Decomposability of orthogonal involutions in degree 12
aa r X i v : . [ m a t h . K T ] N ov DECOMPOSABILITY OF ORTHOGONAL INVOLUTIONS IN DEGREE ANNE QU´EGUINER-MATHIEU AND JEAN-PIERRE TIGNOL
Abstract.
A theorem of Pfister asserts that every 12-dimensional quadratic form with triv-ial discriminant and trivial Clifford invariant over a field of characteristic different from 2decomposes as a tensor product of a binary quadratic form and a 6-dimensional quadraticform with trivial discriminant. The main result of the paper extends Pfister’s result to or-thogonal involutions : every central simple algebra of degree 12 with orthogonal involutionof trivial discriminant and trivial Clifford invariant decomposes into a tensor product of aquaternion algebra and a central simple algebra of degree 6 with orthogonal involutions. Thisdecomposition is used to establish a criterion for the existence of orthogonal involutions withtrivial invariants on algebras of degree 12, and to calculate the f -invariant of the involutionif the algebra has index 2. Every semi-simple algebraic group of classical type can be described in terms of a centralsimple algebra with involution, except for groups of type D in characteristic 2, where theinvolution should be replaced by a so-called quadratic pair [7, § D n are quotients of the Spin groupof a degree 2 n algebra with orthogonal involution. If the algebra is the endormophism ring ofsome 2 n -dimensional vector-space V , the involution is adjoint to a quadratic form q defined on V , unique up to a scalar factor, and the corresponding groups are quotients of the Spin groupof this quadratic form.Algebraic groups of low rank, and the corresponding algebras with involution, which havedegree ≤
14, play a special role in the theory. Indeed, these groups have specific properties,which in turn produce efficient tools to study and describe the underlying algebraic objects.In particular, we may mention the so-called exceptional isomorphisms, with consequences onalgebras with involution explored in [7, § D , see [7, Chapter X], and the existence of an openorbit for some representations of algebraic groups of low rank, allowing to view torsors underthose groups as torsors under the stabilizer, see Garibaldi [3, Th. 9.3].Even though they were first studied independently, these facts are related to the classificationtheorems describing quadratic forms of even dimension ≤
12 with trivial discriminant and trivialClifford invariant, which were proved by Pfister in 1966 [8], see also [6, Th. 8.1.1]. It appearsthat those forms always contain a nontrivial subform of even dimension and trivial discriminant,and admit a diagonalisation of a special shape, depending on the dimension of the form. Inparticular, the number of parameters required to describe such a form in general is less than whatone may expect in view of the dimension. An analogous statement was obtained by Rost [12],more than thirty years later, for quadratic forms of dimension 14 (see also [3, Th. 21.3]),based on the representation argument mentioned above. From the point of view of algebraicgroups, it is clear that those results do not extend to higher dimensional quadratic forms. Thiswas formally proved by Merkurjev and Chernousov in [2], where they compute the essentialdimension of a split spinor group Spin n , for n ≥
15. Roughly speaking, since torsors under
Date : 10 July 2019.2010
Mathematics Subject Classification.
Primary 11E72; Secondary 16W10, 11E81.
Key words and phrases.
Algebra with involution, hermitian form, cohomological invariant.The second author acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grantn ◦ J.0159.19.
Spin n are closely related to n -dimensional quadratic forms with trivial discriminant and trivialClifford invariant, this essential dimension provides a measure of the number of parametersrequired to describe such a form in general. It follows from this computation that a generalquadratic form of dimension ≥
15 does not contain a subform of a given dimension and withtrivial discriminant, with two possible exceptions (see [2, Th. 4.2] for a precise statement).As opposed to this, Pfister’s theorem does extend to algebras with orthogonal involutions.This was already known in dimension ≤
10, and partial results in dimension 12 were discussedin [4] and [10]. The main result of this paper is Theorem 1.3, which is an improved version ofthese dimension 12 analogues, obtained by using the descent theorem for unitary involutionsin degree 6 proven in [11, Th. 1.3]. This new statement is closer to Pfister’s original result,which asserts that every 12-dimensional quadratic form with trivial discriminant and trivialClifford invariant over a field of characteristic different from 2 decomposes as a tensor productof a binary quadratic form and a 6-dimensional quadratic form with trivial discriminant.As a consequence, we characterize in Corollary 2.1 the biquaternion F -algebras D such thatthe matrix algebra M ( D ) carries an orthogonal involution with trivial discriminant and trivialClifford algebra. This property turns out to hold for every biquaternion F -algebra if the 2-cohomological dimension of F is at most 2; we show in Example 2.2 that it fails for certaintotally ramified biquaternion F -algebras.Another use of Theorem 1.3 is for the computation of a certain cohomological invariant.Recall from [10] that a cohomological invariant of degree 3 for orthogonal involutions withtrivial discriminant and trivial Clifford invariant is defined on the model of the Arason invariant e of quadratic forms. The generalized Arason invariant takes its values in a quotient of thethird Galois cohomology group of µ ⊗ ; taking the square of a representative yields an invariant f with values in the cohomology of µ . We show in Theorem 2.3 how this invariant can becalculated from a tensor product decomposition afforded by Theorem 1.3.Throughout, F is a field of characteristic different from 2, and ( A, σ ) is a central simple F -algebra with orthogonal involution. A possible characterization of ( A, σ ) is the existence ofa finite Galois extension
L/F and a quadratic space (
V, ϕ ) over L such that A ⊗ F L ≃ End L ( V ) and σ ⊗ Id = ad ϕ where ad ϕ is the involution adjoint to ϕ (or, more precisely, to its polar bilinear form). Wegenerally follow the notation used in [7], to which we refer for background information oninvolutions on central simple algebras. In particular, for any field K containing F , we write( A, σ ) K for the K -algebra with involution ( A ⊗ F K, σ ⊗ Id). If ϕ is a (nondegenerate) quadraticform on some F -vector space V , we write Ad ϕ for (End F ( V ) , ad ϕ ). The discriminant of aquadratic form and the even part of its Clifford algebra, which are invariant under similitudes,and may therefore be considered as invariants of the involution ad ϕ , extend to non-split algebraswith orthogonal involution [7, § i ≥
1, we let H i ( F ) denote the Galois cohomology group H i ( F, µ ) and identify H ( F )with F × /F × (written additively) and H ( F ) with the 2-torsion subgroup of the Brauer groupof F . For a ∈ F × and A a central simple F -algebra of exponent 1 or 2, we write ( a ) for thesquare-class of a and [ A ] for the Brauer class of A . For every orthogonal involution σ on acentral simple F -algebra A of even degree, we let e ( σ ) ∈ H ( F ) denote the discriminant of σ .If e ( σ ) = 0, the Clifford invariant e ( σ ) ∈ H ( F ) / { , [ A ] } is the coset represented by any ofthe two components of the Clifford algebra C ( A, σ ).1.
Decomposability
Our first decomposition result does not require triviality of the Clifford invariant. It ispremised instead on the existence of a quadratic extension making the involution hyperbolic,i.e., adjoint to a hyperbolic hermitian form.
ECOMPOSABILITY OF ORTHOGONAL INVOLUTIONS IN DEGREE 12 3
Proposition 1.1.
Let ( A, σ ) be a central simple F -algebra with orthogonal involution of de-gree , and let K = F ( √ d ) be a quadratic field extension of F . If A is split, assume additionallythat σ is not adjoint to a quadratic form of odd Witt index. (i) The algebra with involution ( A, σ ) K is hyperbolic if and only if ( A, σ ) decomposes as ( A, σ ) = ( A , σ ) ⊗ ( H, ρ ) where ( A , σ ) is a central simple algebra with orthogonal involution of degree and ( H, ρ ) is a quaternion algebra with orthogonal involution of discriminant ( d ) . (ii) The algebra with involution ( A, σ ) K is split and hyperbolic if and only if ( A, σ ) decom-poses as ( A, σ ) = Ad ϕ ⊗ ( H, ρ ) where ϕ is a quadratic form of dimension and ( H, ρ ) is a quaternion algebra withorthogonal involution of discriminant ( d ) .Proof. (i) The condition is obviously sufficient, since ( H, ρ ) K is hyperbolic. Assume converselythat ( A, σ ) K is hyperbolic. By [1, Th. 3.3], this means A contains a skew-symmetric element δ with square d . Writing ι for the nontrivial automorphism of K , we may then identify ( K, ι )with a subalgebra of (
A, σ ). Let B be the centralizer of K in A . The involution σ induces aninvolution τ of B , which restricts to ι on K . Hence by the descent theorem of [11, Th. 1.3],( B, τ ) = ( A , σ ) ⊗ F ( K, ι ), for some algebra with orthogonal involution ( A , σ ). The centralizerof A in A is a quaternion algebra H , which contains K , and by the double centralizer theorem,we have A = A ⊗ H . Moreover, since A is σ -stable, H also is, and we get a decomposition( A, σ ) = ( A , σ ) ⊗ ( H, ρ ) , with σ and ρ of orthogonal type, and ( H, ρ ) ⊃ ( K, ι ). The latter inclusion shows that e ( ρ ) =( d ), and the proof of (i) is complete.(ii) As in (i), the condition is sufficient because ( H, ρ ) K is hyperbolic. For the converse,we modify the argument in (i), taking into account the additional hypothesis that A K is split.From this hypothesis, it follows that the algebra B is split, hence we may identify B = End K ( V )for some K -vector space V , and τ = ad h for some hermitian form h on V . Fix an orthogonalbasis ( e , . . . , e ) of V . The form h restricts to a symmetric bilinear form on the F -vectorspace V spanned by e , . . . , e , and we may take A = End F ( V ) in the proof of (i). Thus,( A , σ ) = Ad ϕ where ϕ ( x ) = h ( x, x ) on V . (cid:3) Remark . Let (
A, σ ) be a central simple F -algebra with orthogonal involution of degree 4 m for some integer m (excluding the case where A is split and σ is adjoint to a quadratic form ofodd Witt index). We compare the following statements:(a) ( A, σ ) = ( A , σ ) ⊗ ( H, ρ ) for some quaternion algebra with orthogonal involution (
H, ρ );(b) there exists a quadratic field extension K of F such that ( A, σ ) K is hyperbolic;(c) e ( σ ) = 0.The implication (a) ⇒ (b) always holds, for we may take for K the subfield of H generated bya skew-symmetric element. (If the skew-symmetric elements in H do not generate a field, then( H, ρ ) is hyperbolic and (b) clearly holds.) The implication (b) ⇒ (c) can be derived from thefirst step in the proof of Proposition 1.1 as follows: if ( A, σ ) K is hyperbolic, then ( K, ι ) embedsin (
A, σ ) by [1, Th. 3.3], hence A contains a skew-symmetric element α such that α ∈ F × .Let α = a . The reduced norm Nrd A ( α ) is ( − a ) m and by definition e ( σ ) = (cid:0) Nrd A ( α ) (cid:1) , so e ( σ ) = 0.On the other hand, taking for A an indecomposable algebra of degree 8 yields exampleswhere (b) holds but (a) does not (see [9, Ex. 3.6]), whereas Proposition 1.1 shows that (a) and(b) are equivalent when deg A = 12. The implication (c) ⇒ (b) does not hold, even when A issplit of degree 12: for instance, any quadratic form which is the orthogonal sum of a 3-fold anda 2-fold Pfister form is a 12-dimensional quadratic form with trivial discriminant, which neednot be hyperbolic over a quadratic field extension of the base field. For an explicit example, A. QU´EGUINER-MATHIEU AND J.-P. TIGNOL consider for instance ϕ = π ⊕ hh x, y ii over F = k (( x ))(( y )), where π is an arbitrary anisotropic3-fold Pfister form over k .Note also that Tao’s computation in [13] shows that when (a) holds, then e ( σ ) is representedby [ H ] + ( d, d ) where e ( ρ ) = ( d ) and e ( σ ) = ( d ). It is therefore easy to see that (a) doesnot imply e ( σ ) = 0.By contrast, the condition e ( σ ) = e ( σ ) = 0 turns out to be sufficient for the existence ofa quadratic extension K such that ( A, σ ) K is hyperbolic (hence also for a decomposition as inProposition 1.1(i)) when deg A = 12. The following result may be regarded as a generalizationof Pfister’s theorem on 12-dimensional quadratic forms with trivial discriminant and trivialClifford invariant. Theorem 1.3.
Let ( A, σ ) be a central simple algebra with orthogonal involution of degree .The following conditions are equivalent: (a) e ( σ ) = e ( σ ) = 0 ; (b) there exists a central simple algebra with orthogonal involution ( A , σ ) of degree anda quaternion algebra with orthogonal involution ( H, ρ ) such that, writing e ( ρ ) = ( d ) and e ( σ ) = ( d ) , ( A, σ ) = ( A , σ ) ⊗ ( H, ρ ) and H = ( d, d ) . Proof.
That (b) implies (a) follows from the computation of the discriminant and the Cliffordalgebra of decomposable algebras with involution, see [7, (7.3)] and [13].The first part of the argument for the converse is borrowed from [4]. More precisely, assumecondition (a) holds. Then one of the half-spin representations V of Spin( A, σ ) is defined over F . By a classical result in representation theory, since the degree of A is 12, Spin( A, σ ) hasan open orbit in P ( V )( F alg ), where F alg is an algebraic closure of F . Using this open orbit,Garibaldi produced in (loc. cit., proof of Th. 3.1) a quadratic field extension K = F ( √ d ) of F over which σ is hyperbolic. Therefore, Proposition 1.1 applies and yields a decomposition( A, σ ) = ( A , σ ) ⊗ ( H, ρ )for some algebra with orthogonal involution ( A , σ ) of degree 6 and some quaternion algebrawith orthogonal involution ( H, ρ ) such that e ( ρ ) = ( d ). Let e ( σ ) = ( d ). Tao’s computationin [13] shows that the Clifford algebra of ( A, σ ) has two components, which are Brauer-equivalentto [ H ] + ( d, d ) and [ A ] + ( d, d ). Therefore, the triviality of e ( σ ) implies that ( d, d ) = [ H ]or [ A ]. The proof is complete if the first equation holds.For the rest of the proof, assume ( d, d ) = [ A ]. Then K splits A as well as H , hence itsplits A . Therefore, by Proposition 1.1, we may assume ( A , σ ) = Ad ϕ for some 6-dimensionalquadratic form ϕ . Let h λ , . . . , λ i be a diagonalization of ϕ and let q ∈ H be such that ρ ( x ) = qxq − for x ∈ H . Then ( d ) = ( − λ · · · λ ), F ( q ) ≃ K , and ( A, σ ) ≃ Ad h for theskew-hermitian form h = h λ q, . . . , λ q i . Let u ∈ H × be a quaternion that anticommutes with q , and let c = u ∈ F × . Then [ H ] = ( c, d ) and ux · q · ux = x · cq · x for x ∈ H, hence the skew-hermitian forms h q i and h cq i are isometric. Therefore, h ≃ h λ q, . . . , λ q, cλ q i ≃ ϕ ′ ⊗ h q i for ϕ ′ = h λ , . . . , λ , cλ i ,and we have another decomposition( A, σ ) ≃ Ad ϕ ′ ⊗ ( H, ρ ) , with e ( ϕ ′ ) = ( cd ).Since ( d, d ) = [ A ] = 0 and [ H ] = ( c, d ), it follows that (cid:0) e ( ϕ ′ ) , e ( ρ ) (cid:1) = [ H ], hence the latterdecomposition satisfies the conditions in (b). (cid:3) To emphasize the analogy between Theorem 1.3 and Pfister’s result in [8, pp. 123–124],we derive an additive decomposition of (
A, σ ) from the multiplicative decomposition in The-orem 1.3(b). Since deg A = 6 and 2[ A ] = 0, there is a quaternion algebra H ′ such that ECOMPOSABILITY OF ORTHOGONAL INVOLUTIONS IN DEGREE 12 5 A ≃ M ( H ′ ). The involution σ is adjoint to some skew-hermitian form h over ( H ′ , ). Picka diagonalization h = h q , q , q i , for some pure quaternions q i ∈ H ′ . Denote a i = q i , andconsider b i ∈ F × for i = 1, 2, 3 such that H ′ = ( a , b ) = ( a , b ) = ( a , b ). Since e ( σ ) = d ,we have ( a a a ) = ( d ). The algebra with involution ( M ( H ′ ) , ad h ) is an orthogonal sum ofthe ( H ′ , ρ i ), where ρ i = Int( q i ) ◦ has discriminant a i . This yields an additive decompositionof ( A, σ ), namely (in the notation of [10, § A, σ ) ∈ ⊞ i =1 ( H ′ , ρ i ) ⊗ ( H, ρ ) . Each term in this decomposition is a central simple algebra of degree 4 with orthogonal involu-tion of trivial discriminant. It can be rewritten as a tensor product of two quaternion algebraswith canonical involution(2) ( H ′ , ρ i ) ⊗ ( H, ρ ) ≃ ( H i , ) ⊗ ( Q i , )with H i = ( a i d , d ) and Q i = ( a i , b i d ). (This follows from a calculation of Clifford algebras or,more elementarily, from a suitable choice of base change.) We thus recover the decompositionin Corollary 3.3 of [10].If A is split, hence ( A, σ ) = Ad ψ for some 12-dimensional form ψ of trivial discriminant andClifford invariant, then H ≃ H ′ , hence each term on the right side of (1) can be written asAd π i for some 2-fold Pfister form π i , and (1) yields(3) ψ ≃ h α i π ⊥ h α i π ⊥ h α i π for some α , α , α ∈ F × . We thus get a decomposition of ψ as in [8, p. 124]. Note moreoverthat each summand ( H ′ , ρ i ) ⊗ ( H, ρ ) becomes hyperbolic over K = F ( √ d ), hence π i ≃ hh β i , d ii for some β i ∈ F × . Since e ( ψ ) = e ( π ) + e ( π ) + e ( π ) = 0, we may assume ( β β β ) = 0.Equation (3) can be rewritten as(4) ψ ≃ ( h α ihh β ii ⊥ h α ihh β ii ⊥ h α ihh β ii ) ⊗ hh d ii with ( β β β ) = 0 . Applications
Existence of orthogonal involutions with trivial invariants.
As a corollary of The-orem 1.3, we characterize the biquaternion algebras D such that M ( D ) carries an orthogonalinvolution with trivial discriminant and Clifford invariant. Corollary 2.1.
Let D be a biquaternion F -algebra. There exists an orthogonal involutionon M ( D ) having trivial discriminant and trivial Clifford invariant if and only if D admits adecomposition into quaternion algebras D = H ′ ⊗ H such that the reduced norm n H ′ and thepure subform n H of the reduced norm n H (i.e., its restriction to the pure quaternions) have acommon nonzero value.If I F = 0 , this condition holds for every biquaternion F -algebra D .Proof. Assume first there exists an orthogonal involution σ on M ( D ) which has trivial dis-criminant and trivial Clifford invariant. The algebra with involution ( M ( D ) , σ ) admits a de-composition as in Theorem 1.3, with A = M ( H ′ ) for some quaternion algebra H ′ . Considerthe discriminant d of the involution σ . We have d = − Nrd M ( H ′ ) ( s ), where s ∈ M ( H ′ )is any invertible skew-symmetric element, hence d is a value of n H ′ by [5, Lemma 2.6.4]. Inaddition, d = j for some pure quaternion j ∈ H = ( d, d ). Therefore d = − n H ( j ), so thequadratic forms n H ′ and n H share − d as a common nonzero value.To prove the converse, assume D = H ′ ⊗ H for some quaternion algebras H and H ′ , such thatthere exists a quaternion q ∈ H ′ and a pure quaternion j ∈ H satisfying n H ′ ( q ) = n H ( j ) = 0.Let d = j = − n H ′ ( q ), and let H ′ ⊂ H ′ be the vector subspace of pure quaternions. Pick anarbitrary invertible q ∈ H ′ . The vector space qq − H ′ ⊂ H ′ has dimension 3, hencedim( qq − H ′ ∩ H ′ ) ≥ . A. QU´EGUINER-MATHIEU AND J.-P. TIGNOL
Since dim H ′ = 3, the Witt index of n H ′ is at most 1, hence qq − H ′ ∩ H ′ contains anisotropicvectors. Therefore, there exist q , q ∈ H ′ invertible such that qq − q − = q , i.e., q = q q q .Then d = − n H ′ ( q ) is the discriminant of the involution adjoint to the skew-hermitian form h = h q , q , q i over ( H ′ , ). Pick a pure quaternion i which anticommutes with j , and define ρ = Int( i ) ◦ . We get that H = ( d, d ), where d = i = − n H ( i ) is the discriminant of theorthogonal involution ρ on H . The involution σ = ad h ⊗ ρ on M ( H ′ ) ⊗ H = M ( D ) satisfies( M ( D ) , σ ) = ( M ( H ′ ) , ad h ) ⊗ ( H, ρ ) with [ H ] = ( d, d ) . Therefore, Theorem 1.3 shows that e ( σ ) = e ( σ ) = 0.If I ( F ) = 0, then the reduced norm form of every quaternion algebra represents everynonzero element in F , hence the condition holds for every biquaternion F -algebra D . (cid:3) Example . Let F be an arbitrary field of characteristic different from 2, and let F = F (( x ))(( y ))(( x ))(( y )) be the field of iterated Laurent series in four variables over F . The bi-quaternion algebra D = ( x , y ) ⊗ ( x , y ) carries a unique valuation v extending the ( x , . . . , y )-adic valuation on F , and it is totally ramified over F . We claim that M ( D ) does not carryany orthogonal involution with trivial discriminant and trivial Clifford invariant. To see thisas a consequence of Corollary 2.1, consider a decomposition D = H ′ ⊗ H into quaternionsubalgebras. Let Γ D , Γ H ′ , Γ H , Γ F be the value groups of D , H ′ , H , F for the valuation v ,so Γ F = Z and Γ D = ( Z ) . By [14, Cor. 8.11] we have Γ D / Γ F = (Γ H ′ / Γ F ) ⊕ (Γ H / Γ F ),hence Γ H ′ ∩ Γ H = Γ F . For x ∈ H ′× we have v ( x ) = v (cid:0) n H ′ ( x ) (cid:1) by [14, Th. 1.4], hence v (cid:0) n H ′ ( x ) (cid:1) ∈ H ′ . Similarly, v (cid:0) n H ( y ) (cid:1) ∈ H for y ∈ H × . But the valuation on H is an“armature gauge” as defined on [14, p. 339], which means that for every standard quaternionbasis 1, i , j , k of H and λ , . . . , λ ∈ Fv ( λ + λ i + λ j + λ k ) = min { v ( λ ) , v ( λ i ) , v ( λ j ) , v ( λ k ) } . Since H is totally ramified over F , v (1), v ( i ), v ( j ), and v ( k ) are in different cosets of Γ D moduloΓ F . Therefore, if y ∈ H × is a pure quaternion, then v ( y ) / ∈ Γ F , hence v (cid:0) n H ( y ) (cid:1) ∈ H \ F .In conclusion, it is impossible to find x ∈ H ′× and y ∈ H such that n H ′ ( x ) = n H ( y ), because2Γ H ′ ∩ H = 2Γ F .2.2. A formula for the f -invariant. In the situation of Theorem 1.3, the algebras H and A occurring in the decomposition of ( A, σ ) with e ( σ ) = e ( σ ) = 0 are not uniquely determined,even up to Brauer-equivalence. Take for instance an arbitrary quaternion algebra H = ( d, d )with an orthogonal involution ρ of discriminant ( d ). As − d is represented by the reducednorm form n H , we may argue as in the proof of Corollary 2.1 to find pure quaternions q , q , q ∈ H such that n H ( q q q ) = − d . On A = M ( H ), the orthogonal involution σ adjoint tothe skew-hermitian form h q , q , q i has discriminant ( d ). Therefore, ( A, σ ) = ( A , σ ) ⊗ ( H, ρ )satisfies the conditions of Theorem 1.3. But A is split since A and H are Brauer-equivalent,hence ( A, σ ) ≃ Ad ψ for some 12-dimensional quadratic form ψ with e ( ψ ) = e ( ψ ) = 0. ByPfister’s result (see (4)), there is a decomposition ψ ≃ ψ ⊗ β for some 6-dimensional form ψ with e ( ψ ) = 0 and some 2-dimensional form β , hence another decomposition of ( A, σ ) as inTheorem 1.3: ( A , σ ) ⊗ ( H, ρ ) = (
A, σ ) ≃ Ad ψ ⊗ Ad β . Nevertheless, we show in this section that the invariant f ( σ ) defined in [10, Def. 2.4] canbe calculated from any decomposition as in Theorem 1.3, and can thus yield some informationon the possible decompositions. The main ingredient of the proof is Theorem 5.4 in [10],which shows that f ( σ ) is the Arason invariant of the sum of the norm forms of all quaternionalgebras in a given decomposition group of ( A, σ ). Since the f invariant is defined only whenthe underlying central simple algebra carries a hyperbolic involution, we need to assume in thefollowing statement, which is the main result of this section, that the index of A is at most 2. ECOMPOSABILITY OF ORTHOGONAL INVOLUTIONS IN DEGREE 12 7
Theorem 2.3.
Let ( A, σ ) be a central simple algebra of degree and index ≤ with an orthog-onal involution with trivial discriminant and trivial Clifford invariant. Pick a decomposition of ( A, σ ) as in Theorem 1.3, ( A, σ ) ≃ ( A , σ ) ⊗ ( H, ρ ) where ( A , σ ) is a central simple algebra with orthogonal involution of degree and ( H, ρ ) is aquaternion algebra with orthogonal involution, and H = ( d, d ) with e ( ρ ) = d and e ( σ ) = d .Let Q and H ′ be the quaternion algebras that are Brauer-equivalent to A and A respectively,and let n Q , n H ′ , n H be the reduced norm forms of Q , H ′ and H respectively. With this notation, (5) f ( σ ) = e ( n Q − n H − h d i n H ′ ) ∈ H ( F ) . (Note that n Q − n H − h d i n H ′ ∈ I F because [ Q ] + [ H ] + [ H ′ ] = 0 .) Moreover, if c ∈ F × is suchthat H , H ′ and Q are all split by F ( √ c ) , and e ∈ F × is such that H = ( c, e ) , then (6) f ( σ ) = ( de ) · [ Q ] = ( de ) · [ H ′ ] . Proof.
Consider the additive decomposition of (
A, σ ) in (1). Together with (2), it shows that { , [ Q ] , [ Q ] , [ H ] , [ Q ] , [ H ] , [ Q ] , [ H ] } is a decomposition group of ( A, σ ) as defined in [10, Def. 3.6]. As a result, Theorem 5.4 in [10]yields f ( σ ) = e (cid:0) n Q + X i =1 n H i + X i =1 n Q i (cid:1) . In order to compute the Arason invariant of this quadratic form, we use the following identityin the Witt group of F : hh λ, µν ii = hh λ, µ ii + h µ ihh λ, ν ii . In particular, it shows that for i = 1, 2, and 3, we have n H i = hh a i , d ii + h a i i n H and n Q i = hh a i , d ii + h d i n H ′ . Therefore,(7) X i =1 n H i + X i =1 n Q i = h a , a , a i n H + h d, d, d i n H ′ + X i =1 hh− , a i , d ii . Recall that ( d ) = ( a a a ), hence h a , a , a i n H ≡ h− d i n H mod I F. Similarly, h d, d, d i n H ′ ≡ h− d i n H ′ mod I F. Therefore, (7) yields e (cid:0) n Q + X i =1 n H i + X i =1 n Q i (cid:1) = e ( n Q − h d i n H − h d i n H ′ ) + X i =1 ( − , a i , d )= e ( n Q − n H − h d i n H ′ ) + ( d ) · [ H ] + ( − , d , d ) . Now, since H = ( d, d ) and ( d , d ) = ( − , d ), the last two terms on the right side of the lastdisplayed equation cancel, and Formula (5) is proved.To obtain Formula (6), choose c ∈ F × such that F ( √ c ) splits Q , H , and H ′ , and let e , e ′ ∈ F × be such that H = ( c, e ) and H ′ = ( c, e ′ ), hence Q = ( c, ee ′ ). Then n Q − n H − h d i n H ′ = hh c, ee ′ ii − hh c, e ii − h d ihh c, e ′ ii = hh c iih e, − ee ′ , − d, de ′ i = h e ihh c, e ′ , de ii . A. QU´EGUINER-MATHIEU AND J.-P. TIGNOL
Therefore, f ( σ ) = ( c, e ′ , de ) = ( de ) · [ H ′ ]. As H = ( c, e ) = ( d, d ), we have( d ) · [ H ] = ( − · [ H ] = ( e ) · [ H ] , hence ( de ) · [ H ] = 0 and ( de ) · [ H ′ ] = ( de ) · [ Q ]. Formula (6) is thus proved. (cid:3) Corollary 2.4.
With the notation of Theorem 2.3, we have f ( σ ) = 0 if any of the followingconditions holds: (i) A is split; (ii) A is split; (iii) A is split by F ( √ d ) .Proof. Formula (6) readily shows that f ( σ ) = 0 when (i) or (ii) holds. In case (iii) we maytake c = d and e = d in Formula (6) to obtain f ( σ ) = 0.Alternatively, in case (i) we may argue that ( A, σ ) = Ad ψ for some quadratic form ψ ∈ I F ,hence e ( σ ) = e ( ψ ) ∈ H ( F, µ ) and therefore f ( σ ) = 0 by definition. Also, in case (ii) ( A, σ )is split and hyperbolic over F ( √ d ), hence f ( σ ) = 0 by [10, Prop. 5.6]. (cid:3) By contrast, f ( σ ) does not necessarily vanish when H is split. In that case we may choose e = 1 in Formula (6) and derive the following: if ( A, σ ) = ( A , σ ) ⊗ Ad hh d ii and e ( σ ) = ( d )is such that ( d, d ) is split, then f ( σ ) = ( d ) · [ A ] . This also follows from [10, Cor. 2.18].
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