2-dimensional isotropic subspace of Eij · Q, so again Eij · Q is hyperbolic. We have

thus shown that the form T(A,σ) is hyperbolic on EndQ (V ) when σ is orthogonal,

hence sgnP T(A,σ) = 0 in this case.

If σ is symplectic, then qi ∈ F — for all i = 1, . . . , n, hence

’1 ’1

T(A,σ) (Eij q, Eij q) = TrdQ γ(q)q qj qi = 2 NrdQ (q)qj qi

for all i, j = 1, . . . , n. From (??) it follows that

T(A,σ) 2 · N Q · q 1 , . . . , qn · q 1 , . . . , qn

where NQ is the reduced norm form of Q. Since Q = (’1, ’1)F , we have NQ 41,

hence the preceding relation yields

2

sgnP T(A,σ) = 4 sgnP q1 , . . . , qn .

(11.9) Remark. In the last case, the signature of the F -quadratic form q1 , . . . , qn

is an invariant of the hermitian form h on V : indeed, the form h induces a quadratic

form hF on V , regarded as an F -vector space, by

hF (x) = h(x, x) ∈ F,

since h is hermitian. Then

hF 4 q 1 , . . . , qn ,

so sgnP hF = 4 sgnP q1 , . . . , qn . Let

sgnP h = sgnP q1 , . . . , qn .

The last step in the proof of (??) thus shows that if A = EndQ (V ) and σ = σh for

some hermitian form h on V (with respect to the canonical involution on Q), then

sgnP T(A,σ) = 4(sgnP h)2 .

(11.10) De¬nition. The signature at P of an involution σ of the ¬rst kind on A

is de¬ned by

sgnP σ = sgnP T(A,σ) .

138 II. INVARIANTS OF INVOLUTIONS

By (??), sgnP σ is an integer. Since sgnP T(A,σ) ¤ dim A and sgnP T(A,σ) ≡

dim T(A,σ) mod 2, we have

0 ¤ sgnP σ ¤ deg A and sgnP σ ≡ deg A mod 2.

From (??), we further derive:

(11.11) Corollary. Let FP be a real closure of F for the ordering P .

(1) Suppose A is not split by FP ;

(a) if σ is orthogonal, then sgnP σ = 0;

(b) if σ is symplectic and σ — IdFP = σh for some hermitian form h over the

quaternion division algebra over FP , then sgnP σ = 2 |sgnP h|.

(2) Suppose A is split by FP ;

(a) if σ is orthogonal and σ — IdFP = σb for some symmetric bilinear form b

over FP , then sgnP σ = |sgnP b|;

(b) if σ is symplectic, then sgnP σ = 0.

11.B. Involutions of the second kind. In this section we consider the case

of central simple algebras with involution of the second kind (B, „ ) over an arbitrary

¬eld F . Let K be the center of B and ι the nontrivial automorphism of K over F .

The form T(B,„ ) is hermitian with respect to ι. Let T„ be its restriction to the space

of symmetric elements. Thus,

T„ : Sym(B, „ ) — Sym(B, „ ) ’ F

is a symmetric bilinear form de¬ned by

T„ (x, y) = TrdB „ (x)y = TrdB (xy) for x, y ∈ Sym(B, „ ).

Since multiplication in B yields a canonical isomorphism of K-vector spaces

B = Sym(B, „ ) —F K,

the hermitian form T(B,„ ) can be recaptured from the bilinear form T„ :

T(B,„ ) xi ± i , y j βj = ι(±i )T„ (xi , yj )βj

i j i,j

for xi , yj ∈ Sym(B, „ ) and ±i , βj ∈ K. Therefore, the form T„ is nonsingular.

Moreover, there is no loss of information if we focus on the bilinear form T„ instead

of the hermitian form T(B,„ ).

(11.12) Examples. (1) Quaternion algebras. Suppose char F = 2 and let Q0 =

(a, b)F be a quaternion algebra over F , with canonical involution γ0 . De¬ne an

involution „ of the second kind on Q = Q0 —F K by „ = γ0 — ι. (According to (??),

every involution of the second kind on a quaternion K-algebra is of this type for a

suitable quaternion F -subalgebra). Let K F [X]/(X 2 ’ ±) and let z ∈ K satisfy

z 2 = ± (and ι(z) = ’z). If (1, i, j, k) is the usual quaternion basis of Q0 , the

elements 1, iz, jz, kz form an orthogonal basis of Sym(Q, „ ) with respect to T„ ,

hence

T„ = 2 · 1, a±, b±, ’ab± .

If char F = 2, Q0 = [a, b)F and K = F [X]/(X 2 + X + ±), let (1, i, j, k) be the

usual quaternion basis of Q0 and let z ∈ K be an element such that z 2 + z = ±

and ι(z) = z + 1. A computation shows that the elements z + i, 1 + z + i + j,

1 + z + i + kb’1 and 1 + z + i + j + kb’1 form an orthogonal basis of Sym(B, „ ) for

the form T„ , with respect to which T„ has the diagonalization

T„ = 1, 1, 1, 1 .

§11. TRACE FORM INVARIANTS 139

(2) Exchange involution. Suppose (B, „ ) = (E — E op , µ) where µ is the exchange

involution:

µ(x, y op ) = (y, xop ) for x, y ∈ E.

The space of symmetric elements is canonically isomorphic to E:

Sym(B, „ ) = { (x, xop ) | x ∈ E } = E

and since TrdB (x, y op ) = TrdE (x), TrdE (y) , the form T„ is canonically isometric

to the reduced trace bilinear form on E:

T„ (x, xop ), (y, y op ) = TrdE (xy) = TE (x, y) for x, y ∈ E.

As a further example, we consider the case of split algebras. Let V be a (¬nite

dimensional) K-vector space with a nonsingular hermitian form h. De¬ne a K-

vector space ι V by

ι

V = { ιv | v ∈ V }

with the operations

ι

v + ι w = ι (v + w) (ι v)± = ι vι(±) for v, w ∈ V , ± ∈ K.

(Compare with §?? and §??). The hermitian form h induces on the vector space

V —K ι V a nonsingular hermitian form h — ι h de¬ned by

(h — ι h)(v1 — ι v2 , w1 — ι w2 ) = h(v1 , w1 )ι h(v2 , w2 ) for v1 , v2 , w1 , w2 ∈ V .

Let s : V —K ι V ’ V —K ι V be the switch map

s(v1 — ι v2 ) = v2 — ι v1 for v1 , v2 ∈ V .

The norm of V is then de¬ned as the F -vector space of s-invariant elements

(see (??)):

NK/F (V ) = { x ∈ V —K ι V | s(x) = x }.

Since (h — ι h)(v2 — ι v1 , w2 — ι w1 ) = ι (h — ι h)(v1 — ι v2 , w1 — ι w2 ) , it follows that

(h — ι h) s(x), s(y) = ι (h — ι h)(x, y) for x, y ∈ V —K ι V .

Therefore, the restriction of the form h — ι h to the F -vector space NK/F (V ) is a

symmetric bilinear form

NK/F (h) : NK/F (V ) — NK/F (V ) ’ F.

The following proposition follows by straightforward computation:

(11.13) Proposition. Let z ∈ K F and let (ei )1¤i¤n be an orthogonal K-basis

of V . For i, j = 1, . . . , n, let Vi = (ei — ei ) · F ‚ V —K ι V and let

Vij = (ei — ιej + ej — ιei ) · F • ei z — ιej + ej ι(z) — ιei · F ‚ V —K ι V.

There is an orthogonal decomposition of NK/F (V ) for the bilinear form NK/F (h):

⊥ ⊥

⊥

NK/F (V ) = Vi • Vij .

1¤i¤n 1¤i<j¤n

Moreover, Vi 1 for all i. If char F = 2, then Vij is hyperbolic; if char F = 2,

then if K F [X]/(X 2 ’ ±) we have

Vij 2h(ei , ei )h(ej , ej ) · 1, ’± .

140 II. INVARIANTS OF INVOLUTIONS

Therefore, letting δi = h(ei , ei ) for i = 1, . . . , n,

n2 1 if char F = 2,

NK/F (h)

n 1 ⊥ 2 · 1, ’± · ⊥1¤i<j¤n δi δj if char F = 2.

Consider now the algebra B = EndK (V ) with the adjoint involution „ = σh

with respect to h.

∼

(11.14) Proposition. The standard identi¬cation •h : V —K ι V ’ B of (??) is

’

an isometry of hermitian spaces

∼

(V —F ι V, h — ι h) ’ (B, T(B,„ ) )

’

and induces an isometry of bilinear spaces

∼

NK/F (V ), NK/F (h) ’ Sym(B, „ ), T„ .

’

Proof : For x = •h (v1 — ι v2 ) and y = •h (w1 — ι w2 ) ∈ B,

T(B,„ ) (x, y) = TrdB •h (v2 — ι v1 ) —¦ •h (w1 — ι w2 ) = h(v1 , w1 )ι h(v2 , w2 ) ,

hence

T(B,„ ) •h (ξ), •h (·) = (h — ι h)(ξ, ·) for ξ, · ∈ V — ι V .

Therefore, the standard identi¬cation is an isometry

∼

•h : (V — ι V, h — ι h) ’ (B, T(B,„ ) ).

’

Since the involution „ corresponds to the switch map s, this isometry restricts to an

isometry between the F -subspaces of invariant elements under s on the one hand

and under „ on the other.

(11.15) Remark. For u ∈ Sym(B, „ )©B — , the form hu (x, y) = h u(x), y on V is

hermitian with respect to the involution „u = Int(u’1 ) —¦ „ . The same computation

as above shows that •h is an isometry of hermitian spaces

∼

(V — ι V, hu — ι h) ’ (B, T(B,„,u) )

’

where (hu — ι h)(v1 — ι v2 , w1 — ι w2 ) = hu (v1 , w1 )ι h(v2 , w2 ) for v1 , v2 , w1 , w2 ∈ V .

In particular, since the Gram matrix of hu with respect to any basis of V is the

product of the Gram matrix of h by the matrix of u, it follows that det T(B,„,u) =

(det u)dim V det(h — ι h), hence

det T(B,„,u) = (det u)dim V · N (K/F ) ∈ F — /N (K/F ).

(11.16) Corollary. Let (B, „ ) be a central simple algebra of degree n with involu-

tion of the second kind over F . Let K be the center of B.

(1) The determinant of the bilinear form T„ is given by

1 · F —2 if char F = 2,

det T„ =

(’±)n(n’1)/2 · F —2 F [X]/(X 2 ’ ±).

if char F = 2 and K

(2) For u ∈ Sym(B, „ ) © B — , the determinant of the hermitian form T(B,„,u) is

det T(B,„,u) = NrdB (u)deg B · N (K/F ) ∈ F — /N (K/F ).

§11. TRACE FORM INVARIANTS 141

Proof : (??) As in (??), the idea is to extend scalars to a splitting ¬eld L of B in

which F is algebraically closed, and to conclude by (??). The existence of such a

splitting ¬eld has already been observed in (??): we may take for L the function

¬eld of the (Weil) transfer of the Severi-Brauer variety of B if K is a ¬eld, or the

function ¬eld of the Severi-Brauer variety of E if B E — E op .

(??) For the same splitting ¬eld L as above, the extension of scalars map

Br(F ) ’ Br(L) is injective, by Merkurjev-Tignol [?, Corollary 2.12] (see the proof

of (??)). Therefore, the quaternion algebra

K, det T(B,„,u) NrdB (u)deg B F

splits, since Remark (??) shows that it splits over L.

The same reduction to the split case may be used to relate the form T„ to the

discriminant algebra D(B, „ ), which is de¬ned when the degree of B is even. In

the next proposition, we assume char F = 2, so that the bilinear form T„ de¬nes a

nonsingular quadratic form

Q„ : Sym(B, „ ) ’ F

by

Q„ (x) = T„ (x, x) for x ∈ Sym(B, „ ).

(11.17) Proposition. Let (B, „ ) be a central simple algebra with involution of

the second kind over a ¬eld F of characteristic di¬erent from 2, and let K be

F [X]/(X 2 ’ ±). Assume that the degree of (B, „ ) is

the center of B, say K

even: deg(B, „ ) = n = 2m. Then the (full ) Cli¬ord algebra of the quadratic space

Sym(B, „ ), Q„ and the discriminant algebra D(B, „ ) are related as follows:

C Sym(B, „ ), Q„ ∼ D(B, „ ) —F ’±, 2m (’1)m(m’1)/2 ,

F

where ∼ is Brauer-equivalence.

Proof : Suppose ¬rst that K is a ¬eld. By extending scalars to the function ¬eld L

of the transfer of the Severi-Brauer variety of B, we reduce to considering the split

case. For, L splits B and the scalar extension map Br(F ) ’ Br(L) is injective, as

observed in (??).

We may thus assume that B is split: let B = EndK (V ) and „ = σh for some

nonsingular hermitian form h on V . If (ei )1¤i¤n is an orthogonal basis of V and

h(ei , ei ) = δi for i = 1, . . . , n, then (??) yields

D(B, „ ) ∼ ±, (’1)n(n’1)/2 d = ±, (’1)m d

(11.18) F F

where we have set d = δ1 . . . δn . On the other hand, (??) and (??) yield

Q„ n 1 ⊥ 2 · 1, ’± · q,

where q = ⊥1¤i<j¤n δi δj . From known formulas for the Cli¬ord algebra of a direct

sum (see for instance Lam [?, Chapter 5, §2]), it follows that

2(’1)m · 1, ’± · q .

(11.19) C(Q„ ) C n1 —F C

Let IF be the fundamental ideal of even-dimensional forms in the Witt ring W F

and let I n F = (IF )n . Let d ∈ F — be a representative of disc(q). Since n = 2m,

we have

d ≡ (’1)m(m’1)/2 d mod F —2 .

142 II. INVARIANTS OF INVOLUTIONS

hence

mod I 2 F

1, ’d if m is even,

q≡

mod I 2 F

d if m is odd.

Therefore, the form 2(’1)m · 1, ’± · q is congruent modulo I 3 F to

1, ’± · 1, ’d if m is even,

1, ’± · ’2d if m is odd.

Since quadratic forms which are congruent modulo I 3 F have Brauer-equivalent

Cli¬ord algebras (see Lam [?, Chapter 5, Cor. 3.4]) it follows that

(±, d )F if m is even,

2(’1)m · 1, ’± · q ∼

C

(±, ’2d )F if m is odd.

On the other hand,

—m(m’1)/2

C n1 ∼ (’1, ’1)F ,

hence the required equivalence follows from (??) and (??).

To complete the proof, consider the case where K F — F . Then, there is a

central simple F -algebra E of degree n = 2m such that (B, „ ) (E—E op , µ) where µ

is the exchange involution. As we observed in (??), we then have Sym(B, „ ), Q„

(E, QE ) where QE (x) = TrdE (x2 ) for x ∈ E. Moreover, D(B, „ ) »m E ∼ E —m .

Since ± ∈ F —2 and (’1, 2)F is split, the proposition reduces to

—m(m’1)/2

C(E, QE ) ∼ E —m —F (’1, ’1)F .

This formula has been proved by Saltman (unpublished), Serre [?, Annexe, p. 167],

Lewis-Morales [?] and Tignol [?].

Algebras of odd degree. When the degree of B is odd, no discriminant of

(B, „ ) is de¬ned. However, we may use the fact that B is split by a scalar extension

of odd degree, together with Springer™s theorem on the behavior of quadratic forms

under odd-degree extensions, to get some information on the form T„ . Since the

arguments rely on Springer™s theorem, we need to assume char F = 2 in this section.

We may therefore argue in terms of quadratic forms instead of symmetric bilinear

forms, associating to the bilinear form T„ the quadratic form Q„ (x) = T„ (x, x).

(11.20) Lemma. Suppose char F = 2. Let L/F be a ¬eld extension of odd degree

and let q be a quadratic form over F . Let qL be the quadratic form over L derived

from q by extending scalars to L, and let ± ∈ F — F —2 . If qL 1, ’± · h for

some quadratic form h over L, of determinant 1, then there is a quadratic form t

of determinant 1 over F such that

q 1, ’± · t.

√ √

Proof : Let K = F ( ±) and M = L · K = L( ±). Let qan be an anisotropic

form over F which is Witt-equivalent to q. The form (qan )M is Witt-equivalent

to the form 1, ’± · h M , hence it is hyperbolic. Since the ¬eld extension M/K

has odd degree, Springer™s theorem on the behavior of quadratic forms under ¬eld

extensions of odd degree (see Scharlau [?, Theorem 2.5.3]) shows that (qan )K is

hyperbolic, hence, by Scharlau [?, Remark 2.5.11],

qan = 1, ’± · t0

§11. TRACE FORM INVARIANTS 143

for some quadratic form t0 over F . Let dim q = 2d, so that dim h = d, and let w

be the Witt index of q, so that

(11.21) q wH ⊥ 1, ’± · t0 ,

where H is the hyperbolic plane. We then have dim t0 = d ’ w, hence

det q = (’1)w (’±)d’w · F —2 ∈ F — /F —2 .

On the other hand, the relation qL 1, ’± · h yields

det qL = (’±)d · L—2 ∈ L— /L—2 .

Therefore, ±w ∈ F — becomes a square in L; since the degree of L/F is odd, this

implies that ±w ∈ F —2 , hence w is even. Letting t1 = w H ⊥ t0 , we then derive

2

from (??):

q 1, ’± · t1 .

It remains to prove that we may modify t1 so as to satisfy the determinant condition.

Since dim t1 = d, we have

1, ’(’1)d(d’1)/2 det t1 mod I 2 F if d is even,

t1 ≡

(’1)d(d’1)/2 det t1 mod I 2 F if d is odd.

We may use these relations to compute the Cli¬ord algebra of q 1, ’± · t1 (up

to Brauer-equivalence): in each case we get the same quaternion algebra:

C(q) ∼ ±, (’1)d(d’1)/2 det t1 .

F

On the other hand, since det h = 1 we derive from qL 1, ’± · h:

C(qL ) ∼ ±, (’1)d(d’1)/2 .

L

It follows that the quaternion algebra (±, det t1 )F is split, since it splits over the

L/F of odd degree. Therefore, if δ ∈ F — is a representative of det t1 ∈

extension

F — /F —2 , we have δ ∈ N (K/F ). Let β ∈ F — be a represented value of t1 , so that

t1 t 2 ⊥ β for some quadratic form t2 over F , and let t = t2 ⊥ δβ . Then

det t = δ · det t1 = 1.

On the other hand, since δ is a norm from the extension K/F we have 1, ’± · δβ

1, ’± · β , hence

1, ’± · t 1, ’± · t1 q.

(11.22) Proposition. Let B be a central simple K-algebra of odd degree n =

2m ’ 1 with an involution „ of the second kind. Then, there is a quadratic form q „

of dimension n(n ’ 1)/2 and determinant 1 over F such that

Q„ n 1 ⊥ 2 · 1, ’± · q„ .

Proof : Suppose ¬rst K = F — F . We may then assume (B, „ ) = (E — E op , µ) where

QE where QE (x) = TrdE (x2 ), as

µ is the exchange involution. In that case Q„

observed in (??). Since ± ∈ F —2 , we have to show that this quadratic form is Witt-

equivalent to n 1 . By Springer™s theorem, it su¬ces to prove this relation over an

odd-degree ¬eld extension. Since the degree of E is odd, we may therefore assume

E is split: E = Mn (F ). In that case, the relation is easy to check. (Observe that

the upper-triangular matrices with zero diagonal form a totally isotropic subspace).

144 II. INVARIANTS OF INVOLUTIONS

For the rest of the proof, we may thus assume K is a ¬eld. Let D be a division

K-algebra Brauer-equivalent to B and let θ be an involution of the second kind

on D. Let L be a ¬eld contained in Sym(D, θ) and maximal for this property. The

¬eld M = L·K is then a maximal sub¬eld of D, since otherwise the centralizer CD M

contains a symmetric element outside M , contradicting the maximality of L. We

have [L : F ] = [M : K] = deg D, hence the degree of L/F is odd, since D is Brauer-

equivalent to the algebra B of odd degree. Moreover, the algebra B — F L = B —K M

splits, since M is a maximal sub¬eld of D. By (??) and (??) the quadratic form

(Q„ )L obtained from Q„ by scalar extension to L has the form

(11.23) (Q„ )L n 1 ⊥ 2 · 1, ’± · h

where h = ⊥1¤i<j¤n ai aj for some a1 , . . . , an ∈ L— . Therefore, the Witt index

of the form (Q„ )L ⊥ n ’1 is at least n:

w (Q„ )L ⊥ n ’1 ≥ n.

By Springer™s theorem the Witt index of a form does not change under an odd-

degree scalar extension. Therefore,

w Q„ ⊥ n ’1 ≥ n,

and it follows that Q„ contains a subform isometric to n 1 . Let

Q„ n 1 ⊥q

for some quadratic form q over F . Relation (??) shows that

(q)L 2 · 1, ’± · h.

Since det h = 1, we may apply (??) to the quadratic form 2 · q, obtaining a

quadratic form q„ over F , of determinant 1, such that 2 · q 1, ’± · q„ ; hence

Q„ n 1 ⊥ 2 · 1, ’± · q„ .

In the case where n = 3, we show in Chapter ?? that the form q„ classi¬es the

involutions „ on a given central simple algebra B.

The signature of involutions of the second kind. Suppose√ that P is an

ordering of F which does not extend to K; this means that K = F ( ±) for some

± < 0. If (V, h) is a hermitian space over K (with respect to ι), the signature

sgnP h may be de¬ned just as in the case of quadratic spaces (see Scharlau [?,

Examples 10.1.6]). Indeed, we may view V as an F -vector space and de¬ne a

quadratic form h0 : V ’ F by

h0 (x) = h(x, x) for x ∈ V ,

since h is hermitian. If (ei )1¤i¤n is an orthogonal K-basis of V for h and z ∈ F

is such that z 2 = ±, then (ei , ei z)1¤i¤n is an orthogonal F -basis of V for h0 .

Therefore, if h(ei , ei ) = δi , then

h0 = 1, ’± · δ1 , . . . , δn ,

hence the signature of the F -quadratic form δ1 , . . . , δn is an invariant for h, equal

to 1 sgnP h0 . We let

2

1

sgnP h = sgnP h0 .

2

EXERCISES 145

Note that if ± > 0, then sgnP h0 = 0. This explains why the signature is meaningful

only when ± < 0.

(11.24) Proposition. Let (B, „ ) be a central simple algebra with involution of the

second kind over F , with center K. Then, the signature of the hermitian form

T(B,„ ) on B is the square of an integer. Moreover, if B is split: B = EndK (V ) and

„ = σh for some hermitian form h on V , then

sgnP T(B,„ ) = (sgnP h)2 .

Proof : If B is split, (??) yields an isometry

(V —K ι V, h — ι h)

(B, T(B,„ ) )

from which the equation sgnP T(B,„ ) = (sgnP h)2 follows.

In order to prove the ¬rst statement, we may extend scalars from F to a real

closure FP since signatures do not change under scalar extension to a real closure.

However, K — FP is algebraically closed since ± < 0, hence B is split over FP .

Therefore, the split case already considered shows that the signature of T(B,„ ) is a

square.

(11.25) De¬nition. For any involution „ of the second kind on B, we set

sgnP „ = sgnP T(B,„ ) .

The proposition above shows that if FP is a real closure of F for P and if „ —IdFP =

σh for some hermitian form h over FP , then

sgnP „ = |sgnP h| .

Exercises

1. Let (A, σ) be a central simple algebra with involution of any kind over a ¬eld F .

Show that for any right ideals I, J in A,

(I + J)⊥ = I ⊥ © J ⊥ and (I © J)⊥ = I ⊥ + J ⊥ .

Use this observation to prove that all the maximal isotropic right ideals in A

have the same reduced dimension.