Strongly anisotropic involutions on central simple algebras

Amit Kulshrestha

Abstract

The classical theorem of Bröcker and Prestel on quadratic forms over formally real fields determines a valuation theoretic condition under which all totally indefinite forms are weakly isotropic. In this paper, we look for analogues of such results in a more general setting of algebras with involutions. We prove that for involutions of first kind over central simple algebras of index two, one indeed has a Bröcker-Prestel like statement. The connection between two conditions, namely total indefiniteness and weak isotropicity is made via so called gauge functions on central simple algebras.

1 Introduction

Let F be a formally real field and q be a quadratic form over F. Let Ω denote the set of real orderings of F. The form q is said to be totally indefinite if over all real closures F_{P}; P ∈ Ω, it is isotropic. Clearly an isotropic form is totally indefinite, though the converse is not true. In fact, even if an orthogonal sum of copies of q is isotropic then q is totally indefinite. Quadratic forms q such that the m-fold orthogonal sum m × q is isotropic for some integer m ≥ 1 are called weakly isotropic. Forms which are not weakly isotropic are termed as strongly anisotropic forms.

If F satisfies so called Strong Approximation Property (SAP), then weakly isotropic forms are precisely those which are totally indefinite (cf. [ELP],[P2]). This however, is not true for arbitrary formally real fields. For arbitrary formally real fields, there is a classical theorem proved independently by Bröcker and Prestel in 1974 (see [B] and [P1, Th. 8.13]) which connects the two conditions, namely weak isotropicity and total indefiniteness. This connection is via valuations on fields.

Theorem 1.1 (Bröcker - Prestel) Let F be a formally real field and q be a quadratic form over F. Then following statements are equivalent:

(i) The form q is weakly isotropic.

(ii) The form q is indefinite at every ordering of F, and for every valuation v where q has more than one residue and the residue field F_{0} is formally real, at least one residue class of q is weakly isotropic.

Since quadratic forms correspond to orthogonal involutions on split central simple algebras, it is interesting to ask if an analogous result holds for arbitrary central simple algebras with involutions. In this paper, we prove such a result for involutions of first kind over central simple algebras of index at most 2. Our result is equivalent to Theorem 1.1 in the split orthogonal case, and in this sense it generalises Theorem 1.1. We first define some terminology.

Let A be a central simple algebra over F and σ be an involution on A. We say that (A, σ) is weakly isotropic if there exist non-zero x_{i} ∈ A; 1 ≤ i ≤ r such that ∑_{i=1}^{r} σ(x_{i})x_{i} = 0. Let t_{n} denote the transpose involution on the split algebra M_{n}(F). It is easy to check that σ is weakly isotropic if and only if there exists n ≥ 1 such that the involution σ ⊗ t_{n} is isotropic on A ⊗ M_{n}(F). We also remark that if F is not formally real then every involution is weakly isotropic. Thus throughout the paper, we shall be implicitly assuming that F is formally real.

Let (A, σ) a central simple algebra over F with an involution of the first kind. Let Trd_{A} denote the reduced trace of A. The trace form of (A, σ) is the quadratic form T_{σ} : A → F given by x ↦ Trd_{A}(σ(x)x). If ad_{T_{σ}} is the adjoint involution on End_{F} (A) associated to the quadratic form T_{σ} then (End_{F} (A), ad_{T_{σ}}) ≃ (A ⊗_{F} A, σ ⊗ σ). In [U, Ch. 5] there are examples of involutions σ which are weakly isotropic if and only the associated trace form T_{σ} is weakly isotropic.

If F is formally real and P is an ordering on F then by [KMRT, Prop. 11.7], the signature sgn_{P} (T_{σ}) of the quadratic form T_{σ} at P is square. We define the signature of σ at an ordering P as follows:

sgn_{P} (A, σ) = \sqrt{sgn_{P} (T_{σ}}).

We say that (A, σ) is totally indefinite if sgn_{P} (A, σ) < deg(A) for all orderings P on F.

One of the main difficulties in extending Theorem 1.1 is that valuations are very rare on central simple algebras (in fact impossible for non-division algebras), and even if they exist there is no clue of the relationship they have with orderings on the base field. However, an alternate notion of gauges on semisimple algebras was recently discovered by Tignol and Wadsworth [TW, TW1], and we use gauges to generalise Theorem 1.1 in several cases.

Let F be a formally real field and A be a central simple F -algebra. Let σ be an involution of first kind on A. We say that (A, σ) satisfies BP-equivalence if the following conditions are equivalent:

(BP1) The involution σ is weakly isotropic (BP2) The involution σ is totally indefinite and at every gauge j on A which is ramified and has

formally real residue field, the residue of σ is weakly isotropic.

The implication (BP1) ⇒ (BP2) of BP-equivalence always holds (Th. 2.20). Interestingly, it was shown by Lewis, Scheiderer and Unger in [LSU] that if F satisfies Effective Diagonalisation (ED) property and (A, σ) is a central simple algebra over F with σ totally indefinite involution of the first kind then (A, σ) is weakly isotropic. In fact, they showed that this is a criterion to classify ED fields.

The main result of this article is the following:

Theorem 1.2 Let F be a formally real field and A be a central simple F -algebra. Let σ be an involution of first kind on A. Then (A, σ) satisfies BP-equivalence in the following cases:

(i) index(A) ≤ 2.

(ii) (A, σ) ≃ (H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{r}, σ_{1} ⊗ σ_{2} ⊗ ⋯ ⊗ σ_{r}), where each H_{i} is a quaternion algebra over H and σ_{i} is an involution of first kind and either type.

Organisation of the paper is as follows: In section 2, we introduce several value functions on vector spaces and central simple algebras. The crucial notions of gauges and residue involutions are also introduced in this section. The section concludes with a proof of the implication (BP1) ⇒ (BP2) of BP-equivalence without any assumption on the index of A. After section 2, the rest of the paper is devoted to prove the implication (ii) ⇒ (i) of BP-equivalence in several cases. These cases are covered in section 3 (split case), section 4 (symplectic case) and section 5 (orthogonal case). Further, in section 6 we prove that if the trace form of (A, σ) determines its weak isotropicity then (A, σ) satisfies BP-equivalence.

2 Value functions on vector spaces and algebras

2.1 Norms on vector spaces

As already pointed out in the introduction, the absence of valuations on arbitrary central simple algebras needs to be taken care of to address the questions regarding the generalisation of Bröcker-Prestel in the set up of involutions on central simple algebras. In this section, we explain the kind of value functions used in the paper. These value functions were recently defined in [TW] and [TW1].

In what follows, D denotes a division algebra and Γ denotes a totally ordered abelian group. Let ∞ be a symbol such that in the union Γ ∪ {∞}, ∞ > δ for every δ ∈ Γ. A valuation v̄ on D with values in Γ is a map v̄ : D → Γ ∪ {∞} such that:

(i) v̄(a) = ∞ if and only if a = 0.

(ii) v̄(ab) = v̄(a) + v̄(b) for all a, b ∈ D.

(iii) v̄(a + b) ≥ min{v̄(a), v̄(b)} for all a, b ∈ D.

Such maps do not necessarily exist. In fact, by a Theorem of Morandi [M, Theorem 1.2], if D is finite dimensional over its centre Z(D) = F, then D carries a valuation v̄ if and only if there is a valuation v on F such that D ⊗_{F} F_{v}^{h} is a division algebra; here F_{v}^{h} denotes the henselisation of F at v. Moreover by [E] and [W], if these conditions hold then

v̄(a) = \frac{1}{deg_{F} (D)}.v(Nrd(a)) ∈ Γ_{F} ⊗ ℚ,

where Γ_{F} is the value group of v. It is clear from the formula above that v = v̄|_{F}. The image v̄(D^{∗}) is denoted by Γ_{D}. The valuation ring of (D, v̄) is the set D^{≥0} := {a ∈ D : v̄(a) ≥ 0}. It is easy to see that D^{≥0} is a ring and has a unique maximal two sided ideal which is given by D^{>0} := {a ∈ D : v̄(a) > 0}. The quotient D_{0} := D^{≥0}/D^{>0} is called the residue algebra of D at v̄ and the elements in D^{≥0}\D^{>0} are called units. It is evident that Γ_{F} is a subgroup of Γ_{D}. The index [Γ_{D} : Γ_{F}] is called the ramification index of v̄.

Let γ ∈ Γ_{D}. Define D^{≥γ} = {a ∈ D : v̄(a) ≥ γ} and D^{>γ} = {a ∈ D : v̄(a) > γ}. Then the associated graded ring of D is defined as:

grv̄(D) := ⨁_{γ∈Γ_{D}}(D^{≥γ}/D^{>γ}).

For a ∈ D^{∗}, we denote the image of a in D^{≥v̄(a)}/D^{>v̄(a)} by a′. It is easy to see that (ab)′ = a′b′ and grv̄(D) is a graded division ring in the sense that every homogeneous element in grv̄(D) is invertible.

Let V be a (right) vector space over D. By a (D, v̄)-value function (or simply a D-value function) on V with respect to v̄, we mean a map α : V → (Γ_{D} ⊗ ℚ) ∪ {∞} which satisfies:

(i) α(x) = ∞ if and only if x = 0.

(ii) α(xλ) = α(x) + v̄(λ) for all x ∈ V, λ ∈ D.

(iii) α(x + y) ≥ min{α(x), α(y)} for all x, y ∈ V.

The image α(V \{0}) is denoted by Γ_{V}. We remark that Γ_{V} need not be a subgroup of Γ_{D} ⊗ ℚ but a union of coset classes modulo Γ_{D}. Let Γ_{V} /Γ_{D} denote the set of these coset classes. If |Γ_{V} /Γ_{D}| > 1, then α is called ramified.

Example 2.1 Let τ be an involution on D and V be a finite dimensional (right) vector space over D with a basis {f_{i}}_{i=1}^{r}. Let γ_{1}, γ_{2}, ⋯, γ_{r} ∈ Γ_{D} ⊗ ℚ. Then the function α_{γ} defined by α_{γ} (∑_{i} f_{i}λ_{i}) = min_{i}{γ_{i} + v̄(λ_{i})} is a D-value function on V. □

A D-value function α is called a D- norm if there exists a D-basis {e_{i}} of V such that if x = ∑_{i} e_{i}λ_{i} for λ_{i} ∈ D^{∗}, then α(x) = min_{i}{α(e_{i})+v̄(λ_{i})}. Such a basis is called a splitting basis of V for α. In the example above, the value function α_{γ} is indeed a D-norm with {f_{i}} as a splitting basis. If W ⊆ V is a D-vector subspace then the restriction α|_{W} is a D-norm on W [RTW, Prop. 2.5].

2.2 Gauges on semisimple algebras

In this section, we define some value functions on F -algebras. We use these functions for more specific central simple algebras in later sections.

Let F be a field with a valuation v : F → Γ_{F} ∪ {∞} and A be an algebra over F. A surmultiplicative F -value function j on A with respect to v is an F -value function j : A → (Γ_{F} ⊗ℚ)∪{∞} on the F -vector space A such that j(1) = 0 and j(ab) ≥ j(a)+j(b) for all a, b ∈ A.

A surmultiplicative F -value function j defines a gradation on A as follows: Let Γ_{A} = j(A)\{0}. For each γ ∈ Γ_{A}, define A^{≥γ} = {a ∈ A : j(a) ≥ γ}, A^{>γ} = {a ∈ A : j(a) > γ} and A_{γ} = A^{≥γ}/A^{>γ}. Further define

gr_{j}(A) := ⨁_{γ∈Γ_{A}} A_{γ}.

The set Γ_{A} is the grade set of gr_{j}(A). Axioms for j imply the algebra structure on A induces a Γ_{A}-graded algebra structure on gr_{j}(A). For a ∈ A we denote the image of a in A_{j(a)} by a′. For a, b ∈ A, the induced multiplication on gr_{j}(A) is a′b′ = (ab)′. Therefore it follows that a′b′ = 0 if j(ab) > j(a) + j(b).

A surmultiplicative value function j on an F -algebra A (not necessarily central simple), is called an F -gauge if the graded algebra gr_{j}(A) is a graded semisimple algebra over gr_{v}(F) with [gr_{j}(A) : gr_{v}(F)] = [A : F]. The latter condition is equivalent to demand that j is an F -norm on the F -vector space A. We remark that if A is not semisimple then it does not carry a gauge.

Let Z(gr_{j}(A)) denote the center of gr_{j}(A) and Z(A) denote the center of A. We say that j is a tame F -gauge if Z(gr_{j}(A)) is separable over gr_{v}(F) and is equal to gr_{j}(Z(A)). In this paper, we are interested in the gauges over central simple algebras only.

Lemma 2.2 Let A be a central simple algebra over a valued field (F, v) with formally real residue field F_{0}. Let j be a F -gauge on A. Then gr_{j}(A) is a central simple graded algebra over gr_{v}(F). Hence j is a tame F -gauge.

Proof Since F_{0} is formally real, char(F_{0}) = 0 and thus every F -gauge on A is tame [TW1, Cor. 3.6]. Now it follows immediately from [TW1, Cor. 3.7] that gr_{j}(A) is a central simple graded algebra over gr_{v}(F). □

Let A be a central simple algebra over F and σ be an involution of the first kind on F. Let j be a surmultiplicative F -value function on A. We say that σ and j are compatible if j ∘σ = j. If j and σ are compatible then the map defined by σ′(x′) = σ(x)′, x ∈ A induces a graded involution σ′ on gr_{j}(A), also called residue of σ at j. The residue σ′ on gr_{j}(A) is anisotropic if there is no non-zero homogeneous element x′ ∈ gr_{j}(A) such that σ′(x′)x′ = 0 ∈ gr_{j}(A), and strongly anisotropic if ∑_{i} σ′(x′_{i})x′_{i} ≠ 0 for all homogeneous x′_{i} ∈ gr_{j}(A).

Proposition 2.3 [TW, Prop. 1.2] For A, σ as above and a surmultiplicative F -value function j on A, the following are equivalent:

(i) For every x ∈ A, j(σ(x)x) = 2j(x).

(ii) j and σ are compatible and the graded involution σ′ is anisotropic.

□

A surmultiplicative function j satisfying the equivalent conditions of Prop. 2.3 is called σ-special.

Theorem 2.4 [TW, Th. 5.1] Let F_{v}^{h} denote the henselisation of F at v. Then the following are equivalent:

(i) For a central simple algebra A over F with an involution σ, the involution σ⊗id is anisotropic over A ⊗_{F} F_{v}^{h}.

(ii) There exists a σ-special F -gauge j on A.

Further if these conditions hold, there is a unique gauge j satisfying (ii). □

We now mention two examples of gauges over central simple algebras.

Example 2.5 This example is rather a trivial one. On A = M_{r}(F), where r ≥ 1 is an integer, we define the gauge g_{r} : M_{r}(F) → Γ_{F} as follows:

g_{r} ([a_{ij}]) = min_{i,j}{v(a_{ij})}.

Proposition 2.6 Let t_{r} denote the transpose involution on M_{r}(F) and t̃_{r} denote the transpose involution on M_{r} (gr_{v}(F)). Let g_{r} be the gauge on M_{r}(F) as in example 2.5. Then we have the following:

(i) The gauge g_{r} is unramified and g_{r} ∘ t_{r} = g_{r}.

(ii) gr_{g_{r}} (M_{r}(F)) ≃ M_{r} (gr_{v}(F)).

(iii) Under the isomorphism gr_{g_{r}} (M_{r}(F)) ≃ M_{r} (gr_{v}(F)), the graded involution (t_{r})′ induced by g_{r} on gr_{g_{r}} (M_{r}(F)) is the transpose involution t̃_{r} on M_{r} (gr_{v}(F)).

(iv) If the residue field F_{0} is formally real then g_{r} is the t_{r}-special gauge.

Proof The assertion (i) is clear from the definition of g_{r}. To prove (ii), let γ ∈ Γ_{F} and a = [a_{ij}] ∈ M_{r}(F). Then g_{r}(a) > γ (resp. ≥ γ) if and only if v(a_{ij}) > γ (resp. ≥ γ) for all i, j. This suggests that a′ + M_{r}(F)_{γ} ↦ [a′_{ij} + F_{γ}] induces an isomorphism gr_{g_{r}} (M_{r}(F)) ≃ M_{r} (gr_{v}(F)). It is evident from this isomorphism that (t_{r})′ = t̃_{r} and (iii) follows. Now to show (iv), we have to prove that (t_{r})′ is anisotropic. In view of [TW, Prop. 1.6], it is enough to show that t_{r_{0}} = (t_{r})′|_{M_{r}(F)_{0}} is anisotropic. It follows from the isomorphism gr_{g_{r}} (M_{r}(F)) ≃ M_{r} (gr_{v}(F)) that M_{r}(F)_{0} ≃ M_{r}(F_{0}) and t_{r_{0}} corresponds to the transpose involution on M_{r}(F_{0}), which is anisotropic because F_{0} is formally real. This completes the proof. □

Example 2.7 Let D be a division algebra with a valuation v̄, V be a (right) vector space over D and α be a D-norm on V. Let A = End_{D}(V). For ϕ ∈ A, we set

j_{α}(ϕ) = min_{x∈V \{0}}{α(ϕ(x)) − α(x)}.

If {e_{i}}_{i=1}^{r} is a splitting basis of α and ϕ ∈ A then it follows from [TW1, Lemma 1.15] that j_{α}(ϕ) = min_{1≤i≤r}{α(ϕ(e_{i})) − α(e_{i})}. In fact the function j_{α} thus defined is a gauge on A [TW1, Lemma 1.16]. If {e_{i}} is an orthogonal basis of a hermitian form h over (D, τ) for some involution τ then the condition when j_{α} is ad_{h}-special is explained in §2.3. □

Let ϕ(e_{j}) = ∑_{i} e_{i}ϕ_{ij}, where ϕ_{ij} ∈ D^{∗}; then

j_{α}(ϕ) = min_{1≤i,j≤r}{α(e_{i}) + v̄(ϕ_{ij}) − α(e_{j})}.

This description of j_{α} allows us to conclude the following:

Proposition 2.8 With the notation as above, if α is ramified then j_{α} is ramified.

Proof Let {e_{i}} be a splitting basis of α. Since α is ramified, |Γ_{V} /Γ_{D}| ≥ 1 and by a permutation of splitting basis we may assume that α(e_{1}) ≠ α(e_{2}) ∈ Γ_{V} /Γ_{D}. We choose ϕ ∈ A = End_{D}(V) such that ϕ_{12} = 1 and ϕ_{ij} = 0 otherwise. Then j_{α}(ϕ) = α(e_{1}) − α(e_{2}) ∉ Γ_{D}. Thus j_{α}(ϕ) ∉ Γ_{F} and j_{α} is ramified. □

Lemma 2.9 [TW1, Lemma 1.10] The graded algebras gr_{j_{α}}(End_{D}(V)) and End_{gr(D)}(gr_{α}(V)) are gr(F)-isomorphic. □

We briefly explain the isomorphism in lemma 2.9. Let Γ = Γ_{D} ⊗ ℚ. Let 0 ≠ f ∈ End_{D}(V) and γ := j_{α}(f) ∈ Γ. Then for each δ ∈ Γ, f induces maps:

f̃_{δ} : V_{δ} → V_{δ+γ}

given by x′ ↦ f(x) + V^{>δ+γ}, where x ∈ V is such that α(x) = δ. The map f̃ = ⨁_{δ∈Γ} : gr_{α}(V) → gr_{α}(V) shifts graded components by γ and belongs to (End_{gr(D)}(gr_{α}(V))_{γ}. Further, if f_{1} ∈ (End_{D}(V))^{>γ}, then { \atop f + f_{1}} = f and this confirms that f′ ↦ f̃ is a well-defined graded homomorphism from gr_{j_{α}}(End_{D}(V)) to End_{gr(D)}(gr_{α}(M)). This is in fact an isomorphism [TW1, Lemma 1.10].

Let j_{1} and j_{2} be gauges on central simple F -algebras A_{1} and A_{2} respectively. Then we define the tensor product j_{1} ⊗ j_{2} as follows:

j_{1} ⊗ j_{2}(x) = sup { min_{1≤i≤n}{j_{1}(a_{i}) + j_{2}(b_{i})} : x = ∑_{i=1}^{n} a_{i} ⊗ b_{i}}  (1)

If {e_{i}} and {f_{k}} are splitting bases for j_{1} and j_{2} respectively, then {e_{i} ⊗ f_{k}} is a splitting basis for j_{1} ⊗ j_{2} and j_{1} ⊗ j_{2}(e_{i} ⊗ f_{k}) = j_{1}(e_{i}) + j_{2}(f_{k}). It follows from [TW1, Cor. 1.25] that j_{1} ⊗ j_{2} is a gauge and gr_{j_{1}⊗j_{2}}(A_{1} ⊗ A_{2}) = gr_{j_{1}}(A_{1}) ⊗ gr_{j_{2}}(A_{2}).

Proposition 2.10 Let (A_{1}, σ_{1}) and (A_{2}, σ_{2}) be two central simple algebras with involutions of first kind. Let j be a σ_{1} ⊗ σ_{2}-special gauge on A_{1} ⊗ A_{2}. Let j_{1} = j|_{A_{1}} and j_{2} = j|_{A_{2}}. Then j_{1} is σ_{1}-special gauge on A_{1} and j_{2} is σ_{2}-special gauge on A_{2}. Moreover, j = j_{1} ⊗ j_{2}.

Proof If j is a σ_{1} ⊗ σ_{2}-special gauge, then it is clear that j_{1} and j_{2} are surmultiplicative norms and satisfy the condition (i) of Prop. 2.3. Thus by [TW, Prop. 1.2], j_{1} is the σ_{1}-special gauge and j_{2} is the σ_{2}-special gauge. We now show that j = j_{1} ⊗ j_{2}. If {a_{i}} is a splitting basis for j_{1} and b_{i} ∈ A_{2} then

j (∑_{i} a_{i} ⊗ b_{i}) ≥ min_{i}(j(a_{i} ⊗ b_{i})) ≥ min_{i}(j_{1}(a_{i}) + j_{2}(b_{i})) = (j_{1} ⊗ j_{2}) (∑_{i} a_{i} ⊗ b_{i})

where a_{i} ∈ A_{1} and b_{i} ∈ A_{2}. Thus j ≥ j_{1} ⊗ j_{2}. Therefore, there is a graded ring isomorphism

gr_{j_{1}⊗j_{2}}(A_{1} ⊗ A_{2}) →^{ι} gr_{j}(A_{1} ⊗ A_{2})

By [TW1, Prop. 1.20], it follows that gr_{j_{1}⊗j_{2}}(A_{1} ⊗ A_{2}) ≃ gr_{j_{1}}(A_{1}) ⊗ gr_{j_{2}}(A_{2}). Since both gr_{j_{1}}(A_{1}) and gr_{j_{2}}(A_{2}) are graded central simple, j_{1} and j_{2} being gauges, so is gr_{j_{1}⊗j_{2}}(A_{1} ⊗A_{2}). If the map ι above is not injective then the non-zero kernel contradicts the fact that gr_{j_{1}⊗j_{2}}(A_{1} ⊗ A_{2}) is simple. Thus ι is injective and by [TW1, Prop. 1.21], j = j_{1} ⊗ j_{2}. □

Proposition 2.11 Let A be a central simple algebra over a valued field F with an involution σ of the first kind. Let K/F be a field extension and w be an extension of v to K. Suppose A ⊗_{F} K has a σ ⊗ id-special w-gauge g and let j = g|_{A}. Then j is σ-special v-gauge on A and g = j ⊗ w.

Proof Consider a henselization (K_{w}^{h}, w_{h}) of (K, w). Then K_{w}^{h} contains a henselization (F_{v}^{h}, v_{h}) of (F, v). The product g ⊗ w_{h} is the σ ⊗ id-special w_{h}-gauge on A ⊗_{K} K_{w}^{h}, hence g ⊗ w_{h} is the σ ⊗ id-special v_{h}-gauge on A ⊗ F_{v}^{h}. By [TW, Th. 5.1] it follows that g ⊗ w_{h}|_{A} is the σ-special v-gauge on A. Since j = g ⊗ w_{h}|_{A}, the first claim is proved. The fact that g = j ⊗ w follows from [TW, Prop. 4.3]. □

Let j be a surmultiplicative F -value function on a central simple algebra A over F. Let j be compatible with an involution σ of the first kind on A and σ′ be the induced graded involution on the graded algebra gr_{j}(A). We denote the restriction σ′|_{A_{0}} by σ_{0}. With this notation we have the following

Proposition 2.12 Let j be a surmultiplicative F -value function on a central simple algebra A with an involution σ of first kind. Then the following are equivalent:

(i) For all r ≥ 1 and all x ∈ A ⊗_{F} M_{r}(F), (j ⊗ g_{r})((σ ⊗ t_{r})(x)x) = 2(j ⊗ g_{r})(x).

(ii) j and σ are compatible and the graded involution σ′ is strongly anisotropic.

(iii) j and σ are compatible and the involution σ_{0} on A_{0} is strongly anisotropic.

Proof We first show (i) ⇔ (ii). With the notation as in example 2.5, (i) ⇔ (ii) follows immediately in view of Prop. 2.3 provided the graded involution (σ ⊗ t_{r})′ on the graded algebra gr_{j⊗g_{r}}(A ⊗_{F} M_{r}(F)) goes to σ′ ⊗ t̃_{r} under the isomorphism gr_{j⊗g_{r}}(A ⊗_{F} M_{r}(F)) ≃ gr_{j}(A) ⊗ gr_{g_{r}}(M_{r}(F)) ≃ gr_{j}(A) ⊗ M_{r}(gr_{v}(F)). Let x ∈ A and a = [a_{ij}] ∈ M_{r}(F) with g_{r}(a) = γ ∈ Γ_{F}. Then (σ ⊗ t_{r})′((x ⊗ a)′) = (σ ⊗ t_{r})′(x′ ⊗ [a_{ij}]′) = σ(x)′ ⊗ [a_{ji} + F_{γ}] = σ(x)′ ⊗ t̃_{r}([a_{ij} + F_{γ}]) = (σ′ ⊗ t̃_{r})(x′ ⊗ [a_{ij}]′), where equality is in fact the identification in the three isomorphic graded algebras above. This suggests that (σ ⊗ t_{r})′ = σ′ ⊗ t̃_{r} indeed, and (i) ⇔ (ii) holds.

We now show (ii) ⇔ (iii). The implication (ii) ⇒ (iii) is trivial. To prove the other implication, let σ′ be weakly isotropic. Let g_{r} be the gauge on M_{r}(F) as in example 2.5. Since σ′ is weakly isotropic, it follows that there exists r ≥ 1 such that σ′⊗t̃_{r} is isotropic. As in the proof of (ii) ⇔ (iii) above, we conclude that σ′ ⊗ t̃_{r} = (σ ⊗ t_{r})′ and hence (σ ⊗ t_{r})′ is isotropic on gr_{j⊗g_{r}}(A ⊗ M_{r}(F)). Now by [TW, Prop. 1.6], the involution (σ ⊗ t_{r})_{0} is isotropic. Let {a_{i}} be a splitting basis of A for j and {e_{kl}} be the standard basis of M_{r}(F), then

j ⊗ g_{r} (∑ a_{i} ⊗ e_{kl}) = min_{i,k,l}{j(a_{i}) + g_{r}(e_{kl})} = min_{i}{j(a_{i})}

This suggests that (A ⊗ M_{r}(F))_{0} = A_{0} ⊗ M_{r}(F_{0}) and (σ ⊗ t_{r})_{0} = σ_{0} ⊗ t_{r} and it follows that σ_{0} is weakly isotropic. This completes the proof. □

A surmultiplicative F -value function j satisfying the above equivalent conditions is called strongly σ-special.

Corollary 2.13 Let F_{v}^{h} denote the henselisation of F at v. Then the following are equivalent:

(i) For a central simple algebra A over F with an involution σ, the involution σ ⊗ id is strongly anisotropic over A ⊗_{F} F_{v}^{h}.

(ii) There exists a strongly σ-special F -gauge j on A.

Further if these conditions hold, there is a unique gauge j satisfying (ii). □

Proof We prove (i) ⇒ (ii), the other implication is just tracing the proof backwards. Let (A, σ)⊗_{F} F_{v}^{h} be strongly anisotropic. Then for all r ≥ 1, (A⊗_{F} M_{r}(F), σ⊗t_{r})⊗_{F} F_{v}^{h} is anisotropic. Now by Theorem 2.4, there exists a σ⊗t_{r}-special gauge j_{r} on A⊗M_{r}(F). It is clear from Proposition 2.10 that j = j_{r}|_{A} is σ-special and j_{r}|_{M_{r}(F)} is t_{r}-special. Thus by example 2.5 and the uniqueness of t_{r}-special gauges, j_{r}|_{M_{r}(F)} = g_{r} and by Proposition 2.10, j_{r} = j ⊗ g_{r}. Now as in the proof of 2.12, (σ ⊗ t_{r})′ = σ′ ⊗ t̃_{r} and the anisotropicity of (σ ⊗ t_{r})′ confirms the strong anisotropicity of σ′. Further since j is σ-special, it follows from the unicity part of Theorem 2.4 that j is the unique gauge satisfying (ii). □

We conclude the section with the following analogue of Proposition 2.10 for strongly special gauges.

Proposition 2.14 Let {(A_{i}, σ_{i})}_{i=1}^{r} be central simple algebras with involutions of first kind. Let j be a strongly ⊗_{i=1}^{r}σ_{i}-special gauge on ⊗_{i=1}^{r}A_{i}. Let j_{i} = j|_{A_{i}}. Then j_{i} is strongly σ_{i}-special gauge on A_{i}. Moreover, j = ⊗_{i=1}^{r}j_{i}.

The proof is a repeated imitation of the proof of Proposition 2.10 where the use of Proposition 2.3 is replaced with the use of Proposition 2.12.

2.3 Hermitian forms and compatible norms

For a D-value function α on V and a hermitian form h : V × V → D on (D, τ), we write that α ≺ h if for all x, y ∈ V, α(x) + α(y) ≤ v̄(h(x, y)). Further, we write that α ⪯ h if α ≺ h and for each x_{0} ∈ V, there exists y_{0} ∈ V such that α(x_{0}) + α(y_{0}) = v̄(h(x_{0}, y_{0})). Let α be a D-norm such that α ≺ h. We define the dual norm α^{∗} on V as:

α^{∗}(x) = inf_{y∈V \{0}}{v̄(h(x, y)) − α(y)}.

In fact, if {e_{i}} is a splitting basis for α then the dual basis {e_{i}^{∗}} given by h(e_{i}^{∗}, e_{j}) = δ_{ij} is a splitting basis for α^{∗}. Further, α^{∗}(e_{i}^{∗}) = −α(e_{i}), α^{∗∗} = α and α = α^{∗} if and only if α ⪯ h.

The D-value functions on vector spaces too yield some graded objects. For δ ∈ Γ_{V}, denote V^{≥δ} := {x ∈ V : α(x) ≥ δ} and V^{>δ} := {x ∈ V : α(x) > δ}. The quotient V_{δ} := V^{≥δ}/V^{>δ} is called the residue of (V, α) at δ. In fact, V_{δ} is a vector space over the residue algebra D_{0} of D at v̄. Further, the direct sum gr_{α}(V) = ⊕_{δ∈Γ_{V}} V_{δ} is a graded vector space over the graded algebra grv̄(D). The set {δ ∈ Γ_{V} : V_{δ} ≠ 0} is called the grade set of gr_{α}(V). The image of x ∈ V in V_{α(x)} is denoted by x′.

Let α be a value function on a vector space V over D. Let h be a hermitian form over (D, τ) such that α ⪯ h. For x, y ∈ V we define:

h′_{α}(x, y) = \begin{cases}h(x, y)′ & if v̄(h(x, y)) = α(x) + α(y), \\ 0 & if v̄(h(x, y)) > α(x) + α(y). \end{cases}

The map h′_{α} can be extended bi-additively to gr_{α}(V). In fact this extension (also denoted by h′_{α}) is a hermitian form on gr_{α}(V) for the graded involution τ′ on grv̄(D). The form h′_{α} has a substantial information about the isotropicity of h, as reflects from the following:

Theorem 2.15 [RTW, Th. 3.11] Let char(D_{0}) ≠ 2. Then h ↦ h′_{α} gives a surjection Θ of the Witt group W (D, τ) into the graded Witt group W_{g}(grv̄(D), τ′). Further if v is henselian, then Θ is an isomorphism.

Let h : V × V → D be a hermitian form over (D, τ) and α be a D-norm on V such that α ≺ h. The following proposition highlights a relation between j_{α} and j_{α^{∗}}, where α^{∗} is the dual of α.

Proposition 2.16 [TW, Prop. 1.7] If ad_{h} is the involution on End_{D}(V) which is adjoint to the hermitian form h then for all ϕ ∈ End_{D}(V) we have j_{α^{∗}}(ad_{h}(ϕ)) = j_{α}(ϕ).

As a consequence of above proposition one can derive the following

Corollary 2.17 [TW, Prop. 1.7(iii)] If α is a D-norm on V such that j_{α} ∘ ad_{h} = j_{α} then there exists a constant γ ∈ \frac{1}{2}Γ_{V} such that α − γ ⪯ h.

Let V be a D-vector space, α be a D-norm on V and h be a hermitian form over (D, τ) such that α ⪯ h. Let j_{α} be the gauge on End_{D}(V) as in example 2.7. Then we have the following:

Lemma 2.18 Let h′_{α} be the graded hermitian form over gr_{α}(V) and ad_{h′_{α}} be the adjoint involution on End_{gr(D)}(gr_{α}(V)). Further, let ad_{h} be the adjoint involution on End_{D}(V) and (ad_{h})′ be the graded involution on gr_{j_{α}}(End_{D}(V)). Then under the isomorphism of Lemma 2.9, ad_{h′_{α}} = (ad_{h})′.

Proof Let f ∈ End_{D}(V). With the notation as in Lemma 2.9, it suffices to show that ad_{h′_{α}}(f̃) = { \atop (ad_{h})′(f′)}. Let x, y ∈ V. Then using the isomorphism f′ ↦ f̃ we have

h′_{α} ({ \atop (ad_{h})′(f′)}(x′), y′) = h′_{α} ((ad_{h}(f)′(x′), y′)
= (h(ad_{h}(f(x), y)))′
= (h(x, f(y)))′
= h′_{α} (x′, f̃(y′))
= h′_{α} (ad_{h′_{α}}(f̃)(x′), y′)

Thus h′_{α} ({ \atop (ad_{h})′(f′)}(x′), y′) = h′_{α} (ad_{h′_{α}}(f̃)(x′), y′), which implies ad_{h′_{α}}(f̃) = { \atop (ad_{h})′(f′)}. □

We say that a gauge j over an algebra A over F is residually real if the residue field of the valuation j|_{F} = v on F is formally real.

2.4 Consequences of weak isotropicity

We now prove the implication (BP1) ⇒ (BP2) of BP-equivalence. As usual, F denotes a formally real field with valuation v. We begin with the following:

Proposition 2.19 Let (A, σ) be a central simple algebra with an involution σ of the first kind. If σ is weakly isotropic then it is totally indefinite.

Proof Let P be an ordering on F and F_{P} be the real closure of F at P. Let D_{P} be the division algebra Brauer equivalent to A_{P} := A ⊗_{F} F_{P}. Then the involution σ ⊗ id is adjoint to a hermitian form h_{P} : V × V → D_{P} over (D_{P} , τ_{P}), where τ_{P} is an involution on D_{P} of the same type as σ.

First assume that σ is orthogonal. If D_{P} ≠ F_{P} then by [KMRT, Cor. 11.11(1)(a)], sgn_{P} (σ) = 0 < deg(A), otherwise h_{P} corresponds to a weakly isotropic quadratic form over F_{P} and therefore h_{P} is indefinite. Thus by [KMRT, Cor. 11.11(2)(a)] sgn_{P} (σ) = | sgn_{P} (h_{P})| < deg_{F_{P}} (A_{P}) = deg(A). This confirms that σ is totally indefinite.

Now we finish the proof for the case when σ is symplectic. If the algebra D_{P} is split then by [KMRT, Cor. 11.11(2)(b)] sgn_{P} (σ) = 0 < deg(A). If D_{P} is not split then it is the quaternion division algebra over F_{P}. Let t(h_{P}) be the quadratic form over F_{P} defined by t(h_{P})(x) = h_{P} (x, x), x ∈ V. Then dim_{F_{P}} (t(h_{P})) = dim_{F_{P}} (D_{P}).(dim_{D_{P}} (V)) = 4.(\frac{1}{2} deg_{F_{P}} (A_{P})) = 2 deg_{F_{P}} (A_{P}) = 2 deg(A). Further, the weak isotropicity of σ implies the weak isotropicity of t(h_{P}) and thus by [KMRT, Cor. 11.11(1)(b)] we have sgn_{P} (σ) = \frac{1}{2}| sgn_{P} (t(h_{P})| < \frac{1}{2} dim(t(h_{P})) = deg(A). This establishes the total indefiniteness of σ. □

Now we are ready to prove the implication (BP1) ⇒ (BP2) of BP-equivalence.

Proposition 2.20 Let F be a formally real field and A be a central simple algebra over F. Let σ be an involution of first kind on A. If σ is weakly isotropic then it is totally indefinite and at every residually real ramified gauge j on A which is compatible with σ, the residue σ′ is weakly isotropic.

Proof That σ is weakly isotropic implies it is totally indefinite follows from Prop. 2.19. Let j be a residually real ramified gauge on A which is compatible with σ. If σ′ is not weakly isotropic then by Theorem 2.13, (A ⊗_{F} F_{v}^{h}, σ ⊗ id) is strongly anisotropic, which is a contradiction to the hypothesis that σ is weakly isotropic. This completes the proof. □

3 The classical Bröcker-Prestel Theorem

This section is devoted to a brief introduction of the theorem of Bröcker and Prestel. For a detailed account of some parts of the section, one can refer to [S, Chapter 3, §7].

A quadratic semiordering on F is a subset P ⊆ F such that: P + P ⊆ P, F^{2}P ⊆ P, 1 ∈ P, P ∪ −P = F and P ∩ −P = {0}. A quadratic semiordering P ⊆ F is an ordering if P.P ⊆ P. A quadratic semiordering is P is called archimedian if for every a ∈ F, there exists n ∈ ℕ (depending on a) such that n − a ∈ P.

Quadratic semiorderings connect orderings and valuations in the sense that archimedian quadratic semiorderings are in fact orderings, while the non-archimedian ones give rise to a valuation on F. A quadratic form is weakly isotropic if and only if it is indefinite with respect to all quadratic semiorderings [S, Th. 7.6]. Carefully looking at archimedian and non-archimedian quadratic semiorderings, many technical arguments yield the Theorem 1.1 as stated in the introduction.

We translate Theorem 1.1 in the language of value functions defined in the section 2.

Theorem 3.1 Let F be a formally real field and V be a vector space over F. Let q be a quadratic form on V. Let v be a valuation on F with formally real residue field F_{0}. Then we have the following:

1. The quadratic form q is ramified at v if and only if there exists a ramified gauge j on End_{F} (V) which is compatible with ad_{q}.

2. Let j be a gauge on End_{F} (V) which is compatible with ad_{q}. Let (ad_{q})′ be the graded involution with respect to j. Then the residues of q at v are strongly anisotropic over F_{0} if and only if the involution (ad_{q})′ is strongly anisotropic.

Proof We first prove 1. Let q be ramified. Let {e_{i}} be an orthogonal basis of q and a_{i} = q(e_{i}) ∈ F^{∗}. Since q is ramified at v, by a permutation of basis, we may assume that v(a_{1}) ≠ v(a_{2}) ∈ Γ_{F} /2Γ_{F}, that is v(a_{1}) − v(a_{2}) ∉ 2Γ_{F}. Let α = α_{q} be the D-norm on V defined by α(e_{i}) = \frac{1}{2}v(a_{i}). It is clear that α is ramified and α ⪯ q. Let j_{α} be the gauge on End_{F} (V) as defined in 2.7. Since α is ramified, by Prop. 2.8 it follows that j_{α} is ramified. That j_{α} satisfies j_{α} ∘ ad_{q} = j_{α} follows from Proposition 2.16 in view of the fact that α ⪯ q if and only if α = α^{∗}.

Conversely, suppose j is a ramified gauge on End_{F} (V) such that j ∘ ad_{q} = j. By [TW1, Cor. 1.15], j ⊗ v^{h} is a gauge on End_{F_{v}^{h}}(V ⊗_{F} F_{v}^{h}), where v^{h} is the extension of v to the henselisation F_{v}^{h}. Further by [TW1, Prop. 3.3], there exists a F_{v}^{h}-norm α on V ⊗_{F} F_{v}^{h} such that j ⊗ v^{h} = j_{α} where j_{α} is as in the example 2.7. From the hypothesis j ∘ ad_{q} = j, it follows immediately that j_{α} ∘ ad_{q^{h}} = j_{α}. In view of Corollary 2.17 we may choose α such that α ⪯ h. Further, under the composed map W (F) → W (F_{v}^{h}) → W_{gr}(gr_{v}(F)) the class [q] ∈ W (F) of the quadratic form q goes to [(q_{F_{v}^{h}})′_{α}]. Since j is ramified, j ⊗ v^{h} = j_{α} is ramified. From 2.7, it is clear that α is ramified. This, together with the definition of the bilinear form (q_{F_{v}^{h}})′_{α} implies that (q_{F_{v}^{h}})′_{α} has more than one non-zero component. It follows from isomorphisms of [RTW, Prop. 1.5] that W_{gr}(gr_{v}(F)) ≃ ⨁_{Γ_{F} /2Γ_{F}} W (F_{0}) and that under this isomorphism the graded form (q_{F_{v}^{h}})′_{α} corresponds to various residues of q at v. In view of this and the deduction above that (q_{F_{v}^{h}})′_{α} has more than one non-zero component, it follows that q is ramified at v.

We now prove 2. By [TW1, Prop 3.3], j ⊗ F_{v}^{h} = j_{α} for some F_{v}^{h}-norm α on V ⊗ F_{v}^{h} such that α ⪯ q ⊗ F_{v}^{h}. First suppose that all residues of q are strongly anisotropic. Then by Springer’s theorem, q is strongly anisotropic. In view of this and [TW1, Prop. 4.3, Th. 4.6], the graded bilinear form q′_{α} on the graded vector space gr_{α}(V ⊗_{F} F_{v}^{h}) is strongly anisotropic. Thus the adjoint graded involution ad_{q′_{α}} is strongly anisotropic over the graded algebra End_{gr(F_{v}^{h})}(gr_{α}(V ⊗_{F} F_{v}^{h})). By [TW1, Lemma 1.10], we have an isomorphism

gr_{j_{α}}(End_{F_{v}^{h}}(V ⊗_{F} F_{v}^{h})) ≃ End_{gr(F_{v}^{h})}(gr_{α}(V ⊗_{F} F_{v}^{h})).

By Lemma 2.18, the graded involution ad_{q′_{α}} on the graded algebra End_{gr(F_{v}^{h})}(gr_{α}(V ⊗_{F} F_{v}^{h})) corresponds to the graded involution (ad_{q})′ on gr_{j_{α}}(End_{F_{v}^{h}}(V ⊗_{F} F_{v}^{h})) ≃ gr_{j}(End_{F} (V)) under the isomorphism above. Thus (ad_{q})′ is strongly anisotropic. The converse holds by tracing back the same arguments. □

Corollary 3.2 Let (V, q) be a quadratic form over a formally real field F and ad_{q} be the involution on End_{F} (V) which is adjoint to q. Then BP-equivalence holds for (End_{F} (V), ad_{q}).

Proof The implication (BP1) ⇒ (BP2) of BP-equivalence follows from Prop. 2.20. To prove (BP2) ⇒ (BP1), we assume that ad_{q} is strongly anisotropic and totally indefinite. By the classical Bröcker-Prestel Theorem (Theorem 1.1) there exists a valuation v on F with formally real residue field such that q is ramified at v and all residues of q at v are strongly anisotropic. Thus by Theorem 3.1, there exists a ramified gauge j on End_{F} (V) which is residually real and strongly ad_{q}-special. This is a contradiction to the hypothesis (ii), and this completes the proof. □

4 Symplectic case

Let (H, −) be a quaternion algebra with standard involution over F and h : V × V → D be a hermitian form over (H, −). Let ad_{h} denote the adjoint involution on End_{H}(V). The main result of this section is the following

Theorem 4.1 Let A be a central simple algebra over a formally real field F. Let σ be an involution of symplectic type on A and index(A) = 2. Then BP-equivalence holds for (A, σ).

Proof The implication (BP1) ⇒ (BP2) of BP-equivalence follows from Prop. 2.20. We keep the same notation as above. To prove (ii) ⇒ (i), assuming that the involution ad_{h} is strongly anisotropic and totally indefinite, we show the existence of a residually real ramified gauge j which is strongly ad_{h}-special.

To the hermitian form h over (H, −), we associate the quadratic form t(h) : V → F given by t(h)(x) = h(x, x). Let H = (a, b)_{F}; a, b ∈ F^{∗}. Let {e_{i}} be an orthogonal basis of V and h(e_{i}, e_{i}) = a_{i}. Since h(e_{i}, e_{i}) are all symmetric, each a_{i} ∈ F^{∗} and the hermitian form h is represented by the diagonal ⟨a_{1}, a_{2}, ⋯, a_{r}⟩; a_{i} ∈ F^{∗}. Further, q_{h} ≃ ⟨1, −a, −b, ab⟩ ⊗ ⟨a_{1}, a_{2}, ⋯, a_{r}⟩.

Since h is strongly anisotropic and totally indefinite, by Theorem 1.1, there exists a valuation v on F with respect to which q_{h} has at least two residues and all of them are strongly anisotropic. Thus by Springer’s theorem, the quadratic form q_{h} is strongly anisotropic over the henselisation F_{v}^{h} of F with respect to v. Since ⟨1, −a, −b, ab⟩ is a subform of q_{h}, it follows that the algebra H_{v} = H ⊗_{F} F_{v}^{h} is division. Thus by [M, Th. 1.2], the valuation v : F → Γ_{F} ∪ {∞} on F extends to a valuation on H ⊗_{F} F_{v}^{h}. We denote this valuation on H_{h} = H ⊗_{F} F_{v}^{h} by v̄. Now, with {e_{i} ⊗ 1} as a splitting basis we define a H_{h}-norm α : V ⊗_{F} F_{v}^{h} → Γ_{F} ⊗ ℚ ∪ {∞} as follows:

α (∑_{i}(e_{i} ⊗ λ_{i})) = min_{i}\{\frac{1}{2}v̄(h(e_{i}, e_{i})) + v̄(λ_{i})\}

It is clear that α ⪯ h ⊗ F_{v}^{h}. Further, that α is ramified is immediate by the definition and the fact that q_{h} ⊗ F_{v}^{h} is ramified. Since h ⊗ F_{v}^{h} is strongly anisotropic, it follows from [RTW, Prop 4.3 and Prop 4.6] that (h ⊗ F_{v}^{h})′_{α} is strongly anisotropic. Therefore by [RTW, Prop. 4.2], α(x) = \frac{1}{2} v̄(h(x, x)) for all x ∈ V ⊗_{F} F_{h}. Now by the proofs of [TW, Th. 2.1] and Cor. 2.13, the gauge j_{α} on End_{H}(V) ⊗_{F} F_{v}^{h} is strongly ad_{h⊗F_{v}^{h}}-special. Let j = j_{α}|_{End_{H}(V)}. Then j ⊗ v_{h} is strongly ad_{h}⊗id-special gauge on End_{H}(V)⊗_{F} F_{v}^{h} and by the unicity statement in 2.13, j_{α} = j⊗v_{h}. Now it follows from Prop. 2.14 that j is strongly ad_{h}-special, and hence the residue (ad_{h})′ of ad_{h} at j is strongly anisotropic. □

5 Orthogonal case

Let h be a hermitian form over a quaternion algebra H with an orthogonal involution τ. Let ad_{h} be the adjoint involution on End_{H}(V). Then ad_{h} is an involution of orthogonal type and every involution on End_{H}(V) is obtained in this way. In this section we prove Theorem 1.2 for (End_{H}(V), ad_{h}).

Let C denote the associated conic of the quaternion algebra H and K = F (C) denote the function field of C. If H = (a, b)_{F} then K = F (C) = \frac{F [u,v]}{(au^{2}+bv^{2}−1)} and H ⊗_{F} K is split. Further, the hermitian form h_{K} = h ⊗ K corresponds by Morita theory to a quadratic form q over K. We recall the following:

Theorem 5.1 [PSS] With the notation as above, if h is (strongly) anisotropic over (H, σ) then q is (strongly) anisotropic over K.

Remark Let h be totally indefinite. Then the hermitian form h_{K} is also totally indefinite, because h_{K} is defined over F and is totally indefinite at F. Thus the quadratic form q is totally indefinite over K as well.

Lemma 5.2 A gauge j on a central simple F -algebra A is ramified if and only if [A_{0} : F_{0}] < [A : F].

Proof This follows from the dimension computation of [gr_{j}(A) : gr(F)]. Let the cosets of Γ_{A} modulo Γ_{F} be represented by γ_{A} = {γ_{i}} ⊆ Γ_{A}. We choose γ_{A} so that 0 ∈ γ_{A}. Now by [TW, §2, eq. (2.4)] we have:

[gr_{j}(A) : gr_{v}(F)] = ∑_{γ_{i}∈γ_{A}}[A_{γ_{i}} : F_{0}]

By definition j is ramified if and only if |γ_{A}| ≥ 2, which in view of the above equation is equivalent to [A_{0} : F_{0}] < [gr_{j}(A) : gr_{v}(F)]. Since j is a gauge, we have [gr_{j}(A) : gr_{v}(F)] = [A : F] and the lemma follows immediately. □

Lemma 5.3 Let A be a central simple algebra over F and K/F be a field extension. Let j be a gauge on A and w be any extension of valuation v = j|_{F} to K. If j ⊗ w is ramified on A ⊗_{F} K then j is ramified on A.

Proof Let A_{0} denote the residue {a ∈ A : j(a) ≥ 0}/{a ∈ A : j(a) > 0} and (A ⊗_{F} K)_{0} denote the residue {b ∈ A ⊗_{F} K : (j ⊗ w)(b) ≥ 0}/{b ∈ A ⊗_{F} K : (j ⊗ w)(b) > 0}. Since j ⊗ w is ramified, by Lemma 5.2 we have

[A ⊗_{F} K : K] > [(A ⊗_{F} K)_{0} : K_{0}]

We now observe that A_{0} ⊗_{F_{0}} K_{0} ⊆ (A ⊗_{F} K)_{0}. Thus

[A : F] = [A ⊗_{F} K : K] > [(A ⊗_{F} K)_{0} : K_{0}] ≥ [A_{0} ⊗_{F_{0}} K_{0} : K_{0}] = [A_{0} : F_{0}]

and hence [A : F] > [A_{0} : F_{0}]. This in view of Lemma 5.2 confirms that j is ramified. □

Theorem 5.4 Let A be a central simple algebra over a formally real field F. Let σ be an involution of orthogonal type on A and index(A) = 2. Then BP-equivalence holds for (A, σ).

Proof The proof of the implication (BP1) ⇒ (BP2) of BP-equivalence follows from Proposition 2.20. We prove the other implication (using the same notation as above). Let ad_{h} be totally indefinite and strongly anisotropic. It is clear from the remark above and Theorem 5.1 that the involution ad_{h} ⊗ id on End_{H}(V) ⊗_{F} K is totally indefinite and strongly anisotropic. Thus by Theorem 3.2 there exists valuation w on K and a ramified w-gauge g on (End_{H}(V), ad_{h}) ⊗_{F} K which has formally real residue field and is strongly (ad_{h} ⊗ id)-special. It is clear from Theorem 2.13 that (End_{H}(V), ad_{h}) ⊗_{F} K_{w}^{h} is strongly anisotropic, where K_{w}^{h} denotes the henselisation of K at the valuation w. Let v = w|_{F}. Then v is a valuation on F with F_{v}^{h} ⊆ K_{w}^{h} and clearly (End_{H}(V), ad_{h}) ⊗_{F} F_{v} is strongly anisotropic.

Now again by Theorem 2.13, it follows that there exists a strongly ad_{h}-special gauge on j on End_{H}(V). Further, from the proof of Theorem 2.4 and its unicity statement, it follows that j = g|_{End_{H}(V)} and by Prop. 2.11, g = j ⊗ w. Now by Lemma 5.3, we conclude that j is ramified. Thus j is a ramified gauge on End_{H}(V) which has formally real residue field and the residue ad′_{h} is strongly anisotropic. □

6 Trace forms and involutions

It was pointed out by Unger in his thesis that there are central simple algebras A with involution σ which satisfy the following: ℭ:‘σ is weakly isotropic if and only if its trace form T_{σ} is weakly isotropic’. Such examples include multiquaternion algebras with totally decomposable involutions of first kind. For the algebras with involutions (A, σ) belonging to this class, we prove the following:

Theorem 6.1 If (A, σ) satisfies ℭ then BP-equivalence holds for (A, σ).

Proof The implication (BP1) ⇒ (BP2) follows from Theorem 2.20. For the rest, we first make a claim: Claim: Let (A, σ) be strongly anisotropic and totally indefinite. Then A carries a residually real ramified gauge which is strongly σ-special. Proof of the claim: The trace form T_{σ} of σ is adjoint to the involution σ⊗σ on A⊗_{F} A ≃ End_{F} (A). Since σ is totally indefinite and sgn(σ ⊗ σ) = sgn(σ)^{2}, it follows that σ ⊗ σ is totally indefinite. Further, since (A, σ) satisfies ℭ, the form T_{σ} is strongly anisotropic. Thus, as in the proof of 3.1, there exists a ramified gauge g on A ⊗ A which is strongly σ ⊗ σ-special.

Let A be embedded into A ⊗_{F} A via embeddings a ↦ a ⊗ 1 and a ↦ 1 ⊗ a. Through these embedding, we restrict g to functions j_{1} = g|_{A⊗1} and j_{2} = g|_{1⊗A}, respectively. It follows from Prop. 2.10 that j = j_{1} ⊗ j_{2}. Now that one of j_{1} and j_{2} is ramified follows from [§2.2,(1)]:

g(x) = j_{1} ⊗ j_{2}(x) = sup{ min_{1≤i≤r} (j_{1}(a_{i}) + j_{2}(b_{i})) : x = ∑_{i=1}^{r} a_{i} ⊗ b_{i}}

and the fact that g is ramified.

We now prove (ii) ⇒ (i). Let (A, σ) be strongly anisotropic and totally indefinite. By the above discussion, there exists a ramified gauge j on A which is strongly σ-special and has formally real residue field. We consider the residue of σ at j. Now by Theorem 2.13 this residue σ′ is strongly anisotropic. □

7 An induction step

Lemma 7.1 Let (A, σ) be a central simple algebra with an involution of first kind and (H, θ) be a quaternion algebra with an involution of first kind, then (A ⊗_{F} H, σ ⊗ θ) is strongly anisotropic if and only if (A ⊗_{F} H ⊗_{F} H, σ ⊗ θ ⊗ θ) ≃ (A ⊗ End_{F} (H), σ ⊗ ad_{T_{θ}}) is strongly anisotropic.

Proof It is clear that if (A ⊗_{F} H, σ ⊗ θ) is weakly isotropic, then (A ⊗_{F} H ⊗_{F} H, σ ⊗ θ ⊗ θ) is so. For the converse, assume that (A ⊗_{F} H ⊗_{F} H, σ ⊗ θ ⊗ θ) is weakly isotropic. We make the following identification:

A ⊗_{F} H ⊗_{F} H = A ⊗_{F} End_{F} (H) ≃ End_{A}(H ⊗ A)

via a ⊗ f ↦ f_{a}, where f_{a}.(u ⊗ b) = f(u) ⊗ ab. Via this identification, let σ ⊗ θ ⊗ θ = σ ⊗ ad_{T_{θ}} correspond to an involution τ on End_{A}(H ⊗ A). It follows that τ = ad_{h} for a weakly isotropic hermitian form h : (H ⊗ A) × (H ⊗ A) → A over (A, σ). Thus ad_{h} = σ ⊗ ad_{T_{θ}}. We claim that the restriction h|_{H⊗1} : (H ⊗ 1) × (H ⊗ 1) → A takes values in F. Indeed, if a ∈ A and x, y ∈ H are arbitrary, then we have:

ah(x ⊗ 1, y ⊗ 1) = h(x ⊗ σ(a), y ⊗ 1)
= h((σ(a) ⊗ id).(x ⊗ 1), y ⊗ 1)
= h(x ⊗ 1, ad_{h}(σ(a) ⊗ id)(y ⊗ 1))
= h(x ⊗ 1, σ ⊗ ad_{T_{θ}}(σ(a) ⊗ id)(y ⊗ 1))
= h(x ⊗ 1, (a ⊗ id).(y ⊗ 1))
= h(x ⊗ 1, y ⊗ a)
= h(x ⊗ 1, y ⊗ 1)a

Thus h(x ⊗ 1, y ⊗ 1) ∈ Z(A) = F. Further, for an arbitrary f ∈ End_{F} (H) we have

h|_{H⊗1}((1 ⊗ f)(x ⊗ 1, y ⊗ 1)) = h|_{H⊗1}(x ⊗ 1, (1 ⊗ T_{θ}(f))(y ⊗ 1))

and it follows that ad_{T_{θ}} = ad_{h}|_{End_{F} (H)} = ad_{h|_{H⊗1}}. Thus h|_{H⊗1} and T_{θ} are similar forms. Now let {1 = i_{0}, i_{1}, i_{2}, i_{3}} be a quaternion basis of H such that θ(i_{k}) = ±i_{k} for 1 ≤ k ≤ 4. Let x = ∑_{r} i_{r} ⊗ a_{r}, where a_{0}, a_{1}, a_{2}, a_{3} ∈ A are arbitrary. Then (upto a scalar factor) we have:

h(x, x) = ∑_{r,s} σ(a_{r})h(i_{r} ⊗ 1, i_{s} ⊗ 1)a_{s}

= ∑_{r,s} σ(a_{r})T_{θ}(i_{r}, i_{s})a_{s}

Since T_{θ}(i_{r}, i_{s}) = Trd_{H}(θ(i_{r})i_{s}), we get

h(x, x) = 2σ(a_{0})a_{0} + 2θ(i_{1})i_{1}σ(a_{1})a_{1} + 2θ(i_{2})i_{2}σ(a_{2})a_{2} + 2θ(i_{3})i_{3}σ(a_{3})a_{3}
= 2((σ ⊗ θ)(a_{0} ⊗ 1))(a_{0} ⊗ 1) + 2((σ ⊗ θ)(a_{1} ⊗ i_{1}))(a_{1} ⊗ i_{1})
+ 2((σ ⊗ θ)(a_{2} ⊗ i_{2}))(a_{2} ⊗ i_{2}) + 2((σ ⊗ θ)(a_{3} ⊗ i_{3}))(a_{3} ⊗ i_{3})

Since h is weakly isotropic, ∑_{i} h(x_{i}, x_{i}) = 0 for some x_{i} ∈ H ⊗ A. From the above computation it is clear that σ ⊗ θ is weakly isotropic. This completes the proof. □

Theorem 7.2 If BP-equivalence holds for all central simple algebras of index n with an involution of the first kind then it holds for every (A ⊗ H, σ ⊗ θ) where (A, σ) is an index n central simple algebra with σ of the first kind and (H, θ) is a quaternion algebra with θ of the first kind.

Proof Let (A, σ) be an index n central simple algebra with σ of the first kind and (H, θ) be a quaternion algebra with θ of the first kind. We prove that (A ⊗ H, σ ⊗ θ) satisfies BP-equivalence. The implication (i) ⇒ (ii) of BP-equivalence holds by Theorem 2.20. To prove (ii) ⇒ (i), we assume that (A ⊗ H, σ ⊗ θ) is strongly anisotropic and totally indefinite and prove the existence of a residually real and ramified strongly σ ⊗ θ-special gauge. It is clear that (A ⊗ H ⊗ H, σ ⊗ θ ⊗ θ) is totally indefinite and in view of Lemma 7.1, it is strongly anisotropic as well. Since index(A ⊗ H ⊗ H) = index(A) = n, the pair (A ⊗ H ⊗ H, σ ⊗ θ ⊗ θ) satisfies BP-equivalence by hypothesis and hence A ⊗ H ⊗ H carries a strongly σ ⊗ θ ⊗ θ-special gauge g. Let j = g|_{A⊗H}. Let A ⊗ H be embedded into A⊗H ⊗H via a⊗x ↦ a⊗x⊗1 and a⊗x ↦ a⊗1⊗H. Let j_{1} and j_{2} be restrictions of g to A ⊗ H via these embeddings respectively. By Prop. 2.14, j is strongly σ-special and j_{1} and j_{2} are strongly θ-special and j ⊗ j_{1} is strongly σ ⊗ θ-special. Thus by Prop. 2.13, j_{1} = j_{2}. Further by Prop. 2.14, g = j ⊗ j_{1} ⊗ j_{1}. From this and the fact that g is ramified, we conclude that j ⊗ j_{1} is ramified. □

Corollary 7.3 Let {(H_{i}, σ_{i})}_{i=1}^{r} be quaternion algebra over a formally real field with involutions of the first kind. Then (⊗_{i=1}^{r}H_{i}, ⊗_{i=1}^{r}σ_{i}) satisfies BP-equivalence.

Proof Since quaternion algebras with an involution of first kind satisfy BP-equivalence (Theorems 4.1 and 5.4), the corollary follows from Theorem 7.2. □

Remark Since a totally decomposible involution on a multiquaternion algebra is weakly isotropic if and only if its trace form is so [U, Th. 5.20], the corollary 7.3 follows from Theorem 6.1 as well. Also the symplectic case of index 2 (Theorem 4.1) follows from Theorem 7.2 and Theorem 3.2.

Acknowledgement I sincerely thank Jean-Pierre Tignol for many helpful discussions. He brought to my attention his recent work with Adrian Wadsworth on gauges which played key role in this paper. He patiently read many preliminary versions of this paper and gave various suggestions to improve it.

References

[B] Bröcker L., Zur Theorie der quadratischen Formen über formal reelle Körper, Math. Ann. 210(1974), pp 233 – 256.

[E] Ershov Yu.L.; Valued division rings, Fifth All Union Symposium, Theory of Rings, Algebras and Modules, Akad. Nauk. SSSR Sibirsk, Inst. Mat; Novosibirsk, pp 53 – 55.

[ELP] Elman R., Lam T.Y. and Prestel A., On some Hasse principles over formally real fields, Math. Z. 134(1973), pp 291 – 301.

[KMRT] Knus M.-A., Merkurjev A.S., Rost M. and Tignol J.-P., The Book of Involutions, AMS Colloquium Publication 44, 1998.

[L] Lam T.Y., Algebraic Theory of Quadratic Forms, W.A. Benjamin, 1973.

[LSU] Lewis D. W., Scheiderer C. and Unger T.; A weak Hasse principle for central simple algebras with an involution. Documenta Math., Extra Vol. (2001), pp 241 – 251 (electronic).

[M] Morandi P.J.; The henselisation of a valued division algebra, J. Algebra 122(1989) pp 232 – 243.

[P1] Prestel A., Quadratische Semi-Ordnungen und quadratische Formen, Math. Z. 133(1973), pp 319 – 342.

[P2] Prestel A., Lectures on Formally Real Fields, Lecture Notes in Mathematics 1083, Spriner-Verlag, 1984.

[PSS] Parimala R., Suresh V. and Sridharan R.; Hermitian analogue of a theorem of Springer, J. Algebra 243(2001), pp 780 – 789.

[RTW] Renard J-F., Tignol J-P. and Wadsworth A.; Graded hermitian forms and Springer’s theorem, Indag. Math., N.S. 18(2007), pp 97 – 134.

[S] Scharlau W., Quadratic and Hermitian forms, Springer-Verlag, 1985.

[TW] Tignol J-P. and Wadsworth A.; Valuations on algebras with involution, (unpublished).

[TW1] Tignol J-P. and Wadsworth A.; Value functions and associated graded rings for semisimple algebras, (submitted for publication).

[U] Unger T.; Quadratic forms and central simple algebras with involutions (PhD Thesis), 2001.

[W] Wadsworth A., Extending valuations to finite dimensional divison algebras, Proc. Amer. Math. Soc. 98(1986), pp 20 – 22.

        amit@math.ucl.ac.be

Département de mathématique Université catholique de Louvain B-1348 Louvain-la-Neuve, Belgium.