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On the Hermitian U-Invariant of a Quaternion Algebra

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On the Hermitian U-Invariant of a Quaternion Algebra On the hermitian u-invariant of a quaternion algebra Karim Johannes Becher ∗, Mohammad G. Mahmoudi † Abstract The u-invariant of a field is the smallest cardinal u such that every quadratic form in more than u variables over the field has a nontrivial zero. To determine this invariant for particular fields is in the focus of current quadratic form theory. Considering hermitian forms over a division algebra with involution gives rise to a corresponding notion of u-invariant. Here, we investigate this invariant for a quaternion algebra with involution. For several cases, we obtain upper and lower bounds on this invariant. Keywords: hermitian form, isotropy, dimension, algebra with involu- tion, u-invariant Classification (MSC 2000): 11E04, 11E39, 11E81 1 Summary In this article we consider hermitian and skew-hermitian forms over a division algebra with involution over field of characteristic different from 2. The basics about hermitian forms over a division algebra with involution will be recalled in Section 2. In Section 3, we introduce the notion u-invariant of a division algebra with involution. For a division algebra of exponent two we define the orthogonal and the symplectic u-invariant. Section 4 shows how ∗Supported by the Swiss National Science Foundation (Grant No. 200020-100229/1) †Supported by the Irish Research Council for Science (Basic Research Grant SC/02/265) 1 Kneser’s Theorem, relating the classical u-invariant to the number of square classes of a given nonreal field, can be extended to our setting. In Section 5 we investigate the equivalence relation on pure quaternions induced by isom- etry of 1-dimensional skew-hermitian forms on a quaternion division algebra. We obtain an upper bound on the number of isometry classes of those forms. On the way to this result, we retrieve an exact sequence due to Lewis [10], which involves the set of 1-dimensional skew-hermitian forms over a quater- nion algebra and and the square class group of a maximal subfield. Section 6 provides a fairly general method to prove the existence of 3-dimensional skew-hermitian forms over a quaternion division algebra with its canonical involution, showing that the orthogonal u-invariant can be 1 or 2 only in very special cases. In Section 7, we consider fields with a unique quaternion division algebra and obtain a complete classification of skew-hermitian forms in this case. These are straightforward generalisations classical results due to Jacobson [4] and Tsukamoto [14] concerning skew-hermitian forms over quaternion algebras over real closed fields and over local fields. In the final Section 8, we give general upper bounds for the hermitian u-invariants over a division algebra which is defined over a Ci-field, which is sharp for quaternion algebras. 2 Involutions and hermitian forms Throughout this article K denotes a field of characteristic different from 2. Let D be a division ring whose center is equal to K and such that dimK (D) is finite; we refer to such D as a K-central division algebra, for short. We further assume that D is endowed with an involution σ, by which we mean a map σ : D → D such that σ ◦ σ is the identity on D and such that, for any a, b ∈ D, one has σ(a + b) = σ(a) + σ(b) and σ(ab) = σ(b)σ(a) Note that σ restricts to an involution of the center K. There are two cases to be distinguished. If σ|K is the identity on 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 D. 2 Note that σ : D → D is k-linear. If σ is of the second kind, then K/k is a quadratic extension. Let ε ∈ K× with σ(ε)ε = 1. We are mostly interested in the cases where ε = ±1 and in fact, if σ is of the first kind then these are the only possibilities for ε. An ε-hermitian form over (D, σ) is –strictly speaking– a pair (V, h) where V is a finite-dimensional D-right module and h is a map h : V ×V → D which is K-linear in the second argument and which satisfies σ(h(x, y)) = ε · (h(y, x)) for any x, y ∈ V. It follows then that h is ‘sesquilinear’ in the sense that h(xa, yb) = σ(a) · h(x, y) · b for any x, y ∈ V , a, b ∈ D. We will, however, refer in this setting to h as the ε-hermitian form and to V as its underlying space. We simply say that h is hermitian (resp. skew- hermitian) if it is ε-hermitian for ε = 1 (resp. ε = −1). The simplest case occurs when D = K is a field, σ = idK , and ε = 1. In that case, a 1-hermitian form over K is a symmetric bilinear form b : V × V → K on a finite dimensional vector space V over K, and it can be identified (via the choice of a basis) to a quadratic form (i.e. a homogeneous polynomial of degree 2) over K in n = dimK (V ) variables. An ε-hermitian form (V, h) on (D, σ) is regular if, for any x ∈ V \{0}, the associated K-linear form V → K, y 7→ h(x, y) is nontrivial. We say that h is isotropic if there exists x ∈ V \{0} such that h(x, x) = 0. Let h1, h2 be two ε-hermitian forms over (D, σ), with underlying spaces V1,V2. The orthogonal sum h1 ⊥ h2 is the ε-hermitian form on the D-vector space V = V1 × V2 defined by h(x, y) = h1(x1, y1) + h2(x2, y2) for any x = (x1, x2), y = (y1, y2) ∈ V . An isometry between h1 and h2 is an isomorphism of D-vector spaces τ : V1 → V2 such that h1(x, y) = h2(τ(x), τ(y)). If such τ exists, then h1 and h2 are isometric and we may write h1 ' h2 to indicate this; note that τ −1 then is an isometry, too. Witt’s Cancellation Theorem (cf. [7, (6.3.4) Theorem]) says that, whenever h1, h2 and h are ε-hermitian forms on (D, σ) such that h1 ⊥ h ' h2 ⊥ h, then one has an isometry h1 ' h2 as well. For a ε-hermitian form on (D, σ) we write ∆(h) = {h(x, x) | x ∈ V \{0}} ⊂ D. Note that this set contains 0 if and only if h is isotropic. We further denote Symε(D, σ) = {x ∈ D | σ(x) = εx} . 3 Elements of Symε(D, σ) are said to be ε-symmetric. For any ε-hermitian form ε h over (D, σ) we have ∆(h) ⊂ Sym (D, σ). Conversely, given a1, . , an ∈ Symε(D, σ), we define an ε-hermitian form h on the D-vector space V = Dn by h(x, y) = σ(x1)a1y1 + ··· + σ(xn)anyn n for any x = (x1, . , xn), y = (y1, . , yn) ∈ D = V . We denote this form h by ha1, . , ani and call this a diagonal form. Note that it is regular if and only if ai 6= 0 for i = 1, . , n. Since we assumed that char(K) 6= 2, any ε-hermitian form is isometric to some diagonal form (cf. [7, (6.2.4) Proposition]). ε We denote by Hermn(D, σ) the set of isometry classes of regular n- dimensional ε-hermitian forms over (D, σ). We have the surjection ε ε Sym (D, σ) \{0} −→ Herm1(D, σ) , which maps a to the class of hai. Two elements a, b ∈ Symε(D, σ) are congruent if there exists c ∈ D such that a = σ(c)bc; we write a ∼σ b to indicate this. In fact, a and b are congruent if and only if the ε-hermitian forms hai and hbi over (D, σ) are isometric. 2.1 Remark. In the case where D = K and ε = 1, then the elements of ε Herm1(D, σ) are in natural one-to-one correspondence to the elements of the square class group K×/K×2, hence we may identify the two sets and thus 1 endow Herm1(D, σ) with a natural group structure. One can proceed in a similar way in the two (similarly simple) cases, first when D is a quaternion algebra and σ is the canonical involution on D, and second when σ is a unitary involution on a field D = K. As in [14], we define the discriminant of a skew-hermitian form (V, h) over a quaternion division algebra D = (a, b)K as follows. Let (x1, ··· , xn) be a × × D-basis of V . Denoting by N the reduced norm from Mn(D) to K , we n ×2 put disc(h) = (−1) N(h(xi, xj)) mod K . For a more general definition of the discriminant of a hermitian form, see [1, Sect. 3]. 2.2 Remark. Assume that D is a K-quaternion division algebra and let γ denote its canonical involution. For a ∈ K×, there exists a skew-hermitian form of dimension 1 and discriminant a over (D, γ) if and only if −a is represented by the pure part of the norm form of D. In particular, any skew-hermitian form of dimension 1 over (D, γ) has nontrivial discriminant. 4 Given an ε-hermitian forms h over (D, σ) and an element a ∈ k×, we define the scaled ε-hermitian form ah in the obvious way.
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