 mod 4. \end{cases}

In particular, if σ has trivial discriminant, we have for any λ ∈ F^{×}

e_{3}(ad_{⟨1,−λ⟩} ⊗σ) = λ · e_{2}(σ)

and

f_{3}(ad_{⟨1,−λ⟩} ⊗σ) = \begin{cases}0 & if n ≡ 0 mod 4, \\ λ · [A] & if n ≡ 2 mod 4. \end{cases}

Hence, the formula given in [35, Th. 5.5] for algebras of degree 8 is actually valid in arbitrary degree. With this in hand, one may easily check that the f_{3} invariant is trivial up to degree 10. Indeed, since the co-index of the algebra is supposed to be even, the algebra is possibly non-split only when its degree is divisible by 4. In degree 4, any involution with trivial discriminant and Clifford invariant is hyperbolic, hence has trivial invariants. In degree 8, any involution with trivial discriminant and Clifford invariant admits a decomposition as in Corollary 2.18 by [35, Th. 5.5], hence its f_{3} invariant is trivial. In degree 12, one may construct explicit examples of (A, σ) with f_{3}(σ) ≠ 0 as follows. Suppose E is a central simple F -algebra of degree 4. Recall from [26, §10.B] that the second λ-power λ^{2}E is a central simple F -algebra of degree 6, which carries a canonical involution γ of orthogonal type with trivial discriminant, and which is Brauer-equivalent to E ⊗_{F} E.

Corollary 2.19. Let E be a central simple F -algebra of degree and exponent 4. Pick an indeterminate t, and consider the algebra with involution

(A, σ) = Ad_{⟨1,−t⟩} ⊗(λ^{2}E, γ)_{F (t)}.

We have

f_{3}(σ) = t · [A] ≠ 0 ∈ H^{3}(F (t)).

Proof. The formula f_{3}(A, σ) = t · [A] readily follows from Corollary 2.18. The algebra E has exponent 4, therefore [A] = [E ⊗_{F} E] ≠ 0. Since t is an indeterminate, we get t · [A] ≠ 0. □

3. Additive decompositions in degree 12

In the next three sections, we concentrate on degree 12 algebras (A, σ) with orthogonal involution of trivial discriminant and trivial Clifford invariant. The main result of this section is Theorem 3.2, which generalizes a theorem of Pfister on 12-dimensional quadratic forms.

3.1. Additive decompositions. Given three algebras with involution, (A, σ), (A_{1}, σ_{1}) and (A_{2}, σ_{2}), we say that (A, σ) is a direct sum of (A_{1}, σ_{1}) and (A_{2}, σ_{2}), and we write

(A, σ) ∈ (A_{1}, σ_{1}) ⊞(A_{2}, σ_{2}),

if there exist a division algebra with involution (D, θ) and hermitian modules (V_{1}, h_{1}) and (V_{2}, h_{2}) over (D, θ), which are both hermitian or both skewhermitian, such that (A_{1}, σ_{1}) = Ad_{h_{1}}, (A_{2}, σ_{2}) = Ad_{h_{2}} and (A, σ) = Ad_{h_{1}⊥h_{2}}. In particular, this implies A, A_{1} and A_{2} are all three Brauer-equivalent to D, and the involutions σ, σ_{1} and σ_{2} are of the same type. This notion of direct sum for algebras with involution was introduced by Dejaiffe in [10]. As explained there, the algebra with involution (A, σ) is generally not uniquely determined by the data of the two summands (A_{1}, σ_{1}) and (A_{2}, σ_{2}). Indeed, multiplying the hermitian forms h_{1} and h_{2} by a scalar does not change the adjoint involutions, so the adjoint of λ_{1}h_{1} ⊥ λ_{2}h_{2} also is a direct sum of (A_{1}, σ_{1}) and (A_{2}, σ_{2}) for any λ_{1}, λ_{2} ∈ F^{×}. If one of the two summands, say (A_{1}, σ_{1}) = (A_{1}, hyp) is hyperbolic, then all hermitian forms similar to h_{1} actually are isomorphic to h_{1}. Hence in this case, there is a unique direct sum, and we may write

(A, σ) = (A_{1}, hyp) ⊞ (A_{2}, σ_{2}).

The cohomological invariants we consider, when defined, have the following additivity property:

Proposition 3.1. Suppose σ, σ_{1}, σ_{2} are orthogonal involutions such that (A, σ) ∈ (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2}). We have:

(i) deg A = deg A_{1} + deg A_{2}.

(ii) If deg A_{1} ≡ deg A_{2} ≡ 0 mod 2, then e_{1}(σ) = e_{1}(σ_{1}) + e_{1}(σ_{2}).

(iii) If deg A_{1} ≡ deg A_{2} ≡ 0 mod 2 and e_{1}(σ_{1}) = e_{1}(σ_{2}) = 0, then

e_{2}(σ) = e_{2}(σ_{1}) + e_{2}(σ_{2}).

(iv) If the co-indices of A_{1} and A_{2} are even and e_{i}(σ_{1}) = e_{i}(σ_{2}) = 0 for i = 1, 2, then

e_{3}(σ) = e_{3}(σ_{1}) + e_{3}(σ_{2}) and f_{3}(σ) = f_{3}(σ_{1}) + f_{3}(σ_{2}).

Proof. Assertion (i) is clear by definition, and (ii) was established by Dejaiffe [10, Prop. 2.3]. Assertion (iii) follows from [10, § 3.3] (see also the proof of the “Orthogonal Sum Lemma” in [14, §3]). To prove (iv), let D be the division algebra Brauer-equivalent to A, A_{1}, and A_{2}, and let θ be an F -linear involution on D. We may find hermitian forms of orthogonal type h_{1}, h_{2} over (D, θ) such that (A_{i}, σ_{i}) ≃ Ad_{h_{i}} for i = 1, 2, and (A, σ) ≃ Ad_{h_{1}⊥h_{2}}. By Lemma 2.2(ii) we have

e_{3}(h_{1} ⊥ h_{2}) = e_{3}(h_{1}) + e_{3}(h_{2}).

By definition of the e_{3}-invariant of orthogonal involutions (see § 2.5), e_{3}(σ) (resp. e_{3}(σ_{i}) for i = 1, 2) is represented by e_{3}(h_{1} ⊥ h_{2}) (resp. e_{3}(h_{i})), hence e_{3}(σ) = e_{3}(σ_{1})+e_{3}(σ_{2}). Likewise, the additivity of e_{3} induces f_{3}(σ) = f_{3}(σ_{1})+ f_{3}(σ_{2}), by definition of the f_{3}-invariant (see 2.15). □

By a result of Pfister [33, p.123-124], any 12-dimensional quadratic form φ in I^{3}F decomposes as φ = ⟨α_{1}⟩n_{1} ⊥ ⟨α_{2}⟩n_{2} ⊥ ⟨α_{3}⟩n_{3}, where n_{i} is a 2-fold Pfister form and α_{i} ∈ F^{×}, for 1 ≤ i ≤ 3. This can be rephrased as

Ad_{φ} ∈ Ad_{n_{1}} ⊞ Ad_{n_{2}} ⊞ Ad_{n_{3}},

where each summand Ad_{n_{i}} has degree 4 and discriminant 1. We now extend this result to the non-split case.

Theorem 3.2. Let (A, σ) be a central simple F -algebra of degree 12 with orthogonal involution. Assume the discriminant and the Clifford invariant of σ are trivial. There is a central simple F -algebra A_{0} of degree 4 and orthogonal involutions σ_{1}, σ_{2}, σ_{3} of trivial discriminant on A_{0} such that

(A, σ) ∈ (A_{0}, σ_{1}) ⊞(A_{0}, σ_{2}) ⊞(A_{0}, σ_{3}).

Note that since deg A_{0} = 4 we have e_{2}(σ_{i}) = 0 if and only if σ_{i} is hyperbolic (see [41, Th. 3.10]); therefore, even when the index of A is 2 we cannot use Proposition 3.1(iv) to compute e_{3}(σ) (unless each σ_{i} is hyperbolic).

Proof of Theorem 3.2. The index of A is a power of 2 since 2[A] = 0 in Br(F), and it divides deg A = 12, so ind A = 1, 2 or 4. As we just pointed out, the index 1 case is Pfister’s theorem. We consider separately the two remaining cases. If ind A = 2, we have (A, σ) = Ad_{h} for some skew-hermitian form h of relative rank 6 over a quaternion division algebra (Q, ) with its canonical involution. Let q_{1} ∈ Q be a nonzero pure quaternion represented by h, and write h = ⟨q_{1}⟩ ⊥ h′. Over the quadratic extension K_{1} = F (q_{1}), the algebra Q splits and the form ⟨q_{1}⟩ becomes hyperbolic (because its discriminant becomes a square). Therefore, h_{K_{1}} and h′_{K_{1}} are Witt-equivalent, and (ad_{h′})_{K_{1}} is adjoint to a 10dimensional form φ. The discriminant and Clifford invariant of σ are trivial, hence φ ∈ I^{3}K_{1}. Since there is no anisotropic 10-dimensional quadratic forms in I^{3} (see [22, Th. 8.1.1]), it follows that h′_{K_{1}} is isotropic, hence by [34, Prop., p. 382], h′ = ⟨−λ_{1}q_{1}⟩ ⊥ k for some λ_{1} ∈ F^{×} and some skew-hermitian form k of relative rank 4. We thus have

h = ⟨q_{1}⟩⟨1, −λ_{1}⟩ ⊥ k,

and computation shows that e_{1}(⟨q_{1}⟩⟨1, −λ_{1}⟩) = 0. Therefore, e_{1}(k) = 0, and e_{2}(k) = e_{2}(⟨q_{1}⟩⟨1, −λ_{1}⟩) because e_{2}(h) = 0. Now, let q_{2} ∈ Q be a nonzero pure quaternion represented by k, and let K_{2} = F (q_{2}), so k = ⟨q_{2}⟩ ⊥ k′ for some skew-hermitian form k′ of relative rank 3. The forms k_{K_{2}} and k′_{K_{2}} are Witt-equivalent, and (ad_{k′})_{K_{2}} is adjoint to a 6-dimensional form ψ ∈ I^{2}K_{2}, i.e., to an Albert form ψ. We have

e_{2}(ψ) = e_{2}(k′)_{K_{2}} = e_{2}(k)_{K_{2}} = e_{2}(⟨q_{1}⟩⟨1, −λ_{1}⟩)_{K_{2}},

hence the index of e_{2}(ψ) is at most 2, and it follows that ψ is isotropic. Therefore, k′_{K_{2}} is isotropic, and k′ = ⟨−λ_{2}q_{2}⟩ ⊥ ℓ for some λ_{2} ∈ F^{×} and some skew-hermitian form ℓ of relative rank 2. Thus, we have

h = ⟨q_{1}⟩⟨1, −λ_{1}⟩ ⊥ ⟨q_{2}⟩⟨1, −λ_{2}⟩ ⊥ ℓ.

Since e_{1}(⟨q_{1}⟩⟨1, −λ_{1}⟩) = e_{1}(⟨q_{2}⟩⟨1, −λ_{2}⟩) = 0 and e_{1}(h) = 0, we also have e_{1}(ℓ) = 0. We thus obtain the required decomposition, with

A_{0} = M_{2}(Q),  σ_{1} = ad_{⟨q_{1}⟩⟨1,−λ_{1}⟩},  σ_{2} = ad_{⟨q_{2}⟩⟨1,−λ_{2}⟩},  σ_{3} = ad_{ℓ} .

Suppose now ind A = 4, and let D be the division algebra of degree 4 Brauerequivalent to A. By [17, Th. 3.1], there exists a quadratic extension K of F such that (A, σ)_{K} is hyperbolic. The co-index of A_{K} is therefore even, so the index of A_{K} is 2, hence we may identify K with a subfield of D. The following construction is inspired by the Parimala–Sridharan–Suresh exact sequence in Appendix 2 of [5]. We have D = D̃ ⊕ D′, where D̃ is the centralizer of K in D and, writing ι for the nontrivial F -automorphism of K,

D′ = {x ∈ D | xy = ι(y)x for all y ∈ K}.

Let θ be an orthogonal involution on D that fixes K (such involutions exist by [26, (4.14)]). We may represent (A, σ) = (End_{D} V, ad_{h}) for some hermitian form h of relative rank 3 over (D, θ). In view of the decomposition D = D̃⊕D′, we have for x, y ∈ V

h(x, y) = h̃(x, y) + h′(x, y) with h̃(x, y) ∈ D̃ and h′(x, y) ∈ D′.

Since h is a hermitian form over (D, θ), it follows that h̃ is a hermitian form on V viewed as a D̃-vector space, with respect to the restriction of θ to D̃. Clearly, End_{D} V ⊂ EndD̃ V. We may also embed K into EndD̃ V by identifying α ∈ K with the scalar multiplication x ↦ xα for x ∈ V. Thus, we have a K-algebra homomorphism

(End_{D} V) ⊗_{F} K → EndD̃ V.

This homomorphism is injective because the left side is a simple algebra, hence it is an isomorphism by dimension count. For f ∈ End_{D} V we have ad_{h}(f) = adh̃(f), so the isomorphism preserves the involution, and therefore (Ad_{h})_{K} = Adh̃. Since σ becomes hyperbolic over K, the form h̃ is hyperbolic. Therefore, there is an h-orthogonal base of V consisting of h̃-isotropic vectors, which yields a diagonalization

h = ⟨a_{1}, a_{2}, a_{3}⟩ with a_{1}, a_{2}, a_{3} ∈ D′ ∩ Sym(θ).

We thus have (A, σ) ∈ (D, σ_{1}) ⊞(D, σ_{2}) ⊞(D, σ_{3}) with σ_{i} = Int(a_{i}^{−1}) ∘ θ for i = 1, 2, 3. To complete the proof, we show that the discriminant of each σ_{i} is trivial. Recall from [26, (7.2)] that the discriminant is the square class of any skew-symmetric unit. Let α ∈ K^{×} be such that ι(α) = −α. Since a_{i} ∈ D′ we have σ_{i}(α) = −α, so disc σ_{i} = Nrd_{D}(α) = N_{K/F} (α)^{2}. □

Recall that a central simple algebra of degree 4 with orthogonal involution (A_{0}, σ_{0}) of trivial discriminant decomposes as (A_{0}, σ_{0}) ≃ (Q, )⊗(H, ) where the quaternion algebras Q, H are the two components of the Clifford algebra C(A_{0}, σ_{0}) (see [26, (15.12)]). Therefore, Theorem 3.2 can be rephrased as follows:

Corollary 3.3. Let (A, σ) be a central simple algebra of degree 12 with orthogonal involution of trivial discriminant and Clifford invariant. There exist quaternion F -algebras Q_{i}, H_{i} for i = 1, 2, 3 such that [A] = [Q_{i}] + [H_{i}] for i = 1, 2, 3, [H_{1}] + [H_{2}] + [H_{3}] = 0, and

(A, σ) ∈ ((Q_{1}, ) ⊗ (H_{1}, )) ⊞((Q_{2}, ) ⊗ (H_{2}, )) ⊞((Q_{3}, ) ⊗ (H_{3}, )).

Proof. Theorem 3.2 yields orthogonal involutions σ_{1}, σ_{2}, σ_{3} of trivial discriminant on the central simple F -algebra A_{0} of degree 4 Brauer-equivalent to A such that

(A, σ) ∈ (A_{0}, σ_{1}) ⊞(A_{0}, σ_{2}) ⊞(A_{0}, σ_{3}).

Each (A_{0}, σ_{i}) has a decomposition

(A_{0}, σ_{i}) ≃ (Q_{i}, ) ⊗ (H_{i}, )

for some quaternion F -algebras Q_{i}, H_{i} such that

e_{2}(σ_{i}) = [Q_{i}] + {0, [A]} = [H_{i}] + {0, [A]} ∈ M_{A}^{2}(F).

From this decomposition, it follows that

[Q_{i}] + [H_{i}] = [A_{0}] = [A] for i = 1, 2, 3.

Moreover, we have ∑_{i=1}^{3} e_{2}(σ_{i}) = e_{2}(σ) by Proposition 3.1, and e_{2}(σ) = 0, so

∑_{i=1}^{3}[Q_{i}] = (∑_{i=1}^{3}[H_{i}]) + [A] ∈ {0, [A]} ⊂ _{2}H^{3}(F).

Therefore, interchanging Q_{i} and H_{i} if necessary, we may assume ∑_{i=1}^{3}[H_{i}] = 0. □

Example 3.4. For any quaternion algebra Q with norm form n_{Q}, we have Ad_{n_{Q}} ≃ (Q, ) ⊗ (Q, ), see for instance [26, (11.1)]. Therefore, if A is split, and σ is adjoint to the 12-dimensional form φ = ⟨α_{1}⟩n_{1} ⊥ ⟨α_{2}⟩n_{2} ⊥ ⟨α_{3}⟩n_{3}, where n_{i} is the norm form of a quaternion algebra Q_{i} for 1 ≤ i ≤ 3, then

(A, σ) ∈ ⊞_{i=1}^{3}(Q_{i}, ) ⊗ (Q_{i}, ).

Conversely, any decomposition of a split (A, σ) = Ad_{φ} as in the corollary corresponds to a decomposition φ = ⟨α_{1}⟩n_{1} ⊥ ⟨α_{2}⟩n_{2} ⊥ ⟨α_{3}⟩n_{3}, where n_{i} is the norm form of Q_{i} ≃ H_{i}.

Remark 3.5. In [17], it is proved that any (A, σ) of degree 12 with trivial discriminant and trivial Clifford invariant can be described as a quadratic extension of some degree 6 central simple algebra with unitary involution (B, τ), with discriminant algebra Brauer-equivalent to A. This algebra (B, τ) can be described from the above additive decomposition as follows. Since ∑_{i=1}^{3}[H_{i}] = 0, the algebras H_{i} have a common quadratic subfield K, see [28, Th. III.4.13]. All three products (Q_{i}, ) ⊗ (H_{i}, ) are hyperbolic over K, so σ_{K} is hyperbolic. Moreover, as observed in [17, Ex. 1.3], the tensor product (Q_{i}, ) ⊗ (H_{i}, ) is a quadratic extension of (Q_{i}, ) ⊗ (K, ). Therefore, (A, σ) is a quadratic extension of some (B, τ) ∈ ⊞_{i=0}^{3}(Q_{i}, ) ⊗ (K, ), and the discriminant algebra of (B, τ) is Brauer-equivalent to [Q_{1}] + [Q_{2}] + [Q_{3}] = [Q]. Note that in the case where ind A = 4, we use in our proof the main result of [17], which guarantees the existence of a quadratic extension K such that (A, σ)_{K} is hyperbolic. But for ind A ≠ 4, our proof is independent, and does not use the existence of an open orbit of a half-spin representation as in [17, p. 1220].

3.2. Decomposition groups of (A, σ). Until the end of this section, (A, σ) denotes a central simple F -algebra of degree 12 with an orthogonal involution of trivial discriminant and trivial Clifford algebra.

Definition 3.6. Given an additive decomposition as in Corollary 3.3

(A, σ) ∈ ⊞_{i=1}^{3}((Q_{i}, ) ⊗_{F} (H_{i}, )) with ∑_{i=1}^{3}[H_{i}] = 0,

the subset

U = {0, [A], [Q_{1}], [H_{1}], [Q_{2}], [H_{2}], [Q_{3}], [H_{3}]} ⊂ _{2} Br(F)

is called a decomposition group of (A, σ). It is indeed the subgroup of _{2} Br(F) generated by [Q_{1}], [Q_{2}], and [Q_{3}], since [A] = [Q_{1}] + [Q_{2}] + [Q_{3}] and [H_{i}] = [A] + [Q_{i}] for i = 1, 2, 3.

As the following examples show, a given algebra with involution (A, σ) may admit several additive decompositions, corresponding to different decomposition groups, possibly not all of the same cardinality.

Example 3.7. Assume A is split. Since [A] = 0, we have [H_{i}] = [Q_{i}] for all i. Hence all decomposition groups of (A, σ) have order dividing 4. Consider three quaternion division algebras [Q_{1}], [Q_{2}] and [Q_{3}] such that [Q_{1}]+ [Q_{2}] = [Q_{3}]. By the “common slot lemma” [28, Th. III.4.13], there exist a, b_{1}, b_{2} ∈ F^{×} such that Q_{i} = (a, b_{i}) for i = 1, 2 and Q_{3} = (a, b_{1}b_{2}). An easy computation then shows that the norm forms of Q_{1}, Q_{2}, Q_{3}, respectively denoted by n_{1}, n_{2}, n_{3}, satisfy n_{1} − n_{2} = ⟨b_{2}⟩n_{3} in the Witt group of F. Hence, extending scalars to a rational function fields in two variables over F, one may find scalars α_{i} for 1 ≤ i ≤ 3 such that the form

φ = ⟨α_{1}⟩n_{1} ⊥ ⟨α_{2}⟩n_{2} ⊥ ⟨α_{3}⟩n_{3}

is either anisotropic, or isotropic and non-hyperbolic, or hyperbolic. By Example 3.4, in all three cases, {0, [Q_{1}], [Q_{2}], [Q_{3}]} is a decomposition group of order 4 for the involution σ = ad_{φ}. On the other hand, the adjoint involution of an isotropic or a hyperbolic form also has smaller decomposition groups, as we now proceed to show. If the involution σ is isotropic, it is adjoint to a quadratic form φ which is Wittequivalent to a 3-fold Pfister form π_{3}. Let Q be a quaternion algebra such that the norm form n_{Q} is a subform of π_{3}. There exists α_{1}, α_{2} ∈ F^{×} such that φ = ⟨α_{1}, α_{2}⟩ ⊗ n_{Q} ⊥ 2ℍ, (where ℍ denotes the hyperbolic plane) hence {0, [Q]} is a decomposition group of σ = ad φ. If in addition σ, hence π_{3}, is hyperbolic, we may choose [Q] = 0.

Example 3.8. Assume now that A = M_{6}(Q) has index 2. Since 0 ≠ [Q] ∈ U, all decomposition groups U have order 2, 4 or 8. If σ is isotropic, then it is Witt-equivalent to a degree 8 algebra with involution (M_{4}(Q), σ_{0}) that has trivial discriminant and trivial Clifford invariant, so

(A, σ) = (M_{4}(Q), σ_{0}) ⊞ (M_{2}(Q), hyp) = (M_{4}(Q), σ_{0}) ⊞ ((M_{2}(F), ) ⊗ (Q, )).

Because (M_{4}(Q), σ_{0}) has trivial discriminant and Clifford invariant, by [35, Th. 5.2] we may find λ, μ ∈ F^{×} and an orthogonal involution ρ on Q such that

(M_{4}(Q), σ_{0}) ≃ Ad_{⟨⟨λ,μ⟩⟩} ⊗(Q, ρ).

Let Q_{1} and H_{1} be the two components of the Clifford algebra of Ad_{⟨⟨μ⟩⟩} ⊗(Q, ρ). Then

Ad_{⟨⟨μ⟩⟩} ⊗(Q, ρ) ≃ (Q_{1}, ) ⊗ (H_{1}, ).

Therefore,

Ad_{⟨⟨λ,μ⟩⟩} ⊗(Q, ρ) ≃ Ad_{⟨1,−λ⟩} ⊗(Q_{1}, ) ⊗ (H_{1}, )
∈ ((Q_{1}, ) ⊗ (H_{1}, )) ⊞ ((Q_{1}, ) ⊗ (H_{1}, )),

and finally

(A, σ) ∈ (Q_{1}, ) ⊗ (H_{1}, )) ⊞ ((Q_{1}, ) ⊗ (H_{1}, )) ⊞ ((M_{2}(F), ) ⊗ (Q, )).

It follows that {0, [Q], [Q_{1}], [H_{1}]} is a decomposition group for (A, σ). If in addition σ is hyperbolic, we may choose μ = 1, so that {[Q_{1}], [H_{1}]} = {0, [Q]}. Hence {0, [Q]} is a decomposition group of (A, σ) in this case.

Example 3.9. If A has index 4, then all decomposition groups of (A, σ) have order 8. Indeed, since [A] = [Q_{i}] + [H_{i}], the quaternion algebras Q_{i} and H_{i} all are division algebras. Therefore [Q_{i}] ≠ 0 for i = 1, 2, 3. Moreover, we have [Q_{1}] + [Q_{2}] + [Q_{3}] = [A]. Since A has index 4, this guarantees [Q_{i}] + [Q_{j}] is non zero if i ≠ j. Therefore, [Q_{1}], [Q_{2}], [Q_{3}] are ℤ/2 linearly independent, and they do generate a group of order 8. One may also check that the involution σ is anisotropic in this case. Indeed, A ≃ M_{3}(D) for some degree 4 division algebra D, hence A does not carry any hyperbolic involution. Moreover, its isotropic involutions with trivial discriminant are Witt-equivalent to (Q_{1}, ) ⊗ (H_{1}, ), for some quaternion division algebras Q_{1} and H_{1} such that D ≃ Q_{1} ⊗ H_{1}. Hence isotropic involutions on A with trivial discriminant have non trivial Clifford invariant

[Q_{1}] + {0, [D]} = [H_{1}] + {0, [D]} ≠ 0 ∈ H^{2}(F)/{0, [D]}.

From these examples, we easily get the following characterization of isotropy and hyperbolicity:

Lemma 3.10. Let (A, σ) be a degree 12 algebra with orthogonal involution with trivial discriminant and trivial Clifford invariant.

(i) The involution σ is isotropic if and only if it admits a decomposition group generated by [A] and [Q_{1}] for some quaternion algebra Q_{1}. (ii) The involution σ is hyperbolic if and only if it admits {0, [A]} as a

decomposition group. (iii) The algebra with involution (A, σ) is split and hyperbolic if and only if

it admits {0} as a decomposition group.

Proof. Assertion (iii) is clear from the definition of a decomposition group, since (M_{2}(F), ) ⊗ (M_{2}(F), ) is hyperbolic. For (i) and (ii), the direct implications immediately follow from the previous examples. To prove the converse, let us first assume (A, σ) admits {0, [A]} as a decomposition group. Since this group has order 1 or 2, A cannot have index 4 by Example 3.9. Therefore, it is Brauer-equivalent to a quaternion algebra Q. Moreover, by definition,

(A, σ) ∈ ⊞_{i=1}^{3}((Q, ) ⊗ (M_{2}(F), )).

Since each summand is hyperbolic, this proves σ is hyperbolic. Assume now that U is generated by [A] and [Q_{1}]. The order of U then divides 4, hence by Example 3.9, the algebra A is Brauer-equivalent to some quaternion algebra Q. Thus U = {0, [Q], [Q_{1}], [H_{1}]}, with M_{2}(Q) ≃ Q_{1} ⊗ H_{1}. If [Q_{1}] = 0 or [H_{1}] = 0, then U = {0, [Q]}, and by the previous case, σ is hyperbolic. Assume now that both H_{1} and Q_{1} are non split. We get

(A, σ) ∈ ⊞_{i=1}^{3}(Q_{i}, ) ⊗ (H_{i}, ),

with for i = 2, 3 {[Q_{i}], [H_{i}]} equal either to {0, [Q]} or to {[Q_{1}], [H_{1}]}. Picking an arbitrary element in {[H_{1}], [Q_{1}]} for 1 ≤ i ≤ 3, we get three quaternion algebras whose sum is never 0. Therefore, since by Corollary 3.3 we have [H_{1}] + [H_{2}] + [H_{3}] = 0, at least one summand must be (Q, ) ⊗ (M_{2}(F), ), and this proves σ is isotropic. □

Remark 3.11. Reversing the viewpoint, note that any subgroup U ⊂ _{2} Br(F) of order 8 in which all the nonzero elements except at most one have index 2 is the decomposition group of some central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant. If all the nonzero elements in U have index 2, pick a quaternion algebra D representing a nonzero element in U; otherwise, let D be the division algebra such that [D] ∈ U and ind D > 2. In each case, we may organize the other nonzero elements in U in pairs [Q_{i}], [H_{i}] such that [D] = [Q_{i}] + [H_{i}] for i = 1, 2, 3, and ∑_{i=1}^{3}[H_{i}] = 0. Any algebra with involution (A, σ) in ⊞_{i=1}^{3}((Q_{i}, ) ⊗ (H_{i}, )) has decomposition group U. Modifying the scalars in the direct sum leads to several nonisomorphic such (A, σ). Moreover, when all the nonzero elements in U have index 2, we may select for D various quaternion algebras, and thus obtain various (A, σ) that are not Brauer-equivalent. Similarly, any subgroup U ⊂ _{2} Br(F) of order 4 containing at most one element [D] with ind D > 2, and any subgroup {0, [Q]} where Q is a quaternion algebra, is the decomposition group of some central simple algebra of degree 12 endowed with an isotropic orthogonal involution of trivial discriminant and trivial Clifford invariant.

The decomposition groups of (A, σ) are subgroups of the Brauer group generated by at most three quaternion algebras. Those subgroups were considered by Peyre in [32]. His results will prove useful to study degree 12 algebras with involution. For the reader’s convenience, we recall them in the next section.

3.3. A complex of Peyre. Let F be an arbitrary field, and let U ⊂ Br F be a finite subgroup of the Brauer group of F. We let F^{×} · U denote the subgroup of H^{3}(F) generated by classes λ · α, with λ ∈ F^{×} and α ∈ U; any element in F^{×} · U can be written as ∑_{i=1}^{r} λ_{i} · α_{i} for some λ_{i} ∈ F^{×}, where α_{1},…, α_{r} is a generating set for the group U. Let F_{U} be the function field of the product of the Severi–Brauer varieties associated to elements of U. Clearly, F_{U} splits all the elements of U, hence the subgroup F^{×} ·U vanishes after scalar extension to F_{U}. Therefore, the following sequence is a complex, which was first introduced and studied by Peyre in [32, §4]:

F^{×} · U → H^{3}(F) → H^{3}(F_{U}).

We let H_{U} denote the corresponding homology group, that is

H_{U} = \frac{ker(H^{3}(F) → H^{3}(F_{U}))}{F^{×} · U}.

We now return to our standing hypothesis that the characteristic of F is different from 2. Peyre considers in particular subgroups U ⊂ Br F generated by the Brauer classes of at most three quaternion algebras, and proves:

Theorem 3.12 (Peyre [32, Thm 5.1]). If U is generated by the Brauer classes of two quaternion algebras, then H_{U} = 0.

In the next section, we need to consider only subgroups U such that all the elements of U are quaternion algebras; we call them quaternionic subgroups of the Brauer group. These subgroups have also been investigated by Sivatski [38]. We have:

Theorem 3.13 ([32, Prop. 6.1], [38, Cor. 11]). If U ⊂ Br F is generated by the Brauer classes of three quaternion algebras, then H_{U} = 0 or ℤ/2ℤ. Assume in addition U is quaternionic. Then the following conditions are equivalent:

(a) H_{U} = 0;

(b) U is split by an extension of F of degree 2m for some odd m;

(c) U is split by a quadratic extension of F.

The result that H_{U} = 0 or ℤ/2ℤ and the equivalence (a) ⟺ (b) are due to Peyre [32, Prop. 6.1]. The equivalence (b) ⟺ (c) was proved by Sivatski [38, Cor. 11]. We say that an extension K of F splits a subgroup U ⊂ Br F if it splits all the elements in U. If K splits a decomposition group of a central simple algebra with orthogonal involution (A, σ), then A_{K} is split because [A] ∈ U, and σ_{K} is hyperbolic by Lemma 3.10(iii) because (A_{K}, σ_{K}) has a trivial decomposition group. Therefore, Theorem 3.13 is relevant for the quadratic splitting of (A, σ), as we will see in § 5.2.

4. The Arason invariant and the homology of Peyre’s complex

As in the previous section, (A, σ) is a degree 12 algebra with orthogonal involution of trivial discriminant and trivial Clifford invariant. From now on, we assume in addition that the Arason invariant e_{3}(σ) is well defined. So the algebra A has even co-index, hence index 1 or 2. Under this assumption, any decomposition group of (A, σ) is quaternionic, that is consists only of Brauer classes of quaternion algebras. In this section, we relate the decomposition groups of (A, σ) with the values of the Arason invariant e_{3}(σ). Reversing the viewpoint we then explain how one can use the Arason invariant to find explicit generators of the homology group H_{U} of Peyre’s complex, for any quaternionic subgroup U ⊂ Br(F) of order dividing 8.

4.1. Arason invariant in degree 12. For orthogonal involutions on a degree 12 algebra, isotropy and hyperbolicity can be detected via the Arason invariant as follows:

Theorem 4.1. Let (A, σ) be a degree 12 and index 1 or 2 algebra with orthogonal involution of trivial discriminant and trivial Clifford invariant.

(i) The involution σ is hyperbolic if and only if e_{3}(σ) = 0 ∈ M_{A}^{3}(F).

(ii) The involution σ is isotropic if and only if

e_{3}(σ) = e_{3}(π) + F^{×} · [A] ∈ M_{A}^{3}(F)

for some 3-fold Pfister form π, i.e. f_{3}(σ) = 0 and e_{3}(σ) is represented by a symbol.

Proof. Assume first that A is split, so that σ is adjoint to a 12-dimensional quadratic form φ, and e_{3}(σ) = e_{3}(φ) ∈ _{2}H^{3}(F). Since the Arason invariant for quadratic forms has kernel the 4th power I^{4}F of the fundamental ideal of the Witt ring W (F), the first equivalence follows from the Arason–Pfister Hauptsatz. To prove (ii), note that there is no 10-dimensional anisotropic quadratic form in I^{3}F, see [28, Prop. XII.2.8]. So if φ is isotropic, then it has two hyperbolic planes, and it is Witt-equivalent to a multiple of some 3-fold Pfister form π. Hence, e_{3}(φ) = e_{3}(π). Assume conversely that e_{3}(φ) = e_{3}(π). By condition (i), φ becomes hyperbolic over the function field of π. Therefore, by [28, Th. X.4.11], the anisotropic kernel of φ is a multiple of π. In view of the dimensions, this implies φ = ⟨α⟩π + 2ℍ for some α ∈ F^{×}. In particular, φ is isotropic. Assume now A = M_{6}(Q) for some quaternion division algebra Q. By a result of Dejaiffe [11] and of Parimala–Sridharan–Suresh [31, Prop 3.3], the involution σ is hyperbolic if and only if it is hyperbolic after scalar extension to a generic splitting field F_{Q} of the quaternion algebra Q. Since the restriction map M_{Q}^{3}(F) → H^{3}(F_{Q}) is injective, the split case gives the result in index 2. If (A, σ) is isotropic, it is Witt-equivalent to a degree 8 and index 2 algebra with involution. The explicit description of the Arason invariant in degree 8 given in [35, Th. 5.2] shows it is equal to e_{3}(π) mod F^{×} · [Q] ∈ M_{Q}^{3}(F) for some 3-fold Pfister form π. Assume conversely that e_{3}(σ) = e_{3}(π) + F^{×} · [Q]. After scalar extension to F_{Q}, the split case shows σ_{F_{Q}} is isotropic. By Parimala– Sridharan–Suresh [31, Cor. 3.4], this implies σ itself is isotropic. □

Remark 4.2. In the split isotropic case, the involution can be explicitly described from its Arason invariant: the proof of Theorem 4.1(ii) shows that σ is adjoint to π + 2ℍ if e_{3}(σ) = e_{3}(π) ∈ _{2}H^{3}(F). In index 2, we also get an explicit description of (A, σ) in the isotropic case. Indeed, we have

(A, σ) = (M_{2}(Q), hyp) ⊞ (M_{4}(Q), σ_{0})

for some orthogonal involution σ_{0} with trivial discriminant and trivial Clifford invariant, and e_{3}(σ) = e_{3}(σ_{0}). If (a, b, c) is a symbol representing e_{3}(σ), then by [35, Th. 5.2] we may assume that one of the slots, say a, is such that F (\sqrt{a}) splits Q, hence Q carries an orthogonal involution ρ with discriminant a. Theorem 5.2 of [35] further shows that

(M_{4}(Q), σ_{0}) ≃ (Q, ρ) ⊗ Ad_{⟨⟨b,c⟩⟩} .

Under some additional condition, we also have the following classification result:

Proposition 4.3. Let A = M_{6}(Q) be a degree 12 algebra of index at most 2, and let σ and σ′ be two orthogonal involutions with trivial discriminant and trivial Clifford invariant. We assume either A is split, or σ is isotropic. The involutions σ and σ′ are isomorphic if and only if e_{3}(σ) = e_{3}(σ′).

Proof. It is already known that two isomorphic involutions have the same Arason invariant, so we only need to prove the converse. Assume first that A has index 2, in which case we assume in addition that σ is isotropic. Since e_{3}(σ) = e_{3}(σ′), by Theorem 4.1, the involution σ′ also is isotropic. The result then follows from the explicit description given in Remark 4.2, or equivalently from [35, Cor. 5.3(2)], which shows that the anisotropic parts of σ and σ′ are isomorphic. Assume now A is split, and σ and σ′ are adjoint to φ and φ′ respectively. We have e_{3}(φ) = e_{3}(φ′). If there exists a 3-fold Pfister form π such that e_{3}(φ) = e_{3}(φ′) = e_{3}(π), then φ and φ′ are both similar to π + 2ℍ. Otherwise, they are anisotropic, and the result in this case follows by combining Pfister’s theorem (see for instance [22, Th. 8.1.1]), which asserts that φ and φ′ can be decomposed as tensor products of a 1-fold Pfister form and an Albert form, with Hoffmann’s result [19, Corollary], which precisely says that two such forms are similar if and only if their difference is in I^{4}F.

□

4.2. Arason invariant and decomposition groups. Recall from Example 3.7 that the decomposition groups corresponding to additive decompositions of (A, σ) are quaternionic subgroups of order at most 4 when A is split. Hence, by Peyre’s Theorem 3.12, the corresponding homology group is trivial, H_{U} = 0. Using this, we have:

Proposition 4.4. Let φ be a 12-dimensional quadratic form in I^{3}F, and let U = {0, [Q_{1}], [Q_{2}], [Q_{3}]} ⊂ Br(F) be a quaternionic subgroup of order at most 4. For i = 1, 2, 3, let n_{i} be the norm form of Q_{i}. The following are equivalent:

(a) There exists α_{1}, α_{2}, α_{3} ∈ F^{×} such that φ = ⟨α_{1}⟩n_{1} ⊥ ⟨α_{2}⟩n_{2} ⊥ ⟨α_{3}⟩n_{3};

(b) U is a decomposition group of Ad_{φ};

(c) φ is hyperbolic over F_{U};

(d) e_{3}(φ) ∈ ker(H^{3}(F) → H^{3}(F_{U}));

(e) e_{3}(φ) ∈ F^{×} · U.

Proof. The equivalence between (a) and (b) follows from Example 3.4. Assume φ decomposes as in (a). Since the field F_{U} splits all three quaternion algebras Q_{i}, hence also their norm forms n_{i}, the form φ is hyperbolic over F_{U}, hence assertions (c) and (d) hold. By Peyre’s result 3.12, we also get (e), and it only remains to prove that (e) implies (a). Thus, assume now that e_{3}(φ) ∈ F^{×} · U. Since the subgroup U is generated by [Q_{1}] and [Q_{2}], there exists λ_{1} and λ_{2} ∈ F^{×} such that

e_{3}(φ) = (λ_{1}) · [Q_{1}] + (λ_{2}) · [Q_{2}].

The product Q_{1} ⊗ Q_{2} ⊗ Q_{3} is split, so by the common slot lemma ([28, Th. III.4.13]), we may assume Q_{i} = (a, b_{i})_{F} for some a and b_{i} ∈ F^{×}. A direct computation then shows that n_{1} − n_{2} = ⟨b_{2}⟩n_{3}. Hence the 12dimensional quadratic form ⟨−λ_{1}⟩n_{1} + ⟨λ_{2}⟩n_{2} + ⟨b_{2}⟩n_{3} is Witt-equivalent to ⟨1, −λ_{1}⟩n_{1} + ⟨−1⟩⟨1, −λ_{2}⟩n_{2}, which has the same Arason invariant as φ. By Proposition 4.3, this form is similar to φ, so that φ has an additive decomposition as required. □

Let us consider now the index 2 case. By Lemma 3.10, (A, σ) admits decomposition groups of order 4 if and only if it is isotropic. We prove:

Proposition 4.5. Let A = M_{6}(Q) be an algebra of index ≤ 2, and consider an orthogonal involution σ on A, with trivial discriminant and trivial Clifford invariant. Pick a subgroup U = {0, [Q], [Q_{1}], [H_{1}]} ⊂ Br(F) containing the class of Q. The following are equivalent:

(a) (A, σ) admits an additive decomposition of the following type:

(A, σ) ∈ ((Q_{1}, ) ⊗ (H_{1}, )) ⊞ ((Q_{1}, ) ⊗ (H_{1}, )) ⊞ ((M_{2}(F), ) ⊗ (Q, ));

(b) U is a decomposition group of (A, σ);

(c) σ is hyperbolic over F_{U};

(d) e_{3}(σ) ∈ ker(M_{A}^{3}(F) → H^{3}(F_{U}));

(e) There exists α ∈ F^{×} such that

e_{3}(σ) = (α) · [Q_{1}] mod F^{×} · [Q] ∈ M_{Q}^{3}(F).

Proof. The proof follows the same line as for the previous proposition. By the definition of decomposition groups, (a) implies (b). Conversely, if (b) holds, then (A, σ) has an additive decomposition with summands isomorphic to (Q_{1}, ) ⊗ (H_{1}, ) or to (M_{2}(F), ) ⊗ (Q, ). If Q_{1} or H_{1} is split, then the two kinds of summands are isomorphic, hence (a) holds. If Q_{1} and H_{1} are not split, then the number of summands isomorphic to (Q_{1}, ) ⊗ (H_{1}, ) must be even because e_{2}(σ) = 0 (see Proposition 3.1), and it must be nonzero because U is the corresponding decomposition group. Therefore, (a) holds. Now, assume (A, σ) satisfies (a). Since the field F_{U} splits Q, Q_{1} and H_{1}, (c) holds. Assertion (d) follows since hyperbolic involutions have trivial Arason invariant. By Peyre’s Proposition 3.12, we deduce assertion (e), and it only remains to prove that (e) implies (a). Hence, assume

e_{3}(σ) = (α) · [Q_{1}] mod F^{×} · [Q] ∈ M_{Q}^{3}(F),

for some α ∈ F^{×} and some quaternion algebra Q_{1}. By Theorem 4.1(ii), the involution σ is isotropic. Hence, in view of Proposition 4.3, it is enough to find an involution σ′ satisfying (a) and having e_{3}(σ′) = (α) · [Q_{1}] mod F^{×} · [Q]. Since Q ⊗ Q_{1} = H_{1} has index 2, the quaternion algebras Q and Q_{1} have a common slot (see [28, Th. III.4.13]). Therefore, there exists a, b, b_{1} ∈ F^{×} such that Q = (a, b) and Q_{1} = (a, b_{1}). Let ρ be an orthogonal involution on Q with discriminant a, and let

(A, σ′) = (M_{2}(Q), hyp) ⊞((Q, ρ) ⊗ Ad_{⟨⟨b_{1},α⟩⟩}).

One component of the Clifford algebra of (Q, ρ) ⊗ Ad_{⟨⟨b_{1}⟩⟩} is given by the cup product of the discriminants of ρ and ad_{⟨⟨b_{1}⟩⟩}, that is (a, b_{1}) = Q_{1}. By Remark 4.2, it follows that e_{3}(σ′) = e_{3}(σ), hence (A, σ) ≃ (A, σ′) by Proposition 4.3. Therefore (A, σ) satisfies (a) as required. □

4.3. Generators of the homology H_{U} of Peyre’s complex. Let A = M_{6}(Q) for some quaternion F -algebra Q, and let σ be an orthogonal involution on A with trivial discriminant and trivial Clifford invariant. Consider an additive decomposition of (A, σ) as in Theorem 3.2,

(A, σ) ∈ ⊞_{i=1}^{3}((Q_{i}, ) ⊗ (H_{i}, )),

and let U be the corresponding decomposition group, which is a quaternionic subgroup of Br(F),

U = {0, [Q], [Q_{1}], [H_{1}], [Q_{2}], [H_{2}], [Q_{3}], [H_{3}]}.

Since F^{×} · [Q] ⊂ F^{×} · U, we may consider the canonical map

U: M_{Q}^{3}(F) → H^{3}(F)/F^{×} · U.

As in §3.3, let F_{U} be the function field of the product of the Severi–Brauer varieties associated to elements of U. Since F_{U} splits U, Lemma 3.10 shows that A_{F_{U}} is split and σ_{F_{U}} is hyperbolic, hence e_{3}(σ)_{F_{U}} = 0. Therefore, \overline{e_{3}(σ)}^{U} lies in the homology H_{U} of Peyre’s complex. As explained in Remark 3.11, for any quaternionic subgroup U ⊂ Br(F) of order dividing 8, we may find algebras with involution (A, σ) for which U is a decomposition group. The main result of this section is:

Theorem 4.6. Let U be a quaternionic subgroup of Br F of order dividing 8. For any (A, σ) admitting U as a decomposition group, the class of the Arason invariant \overline{e_{3}(σ)}^{U} is a generator of the homology group H_{U} of Peyre’s complex.

The main tool in the proof is the following proposition:

Proposition 4.7. Let U be a quaternionic subgroup of Br F of order dividing 8, and pick an algebra A = M_{6}(Q) with orthogonal involution σ, admitting U as a decomposition group.

(i) For all involutions σ′ on A such that (A, σ′) also admits U as a decomposition group, we have

\overline{e_{3}(σ′)}^{U} = \overline{e_{3}(σ)}^{U}.

(ii) Conversely, for all ξ ∈ M_{A}^{3}(F) such that \overline{ξ}^{U} = \overline{e_{3}(σ)}^{U}, there exists an involution σ′ on A such that U is a decomposition group of (A, σ′) and e_{3}(σ′) = ξ mod F^{×} · [A].

(iii) There exists a hyperbolic involution σ′ on A admitting U as a decomposition group if and only if \overline{e_{3}(σ)}^{U} = 0.

Proof. (i) Since U is a decomposition group of (A, σ) and (A, σ′), we have

(A, σ) and (A, σ′) ∈ ⊞_{i=1}^{3}((Q_{i}, ) ⊗ (H_{i}, )).

Therefore, σ and σ′ are adjoint to some skew-hermitian forms h and h′ over (Q, ) satisfying

h = h_{1} ⊥ h_{2} ⊥ h_{3} and h′ = ⟨α_{1}⟩h_{1} ⊥ ⟨α_{2}⟩h_{2} ⊥ ⟨α_{3}⟩h_{3},

for some h_{i} such that ad_{h_{i}} ≃ (Q_{i}, ) ⊗ (H_{i}, ), and some α_{i} ∈ F^{×}. Therefore,

e_{3}(σ) − e_{3}(σ′) = e_{3}(⊥_{i=1}^{3} ⟨1, −α_{i}⟩h_{i}).

Since h_{i} has discriminant 1, Proposition 2.6 applies to each summand and shows ⟨1, −α_{i}⟩ ⊗ h_{i} has trivial discriminant and trivial Clifford invariant, and

e_{3}(⟨1, −α_{i}⟩ ⊗ h_{i}) = α_{i} · e_{2}(h_{i}) = α_{i} · [Q_{i}] ∈ M_{Q}^{3}(F).

Therefore, e_{3}(σ) − e_{3}(σ′) is represented modulo F^{×} · {0, [Q]} by ∑_{i=1}^{3} α_{i} · [Q_{i}].

Since this element lies in F^{×} · U, we have \overline{e_{3}(σ)}^{U} = \overline{e_{3}(σ′)}^{U}. (ii) Consider a skew hermitian form h over (Q, ) such that σ = ad_{h}, and a decomposition h = h_{1} ⊥ h_{2} ⊥ h_{3} as in the proof of (i). Since \overline{ξ}^{U} = \overline{e_{3}(σ)}^{U}, the difference e_{3}(σ) − ξ ∈ M_{Q}^{3}(F) is represented by a cohomology class of the form ∑_{i=1}^{3} α_{i} · [Q_{i}] for some α_{i} ∈ F^{×}. The computation in (i) shows that e_{3}(ad_{h′}) = ξ for h′ = ⟨α_{1}⟩h_{1} ⊥ ⟨α_{2}⟩h_{2} ⊥ ⟨α_{3}⟩h_{3}. (iii) It follows from (ii) that \overline{e_{3}(σ)}^{U} = 0 if and only if there exists an involution σ′ with decomposition group U and e_{3}(σ′) = 0. Theorem 4.1(i) completes the proof by showing σ′ is hyperbolic. □

With this in hand, we can now prove Theorem 4.6.

Proof of Theorem 4.6. Since H_{U} is either 0 or ℤ/2ℤ, in order to prove that \overline{e_{3}(σ)}^{U} generates H_{U} it is enough to prove that H_{U} is trivial as soon as \overline{e_{3}(σ)}^{U} = 0. If U has order at most 4, then H_{U} is trivial by Theorem 3.12. Hence, let us assume U has order 8, and \overline{e_{3}(σ)}^{U} = 0. By Proposition 4.7, replacing σ by σ′, we may assume σ is hyperbolic. Recall σ is adjoint to a skew-hermitian form h, which admits a decomposition h = h_{1} ⊥ h_{2} ⊥ h_{3} with Ad_{h_{i}} = (Q_{i}, )⊗(H_{i}, ). Since U has order 8, each summand h_{i} is anisotropic. The hyperbolicity of h says h_{1} ⊥ h_{2} ≃ −h_{3} ⊥ ℍ is isotropic. Therefore, there exists a pure quaternion q such that h_{1} and h_{2} represent q and −q respectively. Over the quadratic extension F (q) of F, the involutions ad_{h_{1}} and ad_{h_{2}} are isotropic. Since they are adjoint to 2-fold Pfister forms, they are hyperbolic. Hence F (q) splits the Clifford algebra of h_{1} and h_{2}, that is the quaternion algebras Q_{1}, Q_{2}. Since the Brauer classes of Q, Q_{1} and Q_{2} generate U, it follows that F (q) is a quadratic splitting field of U. By Peyre’s Theorem 3.13, we get H_{U} = 0 as required. □

5. Quadratic splitting and the f_{3} invariant

The f_{3} invariant of an involution σ vanishes if the underlying algebra A is split, or of degree ≤ 10. We keep focusing on the case of degree 12 algebras, where we have explicit examples with f_{3}(σ) ≠ 0, see Corollary 2.19. Thus, as in § 4, (A, σ) is a degree 12 algebra with orthogonal involution of trivial discriminant and trivial Clifford invariant for which the Arason and the f_{3} invariants are defined. In particular, A has index at most 2. Our first goal is to characterize the vanishing of f_{3}(σ); this is done in Proposition 5.6 below. As pointed out in Proposition 2.5, f_{3}(σ) vanishes if there exists a quadratic extension K/F over which (A, σ) is split and hyperbolic. Note that since A is Brauer-equivalent to a quaternion algebra, there exist quadratic extensions of the base field F over which A is split. Moreover, using the additive decompositions of Corollary 3.3, one may easily find quadratic extensions of the base field over which the involution is hyperbolic: it suffices to consider a common subfield of the quaternion algebras H_{1}, H_{2}, H_{3}, which exists by [28, Th. III.4.13] since [H_{1}] + [H_{2}] + [H_{3}] = 0. Yet, we give in Corollary 5.13 examples showing that the converse of Proposition 2.5 does not hold in degree 12: we may have f_{3}(σ) = 0 even when there is no quadratic extension that simultaneously splits A and makes σ hyperbolic. First, we use quadratic forms to introduce an invariant of quaternionic subgroups of the Brauer group of F, which, as we next prove, coincides with the f_{3}-invariant of involutions admitting this subgroup as a decomposition group.

5.1. The invariants f_{3}(U) and f_{3}(σ). To any quaternionic subgroup U of Br(F), we may associate in a natural way a quadratic form n_{U} by taking the sum of the norm forms n_{H} of the quaternion algebras H with Brauer class in U. We have:

Lemma 5.1. Let U be a quaternionic subgroup of Br F generated by the Brauer classes of three quaternion algebras. The quadratic form n_{U} = ∑_{[H]∈U} n_{H} satisfies n_{U} ∈ I^{3}F.

Proof. Pick three generators [Q_{1}], [Q_{2}] and [Q_{3}] of U, and let H_{1}, H_{2}, H_{3}, Q be quaternion algebras with Brauer classes [H_{1}] = [Q_{2}] + [Q_{3}], [H_{2}] = [Q_{1}] + [Q_{3}], [H_{3}] = [Q_{1}]+[Q_{2}], and [Q] = [Q_{1}]+[Q_{2}]+[Q_{3}]. We have [H_{1}]+[H_{2}]+[H_{3}] = 0, and

U = {0, [Q], [Q_{1}], [H_{1}], [Q_{2}], [H_{2}], [Q_{3}], [H_{3}]}.

Since the difference n_{Q_{i}} − n_{H_{i}} is Witt-equivalent to an Albert form of Q_{i} ⊗ H_{i}, which is Brauer-equivalent to Q, there exists λ_{i} ∈ F^{×} such that in the Witt group of F, we have n_{Q_{i}} − n_{H_{i}} = ⟨λ_{i}⟩n_{Q} ∈ W F. Therefore,

(2)  n_{U} = ⟨1, λ_{1}, λ_{2}, λ_{3}⟩n_{Q} + ⟨1, 1⟩(n_{H_{1}} + n_{H_{2}} + n_{H_{3}}).

Since the right side is in I^{3}F, the lemma is proved. □

In view of Lemma 5.1, we may associate to U a cohomology class of degree 3 as follows:

Definition 5.2. For any quaternionic subgroup U generated by three elements, we let f_{3}(U) be the Arason invariant of the quadratic form n_{U}:

f_{3}(U) = e_{3}(n_{U}) ∈ _{2}H^{3}(F).

We may easily compute f_{3}(U) from formula (2): Since [H_{1}] + [H_{2}] + [H_{3}] = 0, we have n_{H_{1}} + n_{H_{2}} + n_{H_{3}} ∈ I^{3}F, hence ⟨1, 1⟩(n_{H_{1}} + n_{H_{2}} + n_{H_{3}}) ∈ I^{4}F and therefore

(3)  f_{3}(U) = (λ_{1}λ_{2}λ_{3}) · [Q].

With this in hand, we get:

Proposition 5.3. If H_{U} = 0, then f_{3}(U) = 0.

Proof. By Theorem 3.13, if H_{U} = 0 then U admits a quadratic splitting field, i.e. the generators of U have a common quadratic subfield. So there exist a, b_{1}, b_{2}, and b_{3} ∈ F^{×} such that Q_{i} = (a, b_{i})_{F} for i = 1, 2, 3. Thus, we have H_{1} = (a, b_{2}b_{3})_{F} and

n_{Q_{1}} − n_{H_{1}} = ⟨⟨a⟩⟩(⟨⟨b_{1}⟩⟩ − ⟨⟨b_{2}b_{3}⟩⟩) = ⟨⟨a⟩⟩⟨−b_{1}, b_{2}b_{3}⟩ = ⟨−b_{1}⟩n_{Q}.

Similar formulas hold for i = 2, 3, and we get

f_{3}(U) = (−b_{1}b_{2}b_{3}) · Q = (−b_{1}b_{2}b_{3}, a, b_{1}b_{2}b_{3}) = 0 ∈ _{2}H^{3}(F).
□

In [38], Sivatski asks about the converse^{1}, that is: if f_{3}(U) = 0, does the homology group H_{U} vanish, or equivalently by Peyre’s Theorem 3.13, do the generators Q_{1}, Q_{2}, and Q_{3} of the group U have a common quadratic subfield? Corollary 5.11 below shows that this is not the case.

1Sivatski’s invariant has a different definition, but one may easily check the quadratic form he considers is equivalent to n_{U} modulo I^{4}F.

Documenta Mathematica · Extra Volume Merkurjev (2015) 529–576

The relation between f_{3}(U) and the f_{3}-invariant for involutions is given by the following:

Theorem 5.4. Let (A, σ) be a central simple algebra of degree 12 and index ≤ 2, with orthogonal involution of trivial discriminant and trivial Clifford invariant. Let U be a quaternionic subgroup of the Brauer group, generated by three elements. If U is a decomposition group for (A, σ) then f_{3}(σ) = f_{3}(U).

Remark 5.5. (i) It follows that any two decomposition groups of a given algebra with involution have the same f_{3}-invariant, and any two algebras with involution having U as a decomposition group have the same f_{3}-invariant. (ii) Let c be a generator of H_{U}, and pick an arbitrary (A, σ) having U as a decomposition group. In view of Theorem 4.6, we have e_{3}(σ) = c mod F^{×} · U. Hence f_{3}(U) = f_{3}(σ) = 2c ∈ _{2}H^{3}(F). In particular, f_{3}(U) = 0 if and only if the homology group H_{U} is generated by a cohomology class of order 2.

Proof of Theorem 5.4. The result follows from the computation of f_{3}(σ) in Proposition 2.7 and the computation of f_{3}(U) in (3). We use the same notation as in Definition 3.6, and we let h_{i} be a rank 2 skew-hermitian form over (Q, ) such that

Ad_{h_{i}} ≃ (Q_{i}, ) ⊗ (H_{i}, ) and σ = ad_{h_{1}⊥h_{2}⊥h_{3}} .

For i = 1, 2, 3, let q_{i} ∈ Q be a nonzero pure quaternion represented by h_{i}, and let a_{i} = q_{i}^{2} ∈ F^{×}. Let also b_{i} ∈ F^{×} be such that Q = (a_{i}, b_{i})_{F}. Scalar extension to F (q_{i}) makes h_{i} isotropic, hence hyperbolic since the discriminant of h_{i} is trivial. Therefore, we have h_{i} ≃ ⟨q_{i}⟩⟨1, −λ_{i}⟩ for some λ_{i} ∈ F^{×}. The two components of the Clifford algebra of Ad_{h_{i}} are (a_{i}, λ_{i})_{F} and (a_{i}, λ_{i}b_{i})_{F}, therefore

{Q_{i}, H_{i}} = {(a_{i}, λ_{i})_{F} , (a_{i}, λ_{i}b_{i})_{F} } for i = 1, 2, 3.

Since Q contains a pure quaternion which anticommutes with q_{i} and with square b_{i}, the form h_{i} is isomorphic to ⟨q_{i}⟩⟨1, −λ_{i}b_{i}⟩ for i = 1, 2, 3. Replacing some λ_{i} by λ_{i}b_{i} if necessary, we may assume H_{i} = (a_{i}, λ_{i})_{F} for all i. Since [H_{1}] + [H_{2}] + [H_{3}] = 0, we get ∑_{i=1}^{3}(a_{i}, λ_{i})_{F} = 0. By Proposition 2.7 this implies f_{3}(σ) = λ_{1}λ_{2}λ_{3} ·[Q]. On the other hand, since n_{Q_{i}} −n_{H_{i}} = ⟨⟨a_{i}, λ_{i}b_{i}⟩⟩− ⟨⟨a_{i}, λ_{i}⟩⟩ = ⟨λ_{i}⟩n_{Q}, we have f_{3}(U) = λ_{1}λ_{2}λ_{2} · [Q] by (3). □

5.2. Quadratic splitting, the f_{3} invariant, and decomposition groups. By using Theorem 5.4 and Peyre’s Theorem 3.13, we can now translate in terms of decomposition groups the two conditions we want to compare, as follows:

Proposition 5.6. Let (A, σ) be a degree 12 and index ≤ 2 algebra with orthogonal involution of trivial discriminant and trivial Clifford invariant. The following conditions are equivalent:

(a) f_{3}(σ) = 0;

(b) (A, σ) has a decomposition group U with f_{3}(U) = 0;

(c) f_{3}(U) = 0 for all decomposition groups U of (A, σ).

Likewise, the following conditions are equivalent:

(a’) there exists a quadratic extension K of F such that A_{K} is split and σ_{K}

is hyperbolic; (b’) (A, σ) has a decomposition group U with H_{U} = 0.

Moreover, any of the conditions (a’), (b’) implies the equivalent conditions (a), (b), (c).

Proof. The equivalence between conditions (a), (b), (c) follows directly from Theorem 5.4. Moreover, they can be deduced from (a’), (b’) by Proposition 5.3 or Proposition 2.5. Hence, it only remains to prove that (a’) and (b’) are equivalent. Assume first that (A, σ) has a decomposition group U with H_{U} = 0. By Peyre’s characterization of the vanishing of H_{U} for quaternionic groups, recalled in Theorem 3.13, U is split by a quadratic extension K of F. Hence, (A_{K}, σ_{K}) admits {0} as a decomposition group. By Lemma 3.10, this implies (A_{K}, σ_{K}) is split and hyperbolic. To prove the converse, let us assume there exists a quadratic field extension K = F (d), with d^{2} = δ ∈ F^{×}, such that A_{K} is split and σ_{K} is hyperbolic. If A is split, as explained in Example 3.7, all decomposition subgroups U of (A, σ) have order dividing 4, and therefore satisfy H_{U} = 0 by Peyre’s Theorem 3.12. Assume next ind A = 2. Since A_{K} is split, we may identify K = F (d) with a subfield of the quaternion division algebra Q Brauer-equivalent to A, and thus consider d as a pure quaternion in Q. Let h be a skew-hermitian form over (Q, ) such that σ = ad_{h}. Since h_{K} is hyperbolic, it follows from [34, Prop., p. 382] that h ≃ ⟨d⟩φ_{0} for some 6-dimensional quadratic form φ_{0} over F. Decompose

φ_{0} = ⟨α_{1}⟩⟨1, −β_{1}⟩ ⊥ ⟨α_{2}⟩⟨1, −β_{2}⟩ ⊥ ⟨α_{3}⟩⟨1, −β_{3}⟩ for some α_{i}, β_{i} ∈ F^{×},

and let Q_{i} = (δ, β_{i})_{F} be the quaternion F -algebra with norm n_{Q_{i}} = ⟨⟨δ, β_{i}⟩⟩ for i = 1, 2, 3. Computation shows that e_{2}(⟨α_{i}d⟩⟨1, −β_{i}⟩) is represented by Q_{i} in M_{Q}^{2}(F), hence (A, σ) decomposes as

(A, σ) ∈ ⊞_{i=1}^{3} Ad_{⟨α_{i}d⟩⟨1,−β_{i}⟩} .

So, the subgroup U ⊂ Br F generated by [Q_{1}], [Q_{2}] and [Q_{3}] is a decomposition group for (A, σ). Again, U is split by K, hence H_{U} = 0. □

5.3. Trivial f_{3}-invariant without quadratic splitting. We now construct an algebra with involution (A, σ), of degree 12 and index 2, such that f_{3}(σ) = 0, and yet, there is no quadratic extension K of F over which (A, σ) is both split and hyperbolic. In particular, by Peyre’s Theorem 3.13, we have H_{U} ≠ 0 for all decomposition groups U of (A, σ). (See Remark 5.14 for an example where (A, σ) has a decomposition group U whith H_{U} ≠ 0 and another U′ with H_{U′} = 0.)

Remark 5.7. In his paper [32, §6.2], Peyre provides an example of a quaternionic subgroup U ⊂ Br(F) with H_{U} ≠ 0, but the way he proves H_{U} is nonzero is by describing an element c ∈ H^{3}(F) which is not of order 2, hence does not belong to F^{×} · U, and yet is in the kernel of the restriction map H^{3}(F) → H^{3}(F_{U}). Thus, the group U in Peyre’s example satisfies f_{3}(U) ≠ 0 (see Remark 5.5). In this section, we construct an example of a different flavor, namely a subgroup U with H_{U} ≠ 0, but f_{3}(U) = 0. Hence, the homology group in this case is generated by a cohomology class which is of order 2, and in the kernel of H^{3}(F) → H^{3}(F_{U}), but does not belong to F^{×} · U.

Notation 5.8. Until the end of this section, k is a field (of characteristic different from 2), M is a triquadratic field extension of k (of degree 8) and K is a quadratic extension of k in M,

M = k(\sqrt{a}, \sqrt{b}, \sqrt{c}) ⊃ K = k(\sqrt{a}).

We let C be a central simple k-algebra of exponent 2 split by M and we write

[C] ∈ Dec(M/k)

to express the property that there exist α, β, γ ∈ k^{×} such that

[C] = (a, α)_{k} + (b, β)_{k} + (c, γ)_{k}.

The existence of algebras C as above such that [C] ∉ Dec(M/k) is shown in [12, §5]. By contrast, it follows from a theorem of Albert that every central simple algebra of exponent 2 split by a biquadratic extension has a decomposition up to Brauer-equivalence into a tensor product of quaternion algebras adapted to the biquadratic extension (see [27, Prop. 5.2]), so (viewing M as K(\sqrt{bc}, \sqrt{c})) there exist x, y ∈ K^{×} such that

[C_{K}] = (bc, x)_{K} + (c, y)_{K}.

By multiplying x and y by squares in K, we may—and will—assume x, y ∉ k. We have cor_{K/k}[C_{K}] = 2[C] = 0, hence letting N denote the norm map from K to k, the previous equation leads, by the projection formula, to the following: (bc, N(x))_{k} + (c, N(y))_{k} = 0. We may then consider the following quaternion k-algebra:

(4)  H = (bc, N(x))_{k} = (c, N(y))_{k}.

Since N(x), N(y) are norms from K = k(\sqrt{a}) to k, we have (a, N(x))_{k} = (a, N(y))_{k} = 0, hence we may also write

(5)  H = (abc, N(x))_{k} = (ac, N(y))_{k}.

Let B = (bc, x)_{K} ⊗_{K} (c, y)_{K} be the biquaternion algebra Brauer-equivalent to C_{K}, and let ψ be the Albert form of B over K defined by

ψ = ⟨bc, x, −bcx, −c, −y, cy⟩.

Let s: K → k be a nontrivial linear map such that s(1) = 0, and let s_{⋆} denote the corresponding Scharlau transfer. Using the properties of s_{⋆} (see for instance [28, p. 189, p. 198]), we can make the following computation in the Witt group W (k):

s_{⋆}(ψ) =s_{⋆}(⟨x, −bcx, −y, cy⟩) = s_{⋆}(⟨x⟩)⟨⟨bc⟩⟩ − s_{⋆}(⟨y⟩)⟨⟨c⟩⟩
=⟨s(x)⟩⟨⟨bc, N(x)⟩⟩ − ⟨s(y)⟩⟨⟨c, N(y)⟩⟩.

(Recall that we assume x, y ∉ k, so s(x), s(y) ≠ 0.) In view of (4), the last equation yields

s_{⋆}(ψ) = ⟨s(x), −s(y)⟩n_{H},

where n_{H} is the norm form of H. Thus, s_{⋆}(ψ) ∈ I^{3}(k), and we may consider

(6)  e_{3}(s_{⋆}(ψ)) = s(x)s(y) · [H] ∈ _{2}H^{3}(k).

This class represents an invariant of B defined by Barry [2]. It is shown in [2, Prop. 4.4] that e_{3}(s_{⋆}(ψ)) ∈ N(K^{×}) · [C] if and only if the biquaternion algebra B has a descent to k, i.e., there exist quaternion k-algebras A_{1}, A_{2} such that B ≃ A_{1} ⊗_{k} A_{2} ⊗_{k} K. Finally, let t be an indeterminate over k, and let F = k(t). Consider the subgroup U ⊂ Br(F) generated by the Brauer classes (a, t)_{F}, (b, t)_{F} and (c, t)_{F} + [H_{F}]. In view of (4) and (5), one may easily check that U is a quaternionic subgroup of order 8:

U = {0, (a, t)_{F} , (b, t)_{F} , (c, N(y)t)_{F} ,

(ab, t)_{F} , (ac, N(y)t)_{F} , (bc, N(x)t)_{F} , (abc, N(x)t)_{F} }.

We set

ξ = t · [C] + e_{3}(s_{⋆}(ψ)) ∈ _{2}H^{3}(F).

This construction yields examples with trivial f_{3} but with no quadratic splitting mentioned in the introduction to this section, as we proceed to show. First, we describe the group H_{U} and give a criterion for its vanishing:

Theorem 5.9. Use the notation 5.8. Denote by \overline{ξ}^{U} ∈ H^{3}(F)/F^{×} ·U the image of ξ ∈ H^{3}(F). We have

H_{U} = \{0, \overline{ξ}^{U}\}  and  f_{3}(U) = 0.

Moreover, the following conditions are equivalent:

(a) [C] ∈ Dec(M/k);

(b) U is split by some quadratic field extension E/F;

(c) H_{U} = 0;

(d) ξ ∈ F^{×} · U.

The core of the proof is the following technical lemma:

Lemma 5.10. With the notation 5.8, every field extension of F that splits U also splits ξ.

Proof. Let L be an extension of F that splits U. We consider two cases, depending on whether a ∈ L^{×2} or a ∉ L^{×2}. Suppose first a ∈ L^{×2}, so we may identify K with a subfield of L, hence x, y ∈ L^{×} and [C_{L}] = (bc, x)_{L} + (c, y)_{L}. Since L splits (b, t)_{F}, we have (t, bc, x)_{L} = (t, c, x)_{L}, hence t · [C_{L}] = xy · (t, c)_{L}. Since L also splits (t, c)_{F} + [H_{F}], we have

t · [C_{L}] = xy · [H_{L}].

Comparing with (6), we see that it suffices to show xy · [H_{L}] = s(x)s(y) · [H_{L}] to prove that L splits ξ. Let ι be the nontrivial automorphism of K over k. Writing x = x_{0} + x_{1}\sqrt{a} and y = y_{0} + y_{1}\sqrt{a} with x_{i}, y_{i} ∈ k, we have

s(x)s(y) = x_{1}y_{1}s(\sqrt{a})^{2} and (x − ι(x))(y − ι(y)) = 4x_{1}y_{1}a.

Hence s(x)s(y) ≡ (x − ι(x))(y − ι(y)) mod L^{×2}. We also have

(x − ι(x), N(x))_{K} = (x, N(x))_{K} because (x^{2} − N(x), N(x))_{K} = 0.

From the expression H = (bc, N(x))_{k} it then follows that x · [H_{K}] = (x − ι(x)) · [H_{K}]. Similarly, from H = (c, N(y))_{k} we have y · [H_{K}] = (y − ι(y)) · [H_{K}], hence

xy · [H_{L}] = s(x)s(y) · [H_{L}].

Thus, we have proved L splits ξ under the additional hypothesis that a ∈ L^{×2}. For the rest of the proof of (i), assume a ∉ L^{×2}. Let L′ = L(\sqrt{a}) = L ⊗_{k} K, and write again s: L′ → L for the L-linear extension of s to L′ and N: L′ → L for the norm map. If t ∈ L^{×2}, then ξ_{L} = e_{3}(s_{⋆}(ψ))_{L}. Moreover, L splits H because it splits U. Therefore, (6) shows that L splits e_{3}(s_{⋆}(ψ)). For the rest of the proof, we may thus also assume t ∉ L^{×2}. Since (a, t)_{L} = 0, we may find z_{0} ∈ L′ such that t = N(z_{0}). Because L splits (b, t)_{F}, we have (b, N(z_{0}))_{L} = 0, so cor_{L′/L}(b, z_{0})_{L′} = 0. It follows that (b, z_{0})_{L′} has an involution of the second kind, hence also a descent to L by a theorem of Albert (see [26, (2.22)]). We may choose a descent of the form (b, z_{0})_{L′} = (b, ζ)_{L′} for some ζ ∈ L^{×}; see [40, (2.6)]. Let z = z_{0}ζ ∈ L′^{×}. We then have (b, z)_{L′} = 0, hence after taking the corestriction to L

(b, N(z))_{L} = 0.

We also have t = N(z_{0}) ≡ N(z) mod L^{×2}. Since L splits [H_{F}] + (c, t)_{F}, we have

H_{L} = (c, N(z))_{L}.

Since H = (bc, N(x))_{k} = (c, N(y))_{k} by (4), it follows that

(bc, N(xz))_{L} = (c, N(yz))_{L} = 0.

If s(xz) = 0 (i.e., xz ∈ L), then s_{⋆}(⟨xz⟩⟨⟨bc⟩⟩) is hyperbolic. If s(xz) ≠ 0, computation yields

s_{⋆}(⟨xz⟩⟨⟨bc⟩⟩) = ⟨s(xz)⟩⟨⟨bc, N(xz)⟩⟩;

but since the quaternion algebra (bc, N(xz))_{L} is split, the form s_{⋆}(⟨xz⟩⟨⟨bc⟩⟩) is also hyperbolic in this case. Therefore, we may find λ ∈ L^{×} represented by ⟨xz⟩⟨⟨bc⟩⟩; we then have

(7)  ⟨xz⟩⟨⟨bc⟩⟩ = ⟨λ⟩⟨⟨bc⟩⟩,  hence also ⟨x⟩⟨⟨bc⟩⟩ = ⟨λz⟩⟨⟨bc⟩⟩.

Similarly, since the quaternion algebra (c, N(yz))_{L} is split, the form s_{⋆}(⟨yz⟩⟨⟨c⟩⟩) is hyperbolic, and we may find μ ∈ L^{×} such that

(8)  ⟨yz⟩⟨⟨c⟩⟩ = ⟨μ⟩⟨⟨c⟩⟩,  hence also ⟨y⟩⟨⟨c⟩⟩ = ⟨μz⟩⟨⟨c⟩⟩.

As a result of (7) and (8), we have ⟨x, −bcx⟩ = ⟨λz, −λzbc⟩ and ⟨−y, cy⟩ = ⟨−μz, μzc⟩, hence we may rewrite ψ over L′ as

ψ_{L′} = ⟨bc, −c, λz, −λzbc, −μz, μzc⟩.

Note that z ∉ L since t ∉ L^{×2}, hence s(z) ≠ 0. Using the last expression for ψ_{L′} we may now compute

s_{⋆}(ψ)_{L} = s_{⋆}(ψ_{L′}) = s_{⋆}(⟨z⟩)⟨λ, −λbc, −μ, μc⟩ = ⟨s(z)⟩⟨⟨N(z)⟩⟩⟨λ, −λbc, −μ, μc⟩.

Since (bc, N(z))_{L} = (c, N(z))_{L} = H_{L}, we have ⟨⟨N(z), bc⟩⟩ = ⟨⟨N(z), c⟩⟩ = (n_{H})_{L}, hence s_{⋆}(ψ)_{L} = ⟨s(z)⟩⟨λ, −μ⟩(n_{H})_{L}, and therefore

(9)  e_{3}(s_{⋆}(ψ))_{L} = (λμ) · [H_{L}].

On the other hand, we have [C_{K}] = (bc, x)_{K} + (c, y)_{K}, hence since (b, z)_{L′} = 0

[C_{L′}] = (bc, xz)_{L′} + (c, yz)_{L′}.

In view of (7) and (8), we may rewrite the right side as follows:

[C_{L′}] = (bc, λ)_{L′} + (c, μ)_{L′}.

Therefore, [C_{L}] + (bc, λ)_{L} + (c, y)_{L} is split by L′. We may then find ν ∈ L^{×} such that

[C_{L}] = (bc, λ)_{L} + (c, μ)_{L} + (a, ν)_{L}.

Since L splits U, we have (t, a)_{L} = (t, b)_{L} = 0 and (t, c)_{L} = H_{L}. It follows that

(t) · [C_{L}] = (t, c, λμ)_{L} = (λμ) · [H_{L}].

By comparing with (9), we see that ξ vanishes over L. The proof of the lemma is thus complete. □

Proof of Theorem 5.9. Since 2ξ = 0, the assertion f_{3}(U) = 0 follows from H_{U} = \{0, \overline{ξ}^{U}\}, see Remark 5.5(ii). Moreover, the field F_{U} splits U. Therefore, by Lemma 5.10, we have ξ ∈ ker(H^{3}(F) → H^{3}(F_{U})), so that \overline{ξ}^{U} ∈ H_{U}. Since we know from Theorem 3.13 that the order of H_{U} is at most 2, it suffices to show that \overline{ξ}^{U} ≠ 0 when H_{U} ≠ 0 to establish H_{U} = \{0, \overline{ξ}^{U}\}. Therefore, proving the equivalence of (a), (b), (c) and (d) completes the proof. Let us first prove (a) ⇒ (b). Suppose [C] = (a, α)_{k} + (b, β)_{k} + (c, γ)_{k} for some α, β, γ ∈ k^{×}. Since [C_{K}] = (bc, x)_{K} +(c, y)_{K}, it follows that (b, β)_{K} +(c, γ)_{K} = (bc, x)_{K} + (c, y)_{K}, hence

(bc, βx)_{K} = (c, βγy)_{K}.

By the common slot lemma [28, Th. III.4.13], we may find z ∈ K^{×} such that

(10)  (bc, βx)_{K} = (bc, z)_{K} = (c, z)_{K} = (c, βγy)_{K}.

Let E = F (\sqrt{N(z)t} ), a quadratic extension of F. We claim that E splits U. First, observe that N(z)t is represented by the form ⟨t, −at⟩, hence the quaternion algebra (a, t)_{F} contains a pure quaternion with square N(z)t. Therefore, E splits (a, t)_{F}. Likewise, from (10) we see that (b, z)_{K} = 0, hence by taking the corestriction to k we have (b, N(z))_{k} = 0. Therefore, N(z)t is represented by the form ⟨t, −bt⟩, and it follows that E splits (b, t)_{F}. Finally, by taking the corestriction of each side of the rightmost equation in (10), we obtain (c, N(z))_{k} = (c, N(y))_{k}, so N(y)N(z) is represented by ⟨1, −c⟩ and therefore N(z)t is represented by ⟨N(y)t, −cN(y)t⟩. It follows that E splits the quaternion algebra (c, N(y)t)_{F}. We have thus shown that E splits three generators of U, hence E splits U. The implication (b) ⇒ (c) follows immediately from Peyre’s Theorem 3.13. Moreover, (c) ⇒ (d) is clear since \overline{ξ}^{U} ∈ H_{U}. To complete the proof, we show (d) ⇒ (a). Suppose there exist λ_{1}, λ_{2}, λ_{3} ∈ F^{×} such that

(11)  ξ = (λ_{1}, a, t) + (λ_{2}, b, t) + (λ_{3}, c, N(y)t).

Let ∂: H^{i}(F) → H^{i−1}(k) be the residue map associated to the t-adic valuation, for i = 2, 3. Since e_{3}(s_{⋆}(ψ)) ∈ H^{3}(k) we have ∂(e_{3}(s_{⋆}(ψ))) = 0, hence ∂(ξ) = [C]. Therefore, taking the image of each side of (11) under the residue map yields

[C] = a · ∂(λ_{1}, t) + b · ∂(λ_{2}, t) + c · ∂(λ_{3}, N(y)t),

so that [C] ∈ Dec(M/k). □

As a corollary, we get:

Corollary 5.11. Use the notation 5.8, and assume [C] ∉ Dec(M/k). Then U ⊂ Br(F) is a quaternionic subgroup of order 8 such that ∑_{[H]∈U} n_{H} ∈ I^{4}(F) (i.e., f_{3}(U) = 0), which is not split by any quadratic extension of F (i.e., H_{U} ≠ 0).

To obtain an example of a central simple algebra with orthogonal involution of degree 12 without quadratic splitting, we need a more stringent condition on C:

Lemma 5.12. With the notation 5.8, the following conditions are equivalent: (a) there is a quadratic extension E of F that splits (a, t)_{F} and ξ;

(b) the algebra C is Brauer-equivalent to a tensor product of quaternion k-algebras A_{1} ⊗_{k} A_{2} ⊗_{k} A_{3} with A_{3} split by K.

Proof. (a) ⇒ (b): Let F̂ = k((t)) be the completion of F for the t-adic valuation. The field E does not embed in F̂ because F̂ does not split (a, t)_{F}. Therefore, E and F̂ are linearly disjoint over F and we may consider the field Ê = E ⊗_{F} F̂, which is a quadratic extension of F̂ that splits (a, t)F̂ and ξF̂. Each square class in F̂ is represented by an element in k^{×} or an element of the form ut with u ∈ k^{×}, see [28, Cor. VI.1.3]. Therefore, we may assume that either Ê = F̂(\sqrt{u}) or Ê = F̂(\sqrt{ut}) for some u ∈ k^{×}. Suppose first Ê = F̂(\sqrt{u}) with u ∈ k^{×}. Since the quaternion algebra (a, t)F̂ is split by Ê, it must contain a pure quaternion with square u, hence u is represented by ⟨a, t, −at⟩ over F̂. Therefore, u ≡ a mod k^{×2}, and Ê = K((t)). From

ξÊ = t · [CÊ] + e_{3}(s_{⋆}(ψ))Ê = 0,

it follows by taking images under the residue map H^{3}(Ê) → H^{2}(K) associated to the t-adic valuation that [C_{K}] = 0. Then C is Brauer-equivalent to a quaternion algebra A_{3} split by K, and (b) holds with A_{1}, A_{2} split quaternion algebras. Suppose next Ê = F̂(\sqrt{ut}) for some u ∈ k^{×}. Since Ê splits (a, t)F̂, it follows as above that ut is represented by ⟨a, t, −at⟩ over F̂, hence u is represented by ⟨1, −a⟩, which means that u ∈ N(K^{×}). Because ut is a square in Ê, we have t · [CÊ] = u · [CÊ], hence the equation ξÊ = 0 yields

u · [CÊ] + e_{3}(s_{⋆}(ψ))Ê = (u · [C] + e_{3}(s_{⋆}(ψ)))Ê = 0.

Since F̂ = k((t)) = k((ut)) we have Ê = k((\sqrt{ut})), hence the scalar extension map H^{3}(k) → H^{3}(Ê) is injective. Therefore, the last equation yields

u · [C] + e_{3}(s_{⋆}(ψ)) = 0,

which shows that e_{3}(s_{⋆}(ψ)) ∈ N(K^{×}) · [C] because u ∈ N(K^{×}). By Barry’s result [2, Prop. 4.4], it follows that the biquaternion algebra B has a descent to k: there exist quaternion k-algebras A_{1}, A_{2} such that B ≃ A_{1} ⊗_{k} A_{2} ⊗_{k} K. Since C_{K} is Brauer-equivalent to B, it follows that C ⊗ A_{1} ⊗_{k} A_{2} is split by K. It is therefore Brauer-equivalent to a quaternion algebra A_{3} split by K, and C is Brauer-equivalent to A_{1} ⊗_{k} A_{2} ⊗_{k} A_{3}, proving (b).

(b) ⇒ (a): Since B is Brauer-equivalent to C_{K}, condition (b) implies that B ≃ A_{1} ⊗_{k} A_{2} ⊗_{k} K. From Barry’s result [2, Prop. 4.4], it follows that e_{3}(s_{⋆}(ψ)) = u · [C] for some u ∈ N(K^{×}). Let E = F (\sqrt{ut}). Then (a, t)_{E} ≃ (a, u)_{E}, hence E splits (a, t)_{F} because u ∈ N(K^{×}). Moreover, ξ_{E} = (u · [C] + e_{3}(s_{⋆}(ψ)))_{E}, so E also splits ξ. Therefore, (a) holds. □

Examples of algebras C for which condition (b) of Lemma 5.12 does not hold include indecomposable division algebras of degree 8 and exponent 2; other examples are given in [3]. Note that condition (b) is weaker than [C] ∈ Dec(M/k); it is in fact strictly weaker: see Remark 5.14.

Corollary 5.13. Use the notation 5.8, and let Q = (a, t)_{F}. There exists an orthogonal involution ρ on M_{6}(Q) with the following properties:

(i) ρ has trivial discriminant and trivial Clifford invariant; (ii) e_{3}(ρ) = ξ mod F^{×} · [Q]; (iii) U is a decomposition group of ρ; (iv) f_{3}(ρ) = 0.

For any involution ρ satisfying (i) and (ii), there exists a quadratic extension of F over which Q is split and ρ is hyperbolic if and only if the equivalent conditions of Lemma 5.12 hold.

Proof. By Remark 3.11, there is an orthogonal involution ρ on M_{6}(Q) with trivial discriminant and trivial Clifford invariant, and with decomposition group U. By Theorems 4.6 and 5.4, \overline{e_{3}(ρ)}^{U} generates H_{U}, and f_{3}(ρ) = f_{3}(U). Therefore, Theorem 5.9 yields \overline{e_{3}(ρ)}^{U} = \overline{ξ}^{U} and f_{3}(ρ) = 0. By Proposition 4.7(ii), we may assume e_{3}(ρ) = ξ mod F^{×} · [Q]. Thus, ρ satisfies conditions (i)–(iv). Now, let ρ be any orthogonal involution on M_{6}(Q) satisfying (i) and (ii). Because of (ii), condition (a) of Lemma 5.12 holds if and only if there is a quadratic extension E of F such that [Q_{E}] = 0 and e_{3}(ρ)_{E} = 0. By Theorem 4.1(i), the last equation holds if and only if ρ_{E} is hyperbolic. □

Suppose C does not satisfy condition (b) of Lemma 5.12 (e.g., C is an indecomposable division algebra of degree 8 and exponent 2 split by M). Then for any involution ρ on M_{6}(Q) satisfying the properties (i)–(iv) of Corollary 5.13 we have f_{3}(ρ) = 0, and yet there is no quadratic extension of F over which Q is split and ρ is hyperbolic. From (b’) ⇒ (a’) in Proposition 5.6, it follows that H_{U′} ≠ 0 for every decomposition group U′ of ρ.

Remark 5.14. By [40, Cor. 3.2], for any triquadratic extension M/k, any 2torsion Brauer class in Br(k) split by M is represented modulo Dec(M/k) by a quaternion algebra. Therefore, if the triquadratic extension M/k is such that Dec(M/k) does not coincide with the subgroup of _{2} Br(k) split by M (see [12, §5] for examples of such extensions), we may find a quaternion k-algebra C split by M such that [C] ∉ Dec(M/k). The algebra C obviously satisfies condition (b) of Lemma 5.12 (with A_{2} and A_{3} split), so for any involution ρ on M_{6}(Q) satisfying the properties (i)–(iv) of Corollary 5.13 we may find a quadratic extension of F over which Q is split and ρ is hyperbolic. From (a’) ⇒ (b’) in Proposition 5.6, it follows that there exists a decomposition group U′ of ρ such that H_{U′} = 0. Yet, because [C] ∉ Dec(M/k), the decomposition group U of ρ satisfies H_{U} ≠ 0 by Theorem 5.9.

6. Application to degree 8 algebras with involution

The Arason invariant in degree 8 was studied in [35] for orthogonal involutions with trivial discriminant and trivial Clifford algebra. In this section, we extend it to algebras of degree 8 and index 2, when the involution has trivial discriminant and the two components of the Clifford algebra also both have index 2. First, we prove an analogue of Theorem 3.2 on additive decompositions, for degree 8 algebras with orthogonal involution of trivial discriminant.

6.1. Additive decompositions in degree 8. Let (A, σ) be a degree 8 algebra with orthogonal involution of trivial discriminant. We let (C^{+}(A, σ), σ^{+}) and (C^{−}(A, σ), σ^{−}) denote the two components of the Clifford algebra of (A, σ), endowed with the involutions induced by the canonical involution of the Clifford algebra. Both algebras have degree 8, both involutions have trivial discriminant, and by triality [26, (42.3)],

(12)  C(C^{+}(A, σ), σ^{+}) ≃ (C^{−}(A, σ), σ^{−}) × (A, σ)

and

(13)  C(C^{−}(A, σ), σ^{−}) ≃ (A, σ) × (C^{+}(A, σ), σ^{+}).

Assume (A, σ) decomposes as a sum (A, σ) ∈ (A_{1}, σ_{1}) ⊞(A_{2}, σ_{2}) of two degree 4 algebras with orthogonal involution of trivial discriminant. Each summand is a tensor product of two quaternion algebras with canonical involution, and we get

(14)  (A, σ) ∈ ((Q_{1}, ) ⊗ (Q_{2}, )) ⊞ ((Q_{3}, ) ⊗ (Q_{4}, )),

for some quaternion algebras Q_{1}, Q_{2}, Q_{3} and Q_{4} such that A is Brauerequivalent to Q_{1} ⊗ Q_{2} and Q_{3} ⊗ Q_{4}. By [36, Prop. 6.6], the two components of the Clifford algebra of (A, σ) then admit similar decompositions, namely, up to permutation of the two components, we have:

(15)  (C^{+}(A, σ), σ^{+}) ∈ ((Q_{1}, ) ⊗ (Q_{3}, )) ⊞ ((Q_{2}, ) ⊗ (Q_{4}, )),

and

(16)  (C^{−}(A, σ), σ^{−}) ∈ ((Q_{1}, ) ⊗ (Q_{4}, )) ⊞ ((Q_{2}, ) ⊗ (Q_{3}, )).

Mimicking the construction in §3, we associate to every decomposition of (A, σ) as above the subgroup W of the Brauer group of F generated by any three elements among the [Q_{i}] for 1 ≤ i ≤ 4. We call W a decomposition group of (A, σ). It consists of at most 8 elements, and can be described explicitly by

W = {0, [A], [C^{+}(A, σ)], [C^{−}(A, σ)], [Q_{1}], [Q_{2}], [Q_{3}], [Q_{4}]}.

In view of their additive decompositions, W also is a decomposition group of the two components (C^{+}(A, σ), σ^{+}) and (C^{−}(A, σ), σ^{−}) of the Clifford algebra. Note that, in contrast with the decomposition groups of algebras of degree 12 in Definition 3.6, the group W may contain three Brauer classes of index 4 instead of at most one. Nevertheless, it has similar properties; for instance, we prove:

Proposition 6.1. Let (A, σ) be a central simple algebra of degree 8 with an orthogonal involution of trivial discriminant.

(i) Suppose (A, σ) has an additive decomposition as in (14), with decomposition group W. For any extension K/F which splits W, the algebra with involution (A_{K}, σ_{K}) is split and hyperbolic.

(ii) The converse holds for quadratic extensions: if (A, σ) is split and hyperbolic over a quadratic extension K of F, then (A, σ) has an additive decomposition with decomposition group split by K.

Proof. (i) If a field K splits W, then it splits A, and moreover each summand in (14) is split and hyperbolic over K, therefore σ_{K} is hyperbolic. (ii) To prove the converse, suppose K = F (d) with d^{2} = δ ∈ F^{×}, and assume A_{K} is split and σ_{K} is hyperbolic, hence ind A ≤ 2. If A is split, we have as in the proof of Proposition 5.6 (A, σ) ≃ Ad_{φ} with φ an 8-dimensional quadratic form multiple of ⟨1, −δ⟩. We may then find quaternion F -algebras Q_{1}, Q_{2} split by K and scalars α_{1}, α_{2} ∈ F^{×} such that φ ≃ ⟨α_{1}⟩n_{Q_{1}} ⊥ ⟨α_{2}⟩n_{Q_{2}}. As in Example 3.4, we obtain a decomposition

(A, σ) ∈ ((Q_{1}, ) ⊗ (Q_{1}, )) ⊞((Q_{2}, ) ⊗ (Q_{2}, )).

The corresponding decomposition group is {0, [Q_{1}], [Q_{2}], [Q_{1}] + [Q_{2}]}; it is split by K. If ind A = 2, let Q be the quaternion division algebra Brauer-equivalent to A. Since K splits A, we may, again as in the proof of Proposition 5.6, identify K with a subfield of Q and find a skew-hermitian form h of the form ⟨d⟩⟨1, α, β, γ⟩ (with α, β, γ ∈ F^{×}) such that (A, σ) ≃ Ad_{h}. Then

(A, σ) ∈ Ad_{⟨d⟩⟨1,α⟩} ⊞ Ad_{⟨d⟩⟨β,γ⟩}

is a decomposition in which each of the summands becomes hyperbolic over K. The corresponding decomposition group is therefore split by K. □

There exist quadratic forms φ of dimension 8 with trivial discriminant and Clifford algebra of index 4 that do not decompose into an orthogonal sum of two 4-dimensional quadratic forms of trivial discriminant, see [21, Cor. 16.8] or [20, Cor 6.2]. For such a form, neither Ad_{φ} nor the components of its Clifford algebra have additive decompositions as in (14). The next proposition shows, by contrast, that such a decomposition always exist if at least two among the algebras A, C^{+}(A, σ) and C^{−}(A, σ) have index ≤ 2.

Proposition 6.2. Let (A, σ) be a central simple algebra of degree 8 with orthogonal involution of trivial discriminant. We assume at least two among the algebras A, C^{+}(A, σ) and C^{−}(A, σ) have index ≤ 2. Then all three algebras with involution (A, σ), (C^{+}(A, σ), σ^{+}) and (C^{−}(A, σ), σ^{−}) admit an additive decomposition as a sum of two degree 4 algebras with orthogonal involution of trivial discriminant as in (14).

Proof. Assume two among ind A, ind C^{+}(A, σ), ind C^{−}(A, σ) are 1 or 2. By triality, see (12) to (16) above, it is enough to prove that one of the three algebras with involution, say (A, σ) has an additive decomposition. Since A, C^{+}(A, σ), C^{−}(A, σ) are interchanged by triality, we may also assume ind A ≤ 2. If A is split, so (A, σ) ≃ Ad_{φ} for some 8-dimensional quadratic form φ with trivial discriminant and Clifford algebra of index at most 2, then (a) holds by a result of Knebusch [25, Ex. 9.12], which shows that φ is the product of a 2-dimensional quadratic form and a 4-dimensional quadratic form. For the rest of the proof, assume (A, σ) ≃ Ad_{h} for some skew-hermitian form h over a quaternion division algebra (Q, ). Let q ∈ Q be a nonzero quaternion represented by h, and let h ≃ ⟨q⟩ ⊥ h′ for some skew-hermitian form h′ of absolute rank 6. As we saw in the proof of Theorem 3.2, over the quadratic extension K = F (q) the algebra Q splits and the form ⟨q⟩ becomes hyperbolic, hence h_{K} and h′_{K} are Witt-equivalent. In particular, it follows that e_{2}((ad_{h′})_{K}) = e_{2}((ad_{h})_{K}) has index at most 2. But (ad_{h′})_{K} = ad_{ψ} for some Albert form ψ over K, so ψ is isotropic. It follows by [34, Prop., p. 382] that h′ represents some scalar multiple of q; thus h ≃ ⟨q⟩⟨1, −λ⟩ ⊥ h′′ for some λ ∈ F^{×} and some skew-hermitian form h′′ of absolute rank 4. The discriminant of h′′ must be trivial because h and ⟨q⟩⟨1, −λ⟩ have trivial discriminant, and we thus have the required decomposition for (A, σ). □

Remark 6.3. It follows that all trialitarian triples such that at least two of the algebras have index ≤ 2 have a description as in (14) to (16).

6.2. An extension of the Arason invariant in degree 8 and index 2. Throughout this section, (A, σ) is a central simple F -algebra of degree 8 and trivial discriminant. It is known that (A, σ) is a tensor product of quaternion algebras with involution if and only if e_{2}(σ) = 0, see [26, (42.11)]. In this case, the Arason invariant e_{3}(σ) ∈ M_{A}^{3}(F) is defined when A has index at most 4 (see §2.5) , and represented by an element of order 2 in H^{3}(F), see [35]. Here, we extend the definition of the e_{3} invariant under the following hypothesis:

(17)  ind A = ind C^{+}(A, σ) = ind C^{−}(A, σ) = 2.

By Proposition 6.2, this condition implies that (A, σ) decomposes into a sum of two central simple algebras of degree 4 with involutions of trivial discriminant. Moreover, the associated decomposition group W is a quaternionic subgroup of Br(F). Let Q, Q^{+}, Q^{−} be the quaternion division algebras over F that are Brauer-equivalent to A, C^{+}(A, σ), and C^{−}(A, σ) respectively. From the Clifford algebra relations [26, (9.12)], we know [Q^{+}] + [Q^{−}] = [Q]. Therefore, the following is a subgroup of the Brauer group:

V = {0, [Q], [Q^{+}], [Q^{−}]} ⊂ Br(F).

Condition (17) implies that |V | = 4. Moreover, V also is a subgroup of every decomposition group of (A, σ). To (A, σ), we may associate algebras of degree 12 with orthogonal involution with trivial discriminant and trivial Clifford invariant by considering any involution ρ of M_{6}(Q) such that

(18)  (M_{6}(Q), ρ) ∈ (A, σ) ⊞((Q^{+}, ) ⊗ (Q^{−}, )).

Since the two components of the Clifford algebra of σ are Brauer-equivalent to Q^{+} and Q^{−}, the involution ρ has trivial Clifford invariant. Therefore, we may consider its Arason invariant e_{3}(ρ) ∈ M_{Q}^{3}(F). The following lemma compares the Arason invariant of two such involutions:

Lemma 6.4. Let ρ and ρ′ be two involutions of M_{6}(Q) satisfying (18). There exists λ ∈ F^{×} such that

e_{3}(ρ) − e_{3}(ρ′) = (λ) · [Q^{+}] = (λ) · [Q^{−}] ∈ M_{Q}^{3}(F).

Moreover, f_{3}(ρ) = f_{3}(ρ′).

Proof. By definition of the direct orthogonal sum for algebras with involution, we may pick skew-hermitian forms h_{1} and h_{2} over (Q, ) such that σ = ad_{h_{1}},

⊗ ≃ ad_{h_{2}} and ρ = ad_{h_{1}⊥h_{2}}. Moreover, there exists λ ∈ F^{×} such that ρ′ = ad_{h_{1}⊥⟨λ⟩h_{2}}. Therefore, we have (see § 2.4 and 2.5):

e_{3}(ρ) − e_{3}(ρ′) = e_{3}(⟨1, −λ⟩h_{2}) = (λ) · [Q^{+}] mod F^{×} · [Q],

by Proposition 2.6, since e_{2}(h_{2}) = [Q^{+}] = [Q^{−}] mod [Q]. Moreover, if c and c′ ∈ H^{3}(F) are representatives of e_{3}(ρ) and e_{3}(ρ′) respectively, then

c′ − c ∈ (λ) · [Q^{+}] + F^{×} · [Q].

So 2c = 2c′, and this finishes the proof.

□

Let F^{×} · V ⊂ H^{3}(F) be the subgroup consisting of the products λ · v with λ ∈ F^{×} and v ∈ V = {0, [Q], [Q^{+}], [Q^{−}]}. This subgroup contains F^{×} · [Q], so we may consider the canonical map ^{V} : M_{Q}^{3}(F) → H^{3}(F)/F^{×} · V. The

previous lemma shows that the image \overline{e_{3}(ρ)}^{V} of the Arason invariant of ρ does not depend on the choice of an involution ρ satisfying (18). This leads to the following:

Definition 6.5. With the notation above, we set

e_{3}(σ) = \overline{e_{3}(ρ)}^{V} ∈ H^{3}(F)/F^{×} ·V  and  f_{3}(σ) = f_{3}(ρ) ∈ F^{×} ·[Q] ⊂ H^{3}(F)

where ρ is any involution satisfying

(M_{6}(Q), ρ) ∈ (A, σ) ⊞((Q^{+}, ) ⊗ (Q^{−}, )).

This definition functorially extends the definition of the Arason invariant. Indeed, if K is any extension of F that splits Q^{+} or Q^{−} (or both), then the scalar extension map Br(F) → Br(K) carries V to {0, [Q]}, and any involution ρ as in (18) becomes Witt-equivalent to σ over K. Therefore, scalar extension carries e_{3}(σ) ∈ H^{3}(F)/F^{×} · V defined above to e_{3}(σ_{K}) ∈ M_{Q}^{3}(K) as defined in §2.5.

Example 6.6. Consider a central simple algebra (M_{6}(Q), ρ) of degree 12 and index 2 with an orthogonal involution of trivial discriminant and trivial Clifford invariant. By Theorem 3.2 (M_{6}(Q), ρ) admits additive decompositions

(M_{6}(Q), ρ) ∈ ⊞_{i=1}^{3}((Q_{i}, ) ⊗ (H_{i}, )) with ∑_{i=1}^{3}[H_{i}] = 0,

so it contains symmetric idempotents e_{1}, e_{2}, e_{3} such that

(e_{i}M_{6}(Q)e_{i}, ρ|_{e_{i}M_{6}(Q)e_{i}}) ≃ (Q_{i}, ) ⊗ (H_{i}, ).

Consider the restriction of ρ to (e_{1} + e_{2})M_{6}(Q)(e_{1} + e_{2}); we thus obtain an algebra with involution (M_{4}(Q), σ) such that

(M_{6}(Q), ρ) ∈ (M_{4}(Q), σ) ⊞((Q_{3}, ) ⊗ (H_{3}, ))

and (M_{4}(Q), σ) ∈ ⊞_{i=1}^{2}((Q_{i}, ) ⊗ (H_{i}, )).

It is clear that the discriminant of σ is trivial. Since e_{2}(ρ) = 0, we have

e_{2}(σ) = e_{2}((Q_{3}, ) ⊗ (H_{3}, )) = {[Q_{3}], [H_{3}]}.

Therefore, Condition (17) holds for (M_{4}(Q), σ) if Q_{3} and H_{3} are not split. In that case, we have V = {0, [Q], [Q_{3}], [H_{3}]} and, by definition,

e_{3}(σ) = \overline{e_{3}(ρ)}^{V} ∈ H^{3}(F)/F^{×} · V and f_{3}(σ) = f_{3}(ρ) ∈ F^{×} · [Q] ⊂ H^{3}(F).

The condition that Q_{3} and H_{3} are not split holds in particular when the decomposition group U generated by [Q_{1}], [Q_{2}], [Q_{3}] has order 8.

Example 6.7. Take for (M_{6}(Q), ρ) the algebra Ad_{⟨1,−t⟩} ⊗(λ^{2}E, γ)_{F (t)} of Corollary 2.19, with E a division algebra of degree and exponent 4. (Note that λ^{2}E is Brauer-equivalent to E ⊗ E, hence it has index 2.) Since f_{3}(ρ) ≠ 0, every decomposition group of ρ has order 8; indeed, quaternionic subgroups U ⊂ Br(F) of order dividing 4 have H_{U} = 0 by Theorem 3.12, hence trivial f_{3} by Proposition 5.3. The construction in the previous example yields an algebra with involution (M_{4}(Q), σ) of degree 8 satisfying Condition (17), with f_{3}(σ) = t · [Q] ≠ 0.

Example 6.8. Also, we may take for (M_{6}(Q), ρ) the algebra with involution of Corollary 5.13, and obtain an algebra with involution (M_{4}(Q), σ) of degree 8 satisfying Condition (17) such that (with the notation 5.8) e_{3}(σ) = \overline{ξ}^{V} ∈ H^{3}(F)/F^{×}V. Since ξ ∉ F^{×} · U, we have e_{3}(σ) ≠ 0. Yet, we have f_{3}(σ) = f_{3}(ρ) = 0 by Corollary 5.13. Moreover, there is no quadratic extension K of F such that Q_{K} is split and σ_{K} is hyperbolic. Indeed, over such a field, (M_{6}(Q), ρ)_{K} would be Witt-equivalent to an algebra of degree 4, hence it would be hyperbolic because e_{2}(ρ) = 0. Corollary 5.13 shows that such quadratic extensions K do not exist.

The next proposition shows that the e_{3} invariant detects isotropy, for any central simple algebra with involution (A, σ) satisfying Condition (17). As in §6.1, we let σ^{+} and σ^{−} denote the canonical involutions on C^{+}(A, σ) and C^{−}(A, σ).

Proposition 6.9. Let (A, σ) be a central simple F -algebra of degree 8 with orthogonal involution of trivial discriminant satisfying (17). With the notation above, we have e_{3}(σ) = e_{3}(σ^{+}) = e_{3}(σ^{−}) and f_{3}(σ) = f_{3}(σ^{+}) = f_{3}(σ^{−}). Moreover, the following conditions are equivalent:

(a) e_{3}(σ) = 0;

(b) σ is isotropic;

(c) (A, σ) is Witt-equivalent to (Q^{+}, ) ⊗ (Q^{−}, ).

Proof. As in §3.3, let F_{V} denote the function field of the product of the Severi– Brauer varieties associated to the elements of V. Extending scalars to F_{V}, we split Q and e_{2}(σ), hence there is a 3-fold Pfister form π over F_{V} such that

σ_{F_{V}} ≃ ad_{π} .

For Pfister forms, we have ad_{π} ≃ ad_{π}^{+} ≃ ad_{π}^{−} (see [26, (42.2)]), hence σ_{F_{V}}^{+} ≃ σ_{F_{V}}^{−} ≃ σ_{F_{V}}, and therefore

e_{3}(σ)_{F_{V}} = e_{3}(σ^{+})_{F_{V}} = e_{3}(σ^{−})_{F_{V}} = e_{3}(π).

Since V is generated by the Brauer classes of two quaternion algebras, it follows from Theorem 3.12 that F^{×} · V is the kernel of the scalar extension map H^{3}(F) → H^{3}(F_{V}), hence the preceding equations yield e_{3}(σ) = e_{3}(σ^{+}) = e_{3}(σ^{−}). We then have f_{3}(σ) = f_{3}(σ^{+}) = f_{3}(σ^{−}), since f_{3}(σ) (resp. f_{3}(σ^{+}), resp. f_{3}(σ^{−})) is 2 times any representative of e_{3}(σ) (resp. e_{3}(σ^{+}), resp. e_{3}(σ^{−})) in H^{3}(F). To complete the proof, we show that (a), (b), and (c) are equivalent. Clearly,

(c) implies (b). The converse follows easily from [26, (15.12)] if A has index 2, and [26, (16.5)] if A is split. Moreover, in view of the definition of e_{3}(σ), the equivalence between (a) and (c) follows from Proposition 4.5. □

As in §4, we may relate the e_{3} invariant to the homology of the Peyre complex of any decomposition group, as follows:

Proposition 6.10. Let (A, σ) be a central simple algebra of degree 8 with an orthogonal involution of trivial discriminant satisfying (17), and let W be a decomposition group of (A, σ). The image \overline{e_{3}(σ)}^{W} of e_{3}(σ) ∈ H^{3}(F)/F^{×} · V in H^{3}(F)/F^{×} · W generates H_{W}, and f_{3}(σ) = f_{3}(W).

Proof. As above, let Q be the quaternion division algebra Brauer-equivalent to A, so we may identify A with M_{4}(Q). Let ρ be an involution on M_{6}(Q) such that

(M_{6}(Q), ρ) ∈ (A, σ) ⊞((Q^{+}, ) ⊗ (Q^{−}, )).

By definition, we have e_{3}(σ) = \overline{e_{3}(ρ)}^{V} and f_{3}(σ) = f_{3}(ρ). Now, consider a decomposition of (A, σ) with decomposition group W (which necessarily contains V):

(A, σ) ∈ ((C_{1}^{+}, ) ⊗ (C_{1}^{−}, )) ⊞((C_{2}^{+}, ) ⊗ (C_{2}^{−}, )).

We have

(M_{6}(Q), ρ) ∈ ((C_{1}^{+}, )⊗(C_{1}^{−}, )) ⊞((C_{2}^{+}, )⊗(C_{2}^{−}, )) ⊞((Q^{+}, )⊗(Q^{−}, )),

which is a decomposition of (M_{6}(Q), ρ) with decomposition group W. Therefore, Theorem 4.6 shows that \overline{e_{3}(ρ)}^{W} generates H_{W} and f_{3}(ρ) = f_{3}(W). The proposition follows because f_{3}(σ) = f_{3}(ρ) and \overline{e_{3}(σ)}^{W} = \overline{e_{3}(ρ)}^{W} since V ⊂ W. □

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Anne Quéguiner-Mathieu Université Paris 13, Sorbonne Paris Cité LAGA - CNRS (UMR 7539) F-93430 Villetaneuse, France queguin@math.univ-paris13.fr Jean-Pierre Tignol ICTEAM Institute, Box L4.05.01 Université catholique de Louvain B-1348 Louvain-la-Neuve, Belgium jean-pierre.tignol@uclouvain.be