The orthogonal u-invariant of a quaternion algebra

Karim Johannes Becher, Mohammad G. Mahmoudi

Abstract

In quadratic form theory over fields, a much studied field invariant is the u-invariant, defined as the supremum over the dimensions of anisotropic quadratic forms over the field. We investigate the corresponding notions of u-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra. Under certain conditions on the center of the quaternion algebra, we obtain sharp bounds for this invariant.

Classification (MSC 2000): 11E04, 11E39, 11E81

Key words: hermitian form, involution, division algebra, isotropy, system of quadratic forms, discriminant, Tsen-Lang Theory, Kneser’s Theorem, local field, Kaplansky field

1 Involutions and hermitian forms

Throughout this article K denotes a field of characteristic different from 2 and K^{×} its multiplicative group. We shall employ standard terminology from quadratic form theory, as used in [9]. We say that K is real if K admits a field ordering, nonreal otherwise. By the Artin-Schreier Theorem, K is real if and only if −1 is not a sum of squares in K.

Let Δ be a division ring whose center is K and with dim_{K}(Δ) < ∞; we refer to such Δ as a division algebra over K, for short. We further assume that Δ is endowed with an involution σ, that is, a map σ : Δ → Δ such that σ(a + b) = σ(a) + σ(b) and σ(ab) = σ(b)σ(a) hold for any a, b ∈ Δ and such that σ∘σ = id_{Δ}. Then σ|_{K} : K → K is an involution of K, and there are two cases to be distinguished. If σ|_{K} = id_{K}, then we say that the involution σ is of the first kind. In the other case, when σ|_{K} is a nontrivial automorphism of the field K, we say that σ is of the second kind. In general, we fix the subfield k = {x ∈ K | σ(x) = x} and say that σ is a K/k-involution of Δ. Note that σ : Δ → Δ is k-linear. If σ is of the second kind, then K/k is a quadratic extension. Recall that involutions of the first kind on a division algebra Δ over K do exist if and only if Δ is of exponent at most 2, i.e. Δ ⊗_{K} Δ is isomorphic to a matrix algebra over K. Moreover, an involution σ of the first kind over Δ is either of orthogonal or of symplectic type, depending on the dimension of the subspace {x ∈ Δ | σ(x) = x}.

Let ε ∈ K^{×} with σ(ε)ε = 1. We are mainly interested in the cases where ε = ±1; if σ is of the first kind then these are the only possibilities for ε. An ε-hermitian form over (Δ, σ) is a pair (V, h) where V is a finite-dimensional Δ-vector space and h is a map h : V × V → Δ that is Δ-linear in the second argument and with σ(h(x, y)) = ε·h(y, x) for any x, y ∈ V; it follows that h is ‘sesquilinear’ in the sense that h(xa, yb) = σ(a)h(x, y)b holds for any x, y ∈ V and a, b ∈ Δ. In this situation we may also refer to h as the ε-hermitian form and to V as the underlying vector space. We simply say that h is hermitian (resp. skew-hermitian) if h is 1-hermitian (resp. (−1)-hermitian).

In the simplest case we have Δ = K, σ = id_{K}, and ε = 1. A 1-hermitian form over (K, id_{K}) is a symmetric bilinear form b : V × V → K on a finite dimensional vector space V over K; by the choice of a basis it can be identified with a quadratic form over K in n = dim_{K}(V) variables.

An ε-hermitian form h over (Δ, σ) with underlying vector space V is said to be regular or nondegenerate if, for any x ∈ V \{0}, the associated Δ-linear form V → Δ, y ↦ h(x, y) is nontrivial; if this condition fails h is said to be singular or degenerate. We say that h is isotropic if there exists a vector x ∈ V \ {0} such that h(x, x) = 0, otherwise we say that h is anisotropic. Let h_{1} and h_{2} be two ε-hermitian forms over (Δ, σ) with underlying spaces V_{1} and V_{2}. The orthogonal sum of h_{1} and h_{2} is the ε-hermitian form h on the Δ-vector space V = V_{1} × V_{2} given by h(x, y) = h_{1}(x_{1}, y_{1}) + h_{2}(x_{2}, y_{2}) for x = (x_{1}, x_{2}), y = (y_{1}, y_{2}) ∈ V, and it is denoted by h_{1} ⊥ h_{2}. An isometry between h_{1} and h_{2} is an isomorphism of Δ-vector spaces τ : V_{1} → V_{2} such that h_{1}(x, y) = h_{2}(τ(x), τ(y)) holds for all x, y ∈ V_{1}. If an isometry between h_{1} and h_{2} exists, then we say that h_{1} and h_{2} are isometric and write h_{1} ≃ h_{2}. Witt’s Cancellation Theorem [2, (6.3.4)] states that, whenever h_{1}, h_{2} and h are ε-hermitian forms on (Δ, σ) such that h_{1} ⊥ h ≃ h_{2} ⊥ h, then also h_{1} ≃ h_{2} holds. A regular 2n-dimensional ε-hermitian form (V, h) is said to be hyperbolic if there exits an n-dimensional subspace W of V such that h(x, y) = 0 for all x, y ∈ W. The (up to isometry) unique isotropic 2dimensional ε-hermitian form is denoted by ℍ.

Given an ε-hermitian form (V, h) on (Δ, σ) we write

D(h) = {h(x, x) | x ∈ V \ {0}} ⊆ Δ .

Note that this set contains 0 if and only if h is isotropic. We further put

Sym^{ε}(Δ, σ) = {x ∈ Δ | σ(x) = εx} .

For any ε-hermitian form h over (Δ, σ) we have D(h) ⊆ Sym^{ε}(Δ, σ). Given a_{1}, …, a_{n} ∈ Sym^{ε}(Δ, σ), an ε-hermitian form h on the Δ-vector space V = Δ^{n} is given by h(x, y) = σ(x_{1})a_{1}y_{1} +⋯+σ(x_{n})a_{n}y_{n} for x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) ∈ Δ^{n} = V. We denote this form h by ⟨a_{1}, …, a_{n}⟩ and observe that it is regular if and only if a_{i} ≠ 0 for 1 ⩽ i ⩽ n. As char(K) ≠ 2, any ε-hermitian form is isometric to ⟨a_{1}, …, a_{n}⟩ for some n ∈ ℕ and a_{1}, …, a_{n} ∈ Sym^{ε}(Δ, σ) [2, (6.2.4)].

We denote by Herm_{n}^{ε}(Δ, σ) the set of isometry classes of regular ndimensional ε-hermitian forms over (Δ, σ). Mapping a ∈ Sym^{ε}(Δ, σ) to the class of ⟨a⟩ yields a surjection

Sym^{ε}(Δ, σ) \ {0} ⟶ Herm_{1}^{ε}(Δ, σ) .

Two elements a, b ∈ Sym^{ε}(Δ, σ) are congruent if there exists c ∈ Δ such that a = σ(c)bc, which is equivalent to saying that ⟨a⟩ ≃ ⟨b⟩ over (Δ, σ).

1.1 Remark. In the case where Δ = K and ε = 1, there is a natural oneto-one correspondence between Herm_{1}^{ε}(Δ, σ) and K^{×}/K^{×2}. We may then identify the two sets and thus endow Herm_{1}^{1}(Δ, σ) with a natural group structure. One can proceed in a similar way in the two cases, first where Δ is a quaternion algebra and σ its canonical involution, and second when σ is a unitary involution on a field Δ = K.

Given an ε-hermitian form h over (Δ, σ) and an element a ∈ k^{×} where k = {x ∈ K | σ(x) = x}, we define the scaled ε-hermitian form ah in the obvious way. Two ε-hermitian forms h and h′ over (Δ, σ) are said to be similar, if h′ ≃ ah for some a ∈ k^{×}.

2 Hermitian u-invariants

We keep the setting of the previous section. Following [8, Chap. 9, (2.4)] we define

u(Δ, σ, ε) = sup {dim(h) | h anisotropic ε-hermitian form over (Δ, σ)}

in ℕ ∪ {∞} and call this the u-invariant of (Δ, σ, ε). Then

u(K, id_{K}, 1) = sup {dim(φ) | φ anisotropic quadratic form over K}

is the u-invariant of the field K, denoted by u(K). We refer to [8, Chap. 8] for an overview on this invariant for nonreal fields and for the discussion of a different definition of this definition, which is more reasonable when dealing with real fields.

To obtain upper bounds on u(Δ, σ, ε), one can use the theory of systems of quadratic forms. In fact, to every ε-hermitian form h over (Δ, σ) one can associate a system of quadratic forms over k in such a way that the isotropy of h is equivalent to the simultaneous isotropy of this system.

For r ∈ ℕ, one denotes by u_{r}(K) the supremum over the n ∈ ℕ for which there exists a system of r quadratic forms in n variables over K having no nontrivial common zero. The numbers u_{r}(K) are called the system uinvariants of K. Note that u_{0}(K) = 0 and u_{1}(K) = u(K). Leep proved that these system u-invariants satisfy the inequalities

u_{r}(K) ⩽ ru(K) + u_{r−1}(K) ⩽ \frac{r(r+1)}{2} u(K)

for any integer r ⩾ 1. Using systems of quadratic forms, he further showed that u(L) ⩽ \frac{[L:K]+1}{2}u(K) holds for an arbitrary finite field extension L/K. (See [9, Chap. 2, Sect. 16] for these and more facts on systems on quadratic forms.) In the same vein the following result was obtained in [7, (3.6)].

2.1 Proposition. Let Δ be a division algebra over K, σ an involution on Δ, and ε ∈ K with εσ(ε) = 1. Then

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

where k = {x ∈ K | σ(x) = x}, m = deg(Δ) and r = dim_{k}(Sym^{ε}(Δ, σ)). In particular, if u(k) < ∞, then u(Δ, σ, ε) < ∞.

In this article, we are mainly concerned with the u-invariant of an involution of the first kind. Assume that σ is an involution of the first kind on the division algebra Δ over K. In this case Δ ⊗_{K} Δ is isomorphic to a matrix algebra and ε = ±1. In [7] it is explained that u(Δ, σ, ε) only depends on ε and on the type of σ, i.e., whether it is orthogonal or symplectic. More precisely, given two involutions of the first kind σ and τ on Δ one has u(Δ, σ, ε) = u(Δ, τ, ε) if σ and τ are of same type and u(Δ, σ, ε) = u(Δ, τ, −ε) if they are of opposite type. We define

u^{+}(Δ) = u(Δ, σ, +1) and u^{−}(Δ) = u(Δ, σ, −1)

with respect to an arbitrary orthogonal involution σ on Δ, as these numbers do not depend on the choice of σ. We call u^{+}(Δ) the orthogonal and u^{−}(Δ) the symplectic u-invariant of Δ. By the previous, for any symplectic involution τ on Δ one has u(Δ, τ, ε) = u^{−ε}(Δ).

Let us briefly turn to the case of an involution σ of the second kind. It turns out that u(Δ, σ, ε) depends only on the field k = {x ∈ K | σ(x) = x}, and in particular it does not depend on ε at all.

Let i ∈ ℕ. Using (2.1) one can obtain estimates for the u-invariants of division algebras with involution over a C_{i}-field. We recall some facts from Tsen-Lang Theory, following [9, Chap. 2, Sect. 15]. A field K is called a C_{i}-field if every homogeneous polynomial over K of degree d in more than d^{i} variables has a nontrivial zero. The natural examples of C_{i}-fields are extensions of transcendence degree i of an arbitrary algebraically closed field and (for i > 0) extensions of transcendence degree i − 1 of a finite field. A result due to Lang and Nagata states that if K is a C_{i}-field then u_{r}(K) ⩽ r·2^{i} for any r ∈ ℕ (cf. [9, Chap. 2, (15.8)]). In [8, Chap. 5], variations of the C_{i}-property and open problems in this context are discussed.

2.2 Corollary. Let K be a C_{i}-field and let Δ be a division algebra of exponent 2 and of degree m over K. Then u^{+}(Δ) ⩽ 2^{i−1} · \frac{m+1}{m} and u^{−}(Δ) ⩽ 2^{i−1} · \frac{m−1}{m}.

Proof: We use (2.1) and the fact that u_{r}(k) ⩽ 2^{i}r. □

2.3 Corollary. Let K be a C_{i}-field. Let Δ be a quaternion division algebra over K. Then u^{+}(Δ) ⩽ 3 · 2^{i−2} and u^{−}(Δ) ⩽ 2^{i−2}.

Example (5.4) will show that the first bound in (2.3) is sharp. For the second bound, we leave this as an easy exercise. In fact, determining the symplectic u-invariant of a quaternion algebra is a pure quadratic form theoretic problem in view of Jacobson’s Theorem [9, Chap. 10, (1.1)], which relates skew-hermitian forms over a quaternion algebra with canonical involution to quadratic forms over the center. This is why our investigation for quaternion algebras concentrates on the orthogonal u-invariant.

3 Kneser’s Theorem

In this section, we give an upper bound on the u-invariant of a division algebra with involution in terms of the number of 1-dimensional (skew-)hermitian forms, under a condition on the levels of certain subalgebras. This extends an observation due to Kneser [4, Chap. XI, (6.4)] on the commutative case.

From [6] we recall the definition of the level of an involution. Let σ be an involution on a central simple algebra Δ over K. The level of σ is defined as

s(Δ, σ) = sup {m ∈ ℕ | m × ⟨1⟩ is anisotropic over (Δ, σ)}

in ℕ ∪ {∞}. Whenever s(Δ, σ) is finite, it is equal to the smallest number m for which −1 can be written as a sum of m hermitian squares over (Δ, σ).

3.1 Theorem. Let Δ be a division algebra over K equipped with an involution σ. Let ε ∈ K be such that σ(ε)ε = 1. Let ψ be an ε-hermitian form over (Δ, σ) and let α ∈ D^{×} be such that σ(α) = εα. Let C_{D}(α) be the centralizer of K(α) in Δ. Suppose that s(C_{Δ}(α), σ|_{C_{Δ}(α)}) < ∞. If φ = ψ ⊥ ⟨α⟩ is anisotropic then D(ψ) ⊊ D(φ).

Proof: We write 0 = σ(d_{0})d_{0} + ⋯ + σ(d_{s})d_{s} with s = s(C_{Δ}(α), σ|_{C_{Δ}(α)}) and d_{0}, …, d_{s} ∈ C_{Δ}(α) \ {0}. We suppose that D(ψ) = D(φ) and want to conclude that φ is isotropic. We claim that α·(σ(d_{0})d_{0}+⋯+σ(d_{i})d_{i}) ∈ D(φ) for any 0 ⩽ i ⩽ s. For i = s this yields that φ is isotropic.

For i = 0, the elements α and ασ(d_{0})d_{0} are indeed represented by φ. Let now 1 ⩽ i ⩽ s and assume that the claim is established for i − 1. With α(σ(d_{0})d_{0} + ⋯ + σ(d_{i−1})d_{i−1}) ∈ D(φ) = D(ψ), we obtain readily that α(σ(d_{0})d_{0} + ⋯ + σ(d_{i−1})d_{i−1}) + ασ(d_{i})d_{i} ∈ D(φ), finishing the argument. □

3.2 Corollary. Assume that s(C_{Δ}(K(α)), σ) < ∞ for every α ∈ Sym^{ε}(Δ, σ). Then u(Δ, σ, ε) ⩽ |Herm_{1}^{ε}(Δ, σ)|.

Proof: Let h ≃ ⟨a_{1}, …, a_{n}⟩ be an anisotropic ε-hermitian form of dimension n over (Δ, σ). Let h_{i} = ⟨a_{1}, …, a_{i}⟩ for i = 1, …, n. By (3.1) we have D(h_{1}) ⊊ D(h_{2}) ⊊ ⋯ ⊊ D(h_{n}) = D(h). We conclude that h represents at least n pairwise incongruent elements of Sym^{ε}(Δ, σ), i.e. |Herm_{1}^{ε}(Δ, σ)| ⩾ n. Therefore we have |Herm_{1}^{ε}(Δ, σ)| ⩾ u(Δ, σ, ε). □

3.3 Remark. The hypothesis of (3.2) is trivially satisfied if the subfield of K consisting of the elements fixed by σ is nonreal; this is for example the case whenever σ is of the first kind and K is a nonreal field.

3.4 Example. Let p be a prime number different from 2 and let Q denote the unique quaternion division algebra over ℚ_{p}. Then it follows from [9, Chap. 10, (3.6)] that u^{+}(Q) = |Herm_{1}^{−1}(Q, γ)| = 3 (see also (4.9), below). Let now m be a positive integer and K = ℚ_{p}((t_{1})) … ((t_{m})). Then Q_{K} is a quaternion division algebra over K and u^{+}(Q_{K}) = |Herm_{1}^{−1}(Q_{K}, γ)| = 3 · 2^{m}. This follows from the fact that the u-invariant(s) and the number of 1-dimensional ε-hermitian forms over a division algebra defined over a field K both double when the center is extended from K to K((t)).

The upper bound on the u-invariant obtained in (3.2) motivates to look for criteria for the finiteness of Herm_{1}^{ε}(Δ, σ) where Δ is a division algebra over K, σ an involution on Δ, and ε = ±1. We conjecture that |Herm_{1}^{ε}(Δ, σ)| < ∞ is equivalent to |K^{×}/K^{×2}| < ∞. In the next section we shall confirm this in the case of skew-hermitian forms over a quaternion division algebra.

4 Congruence of pure quaternions

From this section on we consider a quaternion division algebra Q over K. Let γ denote the canonical involution of Q, π the norm form of Q and π′ its pure part, so that π = ⟨1⟩ ⊥ π′. By a skew-hermitian form over Q we always mean a regular skew-hermitian form over (Q, γ). In this section we want to describe Herm_{1}^{−1}(Q, γ).

Following [10] the discriminant of a skew-hermitian form h over Q is defined as the square class disc(h) = (−1)^{n} Nrd((h(x_{i}, x_{j})))K^{×2} in K^{×}/K^{×2} where (x_{1}, …, x_{n}) is an arbitrary Δ-basis of the underlying vector space and where Nrd : M_{n}(Δ) → K denotes the reduced norm.

4.1 Remark. For a ∈ K^{×}, there exists a skew-hermitian form of dimension 1 and discriminant a over Q if and only if −a is represented by the pure part of the norm form of Q. In particular, any 1-dimensional skew-hermitian form over Q has nontrivial discriminant.

4.2 Proposition. Skew-hermitian forms of dimension 1 over Q are classified up to similarity by their discriminants.

Proof: More generally, similar skew-hermitian forms over Q have the same discriminant. Assume now that z_{1}, z_{2} ∈ Q^{×} are pure quaternions such that the discriminants of the skew-hermitian forms ⟨z_{1}⟩ and ⟨z_{2}⟩ coincide. Hence there exists d ∈ K^{×} such that z_{2}^{2} = d^{2}z_{1}^{2} = (dz_{1})^{2}. Therefore the pure quaternions z_{2} and dz_{1} are congruent in Q, i.e. there exists α ∈ Q^{×} such that dz_{1} = α^{−1}z_{2}α. Multiplying this equality with Nrd(α) = γ(α)α, if follows that (Nrd(α)d)z_{1} = γ(α)z_{2}α. With c = (Nrd(α)d) ∈ K^{×} we obtain that ⟨cz_{1}⟩ ≃ ⟨z_{2}⟩, so ⟨z_{1}⟩ and ⟨z_{2}⟩ are similar. □

4.3 Remark. A closer look at the above argument yields the following refinement. Let G be a subgroup of K^{×} containing Nrd(Q^{×}). Two 1-dimensional skew-hermitian forms are obtained one from each other by scaling with an element of G if and only if their discriminants coincide in K^{×}/G^{2}.

4.4 Lemma (Scharlau). Let λ, μ ∈ Q^{×} be anticommuting elements, in particular Q = (a, b)_{K} for a = λ^{2}, b = μ^{2} ∈ K^{×}. Let c ∈ K^{×}. The skew-hermitian forms ⟨λ⟩ and ⟨cλ⟩ over Q are isometric if and only if c is represented over K by one of the quadratic forms ⟨1, −a⟩ and ⟨b, −ab⟩ over K.

Proof: See [9, Chap. 10, (3.4)]. □

The following result was obtained in [5], in slightly different terms.

4.5 Proposition (Lewis). Let λ ∈ Q^{×} be a pure quaternion. We consider Herm_{1}^{−1}(Q, γ) as a pointed set with the isometry class of ⟨λ⟩ as distinguished point. With L = K(λ) and a = λ^{2} ∈ K^{×}, one obtains an exact sequence

1 ⟶ ℤ/2ℤ ⟶ K^{×}/N_{L/K}(L^{×}) ⟶^{· λ} Herm_{1}^{−1}(Q, γ) ⟶^{(−a) Nrd} K^{×}/K^{×2} .

Proof: Let b ∈ K^{×} be such that Q = (a, b)_{K}. By (4.4) the group of elements x ∈ K^{×} such that ⟨xλ⟩ ≃ ⟨λ⟩ coincides with N_{L/K}(L^{×}) ∪ bN_{L/K}(L^{×}). This proves the exactness in the first two terms. The exactness at Herm_{1}^{−1}(Q, γ) follows from (4.2). □

4.6 Remark. We sketch an alternative, cohomological argument for the exact sequence in (4.5), which has been pointed out to us by J.-P. Tignol. Let ρ = Int(λ) ∘ γ. First note that Herm_{1}^{−1}(Q, γ) can be identified with Herm_{1}^{1}(Q, ρ) = H^{1}(K, O(ρ)) where O(ρ) = {x ∈ Q | ρ(x)x = 1}. By [3, Chap. VII, §29], there is an exact sequence 1 → O^{+}(ρ) → O(ρ) → μ_{2} → 1. Moreover, we have O^{+}(ρ) = L^{1} = {x ∈ L | N_{L/K}(x) = 1}. This yields the exact sequence 1 → μ_{2} → H^{1}(K, L^{1}) → H^{1}(K, O(ρ)) → K^{×}/K^{×2}. Using that H^{1}(K, L^{1}) ≃ K^{×}/N_{L/K}(L^{×}) we obtain the sequence in (4.5).

4.7 Proposition. Let S = {aK^{×2} | a ∈ D(π′)} ⊆ K^{×}/K^{×2}. For α ∈ S let H_{α} = {h ∈ Herm_{1}^{−1}(Q, γ) | disc(h) = α}. Then Herm_{1}^{−1}(Q, γ) = ⋃_{α∈S} H_{α}, in particular |Herm_{1}^{−1}(Q, γ)| = ∑_{α∈S} |H_{α}|. Moreover, for any α = aK^{×2} ∈ S one has |H_{α}| ⩽ \frac{1}{2} |K^{×}/N_{L/K}(L^{×})| with L = K(\sqrt{−a}).

Proof: The first part is clear. For α ∈ S, there is a pure quaternion λ ∈ Q^{×} with disc(⟨λ⟩) = −α, and (4.5) applied to L = K(λ) yields the last part. □

4.8 Corollary. Let S = {aK^{×2} | a ∈ D(π′)} and let L be the set of maximal subfields of Q. Then

|Herm_{1}^{−1}(Q, γ)| ⩽ \frac{1}{2} sup_{L∈L} |K^{×}/N_{L/K}(L^{×})| · |S| .

Proof: This is immediate from (4.7). □

4.9 Remark. We keep the notation of (4.8). Kaplansky showed in [1] that Q is the unique quaternion division algebra over K if and only if

sup_{L∈L} |K^{×}/N_{L/K}(L^{×})| = 2 .

If this condition holds, then (4.8) yields |Herm_{1}^{−1}(Q, γ)| ⩽ |S|, and as the converse inequality follows from (4.7), we obtain that |Herm_{1}^{−1}(Q, γ)| = |S|. This applies in particular to any local field. Moreover, if K is a non-dyadic local field, then |K^{×}/K^{×2}| = 4 and |S| = 3, so that we obtain immediately that u^{+}(Q) = |Herm_{1}^{−1}(Q, γ)| = |S| = 3.

4.10 Theorem. Herm_{1}^{−1}(Q, γ) is finite if and only if K^{×}/K^{×2} is finite.

Proof: Let S = {aK^{×2} | a ∈ D(π′)}. We fix a pure quaternion λ in Q and put L = K(λ).

Assume that K^{×}/K^{×2} is finite. Then S is finite. For α = aK^{×2}, there is a surjection from H_{α} to the group K^{×}/N_{L/K}(L^{×}), where L = K(\sqrt{−a}), and this group is a quotient of K^{×}/K^{×2}. Therefore H_{α} is finite for any α ∈ S. Since S is also finite, it follows that Herm_{1}^{−1}(Q, γ) = ⋃_{α∈S} H_{α} is finite.

Suppose now that Herm_{1}^{−1}(Q, γ) is finite. Then K^{×}/N_{L/K}(L^{×}) is finite by (4.5). As K^{×}/ Nrd(Q^{×}) is a quotient of this group, it is also finite. Moreover, the image of disc : Herm_{1}^{−1}(Q, γ) ⟶ K^{×}/K^{×2} is finite, which means that S is finite. Since the group of reduced norms Nrd(Q^{×}) is generated by the elements of D(π′), it follows that Nrd(Q^{×})/K^{×2} is finite. Hence, K^{×}/K^{×2} is finite. □

5 Anisotropic forms of dimension three

We keep the setting of the previous section. In this section we show that 3-dimensional anisotropic skew-hermitian forms over Q do exist except for a few exceptional cases.

5.1 Lemma. Let x, y, z ∈ Q^{×} be pure quaternions. If Nrd(xyz) ∉ D(π′), then the skew-hermitian form ⟨x, y, z⟩ over Q is anisotropic.

Proof: If ⟨x, y, z⟩ is isotropic, then ⟨x, y, z⟩ ≃ ℍ ⊥ ⟨w⟩ for some pure quaternion w ∈ Q^{×} and it follows that Nrd(xyz) = Nrd(w) ∈ D(π′). □

Recall that a preordering of a field K is a subset T ⊆ K that is closed under addition and under multiplication and contains all squares in K.

5.2 Theorem. The following are equivalent:

(1) D(π′) ∪ {0} is a preordering of K.

(2) D(π′) is closed under multiplication.

(3) D(π′) = D(π).

(4) For any a, b, c ∈ D(π′) one has abc ∈ D(π′).

If any of these conditions holds, then K is a real field and Q_{K(\sqrt{−1})} is split.

Proof: By the definition of a preordering, (1) implies (2). Since any element of Q is a product of two pure quaternions, the group of nonzero norms D(π) is generated by the elements of D(π′). Therefore (2) implies (3). Since D(π) is always a group, it is clear that (3) implies (4).

Assume now that (4) holds. Take a diagonalisation π′ ≃ ⟨a, b, c⟩. Then a, b, c ∈ D(π′), so (4) yields that abc ∈ D(π′). Since π′ has determinant 1, we have abc ∈ K^{×2} and conclude that 1 ∈ D(π′). Fixing c = 1 ∈ D(π′) we conclude from (4) that D(π′) is closed under multiplication. Hence (2) and (3) are satisfied. For a, b ∈ D(π′), we have a^{−1}b ∈ D(π′), whence 1 + a^{−1}b ∈ D(π) = D(π′) by (3) and a + b = a(1 + a^{−1}b) ∈ D(π′) by (2). Hence D(π′) is closed under addition. Therefore D(π′)∪{0} is a preordering, showing (1). Since π = ⟨1⟩ ⊥ π′ is anisotropic, this preordering does not contain −1, so K is real. Moreover, Q_{K(\sqrt{−1})} is split because 1 ∈ D(π′). □

5.3 Corollary. If K is nonreal or if Q_{K(\sqrt{−1})} is a division algebra or if D(π) ≠ D(π′), then u^{+}(Q) ⩾ 3.

Proof: By (5.2), in each case there are a, b, c ∈ D(π′) with abc ∉ D(π′). With pure quaternions x, y, z ∈ Q such that Nrd(x) = a, Nrd(y) = b, and Nrd(z) = c, the skew-hermitian form ⟨x, y, z⟩ is anisotropic by (5.1). □

5.4 Example. Let k = ℂ(X_{1}, X_{2}), Q = (X_{1}, X_{2}), and K = ℂ(X_{1}, …, X_{n}) for some n ⩾ 2. Then Q_{K} is a division algebra and u^{+}(Q_{K}) ⩽ 3 · 2^{n−2} by (2.3), because K is a C_{n}-field. By (5.3), there is an anisotropic 3-dimensional skew-hermitian form h over Q. Multiplying this form h with the quadratic form ⟨1, X_{3}⟩ ⊗ ⋯ ⊗ ⟨1, X_{n}⟩ over K, we obtain a skew-hermitian form of dimension 3 · 2^{n−2} over Q_{K}. Therefore u^{+}(Q_{K}) = 3 · 2^{n−2}.

6 Kaplansky fields

Kaplansky [1] noticed that most statements about quadratic over local fields remain valid over what he called ‘generalized Hilbert fields’, which are called ‘pre-Hilbert fields’ in [4, Chap. XII, Sect. 6]. As the relation to Hilbert’s work is vague (based on the notion of ‘Hilbert symbol’ for a local field), we use the term ‘Kaplansky field’ instead. To be precise, K is called a Kaplansky field if there is a unique quaternion division algebra over K (up to isomorphism). Natural examples of such fields are local fields and real closed fields. For the construction of other examples we refer to [4, Chap. XII, Sect. 7].

Tsukamoto [10] obtained a classification for skew-hermitian forms over the unique quaternion division algebra over a field K that is either real closed or a local number field. As observed in [10], the same result holds more generally under the condition that the field K satisfies ‘local class field theory’. In this section we show that Tsukamoto’s classification for skewhermitian forms over a quaternion division algebra Q over K is valid whenever K is a Kaplansky field, which is a strictly weaker condition. The proof is adapted from [10] and [9, Chap. 10, (3.6)].

6.1 Lemma. Let K be a Kaplansky field and let Q be the unique quaternion division algebra over K. For any pure quaternion λ ∈ Q^{×} and any d ∈ K^{×} we have ⟨λ⟩ ≃ ⟨dλ⟩ as skew-hermitian forms over Q.

Proof: Let μ ∈ Q^{×} be such that μλ = −λμ. Then Q ≃ (a, b)_{K} for a = λ^{2} and b = μ^{2}. Assume that there exists d ∈ K^{×} with ⟨λ⟩ ≄ ⟨dλ⟩. By (4.4), none of the forms ⟨1, −a⟩ and ⟨b, −ab⟩ represents d. Then (a, d)_{K} is a quaternion division algebra and not isomorphic to Q, contradicting the hypothesis. □

6.2 Theorem (Tsukamoto). Let K be a Kaplansky field and let Q be the unique quaternion division algebra over K.

(a) Any skew-hermitian form of dimension at least 4 over Q is isotropic.

(b) Skew-hermitian forms over Q are classified by their dimension and discriminant.

(c) A 2-dimensional skew-hermitian form over Q is isotropic if and only if it has trivial discriminant.

(d) Any 3-dimensional skew-hermitian form over Q with trivial discriminant is anisotropic.

Proof: Let γ denote the canonical involution on Q. We first show that 1dimensional skew-hermitian forms over Q are classified by the discriminant. Let z_{1}, z_{2} ∈ Sym^{−}(Q, γ) and assume that the skew-hermitian forms ⟨z_{1}⟩ and ⟨z_{2}⟩ over Q have the same discriminant. According to (4.2), then ⟨z_{1}⟩ ≃ ⟨cz_{2}⟩ for some c ∈ K. Since also ⟨z_{2}⟩ ≃ ⟨cz_{2}⟩ by (6.1), we obtain that ⟨z_{1}⟩ ≃ ⟨z_{2}⟩.

(a) Let z_{1}, z_{2} ∈ Sym^{−}(Q, γ) be such that the skew-hermitian form ⟨z_{1}, z_{2}⟩ over Q has trivial discriminant. Then Nrd(z_{1}) and Nrd(z_{2}) represent the same class in K^{×}/K^{×2}. This means that the 1-dimensional forms ⟨z_{1}⟩ and ⟨−z_{2}⟩ have the same discriminant, whence ⟨z_{1}⟩ ≃ ⟨−z_{2}⟩ by what we showed above.

(b) Let φ be a 3-dimensional skew-hermitian form over Q. If φ is isotropic, then φ ≃ ℍ⊥⟨a⟩ where a ∈ Sym^{−}(Q, γ), and it follows that φ has the same discriminant as ⟨a⟩, which cannot be trivial by part (a).

(c) Let φ be a 4-dimensional skew-hermitian form over Q. There exist a_{1}, …, a_{4} ∈ Sym^{−}(Q, γ) with φ ≃ ⟨a_{1}, a_{2}, a_{3}, a_{4}⟩. As dim_{K}(Sym^{−}(Q, γ)) = 3, there exist c_{1}, …, c_{4} ∈ K, not all zero, such that c_{1}a_{1} +c_{2}a_{2} +c_{3}a_{3} +c_{4}a_{4} = 0. By the first paragraph of the proof, for 1 ⩽ i ⩽ 4 there is some d_{i} ∈ Q with c_{i}a_{i} = γ(d_{i})a_{i}d_{i}. Then ∑_{i=1}^{4} γ(d_{i})a_{i}d_{i} = 0 and thus φ is isotropic.

(d) Let φ and ψ be two n-dimensional skew-hermitian forms over Q for some n ⩾ 1, and assume that both forms have the same discriminant. By (b), the 2n-dimensional form φ ⊥ −ψ then splits off n−1 hyperbolic planes. The remaining 2-dimensional form has trivial discriminant and thus is hyperbolic by (a). Therefore φ ⊥ −ψ is hyperbolic, which means that φ ≃ ψ. □

6.3 Corollary. Let Q be a quaternion division algebra over K. Skewhermitian forms over Q are classified by dimension and discriminant if and only if K is a Kaplansky field.

Proof: By (6.2) the condition is sufficient. To show its necessity, suppose that Q is not the unique quaternion division algebra over K. By (4.9), there exists λ ∈ Q \ K such that, for the field L = K(λ) ⊆ Q, the index of N_{L/K}(L^{×}) in K^{×} is at least 4. Let a, b ∈ K^{×} be such that λ^{2} = a and Q ≃ (a, b)_{K}. Now, there exists c ∈ K^{×} such that neither c nor bc is a norm of L/K. Then the two 1-dimensional skew-hermitian forms ⟨λ⟩ and ⟨cλ⟩ over Q have the same discriminant, but they are not isometric by (4.4). □

6.4 Corollary. Let K be a nonreal Kaplansky field and let Q be the unique quaternion division algebra over K. Then u^{+}(Q) = 3.

Proof: we have u^{+}(Q) ⩽ 3 by (6.2) and u^{+}(Q) ⩾ 3 by (5.3). □

The field K is said to be euclidean if K^{×2} ∪ {0} is an ordering of K, or equivalently, if K is real and K^{×} = K^{×2} ∪ −K^{×2} (cf. [4, Chap. VIII, (4.2)]). If K is euclidean, then (−1, −1)_{K} is the unique quaternion division algebra over K, in particular K is a Kaplansky field.

6.5 Proposition. Let Q be a quaternion division algebra over K and γ its canonical involution. The following are equivalent:

(1) u^{+}(Q) = 1.

(2) |Herm_{1}^{−1}(Q, γ)| = 1.

(3) K is euclidean and Q ≃ (−1, −1)_{K}.

Proof: The equivalence of (1) and (2) is clear. If (3) holds, then K is a Kaplansky field and any 1-dimensional skew-hermitian form over Q has trivial discriminant, and by (6.2) this implies (2).

Suppose that (1) and (2) hold. From (2) it follows that D(π′) = K^{×2}, whence π′ ≃ ⟨1, 1, 1⟩ and ∑ K^{×2} = K^{×2}. Therefore Q ≃ (−1, −1)_{K} and −1 ∉ K^{×2} = ∑ K^{×2} as Q is not split. So K is real. To prove (3), it remains to show that K^{×} = K^{×2} ∪ −K^{×2}. We fix i ∈ Q with i^{2} = −1 and L = K(i). For any a ∈ K^{×}, the skew-hermitian form ⟨i, −ai⟩ over Q is isotropic by (1), whence a ∈ N_{L/K}(L^{×}) ∪ −N_{L/K}(L^{×}) = K^{×2} ∪ −K^{×2} by (4.4). □

6.6 Proposition. Let K be a real Kaplansky field and let Q = (−1, −1)_{K}. Then u^{+}(Q) ⩽ 2.

Proof: Let i be a pure quaternion in Q with i^{2} = −1. By (6.2), the skewhermitian form ⟨i, i⟩ over Q is isotropic. We claim that every 2-dimensional skew-hermitian form over Q is isometric to ⟨i, z⟩ for some pure quaternion z ∈ Q^{×}. Once this is shown, it follows that every 3-dimensional skew-hermitian form over Q contains ⟨i, i⟩ and therefore is isotropic.

Let h be a 2-dimensional skew-hermitian form over Q. We write disc(h) = aK^{×2} with a ∈ K^{×}. Then a ∈ Nrd(Q^{×}) and a is a sum of four squares in K. Since K is a real Kaplansky field, the quaternion algebra (−1, a)_{K} is split, because it is not isomorphic to (−1, −1)_{K}. Therefore a is a sum of two squares in K. It follows that there is a pure quaternion z in Q with Nrd(z) = a. Then the skew-hermitian form ⟨i, z⟩ over Q has discriminant a and is therefore isometric to h, by (6.2). □

6.7 Example. Let K be a maximal subfield of ℝ with 2 ∉ K^{×2}. Then K is a real field with four square classes represented by ±1, ±2, and Q = (−1, −1)_{K} is the unique quaternion division algebra over K. Since Q ≃ (−1, −2)_{K}, there are anticommuting pure quaternions α, β ∈ Q with α^{2} = 1 and β^{2} = 2. Then the skew-hermitian form ⟨α, β⟩ over Q has nontrivial discriminant 2K^{×2}, so it is anisotropic. This together with (6.6) shows that u^{+}(Q) = 2.

6.8 Theorem. Let K be a Kaplansky field and let Q be the unique quaternion division algebra over Q. Then

u^{+}(Q) = \{ \begin{matrix}1 if K is real euclidean, \\ 2 if K is real non-euclidean, \\ 3 if K is nonreal. \end{matrix}

Proof: This follows from (6.2), (6.5), (6.6), and (5.3). □

Acknowledgements

We wish to express our gratitude to David Lewis and to Jean-Pierre Tignol for inspiring discussions on the subject of this work. We further wish to acknowledge financial support provided by the European RTN Network ‘Algebraic K-Theory, Linear Algebraic Groups and Related Structures’ (HPRN CT-2002-00287), the Swiss National Science Foundation (Grant No. 200020-100229/1), the Irish Research Council for Science (Basic Research Grant SC/02/265), and the Research Council of Sharif University of Technology.

References

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[3] M.-A. Knus, S. A. Merkurjev, M. Rost, J.-P. Tignol. The Book of Involutions. American Mathematical Society Colloquium Publications, 44. Providence, RI, 1998.

[4] T. Y. Lam, Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics, 67. American Mathematical Society, Providence, RI, 2005.

[5] D. W. Lewis. Quadratic forms, quaternion algebras and Hilbert’s theorem 90. Preprint (1979).

[6] D. W. Lewis. Sums of Hermitian Squares. J. Algebra 115 (1988): 466– 480.

[7] M. G. Mahmoudi. Hermitian forms and the u-invariant. Manuscripta Math. 116 (2005): 493–516.

[8] A. Pfister. Quadratic forms with applications to algebraic geometry and topology. London Mathematical Society Lecture Note Series, 217. Cambridge University Press, Cambridge, 1995.

[9] W. Scharlau. Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Springer-Verlag, Berlin, 1985.

[10] T. Tsukamoto. On the local theory of quaternionic anti-hermitian forms. J. Math. Soc. Japan 13 (1961): 387–400.

Karim Johannes Becher, Fachbereich Mathematik und Statistik, D203, Universität Konstanz, 78457 Konstanz, Germany.

Email: becher@maths.ucd.ie

Mohammad Gholamzadeh Mahmoudi, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box: 11155-9415, Tehran, Iran.

Email: mmahmoudi@sharif.ir

27 February 2008