THE DISCRIMINANT OF A DECOMPOSABLE SYMPLECTIC INVOLUTION

AHMED SERHIR AND JEAN-PIERRE TIGNOL

Abstract. A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro’s “Pfister Factor Conjecture” is given for tensor products of at most five quaternion algebras.

Throughout this paper, the characteristic of the base field F is supposed to be different from 2. Recall from [4, (2.5)] that a symplectic (resp. orthogonal) involution on a central simple F -algebra A is a map A → A which after scalar extension to a splitting field K can be identified with the adjoint involution of a nonsingular alternating (resp. symmetric) bilinear form over a K-vector space. The discriminant of an orthogonal involution ρ on a central simple F -algebra of even degree was defined by Jacobson, Tits, and Knus–Parimala–Sridharan, see [4, §7]. It is an element of the Galois cohomology group H^{1}(F, μ_{2}) ≃ F^{×}/F^{×2} which we denote by disc ρ.

For symplectic involutions σ, σ_{0} on a central simple F -algebra A of degree divisible by 4, a relative discriminant Δ_{σ_{0}}(σ) ∈ H^{3}(F, μ_{2}) related to the Rost invariant of symplectic groups is defined in [3]. (The definition is recalled at the beginning of Section 1.) If the Schur index ind A divides \frac{1}{2} deg A, then A carries a hyperbolic involution σ_{0}, and we write simply Δ(σ) for Δ_{σ_{0}}(σ). Our main result relates the discriminants of symplectic and orthogonal involutions as follows:

Main Theorem. Suppose B is a central simple F -algebra of degree divisible by 4 with orthogonal involution ρ and Q is a quaternion F -algebra with canonical (symplectic) involution γ. Let A = B ⊗_{F} Q and σ = ρ⊗γ, a symplectic involution on A. If ind A divides \frac{1}{2} deg A, then, denoting by [Q] ∈ H^{2}(F, μ_{2}) the cohomology class associated with Q,

Δ(σ) = (disc ρ) ∪ [Q].

In the particular case where A is a tensor product of three quaternion algebras and σ is the tensor product of the canonical involutions, it follows that Δ(σ) = 0. We then use a theorem of Berhuy–Monsurrò–Tignol to derive a decomposition of σ where one of the factors is an orthogonal involution on a split algebra of degree 2, see Section 1.

Research leading to this paper was initiated while the first author was visiting the Université catholique de Louvain with a grant from the “Secrétariat à la coopération internationale,” whose support is gratefully acknowledged. The second author is partially supported by the National Fund for Scientific Research (Belgium).

In [9, (9.17)], Shapiro proposes the following conjecture:

PC(m)

Let (Q_{1}, σ_{1}), …, (Q_{m}, σ_{m}) be quaternion F -algebras with involutions of orthogonal or symplectic type such that σ_{1} ⊗ ⋯ ⊗ σ_{m} is orthogonal. If Q_{1} ⊗ ⋯ ⊗ Q_{m} is split, then σ_{1} ⊗ ⋯ ⊗ σ_{m} is adjoint to an m-fold Pfister form.

As shown in [9, p. 166], this conjecture is equivalent to the “Pfister factor conjecture” on spaces of similarities discussed by Shapiro in [8, §7] and proved (by techniques from the algebraic theory of quadratic forms) for m ≤ 5 and over number fields by Wadsworth–Shapiro in [11]. In the final section, we show how the decomposition result of Section 1 for involutions on triquaternion algebras yields another proof of PC(m) for m ≤ 5 and also for all m over some special fields. A different proof for m ≤ 4 using central simple algebras with involution was found by Bayer-Fluckiger–Parimala [1].

1. Discriminants

Let A be a central simple algebra over a field F of characteristic different from 2. For u ∈ A^{×}, we denote by Int(u) the inner automorphism induced by u,

Int(u)(x) = uxu^{−1} for x ∈ A.

If σ, σ_{0} are symplectic involutions on A, then σ = Int(u) ∘ σ_{0} for some u ∈ A^{×} such that σ_{0}(u) = u (see [4, (2.7)]). We may then consider the pfaffian norm Nrp_{σ_{0}} u ∈ F^{×} (which is a square root of the reduced norm Nrd_{A} u, see [4, p. 19]) and the corresponding cohomology class (Nrp_{σ_{0}} u) ∈ H^{1}(F, μ_{2}) under the canonical isomorphism F^{×}/F^{×2} ≃ H^{1}(F, μ_{2}). If deg A ≡ 0 mod 4, the cup product

(Nrp_{σ_{0}} u) ∪ [A] ∈ H^{3}(F, μ_{2})

depends only on σ and σ_{0}, and not on the choice of u. It is called the discriminant of σ relative to σ_{0} and denoted by Δ_{σ_{0}}(σ), see [3].

Proof of the Main Theorem. The main point is to establish the existence of an orthogonal involution ρ_{0} on B such that disc ρ_{0} = 1 and ρ_{0} ⊗ γ is hyperbolic. We then have

(1)  ρ = Int(u) ∘ ρ_{0}  for some u ∈ B^{×},

hence σ = Int(u ⊗ 1) ∘ (ρ_{0} ⊗ γ) and therefore

(2)  Δ(σ) = Δ_{ρ_{0}⊗γ}(σ) = (Nrp_{ρ_{0}⊗γ}(u ⊗ 1)) ∪ [A].

We have Nrp_{ρ_{0}⊗γ}(u ⊗ 1) = Nrd_{B} u by [3, Lemma 9] and (Nrd_{B} u) ∪ [B] = 0, hence (2) yields

Δ(σ) = (Nrd_{B} u) ∪ [Q].

On the other hand, (1) implies that disc ρ = (Nrd_{B} u) since disc ρ_{0} = 1 (see [4, (7.3)]), and the theorem follows.

To prove the existence of ρ_{0}, we consider two cases: Case 1: If ind B divides \frac{1}{2} deg B, then we may take for ρ_{0} an orthogonal hyperbolic involution. Case 2: If ind B does not divide \frac{1}{2} deg B, then we may identify B = M_{r}(F) ⊗ D for some central division F -algebra D and some odd integer r. Since ind A divides \frac{1}{2} deg A, the tensor product D ⊗ Q is not a division algebra. By a theorem of Risman [6], it follows that D contains a splitting field L of Q. The extension L/F is separable since the exponent of D is 2 and the characteristic is not 2. Since deg B, hence also deg D, is divisible by 4, a theorem of Parimala–Sridharan– Suresh [5, Theorem 2.1] shows that D carries an involution ρ_{1} of discriminant 1 which leaves invariant a generating element of L. Since L splits Q, the involution Id_{L} ⊗γ on L ⊗ Q is hyperbolic, hence there is an idempotent e ∈ L ⊗ Q such that (Id_{L} ⊗γ)(e) = 1 − e, see [2, Theorem 2.1] or [4, (6.7)]. Since L ⊗ Q ⊂ D ⊗ Q, we also have e ∈ D ⊗ Q and (ρ_{1} ⊗ γ)(e) = 1 − e, hence ρ_{1} ⊗ γ is hyperbolic. If t denotes the transpose involution on M_{r}(F), the involution ρ_{0} = t ⊗ ρ_{1} on B = M_{r}(F) ⊗ D satisfies the required conditions and the proof is complete. □

Remark. The formula for Δ(σ) in the main theorem does not hold if deg B is not divisible by 4. For instance, if B is a quaternion algebra, B = (a, b)_{F} and Q = (a, c)_{F} for some a, b, c ∈ F^{×}, then for any orthogonal involution ρ on B,

Δ(ρ ⊗ γ) = (b disc ρ) ∪ [Q].

The proof is omitted.

Corollary. Let A be a central simple F -algebra of degree 8 and let σ be a symplectic involution on A. If A is not a division algebra, the following conditions are equivalent:

(a) Δ(σ) = 0;

(b) there are quaternion F -algebras with involution (Q_{1}, σ_{1}), (Q_{2}, σ_{2}), (Q_{3}, σ_{3}) such that

(A, σ) = (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ⊗ (Q_{3}, σ_{3});

(c) there is a 2-dimensional quadratic form π (which, upon scaling, may be assumed to be a 1-fold Pfister form) with adjoint involution ad_{π} on M_{2}(F), and quaternion algebras B, C with an orthogonal involution ρ and canonical (symplectic) involution γ respectively, such that

(A, σ) = (M_{2}(F), ad_{π}) ⊗ (B, ρ) ⊗ (C, γ).

Proof. The equivalence (a) ⟺ (c) is proved in [3, Theorem 8], and (c) ⇒ (b) is clear. To complete the proof, it suffices to show (b) ⇒ (a).

Suppose (b) holds. Since σ is symplectic, at least one of σ_{1}, σ_{2}, σ_{3} is symplectic. Renumbering, we may assume σ_{3} is symplectic. The main theorem yields

Δ(σ) = (disc σ_{1} ⊗ σ_{2}) ∪ [Q_{3}].

Since disc σ_{1} ⊗ σ_{2} = 1 by [4, (7.3)], condition (a) follows. □

Note that in degree 4 the special decomposition in (c) of the preceding corollary always holds (independently of the value of Δ), as we now show:

Proposition. Let A be a central simple F -algebra of degree 4 and let σ be a symplectic involution on A. If A is not a division algebra, then there is a 2-dimensional quadratic form π and a quaternion algebra with canonical involution (Q, γ) such that

(A, σ) = (M_{2}(F), ad_{π}) ⊗ (Q, γ).

Proof. Since A is not a division algebra, it is Brauer-equivalent to a quaternion algebra Q, and it can be represented as A = End_{Q} V for some free Q-module V of rank 2. The involution σ is adjoint to a hermitian form h on V with respect to the canonical involution γ on Q. Let e_{1}, e_{2} be an orthogonal basis of V. We have h(e_{1}, e_{1}), h(e_{2}, e_{2}) ∈ F, and the F -span of e_{1}, e_{2} is an F -vector space V_{0} on which π(x) = h(x, x) is a quadratic map. We have

A = End_{Q} V = (End_{F} V_{0}) ⊗ Q,

and the involution σ has a corresponding decomposition σ = ad_{π} ⊗γ. □

2. Pfister factor conjecture

If ρ_{1}, ρ_{2} are orthogonal involutions on quaternion F -algebras Q_{1}, Q_{2}, then there are quaternion F -algebras Q′_{1}, Q′_{2} with canonical involutions γ_{1}, γ_{2} such that

(Q_{1}, ρ_{1}) ⊗ (Q_{2}, ρ_{2}) ≃ (Q′_{1}, γ_{1}) ⊗ (Q′_{2}, γ_{2}),

see [4, (15.12)]. (The algebras Q′_{1} and Q′_{2} are the direct factors of the Clifford algebra C(Q_{1} ⊗ Q_{2}, ρ_{1} ⊗ ρ_{2}). If i_{α}, j_{α} ∈ Q_{α}^{×} are such that ρ_{α}(i_{α}) = −i_{α} = j_{α}i_{α}j_{α}^{−1} for α = 1, 2, then the algebras Q′_{1} and Q′_{2} are generated by i_{1} ⊗ 1, j_{1} ⊗ i_{2} and 1 ⊗ i_{2}, i_{1} ⊗ j_{2} respectively.) Therefore, as noticed by Bayer-Fluckiger–Parimala [1], in discussing Conjecture PC(m) we may assume that all the involutions σ_{i}, except at most one, are canonical.

For any quaternion F -algebra Q with canonical involution γ, there is an isomorphism Q⊗Q →^{∼} End_{F} Q which maps x⊗y to the linear transformation z ↦ xzγ(y). Under that isomorphism, γ ⊗ γ corresponds to the adjoint involution with respect to the norm form of Q, see [4, (11.1)]. This proves PC(2) since the norm form is a 2-fold Pfister form. Alternatively, from an isomorphism

(Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ (M_{4}(F), ad_{q})

it follows that disc q = 1, hence q is a 2-fold Pfister form up to a scalar.

Lemma 1. PC(3) holds.

Proof. Suppose (Q_{1}, σ_{1}), (Q_{2}, σ_{2}), (Q_{3}, σ_{3}) are quaternion algebras with involution such that Q_{1} ⊗ Q_{2} ⊗ Q_{3} is split and σ_{1} ⊗ σ_{2} ⊗ σ_{3} is orthogonal. As observed at the beginning of this section, when aiming to prove that σ_{1} ⊗ σ_{2} ⊗ σ_{3} is adjoint to a Pfister form, we may assume that σ_{1} = γ_{1} and σ_{2} = γ_{2} are symplectic, hence σ_{3} is orthogonal. Then Q_{2} ⊗ Q_{3} ≃ M_{2}(Q_{1}) and σ_{2} ⊗ σ_{3} is symplectic, hence the proposition at the end of the preceding section yields a decomposition

(Q_{2}, σ_{2}) ⊗ (Q_{3}, σ_{3}) = (M_{2}(F), ad_{π}) ⊗ (Q_{1}, γ_{1})

for some 2-dimensional form π. Substituting in Q_{1} ⊗ Q_{2} ⊗ Q_{3}, we get

(Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ⊗ (Q_{3}, σ_{3}) ≃ (M_{2}(F), ad_{π}) ⊗ (Q_{1}, γ_{1}) ⊗ (Q_{1}, γ_{1}).

Now, as observed in the proof of PC(2) above, γ_{1} ⊗ γ_{1} is adjoint to the reduced norm form Nrd_{Q_{1}} of Q_{1}, which is a 2-fold Pfister form, hence

σ_{1} ⊗ σ_{2} ⊗ σ_{3} ≃ ad_{π} ⊗ ad_{Nrd_{Q_{1}}} ≃ ad_{π⊗Nrd_{Q_{1}}}

and the proof is complete. □

Remark. A different proof, using triality, was found by David Tao [10].

To prove PC(m) for m ≥ 4, the following lemma is useful.

Lemma 2. Let m ≥ 4 and suppose PC(m − 1) holds over F. Let (Q_{1}, σ_{1}), …, (Q_{m}, σ_{m}) be quaternion F -algebras with involution such that Q_{1} ⊗ ⋯ ⊗ Q_{m} is split and σ_{1} ⊗ ⋯ ⊗ σ_{m} is orthogonal. If Q_{1} ⊗ Q_{2} ⊗ Q_{3} is not a division algebra and σ_{1} ⊗ σ_{2} ⊗ σ_{3} is symplectic, then σ_{1} ⊗ ⋯ ⊗ σ_{m} is adjoint to an m-fold Pfister form.

Proof. By the corollary of Section 1, there exist quaternion F -algebras with involution (B, ρ), (C, γ) and a 2-dimensional quadratic form π with adjoint involution ad_{π} on M_{2}(F) such that

(Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ⊗ (Q_{3}, σ_{3}) ≃ (M_{2}(F), ad_{π}) ⊗ (B, ρ) ⊗ (C, γ).

Since multiplying π by a scalar does not change the adjoint involution, we may assume π is a 1-fold Pfister form.

Now

(Q_{1}, σ_{1}) ⊗ ⋯ ⊗ (Q_{m}, σ_{m}) ≃

(M_{2}(F), ad_{π}) ⊗ (B, ρ) ⊗ (C, γ) ⊗ (Q_{4}, σ_{4}) ⊗ ⋯ ⊗ (Q_{m}, σ_{m}).

The product B ⊗ C ⊗ Q_{4} ⊗ ⋯ ⊗ Q_{m} is split since Q_{1} ⊗ ⋯ ⊗ Q_{m} is split, and by PC(m − 1) the product ρ ⊗ γ ⊗ σ_{4} ⊗ ⋯ ⊗ σ_{m} is adjoint to an (m − 1)-fold Pfister form π′. Therefore, σ_{1} ⊗ ⋯ ⊗ σ_{m} is adjoint to the m-fold Pfister form π ⊗ π′. □

Theorem. PC(m) holds for m ≤ 5. It holds for all m over fields F which have the following property:

(∗)

Every tensor product of three quaternion F -algebras contains zerodivisors.

Proof. We have already seen that PC(m) holds for m = 2 and m = 3. As observed at the beginning of this section, when proving PC(m) we may assume that all the quaternion involutions σ_{1}, …, σ_{m} are symplectic, except at most one. Therefore, for m ≥ 4 we may assume

σ_{1} ⊗ σ_{2} ⊗ σ_{3} is symplectic. If

m ≤ 5, the condition that Q_{1} ⊗⋯⊗Q_{m} is split implies that ind(Q_{1} ⊗Q_{2} ⊗Q_{3}) ≤ 4. Since PC(3) holds, the lemma successively yields PC(4) and PC(5).

If F satisfies (∗), then ind(Q_{1}⊗Q_{2}⊗Q_{3}) ≤ 4 for all tensor products Q_{1}⊗⋯⊗Q_{m}, and the lemma inductively yields PC(m) for all m ≥ 4. □

Number fields and, more generally, linked fields in the sense of Elman–Lam (see [9, (9.14)]), satisfy (∗). Indeed, these fields satisfy the more restrictive condition that no tensor product of two quaternion algebras is a division algebra, hence we may inductively use the arguments in the proof of Lemma 1 instead of applying the theorem.

References

[1] E. Bayer-Fluckiger, R. Parimala, Pfister Involutions, in preparation.

[2] E. Bayer-Fluckiger, D.B. Shapiro, J.-P. Tignol, Hyperbolic involutions, Math. Z. 214 (1993), 461–476.

[3] G. Berhuy, M. Monsurrò, J.-P. Tignol, The discriminant of a symplectic involution, to appear in Pacific J. Math.

[4] M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Coll. Pub. 44, AMS, Providence, RI, 1998.

[5] R. Parimala, R. Sridharan, V. Suresh, A question on the discriminants of involutions of central division algebras, Math. Ann. 297 (1993), 575–580.

[6] L.J. Risman, Zero divisors in tensor product of division algebras, Proc. Amer. Math. Soc. 51 (1975), 35–36.

[7] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren math. Wiss. 270, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.

[8] D.B. Shapiro, Spaces of similarities. II. Pfister factors, J. Algebra 46 (1977), 171–181.

[9] D.B. Shapiro, Compositions of Quadratic Forms, de Gruyter Expositions in Math. 33, Walter de Gruyter, Berlin, New York, 2000.

[10] D. Tao, Pfister-form-like behavior of algebras with involution, unpublished notes, 1995.

[11] A.R. Wadsworth, D.B. Shapiro, Spaces of similarities. III. Function fields, J. Algebra 46 (1977), 182–188.

Département de Mathématiques, Université Chouaïb Doukkali, El Jadida, Morocco

E-mail address: serhir@ucd.ac.ma

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

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