SIGNATURES OF HERMITIAN FORMS

VINCENT ASTIER AND THOMAS UNGER

Abstract. Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. The main result of this paper consists of a method for resolving this problem, using properties of the underlying algebra with involution.

1. Introduction

Signatures of quadratic forms over formally real fields have been generalized in [BP2] to hermitian forms over central simple algebras with involution over such fields. This was achieved by means of an application of Morita theory and a reduction to the quadratic form case. A priori, signatures of hermitian forms can only be defined up to sign, i.e., a canonical definition of signature is not possible in this way. In [BP2] a choice of sign is made in such a way as to make the signature of the form which mediates the Morita equivalence positive. A problem arises when that form actually has signature zero or, equivalently, when the rank one hermitian form represented by the unit element over the algebra with involution has signature zero, for it is not then possible to make a sign choice.

In this paper, after introducing the necessary preliminaries (Section 2), we review the definition of signature of hermitian forms and study some of its properties, before proposing a method to address the problem mentioned above (Sections 3 and 4). Our main result (Theorem 4.6) shows that there exists a finite number of rank one hermitian forms over the algebra with involution, having the property that at any ordering of the base field at least one of them has nonzero signature. These rank one forms are used in an algorithm for making a sign choice, resolving the problem formulated above.

In Section 5 we show that the resulting total signature map associated to any hermitian form is continuous. Finally, in Section 6 we show, using signatures, that in general there is no obvious connection between torsion in the Witt group of an algebra with involution and sums of hermitian squares in this algebra.

Date: 4 March 2010.

2000 Mathematics Subject Classification. 16K20, 11E39, 13J30.

Key words and phrases. Central simple algebras, involutions, quadratic and hermitian forms, signatures, hermitian squares, torsion, Witt groups.

The authors were partially supported by the Science Foundation Ireland Research Frontiers Programme (project no. 07/RFP/MATF191).

2. Preliminaries

2.1. Central Simple Algebras with Involution. The general reference for this section is [KMRT, 2.A, 2.C]. Let F be a field of characteristic not two and let A be a central simple F-algebra, i.e., Z(A) = F and A has no nontrivial two-sided ideals. We always assume that dim_{F}(A) is finite. It can be shown that dim_{F}(A) is a square. We call deg(A) := \sqrt{dim_{F}(A)}the degree of A. Let m = deg(A).

An involution σ on A is an anti-automorphism of A of period two. Throughout the paper we assume that σ is F-linear, i.e., the restriction of σ to F is the identity on F. Such involutions are also said to be of the first kind. Let

Sym(A, σ) = {a ∈ A | σ(a) = a} and Skew(A, σ) = {a ∈ A | σ(a) = −a}.

Then σ is either orthogonal (or, of type +1) if dim_{F} Sym(A, σ) = m(m + 1)/2, or symplectic (or, of type −1) if dim_{F} Sym(A, σ) = m(m − 1)/2. By the Skolem-Noether theorem, two F-linear involutions σ and τ on A differ by an inner automorphism:

τ = Int(u) ∘ σ

for some u ∈ A^{×} such that σ(u) = ±u. Here Int(u)(x) := uxu^{−1} for x ∈ A. The involutions τ and σ are of the same type if and only if σ(u) = u.

We denote by Sym(A, σ)^{×} and Skew(A, σ)^{×} the sets of invertible elements in Sym(A, σ) and Skew(A, σ), respectively.

Examples 2.1.

(1) (F, id_{F}): the field F is trivially a central simple F-algebra. The identity map id_{F} is an orthogonal involution.

(2) (M_{n}(F), t): the algebra of n × n-matrices with entries from F is a central simple F-algebra. The transposition map t is an orthogonal involution.

(3) ((a, b)_{F}, −): the quaternion algebra determined by a, b ∈ F^{×} with F-basis {1, i, j, k} satisfying i^{2} = a, j^{2} = b and ij = −ji = k is a central simple F-algebra. It is a division algebra if and only if the (quadratic) norm form ⟨1, −a, −b, ab⟩ is anisotropic over F. Quaternion conjugation −, determined by \overline{i} = −i, \overline{j} = −j, and thus \overline{k} = −k is the unique symplectic involution on (a, b)_{F}. Quaternion conjugation is often denoted by γ instead of −.

(4) ((a, b)_{F}, ϑ): the involution ϑ defined on the quaternion algebra (a, b)_{F} by ϑ(i) = −i, ϑ(j) = j, ϑ(k) = k is orthogonal.

2.2. ε-Hermitian Spaces and Forms. The general reference for this section is [K, Chap. I]. Treatments of the general and division cases can also be found in [G-B] and [L2], respectively.

Let A be a central simple F-algebra, equipped with an F-linear involution σ. Let ε ∈ {−1, 1}. An ε-hermitian space over (A, σ) is a pair (M, h), where M is a finitely generated right A-module and h : M × M −→ A is a sesquilinear form such that h(y, x) = εσ(h(x, y)) for all x, y ∈ M. We call (M, h) a hermitian space when ε = 1 and a skew-hermitian space when ε = −1. Consider the left A-module M^{∗} = Hom_{A}(M, A) as a right A-module via the involution σ. The form h induces an A-linear map h^{∗} : M −→ M^{∗}, x ⟼→ h(x, ·). We call (M, h) nonsingular if h^{∗} is an isomorphism. All spaces occurring in this paper are assumed to be nonsingular. If it is clear what M is, we simply write h instead of (M, h) and speak of a form instead of a space.

Witt cancellation and Witt decomposition hold for ε-hermitian spaces (M, h) over (A, σ). Furthermore, if A = D is a division algebra (so that M D^{n} for some integer n) such that (D, σ, ε) (F, id_{F}, −1), then h can be diagonalized: there exist elements a_{1}, …, a_{n} ∈ Sym(D, σ)^{×} such that

h(x, y) = ∑_{i=1}^{n} σ(x_{i})a_{i}y_{i}, ∀x, y ∈ D^{n}.

In this case we use the shorthand notation

h = ⟨a_{1}, …, a_{n}⟩_{σ},

which resembles the notation used for diagonalized quadratic forms. If A is central simple we can certainly consider diagonal hermitian forms defined on free A-modules of finite rank, but some hermitian forms over (A, σ) may not be diagonalizable.

Let S_{ε}(A, σ) denote the commutative monoid of isometry classes of ε-hermitian spaces over (A, σ) under orthogonal sum. In this paper we consider ε-hermitian spaces (M, h) up to isometry, and so identify them with their class in S_{ε}(A, σ). Let W_{ε}(A, σ) denote the Witt group (or, more precisely, the W(F)-module) of Witt classes of εhermitian spaces over (A, σ). When ε = 1 we drop the subscript and simply write S(A, σ) and W(A, σ).

2.3. Adjoint Involutions. The general reference for this section is [KMRT, 4.A]. Let A be a central simple F-algebra, equipped with an F-linear involution σ. Let (M, h) be an ε-hermitian space over (A, σ). The algebra End_{A}(M) is again central simple over F since M is finitely generated [KMRT, 1.10]. The involution ad_{h} on End_{A}(M), defined by

h(x, f (y)) = h(ad_{h}(f)(x), y), ∀x, y ∈ M, ∀f ∈ End_{A}(M)

is called the adjoint involution of h. The involution ad_{h} is F-linear and

type(ad_{h}) = ε type(σ).

Furthermore, every F-linear involution on End_{A}(M) is of the form ad_{h} for some εhermitian form h over (A, σ) and the correspondence between ad_{h} and h is unique up to a multiplicative factor in F^{×} in the sense that ad_{h} = ad_{λh} for every λ ∈ F^{×}.

By a theorem of Wedderburn there exists an F-division algebra D (unique up to isomorphism) and a finite-dimensional right D-vector space V such that A End_{D}(V). Thus A M_{m}(D) for some positive integer m. Furthermore, if there is an F-linear involution σ on A, then there is an F-linear involution − on D and an ε_{0}-hermitian form φ_{0} over (D, −) with ε_{0} ∈ {−1, 1} such that (A, σ) and (End_{D}(V), ad_{φ_{0}}) are isomorphic as algebras with involution. In matrix form ad_{φ_{0}} is described as follows:

ad_{φ_{0}}(X) = Φ_{0}\overline{X}^{t}Φ_{0}^{−1}, ∀X ∈ M_{m}(D),

where Φ_{0} ∈ GL_{m}(D) is the Gram matrix of φ_{0}. Thus \overline{Φ_{0}} ^{t} = ε_{0}Φ_{0}.

2.4. Hermitian Morita Theory. We refer to [BP1, §1], [FM], [G-B, Chap. 2–3], [K, Chap. I, §9], or [L1] for more details. Let (M, h) be an ε-hermitian space over (A, σ). One can show that the algebras with involution (End_{A}(M), ad_{h}) and (A, σ) are Morita equivalent: for every μ ∈ {−1, 1} there is an equivalence between the categories ℋ _{μ}(End_{A}(M), ad_{h}) and ℋ _{εμ}(A, σ) of non-singular μ-hermitian forms over (End_{A}(M), ad_{h}) and non-singular εμ-hermitian forms over (A, σ), respectively (where the morphisms are given by isometry), cf. [K, Thm. 9.3.5]. This equivalence respects isometries, orthogonal sums and hyperbolic forms. It induces isomorphisms

S_{μ}(End_{A}(M), ad_{h}) S_{εμ}(A, σ) and W_{μ}(End_{A}(M), ad_{h}) W_{εμ}(A, σ)

of commutative monoids and W(F)-modules, respectively. The Morita equivalence and the isomorphisms are not canonical. One of the reasons is that ad_{h} = ad_{λh} for any λ ∈ F^{×}, as observed above.

The algebras with involution (A, σ) and (D, −) are also Morita equivalent. For future use, it will be convenient to decompose this Morita equivalence into three noncanonical equivalences of categories, the last two of which we will call scaling and collapsing. For computational purposes later on, we describe them in matrix form. We follow the approach of [LU2]:

ℋ _{ε}(A, σ) ⟶ ℋ _{ε}(M_{m}(D), ad_{φ_{0}}) ⟶^{scaling} ℋ _{ε_{0}ε}(M_{m}(D), −^{t}) ⟶^{collapsing} ℋ _{ε_{0}ε}(D, −).

Scaling: Let (M, h) be an ε-hermitian space over (M_{m}(D), ad_{φ_{0}}). Scaling is given by

(M, h) ⟼→ (M, Φ_{0}^{−1}h).  (1)

Note that Φ_{0}^{−1} is only determined up to a scalar factor in F^{×} since ad_{φ_{0}} = ad_{λφ_{0}} for any λ ∈ F^{×}.

Collapsing: Recall that M_{m}(D) End_{D}(D^{m}) and that we always have M (D^{m})^{k} M_{k, m}(D) for some integer k. Let h : M × M −→ M_{m}(D) be an ε_{0}ε-hermitian form with respect to −^{t}. Then

h(x, y) = \overline{x}^{t}By, ∀x, y ∈ M_{k, m}(D),

where B ∈ M_{k}(D) satisfies \overline{B}^{t} = ε_{0}εB, so that B determines an ε_{0}ε-hermitian form b over (D, −). Collapsing is then given by

(M, h) ⟼→ (D^{k}, b).

3. Signatures of Hermitian Forms

In order to introduce signatures of hermitian forms, we deal with a special case first. Let F be a real closed field and let = (−1, −1)_{F} denote Hamilton’s quaternion division algebra over F. Let − be quaternion conjugation on and let h ≃ ⟨a_{1}, …, a_{n}⟩_{−} be a hermitian form over ( , −). Now, a_{1}, …, a_{n} ∈ Sym( , −) = F and so we can consider the quadratic form q = ⟨a_{1}, …, a_{n}⟩. We define the signature of h, denoted signh, to be the signature of the quadratic form q. Note that this definition is independent of the choice of elements a_{1}, …, a_{n} by a theorem of Jacobson [J].

For the general case we follow the approach of [BP2, §3.3,§3.4]: let F be a formally real field and let (A, σ) be a central simple F-algebra with F-linear involution. Let h be a hermitian form over (A, σ). Consider an ordering P ∈ X_{F}, the space of orderings of F. In order to define the signature of h at P we do the following: Extend scalars to the real closure F_{P} of F at P. The extended algebra with involution (A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) is then Morita equivalent to an F_{P}-division algebra with F_{P}-linear involution (D_{P}, ϑ_{P}), where σ⊗id_{F_{P}} is adjoint to an ε_{P}-hermitian form φ_{P} over (D_{P}, ϑ_{P}) and ε_{P} ∈ {−1, 1}. By a famous theorem of Frobenius the only division algebras with center the real closed field F_{P} are F_{P} itself and _{P}, Hamilton’s quaternions over F_{P}. Furthermore, we may choose ϑ_{P} = id_{F_{P}} (often simply denoted id) in the first case and ϑ_{P} = − in the second case by Morita theory (scaling). Thus we may take

(D_{P}, ϑ_{P}) = (F_{P}, id_{F_{P}}) or (D_{P}, ϑ_{P}) = ( _{P}, −).  (2)

The Morita equivalence induces an isomorphism

ℳ_{P} : S(A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) \underrightarrow{∼} S_{ε_{P}}(D_{P}, ϑ_{P})  (3)

which is not canonical (for instance, the form φ_{P} is only determined up to a nonzero scalar factor). Since Morita equivalence preserves isometries of hermitian forms, we may define the signature of h at P to be equal to the signature of ℳ_{P}(h ⊗ F_{P}).

When ε_{P} = 1, the form ℳ_{P}(h ⊗ F_{P}) is either quadratic over F_{P}, in which case its signature is obtained in the usual way, or hermitian over ( _{P}, −), in which case the signature is computed as in the special case above.

When ε_{P} = −1, the forms φ_{P} and ℳ_{P}(h ⊗ F_{P}) are both skew-hermitian over ( _{P}, −) or alternating over F_{P}. Since skew-hermitian forms over ( _{P}, −) are always torsion [S2, Thm. 10.3.7] and alternating forms over F_{P} are always hyperbolic, it makes sense to define sign_{P}ℳ_{P}(h⊗F_{P}) = sign_{P}φ_{P} = 0 in those cases. We call the orderings P ∈ X_{F} for which ε_{P} = −1 the (A, σ)-nil orderings of F, or simply the nil orderings of F if the context is clear. We denote the set of (A, σ)-nil orderings of F by Nil(A, σ).

A different choice of Morita equivalence between (A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) and (D′_{P}, ϑ′_{P}), say, may at most result in a sign change for the signature. This follows from the computations in [G-B, pp. 54–55]. (Note that such a sign change may occur, cf. Remark 3.3 below.) We fix a Morita equivalence for each ordering P ∈ X_{F}.

In light of these remarks we now make the following

Definition 3.1. We define the signature of h at P, denoted sign_{P}^{⋆} h, as follows:

sign_{P}^{⋆} h := \begin{cases}sign_{P}ℳ_{P}(h ⊗ F_{P}) & if ε_{P} = 1 \\ 0 & if ε_{P} = −1 \end{cases} ,

where the superscript ⋆ indicates the dependence on the choice of Morita equivalence discussed earlier.

Remark 3.2. An attempt to make this definition canonical is more problematic than suggested in [BP2, §3.3, §3.4], see Section 4.

The following table summarizes all the possibilities:

[figure: (D_{P}, ϑ_{P}); σ]  orthogonal  symplectic

( _{P}, −) ε_{P} = −1 φ_{P}, ℳ_{P}(h ⊗ F_{P}): skew-hermitian

sign_{P}^{⋆} h := 0

ε_{P} = 1

φ_{P}, ℳ_{P}(h ⊗ F_{P}): hermitian

sign_{P}^{⋆} h := sign_{P}ℳ_{P}(h ⊗ F_{P})

(F_{P}, id_{F_{P}})

ε_{P} = 1

φ_{P}, ℳ_{P}(h ⊗ F_{P}): quadratic

sign_{P}^{⋆} h := sign_{P}ℳ_{P}(h ⊗ F_{P}) ε_{P} = −1 φ_{P}, ℳ_{P}(h ⊗ F_{P}): alternating

sign_{P}^{⋆} h := 0

By the properties of Morita equivalence, the signature of a hyperbolic form will be zero and

sign_{P}^{⋆}(h_{1} ⊥ h_{2}) = sign_{P}^{⋆} h_{1} + sign_{P}^{⋆} h_{2}

for all hermitian forms h_{1}, h_{2} over (A, σ). Thus sign_{P}^{⋆} induces a homomorphism of additive groups W(A, σ) −→ for each P ∈ X_{F}.

Remark 3.3. Let h be a hermitian form over (A, σ), let P ∈ X_{F} and let λ_{P} ∈ F_{P}^{×}. If we replace φ_{P} by λ_{P}φ_{P} in the computation of sign_{P}^{⋆} h above, the final result is multiplied by the sign of λ_{P}. This follows from considering the scaling part of Morita equivalence, cf. (1).

Let h be a hermitian form over (A, σ) and let P ∈ X_{F} \ Nil(A, σ) (so that ε_{P} = 1). Let ℬ be an F-basis of A. The isomorphism ℳ_{P} in (3) can be decomposed into three isomorphisms as follows:

\xymatrix{S(A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) \ar[r]^{ξ_{P}^{∗}} & S(M_{m}(D_{P}), ad_{φ_{P}}) \ar[r]^{scaling} & S(M_{m}(D_{P}), ϑ_{P}^{t}) \ar[r]^{collapsing} & S(D_{P}, ϑ_{P}) \\ h ⊗ F_{P} \ar@{|->}[r] & ξ_{P}^{∗}(h ⊗ F_{P}) \ar@{|->}[r] & Φ_{P}^{−1}ξ_{P}^{∗}(h ⊗ F_{P}) \ar@{|->}[r] & ℳ_{P}(h ⊗ F_{P})}

(4)

Here ξ_{P}^{∗} is the commutative monoid isomorphism induced by the isomorphism

ξ_{P} : (A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) \underrightarrow{∼} (M_{m}(D_{P}), ad_{φ_{P}})

discussed in the context of Wedderburn’s theorem in §2.3. The scaling matrix Φ_{P} is the matrix of the form φ_{P} with respect to the basis ξ_{P}(ℬ) of M_{m}(D_{P}).

Example 3.4. We describe how to compute the signature of a diagonal hermitian form h over (A, σ) at an ordering P ∈ X_{F}. Assume that h = ⟨a_{1}, …, a_{n}⟩_{σ} with respect to some F-basis ℬ of A. Note that a_{1}, …, a_{n} ∈ Sym(A, σ)^{×}. By the properties of the signature we have

sign_{P}^{⋆}⟨a_{1}, …, a_{n}⟩_{σ} = ∑_{i=1}^{n} sign_{P}^{⋆}⟨a_{i}⟩_{σ},

so it suffices to do the computation for a form ⟨a⟩_{σ} of rank 1 (with a ∈ Sym(A, σ)^{×}).

If ε_{P} = −1, the ordering P is (A, σ)-nil and sign_{P}^{⋆}⟨a⟩_{σ} = 0 for that ordering. Thus we assume that ε_{P} = 1.

We push ⟨a⟩_{σ} through the sequence (4). The first step gives us:

⟨a ⊗ 1⟩_{σ⊗id} ⟼→ ξ_{P}^{∗}(⟨a ⊗ 1⟩_{σ⊗id} ) = ⟨ξ_{P}(a ⊗ 1)⟩_{ad_{φ_{P}}},

where

ξ_{P}(a ⊗ 1) ∈ Sym(M_{m}(D_{P}), ad_{φ_{P}}).

For the second step, let Φ_{p} be the matrix of the form φ_{P} with respect to ξ_{P}(ℬ). It is easy to see that

Φ_{P}^{−1}⟨ξ_{P}(a ⊗ 1)⟩_{ad_{φ_{P}}} = ⟨Φ_{P}^{−1}ξ_{P}(a ⊗ 1)⟩_{ϑ_{P}}^{t}.

For the third step, note that Φ_{P}^{−1}ξ_{P}(a ⊗ 1) ∈ Sym(M_{m}(D_{P}), ϑ_{P}^{t}). Since (D_{P}, ϑ_{P}) is either (F_{P}, id_{F_{P}}) or ( _{P}, −), the matrix Φ_{P}^{−1}ξ_{P}(a⊗1) is either symmetric or hermitian and thus corresponds to a quadratic or a hermitian form ψ_{P} of dimension m over (D_{P}, ϑ_{P}). We then have

sign_{P}^{⋆}⟨a⟩_{σ} := sign_{P}ψ_{P}.

We give a simple illustration of this method (more elaborate examples will be given later in Propositions 6.4 and 6.5):

Example 3.5. Let F be the Laurent series field ((x)). Then X_{F} = {P_{1}, P_{2}}, where x >_{P_{1}} 0 and x <_{P_{2}} 0. Consider the quaternion algebra D = (−1, −x)_{F}. This is a division algebra over F since its norm form ⟨1, 1, x, x⟩ is anisotropic over F. Let A := M_{2}(D) be equipped with the conjugate transpose involution σ = −^{t}, where − denotes quaternion conjugation. Then σ is a symplectic involution. Consider the hermitian form h = ⟨(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})⟩_{σ} over (A, σ). Now A ⊗_{F} F_{P_{1}} M_{2}( _{P_{1}}) since x is a square in F_{P_{1}} and A ⊗_{F} F_{P_{2}} M_{4}(F_{P_{2}}) since −x is a square in F_{P_{2}}. We see that the ordering P_{2} is (D, −)- nil, so that sign_{P_{2}}^{⋆} h = 0. Following the steps in Example 3.4 we get sign_{P_{1}}^{⋆} h = ±2 since σ becomes adjoint to the hermitian form φ_{P_{1}} = ⟨1⟩_{−^{t}} over (M_{2}( _{P_{1}}), −^{t}) after scalar extension to the real closure of F at P_{1}. (As observed in Remark 3.3, only knowing φ_{P_{1}} up to sign only gives us the signature up to sign. In Section 4 we will explain how a choice of sign can be made.)

Lemma 3.6. Let P ∈ X_{F} and let φ_{P} be as above. Then sign_{P}^{⋆}⟨1⟩_{σ} = sign_{P}φ_{P}.

Proof. This is trivially true for the (A, σ)-nil orderings of F. Thus assume that P ∈ X_{F} is not nil. We extend scalars to the real closure of F at P, ⟨1⟩_{σ} ⟼→ ⟨1⟩_{σ} ⊗ F_{P} = ⟨1 ⊗ 1⟩_{σ⊗id} and push ⟨1 ⊗ 1⟩_{σ⊗id} through the sequence (4), as illustrated in Example 3.4:

⟨1 ⊗ 1⟩_{σ⊗id} ⟼→ ξ_{P}^{∗}(⟨1 ⊗ 1⟩_{σ⊗id} ) = ⟨ξ_{P}(1 ⊗ 1)⟩_{ad_{φ_{P}}} = ⟨I_{m}⟩_{ad_{φ_{P}}} ⟼→ Φ_{P}^{−1}⟨I_{m}⟩_{ad_{φ_{P}}} = ⟨Φ_{P}^{−1}⟩_{ϑ_{P}}^{t}.

(Note that ξ_{P}(1 ⊗ 1) = I_{m}, the m × m-identity matrix in M_{m}(D_{P}) since ξ_{P} is an algebra homomorphism.) The matrix Φ_{P}^{−1} now corresponds to a quadratic form over F_{P} or a hermitian form over ( _{P}, −). In either case Φ_{P}^{−1} is congruent to Φ_{P}. Thus sign_{P}^{⋆}⟨1⟩_{σ} = sign_{P}φ_{P}.

In [LT], Lewis and Tignol defined the signature of the involution σ at P ∈ X_{F} as follows:

sign_{P}σ := \sqrt{sign_{P}T_{σ}},

where T_{σ} is the involution trace form of (A, σ) which is a quadratic form over F defined by T_{σ}(x) := Trd_{A}(σ(x)x), for all x ∈ A. Here Trd_{A} denotes the reduced trace of A.

Examples 3.7.

(1) Let (A, σ) = (M_{n}(F), t). Then T_{σ} ≃ n^{2} × ⟨1⟩. Hence sign_{P}σ = n for all P ∈ X_{F}.

(2) Let (A, σ) = ((a, b)_{F}, −). Then T_{σ} ≃ ⟨2⟩ ⊗ ⟨1, −a, −b, ab⟩. Hence sign_{P}σ = 2 for all P ∈ X_{F} such that a <_{P} 0, b <_{P} 0 and sign_{P} = 0 for all other P ∈ X_{F}. Note that N = ⟨1, −a, −b, ab⟩ is the norm form of A.

Remark 3.8. Let (A, σ) and (B, τ) be two central simple F-algebras with F-linear involution.

(1) Consider the tensor product (A⊗_{F}B, σ⊗τ). Then T_{σ⊗τ} = T_{σ} ⊗T_{τ} and so sign_{P}(σ⊗ τ) = (sign_{P}σ)(sign_{P}τ) for all P ∈ X_{F}.

(2) If (A, σ) (B, τ), then T_{σ} ≃ T_{τ} so that sign_{P}σ = sign_{P}τ for all P ∈ X_{F}.

Remark 3.9. Pfister’s local-global principle holds for algebras with involution (A, σ) and also for hermitian forms h over such algebras, [LU1]:

sign_{P}σ = 0, ∀P ∈ X_{F} ⇔ (A, σ) is weakly hyperbolic

(i.e., σ is the adjoint involution of a torsion form) and

sign_{P}^{⋆} h = 0, ∀P ∈ X_{F} ⇔ the class of h in W(A, σ) is torsion.

Remark 3.10. The map signσ is continuous from X_{F} (equipped with the Harrison topology, see [Lam, Chapter VIII 6] for a definition) to (equipped with the discrete topology). Indeed: define the map √ on by setting \sqrt{k} = −1 if k is not a square in . Since is equipped with the discrete topology, this map is continuous. Since T_{σ} is a quadratic form, the map signT_{σ} is continuous from X_{F} to (by [Lam, Proposition 6.6]). Thus, by composition, signσ = \sqrt{signT_{σ}} is continuous from X_{F} to .

Lemma 3.11. Let P ∈ X_{F}. Then

sign_{P}σ = λ_{P}|sign_{P}φ_{P}|,

where λ_{P} = 1 if (D_{P}, ϑ_{P}) = (F_{P}, id_{F_{P}}) and λ_{P} = 2 if (D_{P}, ϑ_{P}) = ( _{P}, −). (If we want to indicate the dependence of λ_{P} on (A, σ) we will write λ_{P,A,σ}.) In particular, if P ∈ Nil(A, σ), then sign_{P}σ = sign_{P}φ_{P} = 0.

Proof. This is a reformulation of [KMRT, 11.11] or of [LT, Theorem 1] and part of its proof.

Lemma 3.12. Let (M, h) be a hermitian space over (A, σ). Let P ∈ X_{F}. Then

sign_{P}ad_{h} = λ_{P}|sign_{P}^{⋆} h|,

with λ_{P} as defined in Lemma 3.11. In particular,

sign_{P}^{⋆} h = 0 ⇔ sign_{P}ad_{h} = 0.

Proof. Assume first that P ∈ Nil(A, σ). Then sign_{P}^{⋆} h = 0. Consider the adjoint involution ad_{h} on End_{A}(M). After extension of scalars to F_{P} we have Morita equivalences

ℋ (End_{A}(M) ⊗_{F} F_{P}, ad_{h} ⊗ id_{F_{P}}) −→ ℋ (A ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) −→ ℋ _{−1}(D_{P}, ϑ_{P})

and ad_{h} ⊗ id_{F_{P}} is adjoint to a skew-hermitian form over (D_{P}, ϑ_{P}). Note that ad_{h} and σ are of the same type since h is hermitian. Thus ad_{h} and ϑ_{P} are of opposite type since P ∈ Nil(A, σ). Therefore P ∈ Nil(End_{A}(M), ad_{h}). By Lemma 3.11 we conclude that sign_{P}ad_{h} = 0.

Next, assume that P ∈ X_{F} \ Nil(A, σ). Without loss of generality we may replace F by its real closure at P. Consider the Morita equivalence

ℋ (A, σ) −→ ℋ (D, ϑ)

with (D, ϑ) = ( , −) or (D, ϑ) = (F, id). Let (N, b) be the hermitian space over (D, ϑ) corresponding to (M, h) under this Morita equivalence. Then sign^{⋆} h = signb. By [BP1, Remark 1.4.2] we have (End_{A}(M), ad_{h}) (End_{D}(N), ad_{b}) so that signad_{h} = signad_{b}. By [LT, Theorem 1] or [KMRT, 11.11] we have signad_{b} = λ |signb| with λ = 1 if (D, ϑ) = (F, id) and λ = 2 if (D, ϑ) = ( , −). We conclude that signad_{h} = λ |sign^{⋆} h|.

Remark 3.13. Since by Remark 3.10 the total signature of an involution on A is continuous, it follows from Lemma 3.12 that if h is any hermitian form over (A, σ), then the set {P ∈ X_{F} | sign_{P}h = 0} is clopen.

Corollary 3.14. Let (M, h) be a hermitian space over (A, σ) and let a ∈ Sym(A, σ)^{×}. Consider the hermitian space (M, ah) over (A, Int(a) ∘ σ). Let P ∈ X_{F}. Then

sign_{P}^{⋆}(ah) = ± sign_{P}^{⋆} h.

Proof. An easy computation shows that the involutions ad_{h}and ad_{ah}coincide on End_{A}(M). Hence they have the same signature at P ∈ X_{F}. The conclusion now follows from Lemma 3.12.

In other words, scaling by an invertible element at most changes the sign of the signature. Scaling by −1 gives an instance where a sign change of the signature occurs. This is contrary to what is claimed in [BP2, p. 662].

Lemma 3.15. Let a ∈ Sym(A, σ)^{×}. For any P ∈ X_{F} we have

sign_{P}^{⋆}⟨a⟩_{σ} = ±\frac{1}{λ_{P}} sign_{P}(Int(a^{−1}) ∘ σ),

with λ_{P} as defined in Lemma 3.11.

Proof. The involution (Int(a^{−1}) ∘ σ) ⊗ id_{F_{P}} on A ⊗_{F} F_{P} is adjoint to some form φ_{P}. We have

sign_{P}^{⋆}⟨1⟩_{Int(a^{−1})∘σ} = sign_{P}φ_{P} = ±\frac{1}{λ_{P}} sign_{P}(Int(a^{−1}) ∘ σ),

by Lemmas 3.6 and 3.11. By Corollary 3.14, we have

sign_{P}^{⋆}⟨a⟩_{σ} = ± sign_{P}⟨1⟩_{Int(a^{−1})∘σ}.

The result now follows.

Lemma 3.16. Let (A, σ) and (B, τ) be central simple F-algebras, equipped with Flinear involutions. Let a ∈ Sym(A, σ)^{×} and b ∈ Sym(B, τ)^{×}. For any P ∈ X_{F} we have

sign_{P}⟨a ⊗ b⟩_{σ⊗τ} = ±μ_{P}sign_{P}⟨a⟩_{σ} sign_{P}⟨b⟩_{τ},

where μ_{P} = 4 if A ⊗_{F} F_{P} and B ⊗_{F} F_{P} are both non-split, and μ_{P} = 1 otherwise.

Proof. By Lemma 3.15 and the fact that the signature of involutions is multiplicative we have

sign_{P}⟨a ⊗ b⟩_{σ⊗τ} = ±\frac{1}{λ_{P,A⊗B,σ⊗τ}} sign_{P}(Int((a ⊗ b)^{−1})∘ (σ ⊗ τ))

= ±\frac{1}{λ_{P,A⊗B,σ⊗τ}} sign_{P}((Int(a^{−1}) ∘ σ) ⊗ (Int(b^{−1}) ∘ τ))

= ±\frac{1}{λ_{P,A⊗B,σ⊗τ}} sign_{P}(Int(a^{−1}) ∘ σ) sign_{P}(Int(b^{−1}) ∘ τ)

= ±\frac{λ_{P,A,σ}λ_{P,B,τ}}{λ_{P,A⊗B,σ⊗τ}} sign_{P}⟨a⟩_{σ} sign_{P}⟨b⟩_{τ}.

Letting μ_{P} = λ_{P,A,σ}λ_{P,B,τ}/λ_{P,A⊗B,σ⊗τ}, its value can be determined by a case analysis.

4. An Algorithm for choosing the Sign of the Signature

In quadratic form theory the signature of the form ⟨1⟩ is always 1 at any ordering of the ground field. In contrast, the signature of the hermitian form ⟨1⟩_{σ} over (A, σ) may not even always be positive and could very well be zero, cf. Lemma 3.6.

In order to pursue the analogy with the quadratic forms case, it seems natural to require of the signature map at P from W(A, σ) to that the signature of ⟨1⟩_{σ} be positive. This is precisely the approach taken in [BP2, §3.3, §3.4], where the form in {φ_{P}, −φ_{P}} is chosen, whose signature at P is nonnegative, cf. Remark (3.3). The effect of this choice is to make the signature of ⟨1⟩_{σ} positive by Lemma 3.6.

However, it is possible that the signature of ⟨1⟩_{σ} at P (i.e. sign_{P}φ_{P}) is zero, in which case the approach taken in [BP2] no longer works and the signature map at P from W(A, σ) to remains defined only up to sign.

In order to fill this gap, our approach consists of replacing the form ⟨1⟩_{σ} by a finite number of rank one hermitian forms ⟨b_{1}⟩_{σ}, …, ⟨b_{ℓ}⟩_{σ} over (A, σ), having the property that at any ordering P ∈ X_{F} at least one of them has nonzero signature. We start by proving the existence of the elements b_{1}, …, b_{ℓ}. As our proof makes use of Merkurjev’s theorem [Me] we will deal with the case of multi-quaternion algebras with decomposable involution first.

Remark 4.1.

(1) Let (A, σ), (B, τ) and (C, υ) be central simple F-algebras with F-linear involution such that (A, σ) (B, τ) ⊗_{F} (C, υ), then

type(σ) = type(τ) · type(υ),  (5)

cf. [KMRT, 2.23].

(2) Assume that A is a biquaternion algebra with decomposable involution σ. If σ is orthogonal, then it is not difficult to see that there exist quaternion algebras with orthogonal involution (Q_{1}, σ_{1}), (Q_{2}, σ_{2}) and quaternion algebras with symplectic involution (Q′_{1}, γ_{1}), (Q′_{2}, γ_{2}) such that

(A, σ) (Q_{1}, σ_{1}) ⊗_{F} (Q_{2}, σ_{2}) (Q′_{1}, γ_{1}) ⊗_{F} (Q′_{2}, γ_{2}),

cf. [ST, §2]. On the other hand, if σ is decomposable symplectic, then it follows from (5) that one of the quaternion components has to be endowed with the canonical (symplectic) involution, and the other with an orthogonal involution.

Lemma 4.2. Let (A, σ) = (Q_{1}, σ_{1}) ⊗_{F} ⋯⊗_{F} (Q_{n}, σ_{n}) be a multi-quaternion algebra with decomposable F-linear involution. Let P ∈ X_{F} \ Nil(A, σ). Then the number of indices i ∈ {1, …, n} such that P ∈ Nil(Q_{i}, σ_{i}) is even.

Proof. Recall that P ∈ Nil(Q_{i}, σ_{i}) if and only if Q_{i} ⊗_{F} F_{P} _{P} (resp. M_{2}(F_{P})) in case σ_{i} is orthogonal (resp. symplectic). The statement now follows from an easy, but tedious, case analysis depending on the type of σ and the parity of n.

Proposition 4.3. Let (A, σ) = (Q_{1}, σ_{1})⊗_{F}⋯ ⊗_{F}(Q_{n}, σ_{n}) be a multi-quaternion algebra with decomposable F-linear involution. There exists a finite subset S = {a_{1}, …, a_{ℓ}} of Sym(A, σ)^{×} such that for every P ∈ X_{F} \ Nil(A, σ) there is an index r ∈ {1, …, ℓ} such that

sign_{P}(Int(a_{r}) ∘ σ) 0.

Proof. We will carry out the proof in three steps.

(1) Assume that n = 1, so that A = (a, b)_{F} for certain elements a, b ∈ F^{×}. For a positive integer t and a_{1}, …, a_{t} ∈ F^{×}, recall the Harrison set notation

H(a_{1}, …, a_{t}) := {P ∈ X_{F} | a_{1} >_{P} 0, …, a_{t} >_{P} 0}.

Observe that

X_{F} = H(a, b) ˙∪ H(a, −b) ˙∪ H(−a, −b) ˙∪ H(−a, b).

If σ is symplectic, then X_{F} \ Nil(A, σ) = H(−a, −b) and thus sign_{P}σ = 2 for all P ∈ H(−a, −b) since T_{σ} is the norm form of A (up to a factor ⟨2⟩). Thus we may take S = {1}.

Next assume that σ is orthogonal. In this case

X_{F} \ Nil(A, σ) = H(a, b) ˙∪ H(a, −b) ˙∪ H(−a, b).

Consider the orthogonal involution ϑ, defined by

ϑ(1) = 1, ϑ(i) = −i, ϑ(j) = j, ϑ(k) = k,

where {1, i, j, k} denotes the usual F-basis of A. Since any two involutions differ by an inner automorphism, there exists a q ∈ Sym(A, σ)^{×} such that ϑ = Int(q) ∘ σ. Consider also the involutions τ = Int(j) ∘ ϑ and ω = Int(k) ∘ ϑ. After computing the involution trace forms of ϑ, τ and ω we see that

sign_{P}ϑ = 2 for all P ∈ H(−a, b),
sign_{P}τ = 2 for all P ∈ H(a, b),
sign_{P}ω = 2 for all P ∈ H(a, −b).

Thus we may take S = {q, jq, kq}. This settles the case n = 1.

(2) Next assume that n = 2, so that A = (a, b)_{F} ⊗_{F} (c, d)_{F} for certain elements a, b, c, d ∈ F^{×} and σ = σ_{1} ⊗ σ_{2}.

σ orthogonal (n = 2) . We may assume that σ_{1} is orthogonal on Q_{1} = (a, b)_{F} and that σ_{2} is orthogonal on Q_{2} = (c, d)_{F}, cf. Remark 4.1(2). We have P ∈ X_{F} \ Nil(A, σ) if and only if A ⊗_{F} F_{P} M_{4}(F_{P}). Hence, P ∈ X_{F} \ Nil(A, σ) if and only if

Q_{1} ⊗_{F} F_{P} M_{2}(F_{P}) and Q_{2} ⊗_{F} F_{P} M_{2}(F_{P})

or

Q_{1} ⊗_{F} F_{P}  _{P} and Q_{2} ⊗_{F} F_{P}  _{P}.

Thus

X_{F} \ Nil(A, σ)= [(H(a, b) ˙∪ H(a, −b) ˙∪ H(−a, b))∩ (H(c, d) ˙∪ H(c, −d) ˙∪ H(−c, d))]
˙∪ [H(−a, −b) ∩ H(−c, −d)]
= [(X_{F} \ Nil(Q_{1}, σ_{1}))∩ (X_{F} \ Nil(Q_{2}, σ_{2}))]
˙∪ (Nil(Q_{1}, σ_{1}) ∩ Nil(Q_{2}, σ_{2})).

We first consider (X_{F} \ Nil(Q_{1}, σ_{1}))∩ (X_{F} \ Nil(Q_{2}, σ_{2})). By the n = 1 case there exist involutions π_{i,1}, π_{i,2}, π_{i,3} on Q_{i} for i = 1, 2 such that

π_{i,k} = Int(a_{i,k}) ∘ σ_{i} for some a_{i,k} ∈ Sym(Q_{i}, σ_{i})^{×}

for k = 1, 2, 3 and for every P ∈ X_{F} \ Nil(Q_{i}, σ_{i}), one of sign_{P}π_{i,1}, sign_{P}π_{i,2}, sign_{P}π_{i,3} is equal to 2. We consider all possible products

ϱ_{k,ℓ}= π_{1,k} ⊗ π_{2,ℓ} = Int(a_{1,k} ⊗ a_{2,ℓ}) ∘ (σ_{1} ⊗ σ_{2})

for k, ℓ ∈ {1, 2, 3}. Note that a_{1,k} ⊗ a_{2,ℓ} ∈ Sym(A, σ)^{×}. Then for each P ∈ (X_{F} \

Nil(Q_{1}, σ_{1}))∩ (X_{F} \ Nil(Q_{2}, σ_{2})), one of

sign_{P}ϱ_{k,ℓ}= (sign_{P}π_{1,k})(sign_{P}π_{2,ℓ})

is equal to 4.

Secondly, we consider Nil(Q_{1}, σ_{1}) ∩ Nil(Q_{2}, σ_{2}). For i = 1, 2, let γ_{i} denote the unique symplectic involution on Q_{i}. Then there exist a_{i} ∈ Skew(Q_{i}, σ_{i})^{×} such that γ_{i} = Int(a_{i}) ∘ σ_{i}. Let

γ = γ_{1} ⊗ γ_{2} = Int(a_{1} ⊗ a_{2}) ∘ (σ_{1} ⊗ σ_{2}).

Then a_{1} ⊗ a_{2} ∈ Sym(A, σ)^{×}. Furthermore,

sign_{P}γ = (sign_{P}γ_{1})(sign_{P}γ_{2}) = 4

for all P ∈ Nil(Q_{1}, σ_{1}) ∩ Nil(Q_{2}, σ_{2}). Thus we may take S = {a_{1,k} ⊗ a_{2,ℓ} | k, ℓ = 1, 2, 3} ∪ {a_{1} ⊗ a_{2}}.

σ symplectic (n = 2) . We may assume that σ_{1} is orthogonal on Q_{1} = (a, b)_{F} and that σ_{2} = γ_{2} is the unique symplectic involution on Q_{2} = (c, d)_{F}, cf. Remark 4.1(2). We have P ∈ X_{F}\Nil(A, σ) if and only if A⊗_{F}F_{P} M_{2}( _{P}). Hence, P ∈ X_{F}\Nil(A, σ) if and only if

Q_{1} ⊗_{F} F_{P} M_{2}(F_{P}) and Q_{2} ⊗_{F} F_{P} _{P}

or

Q_{1} ⊗_{F} F_{P} _{P} and Q_{2} ⊗_{F} F_{P} M_{2}(F_{P}).

Thus

X_{F} \ Nil(A, σ)= [(H(a, b) ˙∪ H(a, −b) ˙∪ H(−a, b))∩ H(−c, −d)]
˙∪ [H(−a, −b) ∩ (H(c, d) ˙∪ H(c, −d) ˙∪ H(−c, d))]
= [(X_{F} \ Nil(Q_{1}, σ_{1}))∩ (X_{F} \ Nil(Q_{2}, γ_{2}))]
˙∪ (Nil(Q_{1}, σ_{1}) ∩ Nil(Q_{2}, γ_{2})).

We first consider (X_{F} \ Nil(Q_{1}, σ_{1}))∩ (X_{F} \ Nil(Q_{2}, γ_{2})). By the n = 1 case there exist involutions π_{1}, π_{2}, π_{3} on Q_{1} such that

π_{k} = Int(a_{k}) ∘ σ_{i} for some a_{k} ∈ Sym(Q_{1}, σ_{1})^{×}

for k = 1, 2, 3 and for every P ∈ X_{F} \ Nil(Q_{1}, σ_{1}), one of sign_{P}π_{1}, sign_{P}π_{2}, sign_{P}π_{3} is equal to 2. Also, sign_{P}γ_{2} = 2 for every P ∈ X_{F} \ Nil(Q_{2}, γ_{2}). For k = 1, 2, 3 we consider, as before, all possible products

π_{k} ⊗ γ_{2} = Int(a_{k} ⊗ 1) ∘ (σ_{1} ⊗ γ_{2}).

Note that a_{k} ⊗ 1 ∈ Sym(A, σ)^{×}. We have that for each P ∈ (X_{F} \ Nil(Q_{1}, σ_{1})) ∩ (X_{F} \

Nil(Q_{2}, γ_{2})), one of

sign_{P}(π_{k} ⊗ γ_{2}) = (sign_{P}π_{k})(sign_{P}γ_{2})

is equal to 4.

Secondly, we consider Nil(Q_{1}, σ_{1})∩Nil(Q_{2}, γ_{2}). Let γ_{1} denote the unique symplectic involution on Q_{1}. Then there exists a_{1} ∈ Skew(Q_{1}, σ_{1})^{×} such that γ_{1} = Int(a_{1}) ∘ σ_{1}. Hence sign_{P}γ_{1} = 2 for all P ∈ Nil(Q_{1}, σ_{1}) = X_{F} \ Nil(Q_{1}, γ_{1}).

Let τ_{2} be an orthogonal involution on Q_{2}. Note that Nil(Q_{2}, γ_{2}) = X_{F} \ Nil(Q_{2}, τ_{2}). There exists an element a_{2} ∈ Skew(Q_{2}, γ_{2})^{×} such that τ_{2} = Int(a_{2}) ∘ γ_{2}. By the case n = 1, there are b_{1}, b_{2}, b_{3} ∈ Sym(Q_{2}, τ_{2})^{×} such that for every P ∈ Nil(Q_{2}, γ_{2}) one of sign_{P}Int(b_{i}) ∘ τ_{2} is equal to 2. For k = 1, 2, 3, let

π_{k} := Int(b_{k}) ∘ τ_{2} = Int(b_{k}) ∘ Int(a_{2}) ∘ γ_{2} = Int(b_{k}a_{2}) ∘ γ_{2}

and note that b_{k}a_{2} ∈ Skew(Q_{2}, γ_{2})^{×}. We consider all possible products

γ_{1} ⊗ π_{k} = Int(a_{1} ⊗ b_{k}a_{2}) ∘ (σ_{1} ⊗ γ_{2}).

Observe that a_{1} ⊗ b_{k}a_{2} ∈ Sym(A, σ)^{×}. We have that for each P ∈ Nil(Q_{1}, σ_{1}) ∩ Nil(Q_{2}, γ_{2}), one of

sign_{P}(γ_{1} ⊗ π_{k}) = (sign_{P}γ_{1})(sign_{P}π_{k})

is equal to 4. Thus we may take S = {a_{k} ⊗ 1 | k = 1, 2, 3} ∪ {a_{1} ⊗ b_{k}a_{2} | k = 1, 2, 3}. (3) Assume finally that n ≥ 3. σ orthogonal (n ≥ 3 ). We may assume that σ_{i} is orthogonal on Q_{i} for i = 1, …, n,

cf. Remark 4.1(2). We have X_{F} \ Nil(A, σ) = {P ∈ X_{F} | A ⊗_{F} F_{P} M_{2^{n}}(F_{P})}. For P ∈ X_{F}, let

δ_{P} = |{i ∈ {1, …, n} | Q_{i} ⊗_{F} F_{P} _{P}}|.

Then X_{F} \ Nil(A, σ) = {P ∈ X_{F} | δ_{P} is even}. Since X_{F} \ Nil(A, σ) is a finite union of sets of the form {P ∈ X_{F} | δ_{P} = 2m} for certain m ∈ , it suffices to prove the theorem for a fixed m ∈ and for the set of orderings {P ∈ X_{F} | δ_{P} = 2m}. The general statement will then follow by taking the union of the different sets S obtained in this way. Therefore we only consider orderings in Y = {P ∈ X_{F} | δ_{P} = 2m} for a fixed m ∈ . After relabeling indices we may assume that Q_{i} ⊗_{F} F_{P} _{P} if and only if 1 ≤ i ≤ 2m. After regrouping we can thus write

(A, σ) = (Q_{1} ⊗_{F} Q_{2}, σ_{1} ⊗ σ_{2}) ⊗_{F} ⋯ ⊗_{F} (Q_{2m−1} ⊗_{F} Q_{2m}, σ_{2m−1} ⊗ σ_{2m})
⊗_{F} (Q_{2m+1}, σ_{2m+1}) ⊗_{F} ⋯ ⊗_{F} (Q_{n}, σ_{n}).

Observe now that P ∈ Y implies that P ∈ X_{F} \ Nil(Q_{2i+1} ⊗_{F} Q_{2i+2}, σ_{2i+1} ⊗ σ_{2i+2}) for i = 0, …, m − 1 and P ∈ X_{F} \ Nil(Q_{ℓ}, σ_{ℓ}) for ℓ = 2m + 1, …, n. We now use the cases n = 1 and n = 2 and products of involutions to settle this case.

σ symplectic (n ≥ 3) . We may assume that σ_{i} is orthogonal on Q_{i} for i = 1, …, n−1 and that σ_{n} = γ_{n} is symplectic on Q_{n}, cf. Remark 4.1(2). We have X_{F} \ Nil(A, σ) = {P ∈ X_{F} | A ⊗_{F} F_{P} M_{2^{n−1}}( _{P})}. For P ∈ X_{F}, let

δ_{P} = |{i ∈ {1, …, n} | Q_{i} ⊗_{F} F_{P} _{P}}|.

Then X_{F} \ Nil(A, σ) = {P ∈ X_{F} | δ_{P} is odd}. By an argument similar to the one in the previous case, it suffices to successively consider the following two sets of orderings:

Case a: {P ∈ X_{F} \ Nil(A, σ) | Q_{n} ⊗_{F} F_{P} _{P}}.

Case b: {P ∈ X_{F} \ Nil(A, σ) | Q_{n} ⊗_{F} F_{P} M_{2}(F_{P})}.

In Case a, after relabeling, we may assume that Q_{i} ⊗_{F} F_{P} _{P} if and only if i ∈ {1, …, 2m} ∪ {n}. After regrouping we may write

(A, σ) = (A_{1}, τ_{1}) ⊗_{F} (Q_{2m+1}, σ_{2m+1}) ⊗_{F} ⋯ ⊗_{F} (Q_{n}, σ_{n}),

where (A_{1}, τ_{1}) = (Q_{1} ⊗_{F} Q_{2}, σ_{1} ⊗ σ_{2}) ⊗_{F} ⋯⊗_{F} (Q_{2m−1} ⊗_{F} Q_{2m}, σ_{2m−1} ⊗ σ_{2m}). We conclude by using the orthogonal n = 2 case for (A_{1}, σ_{1}) and the case n = 1 for the other components together with products of involutions.

In Case b, after relabeling, we may assume that Q_{i} ⊗_{F} F_{P} _{P} if and only if i ∈ {1, …, 2m + 1}. After regrouping we may write

(A, σ) = (A_{1}, τ_{1}) ⊗_{F} (Q_{2m+1}, σ_{2m+1}) ⊗_{F} ⋯ ⊗_{F} (Q_{n}, σ_{n}),

where (A_{1}, τ_{1}) = (Q_{1} ⊗_{F} Q_{2}, σ_{1} ⊗ σ_{2}) ⊗_{F} ⋯⊗_{F} (Q_{2m−1} ⊗_{F} Q_{2m}, σ_{2m−1} ⊗ σ_{2m}). We conclude by using the orthogonal n = 2 case for (A_{1}, σ_{1}), the symplectic n = 2 case for (Q_{2m+1} ⊗_{F} Q_{n}, σ_{2m+1} ⊗ σ_{n}) and the orthogonal n = 1 case for the other components together with products of involutions.

Proposition 4.4. Let (A, σ) be a central simple F-algebra equipped with an involution of the first kind. There exists an integer k and a finite subset {b_{1}, …, b_{ℓ}} of Sym(M_{k}(A), σ ⊗ t)^{×} such that for every P ∈ X_{F} \ Nil(M_{k}(A), σ ⊗ t) there is an index r ∈ {1, …, ℓ} such that

sign_{P}(Int(b_{r}) ∘ (σ ⊗ t)) 0.

Proof. Since σ is of the first kind, the exponent of A in the Brauer group of F is at most 2. Thus, by Merkurjev’s theorem [Me], there exist k, m ∈ such that M_{k}(A) M_{m}(Q) Q ⊗_{F} M_{m}(F), where Q = Q_{1} ⊗_{F} ⋯⊗_{F} Q_{n} is a multi-quaternion algebra. Extend σ to the involution σ ⊗ t on M_{k}(A), where t denotes transposition. Then σ ⊗ t Int(u) ∘ (τ ⊗ t) for an involution τ of the same type as σ on Q and an invertible element u ∈ Sym(Q ⊗_{F} M_{m}(F), τ ⊗ t). Without loss of generality we may assume that τ = σ_{1} ⊗ ⋯ ⊗ σ_{n}, where σ_{i} is an involution on Q_{i} for i = 1, …, n.

Consider the elements a_{1}, …, a_{ℓ} ∈ Q whose existence is asserted by Proposition 4.3. For i = 1, …, ℓ let b_{i} be the element in M_{k}(A) which is mapped to (a_{i}⊗I_{m})u^{−1} under the isomorphism M_{k}(A) Q ⊗_{F} M_{m}(F), where I_{m} denotes the identity matrix in M_{m}(F). Then each b_{i} ∈ Sym(M_{k}(A), σ ⊗ t)^{×}.

Let P ∈ X_{F} \ Nil(M_{k}(A), σ ⊗ t). Observe that Nil(M_{k}(A), σ ⊗ t) = Nil(Q, τ) since M_{k}(A) M_{m}(Q) and σ ⊗ t and τ are of the same type. By Proposition 4.3 there exists an index r ∈ {1, …, ℓ} such that

sign_{P}(Int(a_{r}) ∘ τ) 0.

Thus by Remark 3.8(2) we have

sign_{P}(Int(b_{r}) ∘ (σ ⊗ t)) = sign_{P}(Int((a_{r} ⊗ I_{m})u^{−1})∘ (Int(u) ∘ (τ ⊗ t)))
= sign_{P}(Int(a_{r} ⊗ I_{m}) ∘ Int(u^{−1}) ∘ Int(u) ∘ (τ ⊗ t))
= sign_{P}(Int(a_{r} ⊗ I_{m}) ∘ (τ ⊗ t))
= sign_{P}((Int(a_{r}) ∘ τ) ⊗ t)
= sign_{P}(Int(a_{r}) ∘ τ)· sign_{P}(t)
= m sign_{P}(Int(a_{r}) ∘ τ)
0,

which concludes the proof.

Corollary 4.5. The set of (A, σ)-nil orderings of F is clopen.

Proof. We have Nil(A, σ) = Nil(M_{k}(A), σ ⊗ t) for any k ∈ since σ and σ ⊗ t are of the same type. By Proposition 4.4

Nil(M_{k}(A), σ ⊗ t) = ⋂_{r=1}^{ℓ} {P ∈ X_{F} | sign_{P}(Int(b_{r}) ∘ (σ ⊗ t)) = 0},

which is clopen by Remark 3.10.

Theorem 4.6. Let (A, σ) be a central simple F-algebra equipped with an involution of the first kind. There exists a finite subset {b_{1}, …, b_{ℓ}} of Sym(A, σ)^{×} such that for every P ∈ X_{F} \ Nil(A, σ) there is an index r ∈ {1, …, ℓ} such that

sign_{P}(Int(b_{r}) ∘ σ) 0.

Proof. Assume first that A is split, i.e. A M_{n}(F). If σ is symplectic then Nil(A, σ) = X_{F} and there is nothing to prove. If σ is orthogonal, then there exists a ∈ Sym(A, σ)^{×} such that σ = Int(a) ∘ t, where t is the transpose involution. It follows that

sign_{P}(Int(a^{−1}) ∘ σ) = sign_{P}t = n 0

for all P ∈ X_{F}.

Secondly assume that A is not split, so that A M_{n}(D) for some n ∈ and some division algebra D. Since σ is an F-linear involution on A, there exists an F-linear involution ϑ on D. We first show that for every P ∈ X_{F} \ Nil(A, σ) there exists a b_{P} ∈ Sym(A, σ)^{×} such that sign_{P}(Int(b_{P}) ∘ σ) 0.

Let P ∈ X_{F} \ Nil(A, σ). Assume for the sake of contradiction that sign_{P}ω = 0 for every F-linear involution ω on A. Since such involutions are adjoint to n-dimensional hermitian forms over (D, ϑ) and this correspondence is one-to-one (up to a nonzero scalar factor), all hermitian forms of dimension n over (D, ϑ) have signature zero at P by Lemma 3.12. Let d ∈ Sym(D, ϑ)^{×} be arbitrary, then the n-dimensional hermitian form n × ⟨d⟩_{ϑ} has signature zero at P. This implies sign_{P}^{⋆}⟨d⟩_{ϑ} = 0 for all d ∈ Sym(D, ϑ)^{×}. Hence all hermitian forms over (D, ϑ) have signature zero at P.

However, by Proposition 4.4 (with D in the role of A) there exists k ∈ and an involution τ on M_{k}(D) such that sign_{P}τ 0. But τ is adjoint to some hermitian form over (D, ϑ) which should have zero signature at P, a contradiction with Lemma 3.12. We conclude that there exists a b_{P} ∈ Sym(A, σ)^{×} such that sign_{P}(Int(b_{P}) ∘ σ) 0.

For a ∈ Sym(A, σ)^{×} define

U(a) := {P ∈ X_{F} | sign_{P}(Int(a) ∘ σ) 0}.

By Remark 3.10 the set U(a) is clopen in X_{F}. By the previous part of the proof we have

X_{F} \ Nil(A, σ) = ⋃_{P∈X_{F}}U(b_{P}).

Since X_{F} \ Nil(A, σ) is compact by Corollary 4.5, there exists ℓ ∈ and b_{1}, …, b_{ℓ} ∈ Sym(A, σ)^{×} such that

X_{F} \ Nil(A, σ) = ⋃_{i=1}^{ℓ} U(b_{i}).

Corollary 4.7. An ordering P ∈ X_{F} is (A, σ)-nil if and only if sign_{P}^{⋆} h = 0 for every hermitian form h over (A, σ) if and only if sign_{P}^{⋆}⟨a⟩_{σ} = 0 for every a ∈ Sym(A, σ)^{×} .

Proof. By Theorem 4.6,

Nil(A, σ) = ⋂_{i=1}^{ℓ} {P ∈ X_{F} | sign_{P}(Int(b_{i}) ∘ σ) = 0}.

The result then follows from the definition of nil-ordering and Lemma 3.15 (since sign_{P}(Int(b_{i}) ∘ σ) = | sign_{P}^{⋆}⟨b_{i}^{−1}⟩_{σ}|).

The Algorithm. Fix some tuple of elements (b_{1}, …, b_{ℓ}) with properties as described in Theorem 4.6. Observe that for each P ∈ X_{F} we have

|sign_{P}^{⋆}⟨b_{i}⟩_{σ}| = \frac{1}{λ_{P}} sign_{P}(Int(b_{i}) ∘ σ)  (6)

by Lemma 3.15 since ⟨b_{i}^{−1}⟩_{σ} ≃ ⟨b_{i}⟩_{σ}. By Theorem 4.6 this implies that for each P ∈ X_{F} \ Nil(A, σ) at least one of sign_{P}^{⋆}⟨b_{1}⟩_{σ}, …, sign_{P}^{⋆}⟨b_{ℓ}⟩_{σ} is nonzero.

Therefore, for each P ∈ X_{F} \ Nil(A, σ) we decide if the signature computation is performed with φ_{P} or −φ_{P} as follows:

(i) Let i be the least element in {1, …, ℓ} such that sign_{P}^{⋆}⟨b_{i}⟩ 0. (ii) If sign_{P}^{⋆}⟨b_{i}⟩_{σ} >_{P} 0, we keep using φ_{P} for the signature computation at this order-

ing. If sign_{P}^{⋆}⟨b_{i}⟩_{σ} <_{P} 0, we replace φ_{P} by −φ_{P} in the computation of signatures at P (which then makes sign_{P}^{⋆}⟨b_{i}⟩_{σ} >_{P} 0).

Note that we may assume b_{1} = 1, in which case our algorithm extends the algorithm in [BP2, §3.3, §3.4].

This algorithm depends on the choice of the tuple (b_{1}, …, b_{ℓ}) in Theorem 4.6. Once such a choice is made for the algebra with involution (A, σ) we can consider properties such as the continuity of the total signature function sign(h) : X_{F} −→ associated to a hermitian form h over (A, σ).

Notation. In the remainder of the paper when writing sign_{P} instead of sign_{P}^{⋆} we mean that we use the above algorithm for some fixed choice of a tuple (b_{1}, …, b_{ℓ}).

5. Continuity of the Total Signature Map of Hermitian Forms

Lemma 5.1. There is a finite partition of X_{F} into clopens

X_{F} = Nil(A, σ) ˙∪ ˙⋃_{i=1}^{s} Z_{i},

and there are α_{1}, …, α_{s} ∈ Sym(A, σ)^{×} such that sign⟨α_{i}⟩_{σ} is constant non-zero on Z_{i}.

Proof. Let b_{1}, …, b_{ℓ} be as in Theorem 4.6 and, for r = 1, …, ℓ, let

Y_{r} := {P ∈ X_{F} | sign_{P}⟨b_{i}⟩_{σ} = 0, i = 1, …, r}.

Observe that each Y_{r} is clopen since

Y_{r} = ⋂_{i=1}^{r} {P ∈ X_{F} | sign_{P}(Int(b_{i}) ∘ σ) = 0}.

We have Y_{0} := X_{F} ⊇ Y_{1} ⊇ ⋯⊇ Y_{ℓ−1} ⊇ Y_{ℓ} = Nil(A, σ) and therefore,

X_{F} = (Y_{0} \ Y_{1}) ˙∪(Y_{1} \ Y_{2}) ˙∪ ⋯ ˙∪(Y_{ℓ−1} \ Y_{ℓ}) ˙∪ Nil(A, σ).

Let r ∈ {0, …, ℓ − 1} and consider Y_{r} \ Y_{r+1}. By (6) the map sign⟨b_{r+1}⟩_{σ} is never 0 on Y_{r} \ Y_{r+1} and only takes a finite number of values k_{1}, …, k_{m}.

Claim: There exists a λ ∈ {1, 2} such that

sign⟨b_{r+1}⟩_{σ} = \frac{1}{λ} sign(Int(b_{r+1}) ∘ σ)

on Y_{r} \ Y_{r+1}.

Proof of claim: If σ is orthogonal and P Nil(A, σ), then (D_{P}, ϑ_{P}) (F_{P}, id_{F_{P}}). By Lemma 3.11 together with the definition of signature of a hermitian form (since P ∈ Y_{r} \ Y_{r+1}) we have

sign_{P}⟨b_{r+1}⟩_{σ} = sign_{P}(Int(b_{r+1}) ∘ σ).

If σ is symplectic and P Nil(A, σ), then (D_{P}, ϑ_{P}) ( _{P}, −). By Lemma 3.11 together with the definition of signature of a hermitian form (since P ∈ Y_{r} \ Y_{r+1}) we have

sign⟨b_{r+1}⟩_{σ} = \frac{1}{2} sign(Int(b_{r+1}) ∘ σ).

So we simply take λ = 2 if σ is symplectic and λ = 1 if σ is orthogonal.

The claim gives us:

(sign⟨b_{r+1}⟩_{σ})^{−1}(k_{i}) ∩ (Y_{r} \ Y_{r+1}) = (sign(Int(b_{r+1}) ∘ σ))^{−1}(λk_{i}) ∩ (Y_{r} \ Y_{r+1}), which is clopen by Remark 3.10. It follows that Y_{r} \ Y_{r+1} is covered by finitely many disjoint clopen sets on which the map sign⟨b_{r+1}⟩_{σ} has constant non-zero value. The result follows since the sets Y_{r} \ Y_{r+1} for r = 0, …, ℓ − 1 form a partition of X_{F} \ Nil(A, σ).

Proposition 5.2. Let h be a hermitian form over (A, σ). The total signature of h,

signh : X_{F} −→ , P ⟼→ sign_{P}h

is continuous.

Proof. We use the notation and the conclusion of Lemma 5.1. Since Nil(A, σ) and the sets Z_{i} are clopen, it suffices to show that (signh)|_{Z_{i}} is continuous for every i = 1, …, s.

Let i ∈ {1, …, s} and let k_{i} ∈ \ {0} be such that sign⟨α_{i}⟩_{σ} = k_{i} on Z_{i}. Let k ∈ . Then

((signh)|_{Z_{i}})^{−1}(k)= {P ∈ Z_{i} | sign_{P}h = k}
= {P ∈ Z_{i} | k_{i} sign_{P}h = k_{i}k}
= {P ∈ Z_{i} | k_{i} sign_{P}h = k sign_{P}⟨α_{i}⟩_{σ}}
= {P ∈ Z_{i} | sign_{P}(k_{i} × h ⊥ k × ⟨−α_{i}⟩_{σ}) = 0}.

It follows from Lemma 3.12 that

((signh)|_{Z_{i}})^{−1}(k) = {P ∈ Z_{i} | sign_{P}ad_{k_{i}×h⊥k×⟨−α_{i}⟩_{σ}} = 0},

which is clopen by Remark 3.10.

6. Torsion in Witt Groups and Sums of Hermitian Squares

Let F be a formally real field. It is well-known that the Witt ring of F is torsion-free if and only if F is pythagorean (i.e., every sum of squares in F is a square in F), see [Lam, VIII, Theorem 4.1].

Now let (A, σ) be a central simple F-algebra equipped with an F-linear involution. A hermitian square in (A, σ) is an element of A of the form σ(x)x for some x ∈ A. We denote the set of hermitian squares in (A, σ) by (A, σ)^{2} and the set of sums of hermitian squares in (A, σ) by Σ(A, σ)^{2}. It is clear that

(A, σ)^{2} ⊆ Σ(A, σ)^{2} ⊆ Sym(A, σ).

We say that (A, σ) is pythagorean if Σ(A, σ)^{2} = (A, σ)^{2}, i.e., if every sum of hermitian squares in (A, σ) is a hermitian square in (A, σ).

We denote the torsion subgroup of W(A, σ) by W_{t}(A, σ). A fundamental result of Pfister is that W_{t}(F) is 2-primary. The torsion subgroup W_{t}(A, σ) is 2-primary as well, see [S1, Cor. 6.1] or [Ma, Thm. 4.1].

In this section we will show that there is in general no obvious relation between the property ‘torsion-free Witt group’ (W_{t}(A, σ) = 0) and the property ‘pythagorean’.

An unsurprising exception is the following:

Proposition 6.1. Let D be a quaternion division algebra over a formally real field F, equipped with quaternion conjugation −. Let N be the norm form of D.

(1) If F is pythagorean, then (D, −) is pythagorean.

(2) (D, −) is pythagorean and N is strongly anisotropic if and only if every weakly isotropic hermitian form over (D, −) is isotropic and is of dimension at least two.

(3) (D, −) is pythagorean and N is strongly anisotropic if and only if W_{t}(D, −) = 0. (Note that N is strongly anisotropic if and only if ℓ × N is anisotropic for all ℓ ∈ if and only if every sum of nonzero hermitian squares is nonzero.)

Proof. (1) Follows from a computation with the norm form N of D.

(2) Let h = ⟨a_{1}, …, a_{n}⟩_{−} be a hermitian form over (D, −). Note that a_{1}, …, a_{n} ∈ Sym(D, −) = F. Assume that ℓ × h is isotropic for some positive integer ℓ. Then there exist ℓ vectors (x_{11}, …, x_{1n}), …, (x_{ℓ1}, …, x_{ℓn}) in D^{n}, not all zero, such that

a_{1}(\overline{x_{11}}x_{11} + ⋯ + \overline{x_{ℓ1}}x_{ℓ1}) + ⋯ + a_{n}(\overline{x_{1n}}x_{1n} + ⋯ + \overline{x_{ℓn}}x_{ℓn}) = 0.

Thus, by the hypotheses on (D, −) there exist y_{1}, …, y_{n} ∈ D, not all zero, such that

a_{1}\overline{y_{1}}y_{1} + ⋯ + a_{n}\overline{y_{n}}y_{n} = 0,

i.e., h is isotropic. Note that n ≥ 2 since D is a division algebra.

Conversely, let α = \overline{x}x + \overline{y}y with x, y ∈ D^{×} and note that α ∈ F. Then the hermitian form ⟨1, 1, −α, −α⟩_{−} = 2 × ⟨1, −α⟩_{−} is isotropic. By the assumption the form ⟨1, −α⟩_{−} is isotropic, so that there exists a z ∈ D^{×} such that α = \overline{z}z. Note that α 0 since D is a division algebra. Furthermore, the strong anisotropy of N follows at once.

(3) Assume that (D, −) is pythagorean and that N is strongly anisotropic. Let h be a torsion hermitian form over (D, −). Since the torsion in W(D, −) is 2-primary there exists a minimal positive integer ℓ such that 2^{ℓ}×h is hyperbolic. Let f be an anisotropic hermitian form which is in the Witt class of h in W(D, −). Then 2^{ℓ} × f is hyperbolic, and thus in particular isotropic, which implies that f is isotropic by (2), a contradiction.

Conversely, assume that W_{t}(D, −) = 0. Let x_{1}, …, x_{n} ∈ D^{×} and assume for the sake of contradiction that N(x_{1}) + ⋯+ N(x_{n}) = 0. Let k be an integer such that 2^{k} ≥ n. Then 2^{k} × ⟨1⟩_{−} is isotropic, and so the quadratic form N ⊗ (2^{k} × ⟨1⟩) is isotropic and thus hyperbolic since it is a Pfister form. By a theorem of Jacobson [J] the hermitian form 2^{k} × ⟨1⟩_{−} is hyperbolic. Since W_{t}(D, −) = 0 we obtain that ⟨1⟩_{−} is hyperbolic, which is impossible.

Now let α = \overline{x}x + \overline{y}y with x, y ∈ D^{×} and note that α ∈ F^{×}. Then the hermitian form ⟨1, 1, −α, −α⟩_{−} is isotropic. Hence the quadratic form N ⊗ ⟨1, 1, −α, −α⟩ is isotropic, and thus hyperbolic since it is a Pfister form. But this implies that 2 × ⟨1, −α⟩_{−} = ⟨1, 1, −α, −α⟩_{−} is hyperbolic by Jacobson’s theorem. Thus ⟨1, −α⟩_{−} is hyperbolic by our assumption. Therefore α is a norm.

Remark 6.2. The converse of Proposition 6.1(1) is not true. For example, F = is not pythagorean, but ((−1, −1)_{}, −) is pythagorean since every sum of four squares in is again a square in .

Proposition 6.3. Let F be a formally real field. Consider Hamilton’s quaternion algebra = (−1, −1)_{F} equipped with the orthogonal involution ϑ from Example 2.1(4). Then:

(1) W( , ϑ) = W_{t}( , ϑ) 0.

(2) If F is real closed, then Sym( , ϑ) = ( , ϑ)^{2} and ( , ϑ) is pythagorean.

Proof. (1) Let h ∈ W( , ϑ). Since Nil( , ϑ) = X_{F} we have that sign_{P}h = 0 for all P ∈ X_{F}. Thus h is a torsion form by Pfister’s local-global principle (cf. Remark 3.9). Hence W_{t}( , ϑ) = W( , ϑ) 0.

(2) Let u_{0} = α_{0} + γ_{0}j + δ_{0}k ∈ Sym( , ϑ). For u = α + βi + γj + δk ∈ we have

ϑ(u)u = (α^{2} + β^{2} − γ^{2} − δ^{2}) + 2(αγ + βδ)j + 2(−βγ + αδ)k.

We will show that the equation ϑ(u)u = u_{0} has a solution (α, β, γ, δ) (0, 0, 0, 0). Let γ = 0. An easy computation shows that ϑ(u)u = u_{0} if and only if 2αδ = δ_{0}, 2βδ = γ_{0} and

δ^{4} + α_{0}δ^{2} − \frac{1}{4}(δ_{0}^{2} + γ_{0}^{2}) = 0.

The last equation is quadratic in δ^{2} with discriminant Δ = α_{0}^{2} + δ_{0}^{2} + γ_{0}^{2}. Since F is real closed (and thus pythagorean), Δ is a square. Let ε be the positive square root of Δ. Then

δ^{2} = \frac{−α_{0} ± ε}{2} .

Since Δ ≥ α_{0}^{2}, we have ε ≥ α_{0} and so (−α_{0} + ε)/2 ≥ 0. Thus \sqrt{(−α_{0} + ε)/2} exists, since F is real closed.

Finally, ( , ϑ) is pythagorean since Σ( , ϑ)^{2} ⊆ Sym( , ϑ).

This proposition shows that already for algebras with involution over a real closed base field, ‘pythagorean’ does not imply ‘torsion-free Witt group’. The following two propositions describe examples which show that ‘torsion-free Witt group’ does not imply ‘pythagorean’ either.

Proposition 6.4. Let F = ((x))((y))((z))((w)) be the iterated Laurent series field in the unknowns x, y, z, w over the field of real numbers . Consider the quaternion algebras D_{1} = (x, y)_{F} and D_{2} = (z, w)_{F} and the biquaternion algebra D = D_{1} ⊗_{F} D_{2}. For ℓ = 1, 2, let {1, i_{ℓ}, j_{ℓ}, k_{ℓ}} be the usual F-basis for D_{ℓ} and let σ_{ℓ} be the orthogonal involution on D_{ℓ} that sends i_{ℓ} to −i_{ℓ} and that fixes the other basis elements. Let σ = σ_{1} ⊗ σ_{2} be the resulting orthogonal involution on D. Then:

(1) D is a division algebra.

(2) W_{t}(D, σ) = 0.

(3) (D, σ) is not pythagorean.

Proof. (1) Let v be the standard (x, y, z, w)-adic valuation on F (see for instance [W, §3]). Note that F is Henselian with respect to v. An application of Springer’s theorem shows that the Albert form ⟨x, y, −xy, −z, −w, zw⟩ of D is anisotropic (we obtain six residue forms of dimension 1 over , that are necessarily isotropic). Hence D is a division algebra, cf. [Lam, Chap. III, Thm. 4.8].

(2) Since F is Henselian, the valuation v extends uniquely to a valuation on D (see [Mo, Thm. 2]), which we also denote by v. We now claim that the residue division algebra \overline{D} is isomorphic to . The proof of this claim goes as follows: Since char( ) = 0, the division algebras D_{1} and D_{2} are tame (in the sense of [JW, §6]). By [JW, Corollary 6.7] we have Γ_{D} ⊆ Γ_{D_{1}} + Γ_{D_{2}} (this sum takes place in the divisible closure of Γ_{F}). We first compute Γ_{D_{1}}. Since i_{1}^{2} = x and v(x) = (1, 0, 0, 0) we have v(i_{1}) = (1/2, 0, 0, 0). Similarly v(j_{1}) = (0, 1/2, 0, 0) and v(k_{1}) = (1/2, 1/2, 0, 0). Let γ_{1} be the quaternion conjugation on D_{1}. Since v extends uniquely from F to D_{1} and v ∘ γ_{1} is a valuation on D_{1} we have v(a) = v ∘ γ_{1}(a) for every a ∈ D_{1}. In particular v(γ_{1}(a)a) = 2v(a). If we write a = a_{0} + a_{1}i_{1} + a_{2}j_{1} + a_{3}k_{1} we obtain γ_{1}(a)a = a_{0}^{2} − xa_{1}^{2} − ya_{2}^{2} + xya_{3}^{2}. Since the four terms in this sum have different valuation we get

\begin{aligned}v(a) & = \frac{1}{2} min\{ε_{0}0, ε_{1}v(x), ε_{2}v(y), ε_{3}v(xy)\} \\ & = \frac{1}{2} min\{ε_{0}0, ε_{1}(1, 0, 0, 0), ε_{2}(0, 1, 0, 0), ε_{3}(1, 1, 0, 0)\}, \end{aligned}

where ε_{i} = 0 if a_{i} = 0, and 1 otherwise (for i = 0, …, 3; this is to account for the presense or absence of a_{i}).

This yields Γ_{D_{1}} = \frac{1}{2} × \frac{1}{2} × × . A similar argument shows that Γ_{D_{2}} = × × \frac{1}{2} × \frac{1}{2} . Since Γ_{D} ⊆ Γ_{D_{1}} + Γ_{D_{2}} we get Γ_{D} = \frac{1}{2}( × × × ). In particular [Γ_{D} : Γ_{F}] = 16 and by Draxl’s “Ostrowski Theorem” (see [JW, Equation 1.2]), we obtain [\overline{D} : \overline{F}] = 1, i.e. \overline{D} = \overline{F} = . This proves the claim.

Using now that W_{t}( ) = 0 and also W_{−1}( ) = 0, [Lar, Theorem 3.7] implies that W_{t}(D, σ) = 0.

(3) Consider the sum of two hermitian squares

a = σ(j_{1} ⊗ j_{2} + 1 ⊗ 1)(j_{1} ⊗ j_{2} + 1 ⊗ 1) + σ(i_{1} ⊗ j_{2})(i_{1} ⊗ j_{2})
= (j_{1} ⊗ j_{2} + 1 ⊗ 1)^{2} − (i_{1} ⊗ j_{2})^{2}

in (D, σ). We will show that a is not a hermitian square in (D, σ) by means of a signature computation. Let P ∈ X_{F} be the ordering for which x, y, z, w >_{P} 0 and let F_{P} be the real closure of F at P. Then

D ⊗_{F} F_{P} M_{4}(F_{P}).

If a were a hermitian square, then the hermitian forms ⟨a⟩_{σ} and ⟨1⟩_{σ} over (D, σ) would be isometric. We will shortly see, however, that sign_{P}^{⋆}⟨a⟩_{σ} = ±4, while sign_{P}^{⋆}⟨1⟩_{σ} = 0. Thus the forms are not isometric and so a is not a hermitian square.

In order to compute sign_{P}^{⋆}⟨a⟩_{σ} we follow the method of Example 3.4. The algebra D is generated by the basic tensors i_{1} ⊗ 1, j_{1} ⊗ 1, 1 ⊗ i_{2} and 1 ⊗ j_{2}. We extend scalars to the real closure of F at P, D −→ D ⊗_{F} F_{P}, and then apply the splitting isomorphism

ξ_{P} : (D ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) \underrightarrow{∼} (M_{4}(F_{P}), ad_{φ_{P}})

induced by the algebra isomorphisms

η_{ℓ} : D_{ℓ} ⊗_{F} F_{P} \underrightarrow{∼} M_{2}(F_{P})
i_{ℓ} ⊗ 1 ⟼→ [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}]
j_{ℓ} ⊗ 1 ⟼→ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}],

for ℓ = 1, 2. A straightforward computation shows that

ξ_{P}(a ⊗ 1) = [\begin{matrix}1 & 0 & 0 & 2 \\ 0 & 1 & 2 & 0 \\ 0 & 2 & 1 & 0 \\ 2 & 0 & 0 & 1 \end{matrix}].

Since σ ⊗ id_{F_{P}} is an orthogonal involution, φ_{P} is a quadratic form over F_{P}. Let Φ_{P} denote the Gram matrix of φ_{P}. Since ξ_{P} is an isomorphism of algebras with involution we have that

ξ_{P} ∘ (σ ⊗ id_{F_{P}}) = ad_{φ_{P}} ∘ ξ_{P},

from which it follows (by easy, but tedious computations) that we may take

Φ_{P} = ± [\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}].

Since Φ_{P} = Φ_{P}^{−1} we have

Φ_{P}^{−1}ξ_{P}(a ⊗ 1) = ± [\begin{matrix}2 & 0 & 0 & 1 \\ 0 & 2 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 1 & 0 & 0 & 2 \end{matrix}],

from which it follows that ψ_{P} ≃ ±⟨1, 1, 1, 1⟩. We conclude that sign_{P}^{⋆}⟨a⟩_{σ} = ±4.

On the other hand, since φ_{P} is clearly hyperbolic, it follows from Lemma 3.6 that sign_{P}^{⋆}⟨1⟩_{σ} = sign_{P}φ_{P} = 0.

Proposition 6.5. We use the same notation as in the previous proposition, except that we let σ = σ_{1} ⊗ γ_{2}, where γ_{2} denotes quaternion conjugation on D_{2}, so that σ is a symplectic involution on D. Then:

(1) D is a division algebra.

(2) W_{t}(D, σ) = 0.

(3) (D, σ) is not pythagorean.

Proof. (1) & (2): identical to the proof of Proposition 6.4(1) & (2).

(3) The proof is similar to the proof of Proposition 6.4(3). We explain the main differences. Consider the sum of three hermitian squares

a = σ(j_{1} ⊗ 1 + 1 ⊗ 1)(j_{1} ⊗ 1 + 1 ⊗ 1) + 2σ(i_{1} ⊗ i_{2})(i_{1} ⊗ i_{2})
= (j_{1} ⊗ 1 + 1 ⊗ 1)^{2} + 2(i_{1} ⊗ i_{2})^{2}

in (D, σ). Let P ∈ X_{F} be the ordering for which x, y >_{P} 0 and z, w <_{P} 0. Then

D ⊗_{F} F_{P} = (D_{1} ⊗_{F} D_{2}) ⊗_{F} F_{P} M_{2}(F_{P}) ⊗_{F_{P}} _{P} M_{2}( _{P}).

Let η_{1} be as before and let η_{2} be the isomorphism D_{2} ⊗_{F} F_{P} \underrightarrow{∼} _{P} defined by letting η_{2}(i_{2} ⊗ 1) = i and η_{2}(j_{2} ⊗ 1) = j. Let ξ_{P} be the induced isomorphism

(D ⊗_{F} F_{P}, σ ⊗ id_{F_{P}}) \underrightarrow{∼} (M_{2}( _{P}), ad_{φ_{P}}).

This time the form φ_{P} is hermitian over ( _{P}, −) and a computation shows that we may

take Φ_{P} = ± [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]. Another computation shows that ξ_{P}(a ⊗ 1) = [\begin{matrix}0 & 2 \\ 2 & 0 \end{matrix}]. Hence

Φ_{P}^{−1}ξ_{P}(a ⊗ 1) = ± [\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}]

from which it follows that sign_{P}^{⋆}⟨a⟩_{σ} = ±2. Since again sign_{P}^{⋆}⟨1⟩_{σ} = 0 it follows that a cannot be hermitian square.

Acknowledgements

The authors thank Karim Becher and David Lewis for several discussions on this and related topics.

References

[BP1] E. Bayer-Fluckiger, R. Parimala, Galois cohomology of the classical groups over fields of cohomological dimension ≤ 2, Invent. Math. 122 (1995), no. 2, 195–229.

[BP2] E. Bayer-Fluckiger, R. Parimala, Classical groups and the Hasse principle, Ann. of Math. (2) 147 (1998), no. 3, 651–693.

[FM] A. Fröhlich and A.M. McEvett, Forms over rings with involution. J. Algebra 12 (1969), 79– 104.

[G-B] N. Grenier-Boley, Groupe de Witt d’une algèbre simple centrale à involution, PhD thesis,

Université de Franche-Comté (2004)

[JW] B. Jacob and A. Wadsworth, Division algebras over henselian fields. J. Algebra 128 (1990), 126–179.

[J] N. Jacobson, A note on hermitian forms, Bull. Amer. Math. Soc. 46 (1940), 264–268.

[K] M.-A. Knus, Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften 294, Springer-Verlag, Berlin (1991).

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

[Lam] T.Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, American Mathematical Society, Providence, RI (2005).

[Lar] D. Larmour, A Springer theorem for hermitian forms. Math. Z. 252 (2006), 459–472.

[L1] D.W. Lewis, Forms over real algebras and the multisignature of a manifold, Advances in Math. 23 (1977), no. 3, 272–284.

[L2] D.W. Lewis, The isometry classification of Hermitian forms over division algebras, Linear Algebra Appl. 43 (1982), 245–272.

[LT] D.W. Lewis, J.-P. Tignol, On the signature of an involution, Arch. Math. 60 (1993), 128–135.

[LU1] D.W. Lewis, T. Unger, A local-global principle for algebras with involution and hermitian forms, Math. Z. 244 (2003), 469–477.

[LU2] D.W. Lewis, T. Unger, Hermitian Morita theory: a matrix approach, Irish Math. Soc. Bull. 62 (2008), 37–41.

[Ma] M.G. Mahmoudi, A remark on transfers and hyperbolicity, Comm. Algebra 32 (2004), no. 12, 4733–4739.

[Me] A.S. Merkurjev, On the norm residue symbol of degree 2 (Russian), Dokl. Akad. Nauk. SSSR 261 (1981), 542–547. English translation: Soviet Math. Dokl. 24 (1981), 546–551.

[Mo] P. Morandi, The Henselization of a valued division algebra, J. Algebra 122 (1989), no. 1, 232–243.

[S1] W. Scharlau, Induction theorems and the structure of the Witt group, Invent. Math. 11 (1970), 37–44.

[S2] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften 270, Springer-Verlag, Berlin (1985).

[ST] A. Serhir, J.-P. Tignol, The discriminant of a decomposable symplectic involution, J. Algebra 273 (2004), no. 2, 601–607.

[W] A. Wadsworth, Valuation Theory on Finite Dimensional Division Algebras, Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 385–449, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI (2002).

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

  E-mail address: vincent.astier@ucd.ie, thomas.unger@ucd.ie