THE DISCRIMINANT OF A SYMPLECTIC INVOLUTION

GRÉGORY BERHUY, MARINA MONSURRÒ, AND JEAN-PIERRE TIGNOL

Abstract. An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost’s cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras with symplectic involution.

1. Introduction and Statement of Results

In contrast with orthogonal involutions, for which invariants corresponding to the discriminant and Clifford algebras of quadratic forms are defined, no “classical” invariant is known for symplectic involutions on central simple algebras, besides the signature (see [6, (11.10)]). Using the cohomological invariant of degree 3 defined by Rost for torsors under simply connected absolutely simple linear algebraic groups, we introduce an invariant of symplectic involutions on central simple algeras of degree multiple of 4 with values in the third Galois cohomology group of the center with coefficients {±1} and give an alternative description in terms of trace forms. We call this invariant discriminant since it is the first nontrivial invariant, and because it is directly linked to the discriminant of hermitian forms, see Example 2. Even though its definition is elementary, Rost’s computation of the invariants of torsors under symplectic groups is needed to prove that there is no other cohomological invariant of degree 3 and to establish the relation with trace forms. In the final section, we prove that symplectic involutions of trivial discriminant on central simple algebras of degree 8 and index 4 afford a special type of decomposition. In a sequel to this paper, the discriminant is used to give examples of non R-trivial adjoint symplectic groups of even index.

1.1. Definition of the discriminant. Throughout this paper, F denotes a field of characteristic different from 2. Let A be a finite-dimensional central simple F - algebra, and θ: A → A be an anti-automorphism of order 2. We recall that θ is called a symplectic involution on A if, after scalar extension to a splitting field, θ is adjoint to an alternating form, see [6, (2.5)]. From now on, we suppose the involution θ is of this type. In this case, the degree deg A is necessarily an even integer n = 2m.

The symplectic group Sp(A, θ) is the group scheme over F defined by

Sp(A, θ)(E) = {x ∈ A ⊗_{F} E | θ(x)x = 1}

for any commutative F -algebra E.

The authors gratefully acknowledge support from the TMR network “K-theory and linear algebraic groups” (ERB FMRX CT97-0107). The third author is also grateful to the National Fund for Scientific Research (Belgium) for partial support.

Let Sym(A, θ) be the F -vector space of elements in A fixed by θ,

Sym(A, θ) = {x ∈ A | θ(x) = x}.

We denote by Sym(A, θ)^{×} the set of units in Sym(A, θ),

Sym(A, θ)^{×} = Sym(A, θ) ∩ A^{×}.

We recall that the pfaffian reduced norm is the homogeneous polynomial function of degree m

Nrp_{θ}: Sym(A, θ) → F

uniquely determined by the following conditions:

Nrp_{θ}(1) = 1  and  Nrp_{θ}(x)^{2} = Nrd_{A}(x)  for x ∈ Sym(A, θ),

see [6, (2.10)].

The cohomology set H^{1}(F, Sp(A, θ)) can be represented as

H^{1}(F, Sp(A, θ)) ≃ Sym(A, θ)^{×}/∼ (1)

where ∼ is the equivalence relation defined by x ∼ y if and only if there exists u ∈ A^{×} such that y = uxθ(u), see [6, (29.24)].

Let G_{m} be the multiplicative group. The Kummer exact sequence

1 → μ_{2} → G_{m} ⟶^{2} G_{m} → 1

allows us to identify the cohomology sets H^{1}(F, μ_{2}) and H^{2}(F, μ_{2}) respectively with the quotient F^{×}/F^{×2} and with the 2-torsion subgroup of the Brauer group. For all x ∈ F^{×}, we denote by (x)_{2} ∈ H^{1}(F, μ_{2}) the cohomology class associated to xF^{×2}. Similarly, we denote by [A] ∈ H^{2}(F, μ_{2}) the cohomology class associated to the Brauer class of A. We define

Δ_{θ}: Sym(A, θ)^{×} → H^{3}(F, μ_{2})

as the map given by the cup-product

Δ_{θ}(s) = (Nrp_{θ}(s))_{2} ∪ [A].

It follows from the properties of Nrp_{θ} (see the proof of Proposition 1 below) that Δ_{θ} is well defined on the set of equivalence classes under the relation ∼. The induced map on the quotient can be interpreted under the bijection (1) as the Rost invariant of H^{1}(F, Sp(A, θ)), see [6, p. 440].

Since Nrp_{θ} is homogeneous of degree m, we obtain, for α ∈ F^{×} and s ∈ Sym(A, θ)^{×}, the following relation

(2) Δ_{θ}(αs) = (α^{m} Nrp_{θ}(s))_{2} ∪ [A] = \begin{cases}Δ_{θ}(s) & if m is even, \\ Δ_{θ}(s) + (α)_{2} ∪ [A] & if m is odd. \end{cases}

Therefore, if m is even one can define a relative invariant for symplectic involutions on A as follows:

Definition. Let A be a central simple algebra over F of degree n = 2m ≡ 0 mod 4. Let θ and σ be symplectic involutions on A. There exists (see [6, (2.7)]) s ∈ Sym(A, θ)^{×} such that

σ = Int(s) ∘ θ

where Int(s) denotes the inner automorphism associated with s,

Int(s)(x) = sxs^{−1} for x ∈ A.

The element s is uniquely determined up to multiplication by an element of F^{×}. By (2), it follows that Δ_{θ}(s) ∈ H^{3}(F, μ_{2}) only depends on σ, since m is even. We call this element the discriminant of σ with respect to θ and denote it by Δ_{θ}(σ). Thus,

Δ_{θ}(σ) = (Nrp_{θ}(s))_{2} ∪ [A] ∈ H^{3}(F, μ_{2}).

In the case m = 2, an analogue of this invariant has been studied in [6, §16.B], where it is denoted by j_{θ}(σ). Theorem (16.19) of [6] shows that this invariant classifies, up to conjugation, symplectic involutions on a central simple algebra of degree 4.

In section 2 we establish the following elementary result:

1. Proposition. (a) The discriminant Δ_{θ}(σ) only depends on the conjugacy classes of θ and σ, namely, if u, v ∈ A^{×} and

θ′ = Int(u) ∘ θ ∘ Int(u)^{−1}, σ′ = Int(v) ∘ σ ∘ Int(v)^{−1},

then

Δ_{θ′}(σ′) = Δ_{θ}(σ).

In particular, if σ and θ are conjugate, then Δ_{θ}(σ) = 0.

(b) Let ρ, σ and θ be symplectic involutions on A; then

Δ_{ρ}(σ) = Δ_{ρ}(θ) + Δ_{θ}(σ)  and  Δ_{θ}(σ) = Δ_{σ}(θ).

If the Schur index ind A divides \frac{1}{2} deg A, i.e., if A ≃ M_{2}(A_{0}) for some central simple F -algebra A_{0}, then A carries hyperbolic symplectic involutions, such as γ⊗θ_{0}, where γ is the (unique) symplectic involution on M_{2}(F) and θ_{0} is an arbitrary orthogonal involution on A_{0}. Since all hyperbolic involutions are pairwise conjugate, we may set Δ = Δ_{θ} for any hyperbolic symplectic involution θ.

2. Example. Consider the algebra A = End_{Q} V, where Q is a quaternion division F -algebra and V is an m-dimensional Q-vector space. Symplectic involutions on A are then adjoint to hermitian forms on V with respect to the conjugation involution on Q. Suppose that m is even and let σ be the involution adjoint to a fixed hermitian form h on V. Let

h = ⟨α_{1}, …, α_{m}⟩

be the diagonalization of h relative to some orthogonal basis e of V (α_{1}, …, α_{m} ∈ F^{×}), then

Δ(σ) = ((−1)^{m/2}α_{1} ⋯ α_{m})_{2} ∪ [Q].

Indeed, let θ be the hyperbolic involution adjoint to the hermitian form with diagonalization ⟨1, −1, …, 1, −1⟩ relative to the basis e. Then, identifying A with M_{m}(Q) by e, we get

σ = Int diag(α_{1}, −α_{2}, …, α_{m−1}, −α_{m}) ∘ θ,

and we can compute Δ(σ) = Δ_{θ}(σ) by Lemma 10(e) below.

Note that if V_{0} ⊂ V is the F -subspace spanned by e, then A = (End_{F} V_{0}) ⊗ Q, and we obtain a decomposition σ = σ_{0} ⊗ γ where σ_{0} is the involution adjoint to the bilinear form on V_{0} with diagonalization ⟨α_{1}, …, α_{m}⟩ relative to e, and γ is the canonical (conjugation) involution on Q.

This example can be slightly generalized:

3. Example. Consider the algebra A = A_{0} ⊗_{F} Q where Q is a quaternion F - algebra and A_{0} is a central simple F -algebra. Let σ_{0} be an orthogonal involution on A_{0}, γ the canonical involution on Q, and

σ = σ_{0} ⊗ γ.

Suppose that ind A_{0} divides \frac{1}{2} deg A_{0}. Then

Δ(σ) = (disc σ_{0})_{2} ∪ [Q], (3)

where disc σ_{0} ∈ F^{×}/F^{×2} is the discriminant of the orthogonal involution σ_{0} (see [6, §7]). Indeed, let θ_{0} be a hyperbolic orthogonal involution on A_{0} and let x_{0} ∈ Sym(A_{0}, θ_{0})^{×} be such that σ_{0} = Int(x_{0})∘θ_{0}. The involution θ = θ_{0}⊗γ is hyperbolic, and we have σ = Int(x_{0} ⊗ 1) ∘ θ, so that

Δ(σ) = (Nrp_{θ}(x_{0} ⊗ 1))_{2} ∪ [A].

Thus, by Lemma 10(d), Nrp_{θ}(x_{0} ⊗ 1) = Nrd_{A_{0}}(x_{0}). Equation (3) follows, since disc σ_{0} is represented by Nrd_{A_{0}}(x_{0}), and (Nrd_{A_{0}}(x_{0}))_{2} ∪ [A_{0}] = 0.

1.2. Trace forms. Let A be an arbitrary central simple F -algebra. For every involution σ : A → A, the associated trace form T_{σ}: A → F is defined as follows:

T_{σ}(x) = Trd_{A}(σ(x)x)

where Trd_{A} denotes the reduced trace. Denote by T_{σ}^{+} the restriction of T_{σ} to Sym(A, σ); this form can also be seen as the restriction to Sym(A, σ) of the form T_{A}: A → F defined by

T_{A}(x) = Trd_{A}(x^{2}).

As is the case with involutions of the other types (see [6, §11]), the discriminant of symplectic involutions can be expressed in terms of trace forms; indeed we have the following result:

4. Theorem. Let A be a central simple algebra over F and let θ and σ be symplectic involutions on A. The class in the Witt ring W F of the difference T_{σ}^{+} − T_{θ}^{+} lies in the third power of the fundamental ideal, namely

T_{σ}^{+} − T_{θ}^{+} ∈ I^{3}F.

Moreover, if e_{3} : I^{3}F → H^{3}(F, μ_{2}) denotes the Arason invariant, we obtain

e_{3}(T_{σ}^{+} − T_{θ}^{+}) = \begin{cases}Δ_{θ}(σ) & if deg A ≡ 0 mod 4, \\ 0 & if deg A ≡ 2 mod 4. \end{cases}

A proof of this result is given in section 3 below. For the trace forms T_{σ}, we have the following result:

5. Corollary. Keeping the notation of the previous theorem, we have T_{σ}−T_{θ} ∈ I^{4}F and

e_{4}(T_{σ} − T_{θ}) = \begin{cases}(−1)_{2} ∪ Δ_{θ}(σ) & if deg A ≡ 0 mod 4, \\ 0 & if deg A ≡ 2 mod 4, \end{cases}

where e_{4} : I^{4}F → H^{4}(F, μ_{2}) denotes the degree 4 invariant.

Proof. Let T_{σ}^{−} be the restriction of T_{σ} (or of −T_{A}) to the space of skew-symmetric elements in A. We have

T_{σ} = T_{σ}^{+} + T_{σ}^{−}  and  T_{A} = T_{σ}^{+} − T_{σ}^{−},

so that T_{σ} = 2T_{σ}^{+} − T_{A}. Similarly, T_{θ} = 2T_{θ}^{+} − T_{A}, so that

T_{σ} − T_{θ} = 2(T_{σ}^{+} − T_{θ}^{+}),

hence the corollary is a direct consequence of the previous theorem.

In the special case where θ is hyperbolic we get:

6. Proposition. Suppose A = M_{2}(A_{0}) for some central simple F -algebra A_{0}, and let θ be a hyperbolic symplectic involution on A. Then T_{θ}^{+} is Witt-equivalent to ⟨2⟩ · T_{A_{0}}, and T_{θ} is hyperbolic. If deg A ≡ 2 mod 4, then A is split, hence every symplectic involution on A is hyperbolic. If deg A ≡ 0 mod 4, then, for any symplectic involution σ on A, we have T_{σ} ∈ I^{4}F and

e_{4}(T_{σ}) = (−1)_{2} ∪ Δ(σ).

1.3. Decomposability of symplectic involutions. Section 4 below will be devoted to the relations between the discriminant and the decomposability of symplectic involutions as tensor products of involutions defined on subalgebras. Our main result is concerned with degree 8 algebras with index dividing 4. Such algebras can be written in the form A = M_{2}(A_{0}), where A_{0} is a central simple algebra of degree 4, hence they carry hyperbolic symplectic involutions. The case ind A = 1 is trivial, since every symplectic involution on a split algebra is hyperbolic, and is omitted in the following theorem.

7. Theorem. Let A be a central simple F -algebra of degree 8 having index 2 or 4. For any symplectic involution σ on A, there is a quaternion subalgebra Q ⊂ A such that

(1) Q is stable under σ, and σ restricts to the conjugation involution γ on Q,

(2) the index of the centralizer A_{0} of Q in A is \frac{1}{2} ind A.

We thus have

A = A_{0} ⊗_{F} Q  and  σ = σ_{0} ⊗ γ,

where σ_{0} is the restriction of σ to A_{0} (which is an orthogonal involution on A_{0}).

When ind A = 2, this theorem is easily proved and can be readily generalized to any degree, see Example 2 or [1, Proposition 3.4]. The case ind A = 4 is treated in Section 4.

Theorem 7 shows that the discriminant of a symplectic involution σ on a central simple F -algebra of degree 8 and index 2 or 4 can be computed as in Example 3 above. The following theorem gives a necessary and sufficient condition for the discriminant to be trivial.

8. Theorem. Let A be a central simple F -algebra of degree 8 with index dividing 4. For any symplectic involution σ on A, Δ(σ) = 0 if and only if σ leaves invariant a subalgebra A_{1} ⊂ A isomorphic to M_{2}(F) and restricts to an orthogonal involution on A_{1}.

A proof is given in Section 4.

When the condition of Theorem 8 holds, the centralizer A′_{1} of A_{1} in A is also invariant under σ, and the restriction of σ to A′_{1} must be symplectic. As any symplectic involution on a degree 4 algebra decomposes as a tensor product of involutions on quaternion subalgebras (see [6, (16.16)]), it follows that A′_{1} = A_{2} ⊗ A_{3} for some quaternion subalgebras A_{2}, A_{3} ⊂ A′_{1} invariant under σ. Therefore, Theorem 8 can be reformulated as follows:

9. Corollary. Let A be a central simple F -algebra of degree 8 with index dividing 4. Then, for every symplectic involution σ on A, we have Δ(σ) = 0 if and only if there exist quaternion subalgebras A_{1}, A_{2}, A_{3} ⊂ A, with A_{1} ≃ M_{2}(F), and orthogonal involutions σ_{1} and σ_{2} on A_{1} and A_{2} respectively, such that

A = A_{1} ⊗ A_{2} ⊗ A_{3}  and  σ = σ_{1} ⊗ σ_{2} ⊗ γ_{3}

where γ_{3} denotes the canonical involution on A_{3}.

2. Discriminants and pfaffian Norms

The goal of this section is to prove Proposition 1. Throughout the section, A denotes a central simple F -algebra of degree n = 2m.

10. Lemma. Let σ and θ be symplectic involutions on A and let s ∈ Sym(A, θ)^{×} be an element such that σ = Int(s) ∘ θ. Then,

(a) for every x ∈ Sym(A, σ) ∩ Sym(A, θ),

Nrp_{σ}(x) = Nrp_{θ}(x);

(b) for every x ∈ Sym(A, θ), the product sx lies in Sym(A, σ) and

Nrp_{σ}(sx) = Nrp_{θ}(s) Nrp_{θ}(x);

(c) for every x ∈ Sym(A, θ)^{×},

Nrp_{θ}(x^{−1}) = Nrp_{θ}(x)^{−1}.

(d) Suppose A = A_{1} ⊗_{F} A_{2} for some central simple F -algebras A_{1}, A_{2} ⊂ A of degree n_{1} = 2m_{1} and n_{2} = 2m_{2} respectively; if x_{1} ∈ A_{1} and x_{2} ∈ A_{2} are such that x_{1} ⊗ x_{2} ∈ Sym(A, θ), then

Nrp_{θ}(x_{1} ⊗ x_{2}) = Nrd_{A_{1}}(x_{1})^{m_{2}} Nrd_{A_{2}}(x_{2})^{m_{1}}.

(e) Suppose A = M_{r}(A_{0}) and θ((a_{ij})_{1≤i,j≤r}) = (θ_{0}(a_{ij}))_{1≤i,j≤r}^{t} for some symplectic involution θ_{0} on the central simple F -algebra A_{0}. For the diagonal matrix x = diag(x_{1}, …, x_{r}) with x_{i} ∈ Sym(A_{0}, θ_{0}) for i = 1, …, r, we have

Nrp_{θ}(x) = Nrp_{θ_{0}}(x_{1}) ⋯ Nrp_{θ_{0}}(x_{r}).

Proof. (a) Let t be an indeterminate over F. Define

Prp_{σ,x}(t) = Nrp_{σ}(t − x) ∈ F [t], Prp_{θ,x}(t) = Nrp_{θ}(t − x) ∈ F [t].

Those polynomials, called pfaffian characteristic polynomials in [6, p. 19], are monic and satisfy

Prp_{σ,x}^{2} = Pcrd_{A,x} = Prp_{θ,x}^{2},

where Pcrd_{A,x}(t) = Nrd_{A(t)}(t − x) is the reduced characteristic polynomial of x. Therefore, Prp_{σ,x}(t) = Prp_{θ,x}(t), and evaluation at t = 0 yields Nrp_{σ}(x) = Nrp_{θ}(x).

(b) Straightforward calculations show that σ(sx) = sx if θ(x) = x. Let us consider the two sides of the equality we aim to prove as polynomial functions of x. The squares of the two sides are equal since the reduced norm is multiplicative, hence they are equal up to sign. Moreover, they are equal and nonzero for x = 1 in view of part (a). Hence, they are equal for all x.

(c) We apply (b) with x = s^{−1} and use the relation Nrp_{σ}(1) = 1.

(d) By taking the square root on both sides of the equation

Pcrd_{A,x_{1}⊗x_{2}} = (Pcrd_{A_{1},x_{1}})^{n_{2}}(Pcrd_{A_{2},x_{2}})^{n_{1}},

we obtain

Prp_{θ,x_{1}⊗x_{2}} = (Pcrd_{A_{1},x_{1}})^{m_{2}}(Pcrd_{A_{2},x_{2}})^{m_{1}}.

The property follows by considering the constant terms.

(e) As in the preceding case, the property follows by extracting the monic square root of each side of the equation

Pcrd_{A,x} = Pcrd_{A_{0},x_{1}} ⋯ Pcrd_{A_{0},x_{r}} .

Proposition 1 easily follows from the lemma above. Indeed, if σ = Int(s) ∘ θ, so that Δ_{θ}(σ) = (Nrp_{θ}(s))_{2} ∪ [A], then θ = Int(s^{−1}) ∘ σ and hence Δ_{σ}(θ) = (Nrp_{σ}(s^{−1}))_{2} ∪ [A]. Now Lemma 10 shows that

Nrp_{σ}(s^{−1}) = Nrp_{θ}(s)^{−1},

hence (Nrp_{σ}(s^{−1}))_{2} = (Nrp_{θ}(s))_{2}, and so Δ_{σ}(θ) = Δ_{θ}(σ). If ρ is another symplectic involution, and if t ∈ Sym(A, ρ) is such that θ = Int(t) ∘ ρ, then σ = Int(st) ∘ ρ. Part (b) of Lemma 10 yields

Nrp_{σ}(st) = Nrp_{θ}(s) Nrp_{θ}(t).

Moreover, part (a) shows that Nrp_{σ}(st) = Nrp_{ρ}(st) and Nrp_{θ}(t) = Nrp_{ρ}(t). Therefore, the preceding equality can be written as

Nrp_{ρ}(st) = Nrp_{θ}(s) Nrp_{ρ}(t).

It follows that

Δ_{ρ}(σ) = (Nrp_{ρ}(st))_{2} ∪ [A] =

(Nrp_{θ}(s))_{2} ∪ [A] + (Nrp_{ρ}(t))_{2} ∪ [A] = Δ_{θ}(σ) + Δ_{ρ}(θ),

which completes the proof of part (b) of Proposition 1.

Let now v ∈ A^{×} and σ′ = Int(v) ∘ σ ∘ Int(v)^{−1}, so that σ′ = Int(vsθ(v)) ∘ θ. Then,

Δ_{θ}(σ′) = (Nrp_{θ}(vsθ(v)))_{2} ∪ [A].

By [6, (2.13)], Nrp_{θ}(vsθ(v))= Nrd_{A}(v) Nrp_{θ}(s). Since (Nrd_{A}(v))_{2} ∪ [A] = 0, it follows that

Δ_{θ}(σ′) = Δ_{θ}(σ).

Similarly, if θ′ is a symplectic involution conjugate to θ, then Δ_{σ′}(θ′) = Δ_{σ′}(θ). Now, part (b) of Proposition 10 shows that Δ_{θ′}(σ′) = Δ_{σ′}(θ′) and Δ_{θ}(σ′) = Δ_{σ′}(θ). Therefore,

Δ_{θ′}(σ′) = Δ_{θ}(σ′).

We already observed that Δ_{θ}(σ′) = Δ_{θ}(σ), hence

Δ_{θ′}(σ′) = Δ_{θ}(σ)

and the proof of Proposition 10 is complete.

3. Discriminant and trace form

In this section we prove Theorem 4 and Proposition 6.

Let Fields_{F} be the category of fields containing F and let G be any algebraic group over F. We consider the functor

H^{1}(G): Fields_{F} → Sets*

where Sets* denotes the category of pointed sets, associating to every L ∈ Fields_{F} the Galois cohomology set H^{1}(L, G).

Similarly, the “cohomological cycle module” M = ⨁_{d≥0} H^{d}(•, ℚ/ℤ(d − 1)), introduced by Rost in [9], yields a sequence of functors

M_{d} = H^{d}(•, ℚ/ℤ(d − 1)): Fields_{F} → Sets* .

The natural transformations H^{1}(G) → M_{d} are called cohomological invariants of degree d in Merkurjev’s paper [7]. These invariants form a group denoted Inv^{d}(H^{1}(G), M). (The group Inv^{d}(H^{1}(G), M) is denoted by Inv^{d}(G, ℚ/ℤ(d − 1)) in [6, §31.B].)

Let now A be a central simple F -algebra of degree n = 2m and let θ be a symplectic involution on A. We take for G the group GSp(A, θ) of symplectic similitudes; this is the algebraic group scheme defined by

GSp(A, θ)(E) = {g ∈ A ⊗_{F} E | θ(g)g ∈ E^{×}}

for any commutative F -algebra E. The set H^{1}(L, GSp(A, θ)) is in one to one correspondence with the set of conjugacy classes of symplectic involutions defined on A_{L} = A ⊗_{F} L, the class of θ being the distinguished one (see [6, (29.23)]). The following proposition shows that for symplectic involutions there is no (nontrivial) cohomological invariant of degree 1 or 2.

11. Proposition. Suppose M = ⨁_{d≥0} H^{d}(•, ℚ/ℤ(d − 1)). If the algebra A is split, we get H^{1}(L, GSp(A, θ)) = 1, for every L ∈ Fields_{F} , so that

Inv^{d}(H^{1}(GSp(A, θ)), M) = 0 for all d.

If A is not split, we have

Inv^{d}(H^{1}(GSp(A, θ)), M) = 0 for d = 1, 2

and

Inv^{3}(H^{1}(GSp(A, θ)), M) = \begin{cases}0 & if deg A ≡ 2 mod 4, \\ ℤ/2ℤ & if deg A ≡ 0 mod 4. \end{cases}

Proof. Every symplectic involution on a split algebra is hyperbolic. Therefore, when A is split, H^{1}(L, GSp(A, θ)) = 1 for all L ∈ Fields_{F} .

For the rest of the proof, we may thus assume that A is not split. Let

μ: GSp(A, θ) → G_{m}

be the homomorphism which associates to each similitude g its multiplier μ(g) = θ(g)g. The cohomology sequence induced by the exact sequence

1 → Sp(A, θ) → GSp(A, θ) ⟶^{μ} G_{m} → 1

yields for every L ∈ Fields_{F} the exact sequence

L^{×} → H^{1}(L, Sp(A, θ)) → H^{1}(L, GSp(A, θ)) → 1

since H^{1}(L, G_{m}) = 1 by Hilbert’s Theorem 90. Therefore, for every d, we have an exact sequence

0 → Inv^{d}(H^{1}(GSp(A, θ)), M) → Inv^{d}(H^{1}(Sp(A, θ)), M) → Inv^{d}(G_{m}, M).

For d = 1 or 2, we obtain, by [6, (31.15)], Inv^{d}(H^{1}(Sp(A, θ)), M) = 0 and hence

Inv^{d}(H^{1}(GSp(A, θ)), M) = 0.

The group Inv^{3}(H^{1}(Sp(A, θ)), M) is of order 2, the nontrivial element being the Rost invariant Δ_{θ} defined in the introduction. Equation (2) shows that this invariant is zero in Inv^{3}(G_{m}, M) if and only if deg A ≡ 2 mod 4.

When deg A ≡ 0 mod 4, the unique nontrivial invariant of degree 3 is the discriminant. Our next goal is to give an explicit description of this invariant in terms of trace forms.

Let T_{θ}^{+} : Sym(A, θ) → F be the quadratic form

T_{θ}^{+}(x) = Trd_{A}(θ(x)x) = Trd_{A}(x^{2}).

This forms only depends, up to isometry, on the conjugacy class of θ since, if θ′ = Int(v) ∘ θ ∘ Int(v)^{−1} for some v ∈ A^{×}, then Int(v) defines an isometry between T_{θ}^{+} and T_{θ′}^{+}. Consider L ∈ Fields_{F} . The map sending every symplectic involution σ: A_{L} → A_{L} to the discriminant

disc(T_{σ}^{+} − T_{θ}^{+}) ∈ L^{×}/L^{×2} = H^{1}(L, μ_{2})

defines a cohomological invariant H^{1}(GSp(A, θ)) → H^{1}(μ_{2}). By Proposition 11, this invariant is trivial, hence T_{σ}^{+} − T_{θ}^{+} ∈ I^{2}L. Similarly, the map sending every symplectic involution σ to the Witt (–Clifford) invariant

e_{2}(T_{σ}^{+} − T_{θ}^{+}) ∈ H^{2}(L, μ_{2})

defines a cohomological invariant of degree 2. Again, by Proposition 11, we get e_{2}(T_{σ}^{+} − T_{θ}^{+}) = 0, and hence T_{σ}^{+} − T_{θ}^{+} ∈ I^{3}L using Merkurjev’s theorem. This proves the first part of Theorem 4. Note that the equality e_{2}(T_{σ}^{+} − T_{θ}^{+}) = 0 can also be derived from Quéguiner’s explicit calculation of the Hasse invariant of trace forms in [8, p. 307].

Consider the map associating to every symplectic involution σ: A_{L} → A_{L} the Arason invariant

e_{3}(T_{σ}^{+} − T_{θ}^{+}) ∈ H^{3}(L, μ_{2}).

Using Proposition 11, we see that this invariant is trivial either if deg A ≡ 2 mod 4 or if A is split. We claim that it coincides with the discriminant Δ_{θ}(σ) if A is nonsplit and deg A ≡ 0 mod 4. To prove this, it suffices to show that it is nontrivial because, by Proposition 11, there is an unique nontrivial invariant in Inv^{3}(H^{1}(GSp(A, θ)), M). Therefore, our goal is to find a field L ∈ Fields_{F} and two symplectic involutions σ_{1}, σ_{2} on A_{L} such that e_{3}(T_{σ_{1}}^{+} − T_{σ_{2}}^{+}) ≠ 0. If θ is any involution on A, the equality

e_{3}(T_{σ_{1}}^{+} − T_{θ}^{+}) = e_{3}(T_{σ_{1}}^{+} − T_{σ_{2}}^{+}) + e_{3}(T_{σ_{2}}^{+} − T_{θ}^{+})

shows that at least one of the terms e_{3}(T_{σ_{1}}^{+} − T_{θ}^{+}) and e_{3}(T_{σ_{2}}^{+} − T_{θ}^{+}) is nonzero, hence the invariant σ ↦ e_{3}(T_{σ}^{+} − T_{θ}^{+}) is nontrivial.

12. Lemma. Let A be a central simple F -algebra of exponent 2. There exists a field L ∈ Fields_{F} such that ind A_{L} = 2.

Proof. We may take for L the fonction field of a suitable generalized Severi–Brauer variety (cf [2]). Alternatively, we may consider a multiquadratic extension M/F of minimal degree splitting A (such extensions exist, because A is Brauer–equivalent to a tensor product of quaternion algebras by a theorem of Merkurjev) and take for L a proper maximal subfield of M. The algebra A_{L} is not split, because M is of minimal degree, and it is of index 2 because it is split by the quadratic extension M/L.

Thus we may suppose, after possibly replacing F by a suitable extension, that ind A = 2 i.e. that A is Brauer–equivalent to a quaternion division algebra Q over F. Then, denoting by V an F -vector space of dimension m, we obtain

A ≃ Q ⊗_{F} End_{F} V.

For the rest of this section, we fix an isomorphism identifying A with Q ⊗ End_{F} V. Let b be a symmetric non-degenerate bilinear form on V. The symmetric square S^{2}V and the exterior square ⋀^{2} V are endowed with symmetric bilinear forms b^{S^{2}} and b^{∧2} respectively, defined by

b^{S^{2}}(x_{1} · x_{2}, y_{1} · y_{2}) = b(x_{1}, y_{1})b(x_{2}, y_{2}) + b(x_{1}, y_{2})b(x_{2}, y_{1})

and

b^{∧2}(x_{1} ∧ x_{2}, y_{1} ∧ y_{2}) = b(x_{1}, y_{1})b(x_{2}, y_{2}) − b(x_{1}, y_{2})b(x_{2}, y_{1}).

13. Lemma. Let Q = (α, β)_{F}. On A = Q ⊗_{F} End_{F} V, consider the symplectic involution σ = γ ⊗ ad_{b}, where γ is the quaternion conjugation on Q and ad_{b} is the (orthogonal) involution adjoint to b. Then, the bilinear form B_{σ}^{+}(x, y) = Trd_{A}(xy) on Sym(A, σ) (which is the polar form of the quadratic form T_{σ}^{+}) decomposes as an orthogonal sum

B_{σ}^{+} = b^{S^{2}} ⊥ ⟨−α, −β, αβ⟩ · b^{∧2}.

Proof. Let Skew(End_{F} V, ad_{b}) be the F -vector space of endomorphisms f of V such that ad_{b}(f) = −f, and let Q^{0} be the F –vector space of pure quaternions in Q. Straightforward calculation shows that the decomposition

Sym(A, σ) = (F ⊗ Sym(End_{F} V, ad_{b})) ⊕ (Q^{0} ⊗ Skew(End_{F} V, ad_{b}))

is orthogonal with respect to the form B_{σ}^{+}. Let B_{b}^{+} and B_{b}^{−} be the restrictions of the bilinear trace form B(f, g) = tr(fg) to Sym(End_{F} V, ad_{b}) and Skew(End_{F} V, ad_{b}) respectively. The decomposition above yields

B_{σ}^{+} = ⟨2⟩ · B_{b}^{+} ⊥ ⟨2α, 2β, −2αβ⟩ · B_{b}^{−}.

The lemma follows, since, by [6, (11.4)] B_{b}^{+} ≃ \frac{1}{2}b^{S^{2}} and B_{b}^{−} ≃ −\frac{1}{2}b^{∧2}.

If b = ⟨a_{1}, …, a_{m}⟩ is a diagonalization of b, it is easily verified that

b^{S^{2}} ≃ m⟨2⟩ ⊥ (⊥_{i<j} ⟨a_{i}a_{j}⟩)  and  b^{∧2} ≃ ⊥_{i<j} ⟨a_{i}a_{j}⟩

(cf [6, p. 135]). The formula of the preceding lemma can then be written as

B_{σ}^{+} ≃ m⟨2⟩ ⊥ ⟨1, −α, −β, αβ⟩ · b^{∧2},

hence, in terms of quadratic forms,

T_{σ}^{+} = m⟨2⟩ + n_{Q} · q^{∧2}

where n_{Q} denotes the norm form of Q and q^{∧2} is the quadratic form defined by q^{∧2}(x) := b^{∧2}(x, x).

Let b_{1} and b_{2} be two nonsingular symmetric bilinear forms on V, and let

σ_{1} = γ ⊗ ad_{b_{1}}, σ_{2} = γ ⊗ ad_{b_{2}}

be the symplectic involutions on A = Q ⊗ End_{F} V constructed as in Lemma 13. Observe that T_{σ_{1}}^{+} − T_{σ_{2}}^{+} = n_{Q} · (q_{1}^{∧2} − q_{2}^{∧2}), hence

e_{3}(T_{σ_{1}}^{+} − T_{σ_{2}}^{+}) = [Q] ∪ disc(q_{1}^{∧2} − q_{2}^{∧2}). (4)

Explicit calculation shows that

(5) disc(q_{1}^{∧2} − q_{2}^{∧2}) = det b_{1}^{∧2} · det b_{2}^{∧2} = (det b_{1} · det b_{2})^{m−1}.

Adjoining an indeterminate to F if necessary, we may assume that there exists an element t ∈ F^{×} not belonging to Nrd(Q). By a theorem of Merkurjev, this element satisfies [Q] ∪ (t)_{2} ≠ 0. It is easy to find two bilinear forms b_{1} and b_{2} on V such that det b_{1} · det b_{2} = tF^{×2}. Since m is even, it follows from (4) and (5) that the corresponding involutions σ_{1} and σ_{2} satisfy

e_{3}(T_{σ_{1}}^{+} − T_{σ_{2}}^{+}) ≠ 0.

This completes the proof of Theorem 4.

We now turn to Proposition 6 and assume A = M_{2}(F)⊗_{F} A_{0}. Since all hyperbolic involutions are conjugate, we may assume moreover θ = γ ⊗ θ_{0} for some orthogonal involution θ_{0} on A_{0}, where γ is the unique symplectic involution on M_{2}(F) (which is hyperbolic). As in Lemma 13, we have an orthogonal decomposition

Sym(A, θ) = (F ⊗ Sym(A_{0}, θ_{0})) ⊕ (Skew(M_{2}(F), γ) ⊗ Skew(A_{0}, θ_{0}))

which yields

T_{σ}^{+} = ⟨2⟩ · T_{θ_{0}}^{+} ⊥ ⟨2, −2, −2⟩ · T_{θ_{0}}^{−}.

Therefore, T_{σ}^{+} is Witt-equivalent to ⟨2⟩ · (T_{θ_{0}}^{+} − T_{θ_{0}}^{−}) = ⟨2⟩ · T_{A_{0}}. Since the adjoint involution to T_{θ} is θ ⊗ θ, by [6, (11.1)], it is clear that T_{θ} is hyperbolic when θ is hyperbolic. If deg A ≡ 2 mod 4, then deg A_{0} is odd, hence A_{0} is split. Therefore, A is also split. The other statements in Proposition 6 follow from Corollary 5.

4. Discriminant and decomposability of involutions

Our first goal in this section is to give a proof of Theorem 7. As observed in Section 1.3, the theorem is easy if ind A = 2. Therefore, we assume ind A = 4. We may then represent A as

A = End_{D} V

where D is a division algebra of degree 4 and V is a 2-dimensional D-vector space. Let θ_{0} be an arbitrary symplectic involution on D. The involution σ is adjoint to a hermitian form h on V (with respect to θ_{0}). Using an orthogonal basis of V relative to h, we may identify

A = M_{2}(D)  and  σ = Int diag(u_{1}, u_{2}) ∘ θ̂_{0}

for some u_{1}, u_{2} ∈ Sym(D, θ_{0})^{×}, where

θ̂_{0}((a_{ij})_{1≤i,j≤2}) = (θ_{0}(a_{ij}))_{1≤i,j≤2}^{t},

i.e., θ̂_{0} = t⊗θ on A = M_{2}(F)⊗_{F} D. Substituting Int(u_{1})∘θ_{0} for θ_{0}, we may assume u_{1} = 1. By [6, (16.16)], we may find a decomposition of D into a tensor product of quaternion subalgebras stable under θ_{0},

D = Q_{1} ⊗_{F} Q, θ_{0} = θ_{1} ⊗ γ

where θ_{1} is an orthogonal involution on Q_{1} and γ is the canonical involution on Q. Moreover, we may assume u_{2} ∈ Q_{1}. Then

σ = Int diag(1, u_{2}) ∘ (t ⊗ θ_{1} ⊗ γ) = σ_{0} ⊗ γ

with σ_{0} = Int diag(1, u_{2}) ∘ t ⊗ θ_{1} on M_{2}(F) ⊗ Q_{1}. Theorem 7 is thus proved. Note that the quaternion algebra Q is not uniquely determined by [6, (16.16)].

Let us now prove Theorem 8, starting with the following general remark:

14. Lemma. Let σ and θ be two symplectic involutions on a central simple algebra A having degree a multiple of 4. Assume that the algebra A decomposes as A = A_{1} ⊗ A_{2} for some subalgebras A_{1}, A_{2} ⊂ A and, furthermore, that there exist involutions σ_{1}, θ_{1} on A_{1} and σ_{2}, θ_{2} on A_{2} such that σ = σ_{1} ⊗ σ_{2} and θ = θ_{1} ⊗ θ_{2}. Moreover assume deg A_{1} ≡ 2 mod 4 and deg A_{2} ≡ 0 mod 4. Then, if σ_{1} and θ_{1} are of orthogonal type, Δ_{θ}(σ) = 0.

Proof. Consider u_{1} ∈ Sym(A_{1}, θ_{1})^{×} and u_{2} ∈ Sym(A_{2}, θ_{2})^{×} such that

σ_{1} = Int(u_{1}) ∘ θ_{1}  and  σ_{2} = Int(u_{2}) ∘ θ_{2},

hence

σ = Int(u_{1} ⊗ u_{2}) ∘ θ.

Then Δ_{θ}(σ) = (Nrp_{θ}(u_{1} ⊗ u_{2}))_{2} ∪ [A], and, by Lemma 10(d),

(6) Nrp_{θ}(u_{1} ⊗ u_{2}) = Nrd_{A_{1}}(u_{1})^{\frac{1}{2} deg A_{2}} Nrd_{A_{2}}(u_{2})^{\frac{1}{2} deg A_{1}}.

Since deg A_{2} ≡ 0 mod 4, the first factor is a square. Moreover, since σ_{1} and θ_{1} are of orthogonal type, σ_{2} and θ_{2} must be of symplectic type. Hence,

Nrd_{A_{2}}(u_{2}) = Nrp_{θ_{2}}(u_{2})^{2}.

Therefore, equation (6) shows that Nrp_{θ}(u_{1} ⊗ u_{2}) ∈ F^{×2}, so that Δ_{θ}(σ) = 0.

Even in the case deg A = 8, the converse of Lemma 14 does not hold, since there are examples of algebras with involution which do not contain any invariant quaternion subalgebra on which the restriction of the involution is of orthogonal type. Indeed, arguing as in the proof of Corollary 9 (see section 1.3), we see that if θ is a symplectic involution on a central simple algebra A of degree 8, and if A_{1} ⊂ A is a quaternion subalgebra on which θ restricts to an orthogonal involution θ_{1}, then A and θ decompose in the form

A = A_{1} ⊗ A_{2} ⊗ A_{3}  and  θ = θ_{1} ⊗ θ_{2} ⊗ γ_{3},

where A_{1}, A_{2} and A_{3} are quaternion algebras, θ_{1} and θ_{2} are orthogonal involutions on A_{1} and A_{2} respectively, and γ_{3} is the canonical involution on A_{3}. This implies, in particular, that the signature of θ with respect to every ordering of the field F is either 0 or 8. For example, if F = ℝ is the field of real numbers and θ is the involution adjoint to the hermitian form ⟨1, 1, 1, −1⟩ on the usual quaternion algebra ℍ, then sgn θ = 4, and so A = M_{4}(ℍ) has no quaternion subalgebras on which θ restricts to an orthogonal involution. Therefore, even though Δ_{θ}(θ) = 0, there is no decomposition as in Lemma 14. (See Example 15 for a subtler example where the converse of Lemma 14 fails.)

Returning to the proof of Theorem 8, we suppose until the end of this section that A is a central simple F -algebra of degree 8, with index dividing 4. Let σ be a symplectic involution on A, and suppose A_{1} ≃ M_{2}(F) is an invariant subalgebra on which the restriction of σ is an orthogonal involution. In this situation, we have a decomposition A = A_{1} ⊗ A′_{1}, where A′_{1} denotes the centralizer of A_{1}, and σ = σ_{1} ⊗ σ′_{1}, where σ_{1} and σ′_{1} are the restrictions of σ to A_{1} and A′_{1} respectively. As A_{1} ≃ M_{2}(F), we can find a hyperbolic symplectic involution θ_{1} on A_{1} and set

θ = θ_{1} ⊗ σ′_{1}.

The involution θ is hyperbolic of symplectic type and, by Lemma 14, we have

Δ(σ) = Δ_{θ}(σ) = 0.

Conversely, let σ be a symplectic involution on A such that Δ(σ) = 0. To prove that σ leaves invariant a subalgebra of A isomorphic to M_{2}(F) on which it restricts to an orthogonal involution, we consider separately various cases, depending on the index of A. If A is split, every symplectic involution is hyperbolic and the property is a consequence of [1, Theorem 2.2]. If ind A = 2, we can always represent A in the form

A = End_{Q} V

where Q is a quaternion algebra and V is a 4-dimensional vector space over Q. The involution σ is then adjoint to a hermitian form h on V (with respect to the canonical involution γ on Q). Let e be an orthogonal basis for h. Since h is determined by σ up to a factor in F^{×}, we may assume that the diagonalization of h with respect to the basis e is ⟨1, α_{1}, α_{2}, α_{3}⟩ with α_{1}, α_{2}, α_{3} ∈ F^{×}. Let V_{0} ⊂ V be the F -subspace with basis e. We have V = V_{0} ⊗_{F} Q and

A = (End_{F} V_{0}) ⊗_{F} Q, σ = σ_{0} ⊗ γ

where σ_{0} is the orthogonal involution on End_{F} V_{0} adjoint to the bilinear symmetric form ⟨1, α_{1}, α_{2}, α_{3}⟩. As in Example 2, Δ(σ) = (α_{1}α_{2}α_{3})_{2} ∪ [Q]. Therefore, the condition Δ(σ) = 0 implies, by a theorem of Merkurjev, that α_{1}α_{2}α_{3} ∈ Nrd_{Q}(Q^{×}). Changing basis if necessary, we may assume that α_{3} = α_{1}α_{2}. Then

⟨1, α_{1}, α_{2}, α_{3}⟩ = ⟨1, α_{1}⟩ ⊗ ⟨1, α_{2}⟩.

This implies σ_{0} = σ_{1} ⊗ σ_{2} on End_{F} V_{0} ≃ M_{2}(F) ⊗ M_{2}(F), where σ_{1} and σ_{2} are the involutions adjoint to the bilinear forms ⟨1, α_{1}⟩ and ⟨1, α_{2}⟩, respectively. This proves the theorem in this case.

Finally, suppose ind A = 4. As in the proof of Theorem 7, given at the beginning of this section, we may then represent A as A = End_{D} V where D is a division algebra of degree 4 and V is a 2-dimensional D-vector space. For the rest of the proof, we use the same notation as in the proof of Theorem 7. We may thus assume A = M_{2}(D) = M_{2}(F)⊗_{F} D and σ = Int diag(1, u_{2})∘θ̂_{0} for some symplectic involution θ_{0} on D and θ̂_{0} = t ⊗ θ_{0}. The involution θ = Int diag(1, −1) ∘ θ̂_{0} is hyperbolic, and σ = Int diag(1, −u_{2}) ∘ θ. By Lemma 10(e), we have

Nrp_{θ}(diag(1, −u_{2})) = Nrp_{θ_{0}}(−u_{2}) = Nrp_{θ_{0}}(u_{2}),

hence

Δ(σ) = Δ_{θ}(σ) = (Nrp_{θ_{0}}(u_{2}))_{2} ∪ [D].

Therefore, the condition Δ(σ) = 0 implies by [6, (16.19)] that the involution Int(u_{2}) ∘ θ_{0} on D is conjugate to θ_{0}. We may then find v ∈ D^{×} such that

Int(u_{2}) ∘ θ_{0} = Int(v) ∘ θ_{0} ∘ Int(v)^{−1} = Int(vθ_{0}(v)) ∘ θ_{0},

hence u_{2} = vθ_{0}(v)λ for some λ ∈ F^{×}. The involution θ is conjugate to

Int diag(1, v)^{−1} ∘ θ ∘ Int diag(1, v) = Int diag(1, λ) ∘ θ̂_{0},

which restricts to Int diag(1, λ) ∘ t on M_{2}(F) ⊂ M_{2}(D). Therefore, θ leaves the subalgebra diag(1, v)M_{2}(F) diag(1, v^{−1}) invariant and restricts to an orthogonal involution on that subalgebra. The proof of Theorem 8 is thus complete.

15. Example. The following is another example where the converse of Lemma 14 fails. Consider three quadratic extensions K_{1}, K_{2}, K_{3} of a field k,

K_{i} = k(\sqrt{a_{i}}) for some a_{i} ∈ k,

such that K_{1} ⊗_{k} K_{2} ⊗_{k} K_{3} is a field, and let F = k(x_{1}, x_{2}, x_{3}) be the field of rational fractions in three indeterminates over k. For i = 1, 2, 3, consider K_{i} as a subfield of the quaternion algebra A_{i} = (a_{i}, x_{i})_{F}. On the tensor product

A = A_{1} ⊗_{F} A_{2} ⊗_{F} A_{3},

consider the symplectic involution

θ = θ_{1} ⊗ θ_{2} ⊗ γ_{3},

where γ_{3} is the conjugation involution on A_{3} and θ_{1} (resp. θ_{2}) is an orthogonal involution on A_{1} (resp. A_{2}) which is the identity on K_{1} (resp. K_{2}).

Let λ ∈ N_{K_{1}/k}(K_{1}^{×}) ∩ N_{K_{2}/k}(K_{2}^{×}). By a well-known property of biquadratic extensions (see for instance [4, 2.13]), we may find u ∈ K_{1} ⊗_{k} K_{2} and v ∈ k^{×} such that

λ = v^{2}N_{K_{1}⊗_{k}K_{2}/k}(u).

Viewing u = u ⊗ 1 in A, we let

σ = Int(u) ∘ θ.

By Lemma 10(d), we have Nrp_{θ}(u) = Nrd_{A_{1}⊗_{F} A_{2}}(u) = N_{K_{1}⊗_{k}K_{2}/k}(u), hence

Δ_{θ}(σ) = (λ) ∪ [A].

Since λ is a norm from K_{1} and K_{2}, hence a reduced norm from A_{1} and A_{2}, it follows that

Δ_{θ}(σ) = (λ) ∪ [A_{3}].

Therefore, Δ_{θ}(σ) = 0 if and only if λ ∈ N_{K_{1}/k}(K_{1}^{×}) ∩ N_{K_{2}/k}(K_{2}^{×}) ∩ N_{K_{3}/k}(K_{3}^{×}).

Now, suppose σ = Int(w) ∘ σ′ ∘ Int(w)^{−1} for some involution σ′ leaving A_{1} invariant. Then σ′ = Int(w^{−1}uθ(w)^{−1}) ∘ θ, and the proof of Lemma 14 shows that Nrp_{θ}(w^{−1}uθ(w)^{−1}) ∈ F^{×2}. Since Nrp_{θ}(w^{−1}uθ(w)^{−1}) = Nrp_{θ}(u) Nrd_{A}(w)^{−1} = λv^{−2} Nrd_{A}(w)^{−1}, it follows that

λ ∈ F^{×2} · Nrd_{A}(A^{×}).

By Proposition 9 of [3], we then have

λ ∈ k^{×2} · N_{M/k}(M^{×}) with M = K_{1} ⊗_{k} K_{2} ⊗_{k} K_{3}.

Therefore, examples of triquadratic extensions M = K_{1} ⊗_{k} K_{2} ⊗_{k} K_{3}/k such that

N_{K_{1}/k}(K_{1}^{×}) ∩ N_{K_{2}/k}(K_{2}^{×}) ∩ N_{K_{3}/k}(K_{3}^{×}) ≠ k^{×2} · N_{M/k}(M^{×})

yield examples of involutions σ for which Δ_{θ}(σ) = 0 even though σ is not conjugate to an involution leaving A_{1} invariant. Triquadratic extensions of this type were constructed in [10] (see also [5, Proposition 3]).

References

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[3] M. Chacron, H. Dherte, J.-P. Tignol, A.R. Wadsworth, V.I. Yanchevskiĭ, Discriminants of involutions on Henselian division algebras, Pacific J. Math. 167 (1995), 49–79.

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[5] P. Gille, Examples of non rational varieties of adjoint groups, J. Algebra 193 (1997), 728–747.

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[7] A.S. Merkurjev, Invariants of algebraic groups, J. reine angew. Math. 508 (1999), 127–156.

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Département de mathématiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

   E-mail address: gregory.berhuy@epfl.ch, monsurro@masg1.epfl.ch

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

   E-mail address: tignol@math.ucl.ac.be