Hermitian forms and the u-invariant

M. G. Mahmoudi

December 2004

Abstract We study the notion of hermitian u-invariant. We give some estimates of the u-invariant of a division algebra with involution in terms of the u-invariant of some subalgebras stable under the involution. We also find some finiteness results for comparing the u-invariant of a division algebra with involution and that of its centre. Some results about the values of this invariant are also given. A description of the Tits index of some algebraic groups of classical type over ℚ_{p}(t), p ≠ 2 is given as an application.

Mathematics Subject Classification (2000): 11E39, 11E81, 11E57

1 Introduction

The u-invariant is one of the most interesting invariants in the algebraic theory of quadratic forms. This invariant was introduced by Kaplansky over non formally real fields and by Elman and Lam [2] over formally real fields. (In this work we always assume that characteristic ≠ 2).

Determination of this invariant for a given field, has been of great importance in the literature. For a wide class of fields, the values of this invariant are not known and sometimes it is not clear if the u-invariant of such fields is finite. Obtaining lower and upper bounds for this invariant and comparing the uinvariant of a given field K with the u-invariant of its subfields or the fields containing K are good approaches to the problem.

Systems of quadratic forms and exact sequences of Witt groups are two powerful tools used in the literature to obtain good lower and upper bounds for the u-invariant. For example, systems of quadratic forms have been used in [8] by Leep to obtain the bound u(L) ⩽ \frac{n+1}{2} u(K) for a field extension L/K of degree n.

Using exact sequences of Witt groups is however less common than using systems of quadratic forms, but one can mention an exact triangle of Witt groups used by Elman and Lam (cf. [2], [3]) to compare the u-invariant of a field with the u-invariant of a quadratic extension of this field.

In this work, we try to adapt these tools in the context of hermitian forms over a division algebra with involution. To our best knowledge, the notion of the u-invariant of a division algebra with involution appears, for the first time, in Pfister’s paper [13] in connection with systems of quadratic forms over a formally real field and with exception of [13], this notion has not been further studied in the literature.

Our initial motivation for studying this notion was to describe the Tits index of some classical groups over ℚ_{p}(t), p ≠ 2 for which we would have need to obtain some information about the maximal dimension of anisotropic hermitian forms over some division algebras with involution over ℚ_{p}(t), p ≠ 2. The results concerning Tits indices are given in the last section of this paper as applications of our main results. We later come back to this subject.

Let D be a division algebra with an involution σ and let ε be an element of the centre of D with σ(ε)ε = 1. The u-invariant with respect to ε, is by definition the supremum over the dimension of anisotropic ε-hermitian forms over (D, σ). This number is denoted by u(D, σ, ε) (value ∞ is admitted also). Let E be a subdivision algebra of D stable under σ. We are interested in the following questions:

Question 1.1. (Going up)

Under which conditions does the finiteness of u(E, σ|_{E}, ε) imply that of u(D, σ, ε) and can one obtain an upper bound for u(D, σ, ε) in terms of u(E, σ|_{E}, ε)?

Question 1.2. (Going down)

Under which conditions does the finiteness of u(D, σ, ε) imply that of u(E, σ|_{E}, ε) and can one obtain a lower bound for u(D, σ, ε) in terms of u(E, σ|_{E}, ε)?

A substantial part of this work is devoted to these questions. In certain situations, we are able to give precise answers. We will especially deal with the following cases:

• D/E is a quadratic extension of fields (cf. Remark 3.5, Proposition 5.1, Proposition 5.10).

• D is a quaternion algebra and E is a maximal subfield of D (cf. Corollary 3.4, Remark 3.5, Corollary 5.8).

• more generally if there exist two invertible elements λ and μ in D such that λμ = −μλ, σ(λ) = −λ, σ(μ) = −μ and K(λ) is a quadratic extension of K where K is the centre of D. Let E be the centralizer of K(λ) in D (cf. Proposition 3.1, Remark 3.2, Proposition 5.7).

• E is the fixed field of the restriction of σ to the centre of D (cf. Proposition 3.6).

• D is the tensor product E ⊗_{k} L where k is the fixed field of σ|_{K}, K is the centre of D and L/k is a field extension (cf. Proposition 3.8, Proposition 3.10, Remark 3.11).

In particular the Question 1.1 has an affirmative answer when E is the fixed field of σ|_{K} where K is the centre of D. By contrast, the finiteness of u(D, σ, ε) does not imply, in general, that of u(k), for example if D = (−1, −1)_{ℝ} is the algebra of Hamiltonian quaternions and σ is its canonical involution we have u(D, σ, −1) = 1 and u(ℝ) = ∞. In fact we prove that if dim_{K} D is a power of 2, and if both u(D, σ, ε) and u(D, σ, −ε) are finite then u(K, σ|_{K}) < ∞ (cf. Theorem 6.2). In particular if σ is of the first kind, then u(K) is finite if and only if u(D, σ, 1) and u(D, σ, −1) are finite.

The principal ideas to prove these results are to use some exact sequences of Witt groups, mainly the exact sequence of Milnor-Husemoller [19, Ch.10, 1.2], Lewis [9] and Parimala-Sridharan-Suresh [1, Appendix 2] and to use the results of Leep [8] and Pfister [13] on systems of quadratic and hermitian forms. We also use a variation of the exact sequence of Parimala-Sridharan-Suresh discussed in [4].

On one occasion, we need to compare the hermitian u-invariant of a field with that of some of its subfields. We do this by using an exact sequence of Witt groups of biquadratic extensions. We present this exact sequence in §4, (see Theorem 4.3). This exact sequence may be regarded as a particular case of a general exact sequence of L-groups and projective Witt groups due to Ranicki, see [16] and [17, p.242].

Another natural question is to ask about possible values for the u-invariant for some particular type of division algebra with involution. We consider the case of a quadratic extension L/K together with the nontrivial automorphism ¯ and we prove that u(L,¯ ) ≠ 3, 5, 7 (cf. Proposition 7.3).

This result is a hermitian analogue of a classical theorem in the theory of quadratic forms which states that the u-invariant cannot be equal to 3, 5 and 7 (cf. [19, Ch.2, 16.2]). Our proof is of course similar to the proof of this theorem.

One could also define the notion of u-invariant for central simple algebras with involution. But thanks to Morita theory, the problem can be translated to division algebras with involution, so we are only interested in this situation.

As David Lewis pointed out to me, one can also define the notion of the u-invariant for G-equivariant forms, i.e., the forms stable under the action of a finite group G and with the same ideas one can obtain some results that are similar to those stated here.

Patrick Morandi pointed out to me that by using a version of a theorem of Springer for hermitian forms obtained by Larmour [7], one can calculate the u-invariant of a valued division algebra over a henselian valued field, in terms of the u-invariant of its residue division algebra.

Finally in section §8, we give some applications of these results to describe the Tits index of some classical groups over ℚ_{p}(t) with p ≠ 2. We recall that according to a generalized Witt decomposition theorem ([21]), semisimple algebraic groups defined over an arbitrary field K are determined up to isomorphism by their K^{sep} class, their anisotropic kernel and their Tits index, see Tits [21] or Springer [20, Ch. 17] for these notions. Describing the Tits index is therefore important. For the groups of classical type, these indices can be described in terms of algebras with involution. For finite fields, the field of real numbers, number fields and p-adic fields, all possible indices are known (see [21]). We try to describe these indices over ℚ_{p}(t), p ≠ 2 (cf. §8).

In consequence of the results due to Saltman [18] on the structure of the group Br_{2}(ℚ_{p}(t)), the only division algebras over ℚ_{p}(t) with an involution of the first kind are split ones, quaternions and biquaternions. For split case, by a result due to Parimala and Suresh [12] we have 8 ⩽ u(ℚ_{p}(t)) ⩽ 10 (an earlier result due to Hoffmann and Van Geel [5] states that 8 ⩽ u(ℚ_{p}(t)) ⩽ 22). By using these results and by examining anisotropic hermitian and skew hermitian forms over quaternion algebras and biquaternions algebras over ℚ_{p}(t) we obtain some information about possible indices.

2 The u-invariant of a division algebra with involution

Let K be a field of characteristic different from 2. For a division algebra D over K, a K/k-involution σ on D is an involution where k is the fixed field of σ|_{K}.

Let us denote by S^{ε}(D, σ) the semigroup of isometry classes of ε-hermitian forms over (D, σ) and by W^{ε}(D, σ) the Witt group of ε-hermitian forms over (D, σ).

We refer to [19] and [6] for basic notions about quadratic and hermitian forms and algebras with involution.

Every division algebra over a field K considered in this paper is implicitly assumed to be K-central.

Let D be a division algebra over a field K with an involution σ. Let ε ∈ K with εσ(ε) = 1. A system h = (h_{1}, ⋯, h_{r}) : V × V → D^{r} of r, ε-hermitian forms over a right D-vector space V is called anisotropic if x ∈ V, h(x, x) = 0 implies that x = 0.

We consider the u-invariant in the sense of Kaplansky:

Definition 2.1. ([14, Ch. 9, Definition 2.4])

u_{r}(D, σ, ε) = sup{dim_{D} V; there exists an anisotropic ε-hermitian map

h : V × V → D^{r}}

Let us simplify the notation by writing u(D, σ, ε) instead of u_{1}(D, σ, ε) and by u(D, σ) instead of u(D, σ, 1).

We recall that according to a result due to Leep [8], the system u-invariant u_{r} = u(k) satisfies u_{r} ⩽ r u_{1} + u_{r−1} and u_{r} ⩽ \frac{r(r+1)}{2} u_{1} . This result has been generalized by Pfister [13] to the system u-invariant of ε-hermitian forms over a division algebra with involution and also over skew fields with involution not necessarily finite-dimensional over their centres.

As a first observation, we have:

Proposition 2.2. Let D be a division algebra over a field K. Let σ and τ be two involutions on D with the same restriction to K: σ|_{K} = τ|_{K}. Let ε ∈ K with εσ(ε) = 1.

(1) If σ and τ are of the second kind and ε′ ∈ K with ε′τ(ε′) = 1 then u(D, σ, ε) and u(D, τ, ε′) do not depend on the choice of ε and ε′; moreover u(D, σ, ε) = u(D, τ, ε′).

(2) If σ and τ are of the first kind then we have: u(D, σ, ε) = u(D, τ, ε) if σ and τ have the same type, otherwise we have u(D, σ, ε) = u(D, τ, −ε).

Proof. For (1), there exists an element b ∈ D^{∗} such that σ = Int(b) ∘ τ. We have bτ(b)^{−1} = λ ∈ K^{∗}. We obtain then λτ(λ) = 1. Let λ′ = λ^{−1}ε′ε^{−1}. We have λ′τ(λ′) = 1. It follows from Hilbert 90 that there exists μ ∈ K^{∗} such that μτ(μ^{−1}) = λ′.

If we take c = μb, then σ = Int(b) ∘ τ = Int(c) ∘ τ. Now the correspondence φ ↦ c^{−1}φ gives a bijection between the semigroups S^{ε}(D, σ) and S^{ε′}(D, τ). This bijection preserves isometry, orthogonal sum and dimension. So we conclude that u(D, σ, ε) = u(D, τ, ε′).

For (2), we use the same argument. There exists b ∈ D^{∗} such that σ = Int (b) ∘ τ. We have λ := bτ(b)^{−1} ∈ K^{∗}. Moreover λ = 1 if σ and τ are of the same type and λ = −1 if σ and τ are of different type. The correspondence φ ↦ b^{−1}φ from S^{ε}(D, σ) to S^{λε}(D, τ) gives u(D, σ, ε) = u(D, τ, λε).

Remark 2.3. As pointed out to me by the Referee, the preceding proposition actually says that for a given division algebra D, there are three possible uinvariants, let us say a unitary, an orthogonal and a symplectic one; namely for any σ and ε, the u-invariant of (D, σ, ε) coincides with one of them, depending on the type of σ and the value of ε. Note that for unitary case, the preceding proposition states that for a given D, u(D, σ, ε) depends only to the restriction of σ to the centre of D. This leads us to introduce the notation u^{+}(D) for the orthogonal u-invariant, u^{−}(D) for the symplectic u-invariant (this was also suggested to me by Karim Becher). In this way, if τ_{1} is a symplectic involution and τ_{2} is an orthogonal involution on D we have u^{+}(D) = u(D, τ_{1}, −1) = u(D, τ_{2}, 1) and u^{−}(D) = u(D, τ_{1}, 1) = u(D, τ_{2}, −1). This point of view might lead to some simplification in the presentation of some parts of this paper, cf. for instance Corollary 3.4 and Proposition 3.6 or Theorem 6.2.

3 Going up results

Let D be a division algebra over a field K with an involution σ. We suppose that there exist two invertible elements λ and μ in D such that λμ = −μλ, σ(λ) = −λ, σ(μ) = −μ and K(λ) is a quadratic extension of K.

Let D̃ be the centralizer of L = K(λ) in D. According to [1, Appendix 2], we have μD̃μ^{−1} = D̃, μ^{2} ∈ D̃, μ^{2} ∈ D̃ and D = D̃ ⊕ μD̃. On D̃ we have two natural involutions σ_{1} = σ|_{D} and σ_{2} = Int(μ^{−1})∘σ_{1}. We have deg D̃ = \frac{1}{2} deg D. The involution σ_{1} is always of the second kind. The involution σ_{2} is of the same kind as σ but of different type if σ is of the first kind. See [1, §3.1] for more details.

Let π_{i} : D → D̃ be the L-linear projections π_{1}(α+μβ) = α and π_{2}(α+μβ) = β. If h : V × V → D is a ε-hermitian space over (D, σ), then h_{i} : V × V → D̃ is defined by h_{i}(x, y) = π_{i}(h(x, y)). It is easily verified that h_{1} is an εhermitian space over (D̃, σ_{1}) and h_{2} is an −ε-hermitian space over (D̃, σ_{2}). See [1, Appendix 2] for more details.

We prove the following proposition which plays an important role in this paper.

Proposition 3.1. Let D be a division algebra over a field K with a K/kinvolution σ. Suppose that there exist λ, μ ∈ D^{∗} such that σ(λ) = −λ, σ(μ) = −μ, λμ = −μλ and L = K(λ) is a quadratic extension of K. Let D̃ be the centralizer of L in D, σ_{1} = σ|D̃ and σ_{2} = Int(μ^{−1}) ∘ σ_{1} and ε ∈ K with εσ(ε) = 1. Then we have:

u(D, σ, ε) ⩽ \frac{1}{2} u(D̃, σ_{2}, −ε) + u(D̃, σ_{1}, ε).

Proof. Let π_{1} and π_{2} be the projections from D to D̃ induced by the decomposition D = D̃ ⊕ μD̃, i.e, π_{i}(d_{1} + μd_{2}) = d_{i}, for i = 1, 2. Let (V, h) be a nondegenerate ε-hermitian space over (D, σ) and h_{1} = π_{1}h, h_{2} = π_{2}h. We have dimD̃(h_{1}) = dimD̃(h_{2}) = 2 dim_{D}(h). If dimD̃(h_{2}) ⩾ u(D̃, σ_{2}, −ε) + 2m − 1 for some positive integer m ⩾ 1, then h_{2} contains an orthogonal sum of m hyperbolic planes. Consequently h_{2} is totally isotropic over a D̃-vector subspace W of V of dimension m. If moreover m ⩾ u(D̃, σ_{1}, ε) + 1 then h_{1} is isotropic over W. In this way, in order that h be isotropic, it is sufficient to have 2n = dimD̃(h_{2}) ⩾ u(D̃, σ_{2}, −ε) + 2 u(D̃, σ_{1}, ε) + 1. This is equivalent to

u(D, σ, ε) ⩽ \frac{1}{2} u(D̃, σ_{2}, −ε) + u(D̃, σ_{1}, ε).

Remark 3.2. In the proof of the previous proposition, one may interchange the role of h_{1} et h_{2}, in this way we obtain:

u(D, σ, ε) ⩽ u(D̃, σ_{2}, −ε) + \frac{1}{2} u(D̃, σ_{1}, ε).

Remark 3.3. One may also give an alternative proof of the previous result by using an exact sequence of Parimala-Sridharan-Suresh. See the proof of Proposition 5.7 which uses this idea.

Corollary 3.4. Let Q = (a, b)_{K} be a quaternion division algebra over a field K. Let ¯ be the canonical involution of Q and ˆ an orthogonal involution of Q and let L = K(\sqrt{a}) ⊂ Q which is stable under ¯ , then we have:

u(Q,ˆ ) = u(Q,¯ , −1) ⩽ min\{\frac{1}{2} u(L) + u(L,¯ ), u(L) + \frac{1}{2} u(L,¯ )\},
u(Q,¯ ) = u(Q,ˆ , −1) ⩽ \frac{1}{2} u(L,¯ ).

Remark 3.5. Let L/K be a quadratic extension and let ¯ be its nontrivial automorphism. We have the bound u(L,¯ ) ⩽ \frac{1}{2}u(K) because to every anisotropic hermitian form over (L,¯ ) of dimension n, one can associate an anisotropic quadratic form over K of dimension 2n. In the same way, if Q is a quaternion algebra over a field K with the canonical involution ¯ , we have u(Q,¯ ) ⩽ \frac{1}{4} u(K). See also Proposition 3.6 which states a more general result.

Proposition 3.6. Let D be a division algebra of degree m over its centre K with a K/k-involution σ and let ε ∈ K with εσ(ε) = 1. Then :

u(D, σ, ε) ⩽ \frac{r(r + 1)}{2m^{2}[K : k]} u(k)  (1)

where r is the dimension of k-vector space of ε-hermitian elements of D. In particular, if u(k) is finite then so is u(D, σ, ε).

Proof. Let (V, h) be an anisotropic ε-hermitian space over (D, σ) of dimension n. Take D^{ε} the k-vector space of ε-hermitian elements of D, in other words:

D^{ε} = {x ∈ D : σ(x) = εx}.

It is well known that D = D^{+1} ⊕D^{−1} and dim_{k}(D^{+1}) = dim_{k}(D^{−1}) = m^{2} when σ is of the second kind and dim_{k}(D^{+1}) = \frac{1}{2}m(m + 1) or \frac{1}{2}m(m − 1) when σ is of the first kind. Let {e_{1}, ⋯, e_{r}} be a k-basis of D^{ε} and {f_{1}, ⋯, f_{s}} a k-basis of D^{−ε}. One can write h in the form:

\begin{aligned}h(x, y) & = & φ_{1}(x, y)e_{1} + ⋯ + φ_{r}(x, y)e_{r} \\ & + & ψ_{1}(x, y)f_{1} + ⋯ + ψ_{s}(x, y)f_{s} \end{aligned}

where φ_{1}, ⋯, φ_{r} are symmetric bilinear forms and ψ_{1}, ⋯, ψ_{s} are skew symmetric bilinear forms over k (the forms φ_{i} (1 ⩽ i ⩽ r) and ψ_{i} (1 ⩽ i ⩽ s) are possibly degenerate). So ψ_{i}(x, x) = 0 for all x ∈ V. We deduce that

h(x, x) = φ_{1}(x, x)e_{1} + ⋯ + φ_{r}(x, x)e_{r}.

As h is anisotropic, φ_{1}, ⋯, φ_{r} have no common isotropic vector. Now by using a result due to Leep (cf. [14, Ch 9, 2.1] or [19, Ch.2, 16.5]) we obtain:

m^{2}[K : k]n = dim_{k}(V) ⩽ u_{r}(k) ⩽ \frac{r(r + 1)}{2} u(k).

Therefore n ⩽ \frac{r(r+1)}{2m^{2}[K:k]} u(k) which implies the claimed inequality (1).

Remark 3.7. The previous result is a finiteness statement. For many situations, one may have better estimates for u(D, σ, ε). For example for quadratic extensions and quaternion algebras, see Proposition 3.1, Remark 3.2, Corollary 3.4, Proposition 5.1, Proposition 3.8, Proposition 3.10, Remark 3.11. Nevertheless this bound is optimal for m = 1 or (m = 2 and σ symplectic).

Proposition 3.8. Let D be a division algebra over a field K with a K/kinvolution σ. Let L/k be an extension of degree n and suppose that D ⊗_{k} L is also a division algebra. Then

u(D ⊗_{k} L, σ ⊗ id, ε) ⩽ \frac{n + 1}{2} u(D, σ, ε).  (2)

In particular the finiteness of u(D, σ, ε) implies that of u(D ⊗_{k} L, σ ⊗ id, ε).

Proof. Let (V, φ) be an ε-hermitian space over (D ⊗_{k} L, σ ⊗ id). We choose a k-basis {e_{1}, ⋯, e_{n}} of L. We can write φ(x, y) = (φ_{1}(x, y) ⊗ e_{1})) + ⋯ + (φ_{n}(x, y) ⊗ e_{n}) where φ_{1}, ⋯, φ_{n} are ε-hermitian forms (possibly degenerate) over (D, σ). If φ is anisotropic and dim(φ) = m, then φ_{1}, ⋯, φ_{n} have no common isotropic vector, therefore mn = dim(φ_{i}) ⩽ u_{n}(D, σ, ε) so:

u(D ⊗_{k} L, σ ⊗ id, ε) ⩽ \frac{1}{n}u_{n}(D, σ, ε).

According to a result due to Pfister (cf. [14, Ch.9, 2.5] or [13]) we have: u_{n}(D, σ, ε) ⩽ \frac{n(n+1)}{2}u(D, σ, ε). This implies (2).

Remark 3.9. If in the previous statement we take D = k, then we retrieve Leep’s estimate u(L) ⩽ \frac{n+1}{2} u(k).

Proposition 3.10. Let D be a division algebra over a field K with a K/kinvolution σ. Let L/k be a quadratic extension and let ¯ : L ⟶ L be the nontrivial k-automorphism of L. Suppose that D ⊗_{k} L is a division algebra. Then we have:

u(D ⊗_{k} L, σ ⊗ ¯ , ε) ⩽ \frac{1}{2} u(D, σ, −ε) + u(D, σ, ε).

Proof. Let L = k(ξ) with ξ^{2} ∈ k and ξ̄ = −ξ. Let (V, φ) be an ε-hermitian space over (D ⊗_{k} L, σ ⊗ ¯ ). We can write φ in the form

φ(x, y) = φ_{1}(x, y) ⊗ 1 + φ_{2}(x, y) ⊗ ξ,

where φ_{1} is an ε-hermitian form and φ_{2} is a −ε-hermitian form over (D, σ). By repeating the argument given in the proof of Proposition 3.1 we conclude the result.

Remark 3.11. In the proof of Proposition 3.10, one can interchange the role of φ_{1} and φ_{2}, in this way we obtain:

u(D ⊗_{k} L, σ ⊗ ¯ , ε) ⩽ u(D, σ, −ε) + \frac{1}{2} u(D, σ, ε).

4 An exact sequence of Witt groups for biquadratic extensions

Let L = K_{1} ⊗_{k} K_{2}/k be a field extension of degree 4 where K_{1}/k and K_{2}/k are two quadratic extensions with nontrivial automorphisms τ_{1} et τ_{2} (resp.). Suppose that K_{2} = k(λ) with λ ∈ K_{2}, λ^{2} ∈ k and τ_{2}(λ) = −λ. Every element α of L can be uniquely written in the form α_{1} ⊗ 1 + α_{2} ⊗ λ where α_{1}, α_{2} ∈ K_{1}. We consider two projections:

\begin{matrix}π_{1} : L ⟶ K_{1} & π_{2} : L ⟶ K_{1} \\ α ↦ α_{1} & α ↦ α_{2} \end{matrix}

For every nondegenerate hermitian space (V, h) over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) (resp. over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id)), we associate the hermitian space (V, π_{1}h) (resp. (V, π_{2}h)) over (K_{1}, τ_{1}) defined by

\begin{matrix}(π_{1}h)(x, y) = π_{1}(h(x, y)), x, y ∈ V \\ (π_{2}h)(x, y) = π_{2}(h(x, y)), x, y ∈ V \end{matrix}

It is easy to check that π_{1}h and π_{2}h are nondegenerate hermitian forms over (K_{1}, τ_{1}).

For every nondegenerate hermitian space (W, f) over (K_{1}, τ_{1}), we associate the hermitian spaces (W ⊗_{k}K_{2}, ρ_{1}f) over (K_{1}⊗_{k}K_{2}, τ_{1}⊗id) and (W ⊗_{k}K_{2}, ρ_{2}f) over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) by

\begin{matrix}(ρ_{1}f)(x ⊗ α, y ⊗ β) = f(x, y) ⊗ αβ, & x, y ∈ W; α, β ∈ K_{2} \\ (ρ_{2}f)(x ⊗ α, y ⊗ β) = f(x, y) ⊗ τ_{2}(α)β, & x, y ∈ W; α, β ∈ K_{2} \end{matrix}

Proposition 4.1. (1) Let (V, h) be a nondegenerate hermitian form of dimension 1 over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) with h ≃ ⟨d⟩ where d ∈ K_{1} ⊗_{k} K_{2} and (τ_{1} ⊗ τ_{2})(d) = d. Then π_{1}h is isometric to

( \begin{matrix}d_{1} & d_{2}λ^{2} \\ −d_{2}λ^{2} & −d_{1}λ^{2} \end{matrix} )

where d_{1} = π_{1}(d) and d_{2} = π_{2}(d).

(2) Let (V, h) be a nondegenerate hermitian space of dimension 1 over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id) with h ≃ ⟨d⟩ where d ∈ K_{1} ⊗_{k} K_{2} and (τ_{1} ⊗ id)(d) = d. Then π_{2}h is isometric to

( \begin{matrix}d_{2} & d_{1} \\ d_{1} & d_{2}λ^{2} \end{matrix} )

where d_{1} = π_{1}(d) and d_{2} = π_{2}(d).

(3) For the one dimensional form f = ⟨a⟩ over (K_{1}, τ_{1}) we have

ρ_{1}(f) ≃ ⟨a ⊗ 1⟩, ρ_{2}(f) ≃ ⟨a ⊗ 1⟩.

Proof. (1) Let 0 ≠ x ∈ V with h(x, x) = d. We have a basis {x, xλ} for the K_{1}-vector space V. In this basis we have:

\begin{matrix}(π_{1}h)(x, x) = d_{1}, & (π_{1}h)(x, xλ) = d_{2}λ^{2} \\ (π_{1}h)(xλ, x) = −d_{2}λ^{2} & (π_{1}h)(xλ, xλ) = −d_{1}λ^{2} \end{matrix}

These relations imply the isometry we are looking for. The proof of (2) is similar and (3) is obvious.

Proposition 4.2. (1) Let f be an anisotropic hermitian form over (K_{1}, τ_{1}) such that ρ_{1}f is isotropic. Then f contains a subform isometric to π_{1}(⟨d⟩) where ⟨d⟩ is a one dimensional form over (K_{1} ⊗ K_{2}, τ_{1} ⊗ τ_{2}).

(2) Let h be an anisotropic hermitian form over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id) such that π_{2}h is isotropic , then there exists a one dimensional form ⟨a⟩ over (K_{1}, τ_{1}) such that f contains a subform isometric to ρ_{1}(⟨a⟩).

(3) Let f be an isotropic hermitian form over (K_{1}, τ_{1}) such that ρ_{2}f is isotropic. Then f contains a subform isometric to π_{2}(⟨d⟩) where ⟨d⟩ is a one dimensional hermitian form over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id).

Proof. (1) Let v = x_{1} ⊗ 1 + y_{1} ⊗ λ ≠ 0 be an isotropic vector for ρ_{1}(f), i.e., ρ_{1}(f)(v, v) = 0. This relation implies that

\{ \begin{matrix}f(x_{1}, x_{1}) + f(y_{1}, y_{1})λ^{2} = 0 \\ f(x_{1}, y_{1}) + f(y_{1}, x_{1}) = 0 \end{matrix}  (3)

The vectors x_{1} and y_{1} are linearly independent over K_{1}. In fact if x_{1} = αy_{1} for some α ∈ K_{1}, the previous system gives

\{ \begin{matrix}ατ_{1}(α) + λ^{2} = 0 \\ τ_{1}(α) + α = 0 \end{matrix}

which implies that α^{2} = λ^{2} so K_{1} ≃ K_{2}, contradiction because K_{1} ⊗_{k} K_{2} is a field. Now consider the K_{1}-vector space W generated by x_{1} and y_{1}. For d_{1} = f(y_{1}, y_{1}) and d_{2} = f(y_{1}, x_{1})λ^{−2}, the representing matrix of f|_{W} in the basis {y_{1}, x_{1}} is

( \begin{matrix}d_{1} & d_{2}λ^{2} \\ −d_{2}λ^{2} & −d_{1}λ^{2} \end{matrix} )

According to Proposition 4.1, for a hermitian element d = d_{1} ⊗ 1 + d_{2} ⊗ λ with respect to (τ_{1} ⊗ τ_{2}) we have

π_{1}(⟨d⟩) ≃ ( \begin{matrix}d_{1} & d_{2}λ^{2} \\ −d_{2}λ^{2} & −d_{1}λ^{2} \end{matrix} )

so f contains a subform isometric to π_{1}(⟨d⟩).

(2) Let x ≠ 0 be an isotropic vector for π_{2}h, i.e., (π_{2}h)(x, x) = 0. This relation implies that h(x, x) = a⊗1 ∈ K_{1} ⊗_{k} K_{2} for some a ∈ K_{1}. Since h is anisotropic, we have a ≠ 0. We deduce then that h contains a subform isometric to ρ_{1}(⟨a⟩).

(3) The argument is similar to that of (1). Let v = x_{1} ⊗ 1 + y_{1} ⊗ λ ≠ 0 be an anisotropic vector for ρ_{2}h, i.e., (ρ_{2}h)(v, v) = 0. This relation implies that

\{ \begin{matrix}f(x_{1}, x_{1}) − f(y_{1}, y_{1})λ^{2} = 0 \\ f(x_{1}, y_{1}) − f(y_{1}, x_{1}) = 0 \end{matrix}

The vectors x_{1} and y_{1} are linearly independent over K_{1}. Let W be the K_{1}- vector space generated by x_{1} and y_{1}. The matrix of the form f|_{W} with respect to the basis {y_{1}, x_{1}} is

( \begin{matrix}d_{2} & d_{1} \\ d_{1} & d_{2}λ^{2} \end{matrix} )

According to Proposition 4.1, for the one dimensional hermitian form ⟨d⟩ over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id) where d = d_{1} ⊗ 1 + d_{2} ⊗ λ we have:

π_{2}(⟨d⟩) ≃ ( \begin{matrix}d_{2} & d_{1} \\ d_{1} & d_{2}λ^{2} \end{matrix} )

which completes the proof.

Theorem 4.3. We have the following exact sequence of Witt groups:

\begin{matrix}W (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) →^{π_{1}} W (K_{1}, τ_{1}) →^{ρ_{1}} W (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id) \\ →^{π_{2}} W (K_{1}, τ_{1}) →^{ρ_{2}} W (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) \end{matrix}

Proof. Thanks to the previous proposition, it is enough to verify that this sequence is a complex. Let ⟨d⟩ be a one dimensional hermitian form over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) where d = d_{1} ⊗ 1 + d_{2} ⊗ λ where d_{1}, d_{2} ∈ K_{1}. According to Proposition 4.1 we have

π_{1}(⟨d⟩) ≃ ( \begin{matrix}d_{1} & d_{2}λ^{2} \\ −d_{2}λ^{2} & −d_{1}λ^{2} \end{matrix} ) .

We obtain then

ρ_{1}π_{1}(⟨d⟩) ≃ ( \begin{matrix}d_{1} ⊗ 1 & d_{2}λ^{2} ⊗ 1 \\ −d_{2}λ^{2} ⊗ 1 & −d_{1}λ^{2} ⊗ 1 \end{matrix} )

A simple calculation shows that

v = [ \begin{matrix}1 ⊗ 1 \\ 1 ⊗ λ^{−1} \end{matrix} ]

is an isotropic vector for ρ_{1}π_{1}(⟨d⟩). Consequently we have ρ_{1}π_{1} = 0.

In order to show that π_{2}ρ_{1} = 0, we consider a one dimensional form ⟨a⟩ over (K_{1}, τ_{1}). We have ρ_{1}(⟨a⟩) ≃ ⟨a ⊗ 1⟩. So

π_{2}ρ_{1}(⟨a⟩) ≃ ( \begin{matrix}0 & a \\ a & 0 \end{matrix} )

which is a hyperbolic form.

In order to show that ρ_{2}π_{2} = 0, we consider a one dimensional form ⟨d⟩ over (K_{1} ⊗_{k} K_{2}, τ_{1} ⊗id) where d = d_{1} ⊗1+d_{1} ⊗λ is a hermitian element with respect to (τ_{1} ⊗ id) with d_{1}, d_{2} ∈ K_{1}. We have:

ρ_{2}π_{2}(⟨d⟩) ≃ ( \begin{matrix}d_{2} ⊗ 1 & d_{1} ⊗ 1 \\ d_{1} ⊗ 1 & d_{2}λ^{2} ⊗ 1 \end{matrix} ) .

This form is hyperbolic because

v = [ \begin{matrix}1 ⊗ 1 \\ 1 ⊗ λ^{−1} \end{matrix} ]

is an isotropic vector.

5 Going down results

Proposition 5.1. Let L/K be a quadratic extension and let ¯ be its nontrivial automorphism. Then u(L) and u(L,¯ ) are finite if and only if u(K) is finite, moreover we have u(K) ⩽ 2 u(L,¯ ) + u(L) and u(K) ⩽ u(L,¯ ) + 2 u(L).

Proof. We write L = K(\sqrt{a}) where a ∈ K^{∗}. Let q be an anisotropic form of dimension n over K. Consider the following exact sequence of Witt groups due to Milnor-Husemoller [19, Ch.10, 1.2]:

0 → W (L,¯ ) →^{π} W (K) →^{r^{∗}} W (L)

In this exact sequence, π is the transfer map induced by the projection π : L → K, x + y\sqrt{a} ↦ x and r^{∗} is the restriction map. If r^{∗}(q) is anisotropic, then q contains a subform isometric to ⟨b, −ab⟩ for some b ∈ K. We can then write (by induction)

q ≃ (⟨1, −a⟩ ⊗ q_{1}) ⊕ q_{2}

where q_{1}, q_{2} are two nondegenerate quadratic forms over K so that r^{∗}(q_{2}) is anisotropic. Let φ be the hermitian form over (L,¯ ) induced by q_{1}. We have π(φ) ≃ ⟨1, −a⟩ ⊗ q_{1}. Therefore we have a Witt r^{∗}-decomposition, i.e., there exists an orthogonal decomposition q ≃ q′ ⊥ q′′ where q′ ≃ π(φ) for some nondegenerate hermitian form φ over (L,¯ ) (in particular r^{∗}(q′) is hyperbolic) and r^{∗}(q′′) is anisotropic. We may suppose that φ is anisotropic, therefore: dim(q) = dim(π(φ)) + dim(q′′) = 2 dim(φ) + dim(q′′) ⩽ 2 u(L,¯ ) + u(L) which implies the result. If we use the same argument with the following exact sequence of Witt groups due to Lewis [9]:

W (L) →^{s_{∗}} W (K) →^{r^{∗}} W (L,¯ ),

we obtain u(K) ⩽ u(L,¯ ) + 2 u(L). In this exact sequence, s_{∗} is the Scharlau transfer map and r^{∗} is the restriction map.

Now suppose that we have u(K) < ∞. According to a result due to Elman and Lam (cf. [2, theorem 4.3]) we have u(L) ⩽ \frac{3}{2} u(K) < ∞. The finiteness of u(L,¯ ) is easy to check. In fact if φ = ⟨a_{1}, ⋯, a_{n}⟩ is a form over (L,¯ ) where a_{i} ∈ K, its trace form is isometric to ψ = ⟨a_{1}, ⋯, a_{n}, −aa_{1}, ⋯, −aa_{n}⟩. Isotropy of φ and ψ are equivalent. The finiteness of u(K) implies therefore that of u(L,¯ ).

Remark 5.2. One can regard the estimates of Proposition 5.1 as an improvement, in some situations, of the estimate u(K) ⩽ 4 u(L) for a non formally real field K due to Elman [3, Theorem 3.1(iii)], because one can find many examples where

min{2 u(L,¯ ) + u(L) , u(L,¯ ) + 2 u(L)} < 4 u(L).

For example if K is a p-adic field then u(K) = u(L) = 4 and u(L,¯ ) ⩽ 2. Our bound is independent of the formally real nature of L or K. To our best knowledge, the going down result of Elman, is the best one for arbitrary fields.

In the sequel, we need the following lemma which is an immediate consequence of a theorem of Springer.

Lemma 5.3. Let L/K be a quadratic extension and let τ be its nontrivial automorphism. Let M/K be an extension of odd degree and let φ be an anisotropic hermitian form over (L, τ). Then φ remains anisotropic over (L ⊗_{K} M, τ ⊗ id).

Proof. For a hermitian space (V, φ) over (L, τ), we denote its trace form by tr(φ), defined by tr(φ)(x, y) = tr_{L/K}(φ(x, y)) for every x, y ∈ V. We have tr(φ|_{(L⊗_{K}M)}) ≃ tr(φ)|_{M}. If φ|_{(L⊗_{K}M)} is isotropic, then tr(φ)|_{M} is also isotropic. The strong version of Springer’s theorem implies that tr(φ) is isotropic. Consequently φ is isotropic.

Corollary 5.4. With the notation of the previous lemma we have:

u(L, τ) ⩽ u(L ⊗_{K} M, τ ⊗ id).

Let D be a division algebra over a field K with an involution σ. Suppose that there exist λ, μ ∈ D^{∗} such that σ(λ) = −λ, σ(μ) = −μ, λμ = −μλ and L = K(λ) is a quadratic extension of K. In this situation, we have the following exact sequence of Witt groups due to Parimala, Sridharan and Suresh [1, Appendix 2]:

W^{ε}(D, σ) →^{π_{1}^{ε}} W^{ε}(D̃, σ_{1}) →^{ρ_{1}^{ε}} W^{−ε}(D, σ) →^{π_{2}^{−ε}} W^{ε}(D̃, σ_{2})  (4)

In this sequence, the map π_{1}^{ε} and π_{2}^{−ε} are transfers induced by the projections π_{1} and π_{2} defined in §3. The map ρ_{1}^{ε} is a restriction map defined by the multiplication by λ and the usual restriction map r^{∗}:

\xymatrix{W^{ε}(D̃, σ_{1}) \ar[rr]^{ρ_{1}} \ar[dr]^{×λ} &  & W^{−ε}(D, σ) \\  & W^{−ε}(D̃, σ_{1}) \ar@{-->}[ur]^{r^{∗}}}

See [1, Appendix 2] for more details. These maps also induce homomorphisms between semigroups of isometry classes of hermitian forms:

S^{ε}(D, σ) →^{π_{1}^{ε}} S^{ε}(D̃, σ_{1}) →^{ρ_{1}^{ε}} S^{−ε}(D, σ) →^{π_{2}^{−ε}} S^{ε}(D̃, σ_{2})  (5)

Now we can reformulate the exact sequence of (4) in the following way:

Proposition 5.5. Let φ ∈ S^{ε}(D̃, σ_{1}) be an anisotropic form. Then φ has a Witt ρ_{1}^{ε}-decomposition, i.e., there exists an orthogonal decomposition φ ≃ φ_{1} ⊕ φ_{2} such that ρ_{1}^{ε}(φ_{1}) is hyperbolic and ρ_{1}^{ε}(φ_{2}) is anisotropic. Moreover there exists ψ ∈ S^{ε}(D, σ) such that φ_{1} ≃ π_{1}^{ε}(ψ).

Proof. If ρ_{1}^{ε}(φ) is anisotropic, we take φ_{2} = φ. If ρ_{1}^{ε}(φ) is isotropic then φ contains a subform φ_{0} which comes from S^{ε}(D, σ), i.e., φ_{0} ≃ π_{1}^{ε}(ψ_{0}) for some ψ_{0} ∈ S^{ε}(D, σ); see the proof of (4) in [1, Appendix 2] where this has been implicitly proved, see also [4, 4.4]. We have then an orthogonal decomposition φ ≃ φ_{0}⊕φ′ for some φ′ ∈ S^{ε}(D̃, σ_{1}). As ρ_{1}^{ε}(φ_{0}) is hyperbolic and dim φ′ < dim φ we can use induction on dim φ to finish the proof.

We have also the following exact sequence of Witt groups:

W^{−ε}(D, σ) →^{π_{2}^{−ε}} W^{ε}(D̃, σ_{2}) →^{ρ_{2}^{ε}} W^{−ε}(D, σ)  (6)

which is a variation of the exact sequence of Parimala, Sridharan and Suresh, see [4]. In this sequence, ρ_{2}^{ε} is the composition of the multiplication by −λ and the usual restriction map r^{∗} and the multiplication by −μ:

\xymatrix{W^{ε}(D̃, σ_{2}) \ar[r]^{ρ_{2}} \ar[d]^{−λ} & W^{−ε}(D, σ) \\ W^{ε}(D̃, σ_{2}) \ar[r]_{r^{∗}} & W^{ε}(D, Int(μ^{−1}) ∘ σ) \ar[u]_{×μ}}

We state a Witt-decomposition-like result for the exact sequence of (6):

Proposition 5.6. Let φ ∈ S^{ε}(D̃, σ_{2}) be an anisotropic form. Then φ has a Witt ρ_{2}^{ε}-decomposition, i.e., there exists an orthogonal decomposition φ ≃ φ_{1} ⊕ φ_{2} such that ρ_{2}^{ε}(φ_{1}) is hyperbolic and ρ_{2}^{ε}(φ_{2}) is anisotropic. Moreover there exists ψ ∈ S^{−ε}(D, σ) such that φ_{1} ≃ π_{2}^{−ε}(ψ).

Proof. We use the same argument as in Proposition 5.5. If ρ_{2}^{ε}(φ) is isotropic then φ contains a subform φ_{0} which comes from S^{−ε}(D, σ), i.e., φ_{0} ≃ π_{2}^{ε}(ψ_{0}) for some ψ_{0} ∈ S^{−ε}(D, σ); see [4, 4.4].

Proposition 5.7. Let D, D̃, σ_{1}, σ and σ_{2} be as in Proposition 3.1, then we have:

(1) u(D̃, σ_{1}, ε) ⩽ u(D, σ, −ε) + 2 u(D, σ, ε).

(2) u(D̃, σ_{2}, ε) ⩽ 3 u(D, σ, −ε).

Proof. (1) Let φ be an anisotropic ε-hermitian form over (D̃, σ_{1}). According to Proposition 5.5, there exists an ε-hermitian form ψ over (D, σ) such that φ ≃ π_{1}(ψ) ⊕ φ′ for some form φ′ over (D̃, σ_{1}) such that ρ(φ′) is anisotropic. As φ is anisotropic, so is ψ, therefore dim(ψ) ⩽ u(D, σ, ε). As ρ(φ′) is anisotropic, dim(φ′) = dim ρ(φ′) ⩽ u(D, σ, −ε), consequently dim(φ) = dim(π_{1}(ψ)) + dim(φ′) = 2 dim(ψ) + dim(φ′) ⩽ 2 u(D, σ, ε) + u(D, σ, −ε) .

For (2) we apply the same argument by using Proposition 5.6.

Corollary 5.8. With the notation of Corollary 3.4 we have:

(1) u(L,¯ ) ⩽ 2 u(Q,¯ ) + u(Q,ˆ ) = 2 u^{−}(Q) + u^{+}(Q)

(2) u(L) ⩽ 3 u(Q,ˆ ) = 3 u^{+}(Q)

Proposition 5.9. With the notation of §4, let φ ∈ S(K_{1}, τ_{1}) be an anisotropic form. Then φ has a Witt ρ_{1}-decomposition, i.e., there exists an orthogonal decomposition φ ≃ φ_{1}⊕φ_{2} such that ρ_{1}(φ_{1}) is hyperbolic and ρ_{1}(φ_{2}) is anisotropic. Moreover there exists ψ ∈ S(K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}) such that φ_{1} ≃ π_{1}(ψ).

Proof. If ρ_{1}(φ) is isotropic then according to Proposition 4.2, φ contains a subform φ_{0} which comes from S(K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}), i.e., φ_{0} ≃ π_{1}(ψ_{0}) for some ψ_{0} ∈ S(K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}), we can then use an induction argument as in Proposition 5.5 Proposition 5.10. With the notation of Theorem 4.3, we have

u(K_{1}, τ_{1}) ⩽ 2 u(K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ id) + u(K_{1} ⊗_{k} K_{2}, τ_{1} ⊗ τ_{2}).

Proof. The proof is similar to that of Proposition 5.7; we use Proposition 5.9.

Remark 5.11. Using [4, 4.4], and Proposition 4.2, one can state similar Wittdecomposition-like results for other maps involved in these exact sequences. For example by a Witt π_{1}^{ε}-decomposition result one can give an alternative proof of Proposition 3.1.

6 A Finiteness result

In the article [1], one can find several useful results about extensions of odd degree and their connections with substructures of codimension 2 of central simple algebras with involution, see ([1], Lemma 3.1.1, Lemma 3.3.1, Lemma 3.3.2, Lemma 3.3.3). From these results we can derive the following proposition which has been proved in [11].

Proposition 6.1. Let D be a noncommutative K-division algebra and σ a K/kinvolution on D. Suppose that the degree of D is a 2-power. Then there exists an extension M/k of odd degree such that D_{M} = D ⊗_{k} M contains the elements λ, μ such that τ(λ) = −λ and τ(μ) = −μ and λμ = −μλ and [F (λ) : F] = 2 where F = KM = K ⊗_{k} M and τ is the involution σ ⊗ id when σ is of second kind or σ is symplectic and D is a quaternion algebra otherwise τ = Int(μ) ∘ (σ ⊗ id).

Theorem 6.2. Let D be a division algebra of dimension a power of 2 over its centre K with a K/k-involution σ. Suppose that u(D, σ, ε) < ∞ for ε = 1 and ε = −1, then u(K, σ|_{K}) < ∞. In particular if both u^{+}(D) and u^{−}(D) are finite then u(K) is finite too.

Proof. We prove this result by induction on dim_{K}(D). For dim_{K}(D) = 1 the conclusion is evident. Suppose that dim_{K}(D) > 1.

First suppose that σ is of the second kind. According to Proposition 6.1, there exists an extension M/k of odd degree such that D_{M} = D ⊗_{k} M contains the elements μ and λ such that τ(λ) = −λ, τ(μ) = −μ and λμ = −μλ for τ = σ ⊗ id and F (λ)/F is a quadratic extension where F = K ⊗_{k} M. As M/k is an extension of odd degree, E = D_{M} is a division algebra. According to Proposition 3.8 we have u(E, τ, ε) < ∞ and u(E, τ, −ε) < ∞. By applying Proposition 5.7 we obtain: u(Ẽ, τ_{1}, ±ε) < ∞ and u(Ẽ, τ_{2}, ±ε) < ∞ where Ẽ = C_{E}(F (λ)). As τ is of the second kind, so are τ_{1} and τ_{2}. By the induction hypothesis, we have u(L, τ_{1}|_{L}) < ∞ and u(L, τ_{2}|_{L}) < ∞ where L = F (λ) is the centre of Ẽ. Let F′ be the fixed field of τ_{2}|_{L}. We have

\begin{matrix}(L, τ_{1}|_{L}) ≃ (F ⊗_{M} F′, τ|_{F} ⊗ τ′) \\ (L, τ_{2}|_{L}) ≃ (F ⊗_{M} F′, τ|_{F} ⊗ id) \end{matrix}

where τ′ is the nontrivial automorphism of F′/M. Proposition 5.10 implies that u(F, τ|_{F}) < ∞ and from Corollary 5.4 we deduce that u(K, σ|_{K}) ⩽ u(F, τ|_{F}) < ∞.

Now consider the case where σ is of the first kind. If D is a quaternion algebra and σ is its canonical involution, then there exist λ, μ ∈ D with σ(λ) = −λ, σ(μ) = −μ, λμ = −μλ et [K(λ) : K] = 2.

Otherwise there exists an extension M/K of odd degree such that E = D ⊗_{K} M contains λ and μ with τ(λ) = −λ, τ(μ) = −μ, λμ = −μλ and [M(λ) : M] = 2 where τ = Int(μ) ∘ (σ ⊗ id) (cf. Proposition 6.1). Them involution τ is of the first kind but of a different type from that of σ. In any case take:

τ = \{ \begin{matrix}σ & dim_{K}(D) = 4, σ symplectic \\ Int(μ) ∘ (σ ⊗ id) & otherwise \end{matrix}

As M/K is an extension of odd degree, E is a division algebra. According to Proposition 3.8 we have u(E, τ, ε) < ∞ and u(E, τ, −ε) < ∞ (note that according to Proposition 2.2, the finiteness hypothesis (which is for σ) is still valid for τ). By applying Proposition 5.7 we obtain: u(Ẽ, τ_{1}, ±ε) < ∞ and u(Ẽ, τ_{2}, ±ε) < ∞ where Ẽ = C_{E}(M(λ)). The involution τ_{1} is unitary and by the first part of the proof the condition u(Ẽ, τ_{1}, ±ε) < ∞ implies that u(L, τ_{1}|_{L}) < ∞ where L = M(λ). The involution τ_{2} is of the first kind. The condition u(Ẽ, τ_{2}, ±ε) < ∞ states that both u^{−}(Ẽ) and u^{+}(Ẽ) are finite. So we conclude by induction that u(L) = u(L, τ_{2}|_{L}) < ∞. Here τ_{1}|_{L} is the nontrivial automorphism of L/M. Now Proposition 5.1 implies that u(M) < ∞ and from the strong version of Springer’s theorem we deduce that u(K) ⩽ u(M) < ∞.

Remark 6.3. Note that in preceding theorem, according to Proposition 2.2 one of the two hypotheses u(D, σ, ε) < ∞ and u(D, σ, −ε) < ∞ is actually enough in the unitary case.

7 Values of the u-invariant, a particular case

Let L/K be a quadratic extension and let ¯ be its nontrivial automorphism. The signed discriminant d_{±} defines a map from W (L,¯ ) to K^{∗}/N(L^{∗}). Unfortunately this map is not a homomorphism. But for forms φ and ψ of even dimension we have d_{±}(φ ⊕ ψ) = d_{±}(φ). d_{±}(ψ). Let I(L,¯ ) ⊂ W (L,¯ ) be the classes of all nondegenerate hermitian forms over (L,¯ ). The group W (L,¯ ) has a natural ring structure.

Proposition 7.1. (1) The map e_{1} : I(L,¯ ) → K^{∗}/N(L^{∗}) defined by e_{1}(φ) = d_{±}(φ) is a surjective homomorphism.

(2) ker e_{1} = I^{2}(L,¯ ).

(3) Via e_{1}, the group I/I^{2} is isomorphic to K^{∗}/N(L^{∗}).

Proof. (1) The map e_{1} is surjective because d_{±}(⟨1, −a⟩) = aN(L^{∗}).

(2) The group I = I(L,¯ ) is generated by hermitian forms ⟨a, b⟩ where a, b ∈ K^{∗}. Thus I^{2} is generated by the hermitian forms:

φ = ⟨a, b⟩ ⊗ ⟨c, d⟩ = ⟨ac, ad, bc, bd⟩

We have d_{±}(φ) = 1. So I^{2} ⊂ ker e_{1}. Conversely suppose that φ ∈ I with e_{1}(φ) = 1. The form φ is represented by ⟨a_{1}, ⋯, a_{2n}⟩ with n ⩾ 1.

For n = 1, we have φ = ⟨a_{1}, a_{2}⟩ with −a_{1}a_{2} ∈ N(L^{∗}). So φ ≃ ⟨a_{1}, −a_{1}⟩, therefore φ is hyperbolic and φ = 0 in W (L,¯ ).

Now suppose that n ≥ 2. We can write: φ = ⟨a_{1}, a_{2}, a_{3}⟩ ⊕ ⟨a_{4}, ⋯, a_{2n}⟩. So φ ∼ ⟨a_{1}, a_{2}, a_{3}, a_{1}a_{2}a_{3}⟩ ⊕ ⟨−a_{1}a_{2}a_{3}, a_{4}, ⋯, a_{2n}⟩. We have

⟨a_{1}, a_{2}, a_{3}, a_{1}a_{2}a_{3}⟩ ≃ ⟨a_{1}, a_{2}⟩ ⊗ ⟨1, a_{1}a_{3}⟩ ∈ I^{2}.

The dimension of φ′ = ⟨−a_{1}a_{2}a_{3}, a_{4}, ⋯, a_{2n}⟩ is 2(n − 1) and d_{±} φ′ = 1. By induction we obtain φ′ ∈ I^{2}.

(3) is deduced from (1) and (2).

Let D be a division algebra over a field K with a K/k-involution τ. Here we call a hermitian form φ over (D, τ) an n-fold Pfister form if φ is the restriction of an n-fold Pfister form q over k to D. This notion appears in [10] for quaternion algebras. A hermitian form φ induced by the n-fold Pfister form q = ⟨⟨a_{1}, ⋯, a_{n}⟩⟩, is still denoted by φ = ⟨⟨a_{1}, ⋯, a_{n}⟩⟩.

Definition 7.2. Let D be a division algebra over a field K with a K/kinvolution τ and let ε be an element of K with ετ(ε) = 1. An ε-hermitian form φ over (D, τ) is called universal if φ represents all nonzero ε-hermitian elements of D.

Proposition 7.3. Let L/K be a quadratic extension and let ¯ be the nontrivial automorphism of L/K. Then we have u(L,¯ ) ≠ 3, 5, 7.

Proof. Suppose that u(L,¯ ) < 4. Every 2-fold hermitian Pfister form ⟨⟨a, b⟩⟩ is hyperbolic. The hermitian form ⟨1, a, b⟩ is a hermitian neighbor of ⟨⟨a, b⟩⟩ and therefore it is isotropic. We deduce that every hermitian form ⟨a, b, c⟩ of dimension 3 over (L,¯ ) is isotropic, so we have u(L,¯ ) ⩽ 2.

Suppose that u(L,¯ ) < 8. We conclude that every 3-fold hermitian Pfister form ⟨⟨a, b, −c⟩⟩ is hyperbolic. Thus for every a, b, c ∈ K^{∗} we have:

⟨⟨a, b⟩⟩ ≃ c⟨⟨a, b⟩⟩

Every form in I^{2} = I^{2}(L,¯ ) is an orthogonal sum of the forms ⟨⟨a_{i}, b_{i}⟩⟩. As for every hyperbolic plane IH and c ∈ K^{∗} we have IH ≃ cIH, the Witt cancellation theorem implies that φ ≃ cφ for every φ ∈ I^{2}. In particular φ is universal over (L,¯ ) (in the sense of Definition 7.2).

Now suppose that u = u(L,¯ ) = 5 or 7. Let φ be an anisotropic hermitian form of dimension u. In particular φ represents its discriminant d = d_{±}(φ). We have then φ = ψ ⊕ ⟨d⟩ where ψ is a form of dimension 4 or 6 and d_{±}(ψ) = 1.

As ψ ∈ I, Proposition 7.1 implies that ψ ∈ I^{2}. We have already shown that ψ is universal, consequently φ is isotropic which is a contradiction to the choice of φ.

Remark 7.4. In a similar way, if D = (a, b)_{K} is a quaternion division algebra with the canonical involution ¯ , then u(D,¯ ) ≠ 3, 5, 7. However the value 3 for the u-invariant u(D,¯ , −1) is possible, for example if K is a p-adic field then u(D,¯ , −1) = 3.

Proposition 7.5. Let L_{0} be a field with u(L_{0}) = n. Let L = L_{0}((x)) be the field of Laurent series over L_{0} and let σ be the L_{0}-automorphism of L induced by x ↦ −x. Then u(L, σ) = n.

Proof. Let K be the fixed field of σ. We have K = L_{0}((x^{2})) so u(K) = 2n. Consequently u(L, σ) ⩽ n (cf. Remark 3.5). Let q = ⟨a_{1}, ⋯, a_{n}⟩ be an anisotropic quadratic form of dimension n over L_{0}. The restriction of q to L is anisotropic. In fact the isotropy of q|_{(L,σ)} is equivalent to that of the quadratic form q ⊕ x^{2}q = ⟨a_{1}, ⋯, a_{n}, −x^{2}a_{1}, ⋯, −x^{2}a_{n}⟩ over K. The anisotropy of this form is equivalent to that of q over L_{0}.

Remark 7.6. Proposition 7.5 state in particular that the possible values for the hermitian u-invariant of commutative fields contain the possible values of the usual u-invariant.

8 Classical groups over ℚ_{p}(t), p ≠ 2

We refer to [21] and [20] for basic notions about Tits’s indices. In the symbol gX_{n,r}^{t}, where X = A, B, C, D, the integers n and r are respectively the absolute and relative rank of the considered classical group G, g denotes the order of the quotient of the Galois group Γ = Gal(k^{sep}/k) which operates effectively on the Dynkin diagram. In case the diagram has no nontrivial automorphism, g is necessarily 1. If g = 1, G is called of inner type, otherwise G is called of outer type. The integer t is the degree of a certain division algebra which occurs in the definition of the considered group. If t or g are omitted in the symbol, they are necessarily 1.

Type A_{n}

Lemma 8.1. Let k be a function field of a p-adic field with p ≠ 2. Let L/k be a quadratic extension and let ¯ be the nontrivial automorphism of L/k. Then u(L,¯ ) ⩽ 4.

Proof. Let φ ≃ ⟨a_{1}, ⋯, a_{5}⟩ be a hermitian form of dimension 5 over (L,¯ ), where a_{i} ∈ k, i = 1, ⋯, 5. Then, φ is isotropic if and only if the quadratic form q = ⟨a_{1}, ⋯, a_{5}, −aa_{1}, ⋯, −aa_{5}⟩ is isotropic over k where L = k(\sqrt{a}). But this form is isotropic over k, because its Hasse invariant is c(q) = (−a, d) where d = det(φ), which has index ⩽ 2 in the Brauer group of k and according to the Theorem 4.6 of [12], q is isotropic. We have in particular u(L,¯ ) ⩽ 4.

Proposition 8.2. Let k = ℚ_{p}(t) with p ≠ 2. The index ^{1}A_{n,r}^{(d)} occurs over k for every positive integers d, r and n satisfying rd = n + 1.

Proof. According to [21] or [20, 17.1.3], ^{1}A_{n,r}^{(d)} occurs over k if and only if there exists a division algebra D over k of degree d with rd = n + 1. The existence of such algebras comes from the fact that there exist division algebras of arbitrary degree over ℚ_{p}.

Proposition 8.3. Let k = ℚ_{p}(t) with p ≠ 2.

(1) If the index ^{2}A_{n,r}^{(1)} occurs over k, then n + 1 − 2r ∈ {0, 1, 2, 3, 4}. All the anisotropic indices ^{2}A_{1,0}^{(1)}, ^{2}A_{2,0}^{(1)} ^{2}A_{3,0}^{(1)} and ^{2}A_{4,0}^{(1)} occur over k.

(2) If the index ^{2}A_{n,r}^{(2)} occurs over k then n + 1 − 4r ∈ {0, 2, 4, 6}.

Proof. According to [21] or [20, 17.1.6], ^{2}A_{n,r}^{(d)} occurs over k if and only if there exist a quadratic extension E/k, a division algebra D over E and an involution σ on D of the second kind such that k is the fixed field of σ|_{E} and a nondegenerate hermitian form h over (D, σ) of dimension d^{−1}(n + 1) and of Witt index r.

(1) For d = 1, D = E. According to Lemma 8.1 we have u(E, σ) ⩽ 4. We conclude that 0 ⩽ n + 1 − 2r ⩽ 4. As for E = ℚ_{p}(t)(\sqrt{t}), we have u(E, σ) = 4, the indices ^{2}A_{1,0}^{(1)}, ^{2}A_{2,0}^{(1)} ^{2}A_{3,0}^{(1)} and ^{2}A_{4,0}^{(1)} occur over k.

(2) For d = 2, D is a quaternion algebra over E. Thank to Proposition 3.6 we have u(D, σ) ⩽ 3. We have then 0 ⩽ \frac{n+1}{2} − 2r ⩽ 3.

Type B_{n}

Proposition 8.4. Let k = ℚ_{p}(t) with p ≠ 2. If the index B_{n,r} occurs over k then we have: n − 4 ⩽ r ⩽ n.

Proof. According to [21] or [20, 17.2.3], B_{n,r} can occur over k if and only if there exists a nondegenerate quadratic form q of dimension 2n + 1 and of Witt index r. Let q_{a} be the unique anisotropic part of q up to isometry. We have:

dim q_{a} = 2n + 1 − 2r  (7)

According to [12], the dimension of q_{a} cannot exceed 10. We obtain then 0 ⩽ 2n − 2r + 1 ⩽ 10 and so n − 4 ⩽ r ⩽ n.

Proposition 8.5. The anisotropic indices B_{1,0}, B_{2,0} and B_{3,0} can occur over k = ℚ_{p}(t) with p ≠ 2.

Proof. According to (7), it is enough to find anisotropic quadratic forms of dimension 3, 5 and 7 (resp.), which is possible because u(ℚ_{p}(t)) ⩾ 8.

Remark 8.6. According to (7), the existence of B_{4,0} over ℚ_{p}(t) is equivalent to u(ℚ_{p}(t)) ⩾ 9. According to a conjecture, one believes that u(ℚ_{p}(t)) = 8 (cf. [14, Chapter 5, 2.5], [5] and [12]).

Type C_{n}

Proposition 8.7. Let k = ℚ_{p}(t) with p ≠ 2. If the index C_{n,r}^{(d)} occurs over k then we have d ∈ {1, 2, 4}. For d = 1 we have n = r. For d = 2 we have n − 2r ∈ {0, 1, 2}. For d = 4, we have n − 4r ∈ {0, 2, 4, 6, 8, 10}.

Proof. According to [21] or [20, 17.2.10], C_{n,r}^{(d)} can occur over k if and only if there exists a division algebra D over k of degree d with an orthogonal involution σ and a skew hermitian form h over (D, σ) of dimension 2d^{−1}n and of Witt index r. As D has an involution of the first kind, D lies in the 2-torsion of the Brauer group Br_{2}(k) (cf. [19, Ch. 8, 8.4]). According to a result due to Saltman [18], we know that d ∈ {1, 2, 4}. Let h_{a} be the anisotropic part of h. We have

dim h_{a} = 2d^{−1}n − 2r  (8)

If d = 1, h is alternating and in this case we have r = n.

If d = 2, D is a quaternion algebra over k. According to Corollary 3.4 and Lemma 8.1 we have u(D, σ, −1) = u(D,¯ ) ⩽ \frac{1}{2}u(L,¯ ) ⩽ 2 where ¯ is the canonical involution of D and L is a maximal subfield of D stable under ¯ . We deduce that 0 ⩽ dim h_{a} ⩽ 2. Now (8) implies that n − 2r ∈ {0, 1, 2}.

If d = 4, according to a result due to Albert (cf. [6, 16.1]), D is isomorphic to a biquaternion algebra. We write D = D_{1}⊗D_{2} where D_{1} and D_{2} are quaternion algebras. According to Proposition 2.2 we have u(D, σ, −1) = u(D_{1} ⊗ D_{2},¯ ⊗ ¯ , −1) where ¯ (resp. ¯ ) is the canonical involution of D_{1} (resp. D_{2}). Let L be a maximal subfield of D_{2} stable under ¯ . Thanks to Proposition 3.1 we have:

u(D_{1} ⊗ D_{2},¯ ⊗¯ − 1) ⩽ u(D_{1} ⊗ L,¯ ⊗ id) + \frac{1}{2} u(D_{1} ⊗ L,¯ ⊗ ¯ , −1)

Thanks to Proposition 3.6 we have u(D_{1} ⊗ L,¯ ⊗ id) ⩽ \frac{1}{4} u(L). According to [12], u(L) ⩽ 10. We conclude then u(D_{1} ⊗ L,¯ ⊗ id) ⩽ 2. Now by using Proposition 3.10 and Proposition 3.1 we obtain

u(D_{1} ⊗ L,¯ ⊗ ¯ , −1) ⩽ 6.

We obtain then u(D_{1} ⊗ D_{2},¯ ⊗ ¯ , −1) ⩽ 5. We conclude that 0 ⩽ dim h_{a} ⩽ 5 and (8) implies that n/2 − 2r ∈ {0, 1, ⋯, 5}.

Proposition 8.8. The anisotropic indices C_{1,0}^{(2)}, C_{2,0}^{(2)} and C_{2,0}^{(4)} occur over k = ℚ_{p}(t) with p ≠ 2.

Proof. According to (8), it is enough to find a skew hermitian form of dimension 2 over (D, σ) where D is a suitable quaternion division algebra over ℚ_{p}(t) and σ is an orthogonal involution of D. This choice is possible because u(D, σ, −1) = 2 for D = (−p, u)_{ℚ_{p}(t)} where u ∈ ℤ_{p}^{∗}\ℤ_{p}^{∗}^{2}. In fact by Proposition 3.6, u(D, σ, −1) ⩽ \frac{1}{4} u(ℚ_{p}(t)). We obtain then u(D, σ, −1) ⩽ 2, because u(ℚ_{p}(t)) ⩽ 10 according to [12]. So it is enough to construct an anisotropic skew hermitian form of dimension 2 over (D, σ) or equivalently an anisotropic hermitian form of dimension 2 over (D,¯ ) where ¯ is the canonical involution of D. We may take for example the hermitian form ⟨1, t⟩.

Type D_{n} inner

Lemma 8.9. Let D be a division algebra over a field K with a K/k-involu-tion σ. Consider the division algebra D(t) = k(t) ⊗_{k} D with the involution σ̂ = id ⊗σ. Let (V, h_{V}) and (W, h_{W}) be two anisotropic ε-hermitian spaces over (D, σ). Let V (t) = k(t) ⊗_{k} V, W (t) = k(t) ⊗_{k} W, ĥ_{V} and ĥ_{W} the restrictions of h_{V} and h_{W} (resp.) to D(t). Then the hermitian form ĥ_{V} ⊕ tĥ_{W} is anisotropic over (D(t), σ̂). In particular u(D(t), σ̂, ε) ⩾ 2 u(D, σ, ε) where ε ∈ K satisfies εσ(ε) = 1.

Proof. Let x_{1}⊕x_{2} ∈ V (t)⊕W (t) be a nonzero anisotropic vector for ĥ_{V} ⊕tĥ_{W}, i.e., ĥ_{V} ⊕ tĥ_{W} (x_{1} ⊕ x_{2}, x_{1} ⊕ x_{2}) = 0. This relation implies that:

ĥ_{V} (x_{1}, x_{1}) + tĥ_{W} (x_{2}, x_{2}) = 0  (9)

By using the embedding V (t) ⊂ V ((t)) we may suppose that x_{1} = Σ_{i=N}^{∞}v_{i}t^{i} and x_{2} = Σ_{i=M}^{∞}w_{i}t^{i} where v_{i} ∈ V, w_{i} ∈ W, v_{N} ≠ 0 and w_{M} ≠ 0. We consider two cases: N ⩽ M and N > M. If N ⩽ M, (9) implies that h_{V} (v_{M}, v_{M}) = 0 which is a contradiction because h_{V} is anisotropic. If N > M, (9) implies that h_{W} (w_{N}, w_{N}) = 0 which is a contradiction because h_{W} is anisotropic.

Proposition 8.10. Let k = ℚ_{p}(t) with p ≠ 2. Then if the index ^{1}D_{n,r}^{(d)} occurs over k then d ∈ {1, 2, 4}. Moreover:

(1) For d = 1 we have n − r ∈ {0, 2, 3, 4, 5}

(2) For d = 2 we have n − 2r ∈ {0, 2, 3, 4, 5, 6, 7, 8, 9}

(3) For d = 4 we have n − 4r ∈ {0, 2, 4, ⋯, 20}.

Proof. According to [21] or [20, 17.3.13],^{1}D_{n,r}^{(d)} occurs over k if and only if there exist a division algebra D over k of degree d with an orthogonal involution σ and a nondegenerate hermitian form h of dimension 2nd^{−1} with trivial discriminant and of Witt index r, and moreover d ⩾ 1, rd ⩽ n and n ≠ rd + 1. As D has an involution of the first kind, according to a result of Albert (cf. [19, Ch. 8, 8.4]), D lies in the 2-torsion of Br(k). By using a result of Saltman [18], we obtain d ∈ {1, 2, 4}. Let h_{a} be the anisotropic part of h. We have:

dim h_{a} = 2d^{−1}n − 2r, d_{±} h_{a} = 1  (10)

(1) If d = 1, we have D = k and h_{a} is a symmetric bilinear form over k and dim h_{a} is even. By using a result due to Parimala-Suresh [12] the dimension of h_{a} cannot exceed 10, we obtain then 2n − 2r = dim h_{a} ∈ {0, 2, 4, 6, 8, 10}.

The case dim h_{a} = 2 is impossible because in this case d_{±} h_{a} = 1 implies the isotropy of h_{a}. Consequently 2n − 2r = 2 is impossible.

The case dim h_{a} = 4 is possible, we can take the anisotropic form h_{a} = ⟨1, p, −u, −pu⟩ which has trivial discriminant where u ∈ ℤ_{p}^{∗}\ℤ_{p}^{∗}^{2}. This form is anisotropic over ℚ_{p}(t). The case 2n − 2r = 4 is then possible.

The case dim h_{a} = 6 is possible. Consider the biquaternion division algebra (u, t) ⊗_{k} (t + 1, p). The fact that this algebra is a division algebra can be found in an appendix of the Saltman’s paper [18] due to W. Jacob and J.-P. Tignol. Let h_{a} = ⟨u, t, −ut, −(t + 1), −p, (t + 1)p⟩ be the Albert form of this algebra. This form is anisotropic and has trivial discriminant. Consequently 2n − 2r = 6 is possible.

The case dim h_{a} = 8 is also possible. We can take the anisotropic form h_{a} = ⟨1, t⟩ ⊗ ⟨1, p, −u, −pu⟩. The anisotropy of h_{a} can be deduced for example from Lemma 8.9.

(2) If d = 2, D is a quaternion division algebra over k. Now (10) implies that d_{±} h = d_{±} h_{a} = 1. We write D = (a, b)_{k}, the quaternion division algebra over k generated by i, j with i^{2} = a ∈ k^{∗}, j^{2} = b ∈ k^{∗}, ij = −ji. Let L = k(i) = k(\sqrt{a}) ⊂ D. According to Proposition 2.2, u(D, σ) = u(D,¯ , −1) where ¯ is the canonical involution of D.

As L is the function field of a p-adic field (p ≠ 2), we have u(L) ⩽ 10 (cf. [12, 4.5]). According to Lemma 8.1, u(L,¯ |_{L}) ⩽ 4. Now by applying Corollary 3.4 we obtain:

u(D,¯ , −1) ⩽ \frac{10 + 8}{2} = 9.

Consequently dim h_{a} ⩽ 9. We have then n − 2r ∈ {0, 1, 2, ⋯, 9}.

The case dim h_{a} = 1 is impossible because the discriminant of every skew hermitian form over D of dimension 1 is different from 1. Consequently the case n − 2r = 1 is impossible.

The case dim h_{a} = 2 is possible. To construct, consider an arbitrary nondegenerate skew hermitian form h_{0} of dimension 1 over (D_{0}, σ_{0}) where D_{0} is the unique quaternion division algebra over ℚ_{p} and σ_{0} is its canonical involution. Consider the skew hermitian form

h = h_{0} ⊕ th_{0}  (11)

According to Lemma 8.9, h is anisotropic, moreover its discriminant is trivial. We conclude that the case n − 2r = 2 is possible.

It is well known that, there exists an anisotropic skew hermitian form of dimension 3 over D_{0} (cf. [19, Ch. 10, 3.6], [15] or [22]). The restriction of this form to (D, σ) is anisotropic. Consequently the case dim h_{a} = 3 or n − 2r = 3 is possible.

The cases dim h_{a} = 4 or 6 are similar; we take an anisotropic skew hermitian form h_{0} over (D_{0}, σ_{0}) of dimension 2 or 3 and we consider the anisotropic form h defined in (11).

For the case dim h_{a} = 5, we first consider a subform of dimension 5 of the anisotropic skew hermitian form of dimension 6 over (D, σ) that we constructed above. By multiplying this form by its discriminant, we obtain an anisotropic form of dimension 5 with trivial discriminant. Therefore n − 2r = 5 is possible.

(3) If d = 4, D is a biquaternion algebra. We write D = D_{1} ⊗ D_{2} where D_{1} and D_{2} are two quaternion algebras. We have u(D, σ) = u(D_{1} ⊗D_{2},¯ ⊗¯ ) where ¯ (resp. ¯ ) is the canonical involution of D_{1} (resp. D_{2}). Let L be a maximal subfield of D_{2} stable under ¯ . According to Proposition 3.1 and Proposition 3.10 we have

\begin{aligned}u(D_{1} ⊗ D_{2},¯ ⊗ ¯ ) & ⩽ \frac{1}{2} u(D_{1} ⊗ L,¯ ⊗ id, −1) + u(D_{1} ⊗ L,¯ ⊗ ¯ ) \\ & ⩽ \frac{9}{2} + 6 = \frac{21}{2}, \end{aligned}

We obtain then u(D, σ) ⩽ 10. Now (10) implies that 0 ⩽ n/2 − 2r ⩽ 10.

Corollary 8.11. The anisotropic indices ^{1}D_{n,0}^{(1)} for 2 ⩽ n ⩽ 4 and ^{1}D_{n,0}^{(2)} for 2 ⩽ n ⩽ 6 occur over k = ℚ_{p}(t) with p ≠ 2.

Type D_{n} outer

Proposition 8.12. Let k = ℚ_{p}(t) with p ≠ 2. If the index ^{2}D_{n,r}^{(d)} occurs over k then d ∈ {1, 2, 4}. Moreover:

(1) For d = 1 we have n − r ∈ {0, 1, 2, 3, 4, 5}

(2) For d = 2 we have n − 2r ∈ {0, 1, 2, 3, 5, 6, 7, 8, 9}

(3) For d = 4 we have n − 4r ∈ {0, 2, ⋯, 20}.

Proof. We use the notation of the proof of Proposition 8.10. The criterion for the existence of ^{2}D_{n,r}^{(d)} is the same as for ^{1}D_{n,r}^{(d)} except d_{±} h ≠ 1 (cf. [21]). Therefore we have d ∈ {1, 2, 4}.

(1) For d = 1, dim h_{a} = 2 is possible; it is enough to choose the anisotropic form ⟨1, p⟩. Consequently n − r = 1 is possible.

The case dim h_{a} = 4 is possible. In fact it is enough to choose a subform of dimension 4 of the anisotropic form ⟨1, t⟩ ⊗ ⟨1, p, −u, −pu⟩ with nontrivial discriminant. We can choose for example ⟨p, −u, −pu, t⟩. Consequently n−r = 2 is possible.

For dim h_{a} = 6 we can choose ⟨p, −u, −pu, tp, −tu, −tpu⟩. Consequently n − r = 3 is possible.

(2) For d = 2, D is a quaternion algebra over k. We have that dim h_{a} ⩽ 9 as in the proof of Proposition 8.10.

For the case dim h_{a} = 1, it is enough to consider a skew hermitian form of dimension 1 which has necessarily nontrivial discriminant. Consequently the case n − 2r = 1 is possible.

For the case dim h_{a} = 2, we consider two arbitrary skew symmetric elements of D_{0} = (−p, u)_{ℚ_{p}} with different reduced norm modulo ℚ_{p}^{∗}^{2}. We denote these elements by a and b. Now consider the skew hermitian form h = ⟨a, tb⟩ over D = D_{0} ⊗_{ℚ_{p}} ℚ_{p}(t). The form h is anisotropic form with nontrivial discriminant. Consequently n − 2r = 2 is possible.

The case dim h_{a} = 3 is possible. In fact, there exists a skew hermitian form h_{0} of dimension 2 with discriminant equal to c ≠ 1 ∈ ℚ_{p}^{∗}/ℚ_{p}^{∗}^{2} (cf. [19, Ch. 10, 3.6], [15] or [22]). We also consider a skew hermitian form h′_{0} over (D_{0}, σ_{0}) of dimension 1 and with the discriminant equal to c′ ≠ 1 ∈ ℚ_{p}^{∗}/ℚ_{p}^{∗}^{2}. We may suppose that c′ ≠ c (this choice is possible because Card(ℚ_{p}^{∗}/ℚ_{p}^{∗}^{2}) = 4). Thanks to Lemma 8.9, the form h_{a} = h_{0} ⊕ th′_{0} is an anisotropic skew hermitian form over (D, σ) of dimension 3 and with nontrivial discriminant. Consequently n − 2r = 3 is also possible.

For dim h_{a} = 5, we consider an anisotropic skew hermitian form h_{0} over (D_{0}, σ_{0}) of dimension 3 and with trivial discriminant and an anisotropic skew hermitian form h′_{0} over (D_{0}, σ_{0}) of dimension 2 and with nontrivial discriminant (According to [19, Ch. 10, 3.6], [15] or [22] these choices are possible). Now consider the form h = h_{0} ⊕ th′_{0} over (D, σ) which is of dimension 5 and with nontrivial discriminant Consequently n − 2r = 5 is possible.

Suppose that h ≃ ⟨a_{1}, a_{2}, ⋯, a_{5}⟩ is an anisotropic skew hermitian form of dimension 5 and with nontrivial discriminant constructed as in the preceding paragraph. At least one of the forms ⟨a_{i_{1}}, ⋯, a_{i4}⟩ has nontrivial discriminant where 1 ⩽ i_{1} < i_{2} < i_{3} < i_{4} ⩽ 5. Consequently dim h_{a} = n − 2r = 4 is possible.

(3) For d = 4, D is a biquaternion algebra. As in the proof of Proposition 8.10 (3), we obtain n/2 − 2r ⩽ 10.

Corollary 8.13. The anisotropic indices ^{2}D_{n,0}^{(1)} for 2 ⩽ n ⩽ 3 and ^{2}D_{n,0}^{(2)} for 1 ⩽ n ⩽ 5 occur over k = ℚ_{p}(t) with p ≠ 2.

Acknowledgements

I would like to thank my supervisor Eva Bayer-Fluckiger for suggesting the topic of this paper and for her guidance. I would also like to thank David Lewis and the members of the department of Mathematics of University College Dublin for their kind hospitality during a predoctoral stay of the author at UCD. I would also like to thank Karim Becher for many helpful suggestions. I would also like to thank the Referee for careful reading of the manuscript and for numerous helpful suggestions that improved the final version of this paper. This research has been carried out in the framework of the European research network HPRN-CT-2002-00287 Algebraic K-Theory, Linear Algebraic Groups and Related Structures.

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Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland.

mohammad.mahmoudi@ucd.ie