Tsukamoto’s Theorem in Characteristic two

In this paper it is proved that hermitian forms over quaternion division algebras over local fields of characteristic two are classified by their dimension and discriminant.


Introduction
The Tsukamoto's theorem classifies skew-hermitian forms over quaternion division algebras over local fields of characteristic different from two.It was generalized by Becher and Mahmoudi to a quaternion division algebra over a Kaplansky field, (see §6 of [2]).In this article we consider non-singular (or regular) hermitian forms over a quaternion division algebra over a local field of characteristic two and we show that Tsukamoto's classification is also valid in this case.We see that these forms correspond to skew-hermitian forms over quaternion division algebras over fields of characteristic different from two.The main theorem (see Theorem 3.1) is very similar to theorem 3.6 of Chapter 10 of [7] and the Theorem 3 of [8].However we can see in the following corollary that the structures of these forms are independent of the characteristic.
In order to state our results we need some notation.Throughout this paper F will always denotes a field of characteristic two, and Ḟ its multiplicative group of nonzero elements.
We denote by Q a quaternion algebra over a field F. There always exists an Fbasis {1, i, j, k} of Q with multiplication given by ij + ji = i, j 2 + j = b ∈ F, i 2 = a ∈ Ḟ , ji = k, (see Chapter 8, Section 11 of [7]).Every basis {1, i, j, k} satisfying the above relations is called standard basis of the quaternion algebra.In this case, we also denote Q by (b, a)/F .
The standard involution σ : Q → Q is given by σ(x) = α + β + βj + γi + δk, for all x = α + βj + γi + δk ∈ Q.The element xσ(x) belongs to F, and is called the norm of x and is denoted by N (x).Considering the standard basis {1, i, j, k} of the F -vector space Q, the norm N : In general, the two-dimensional quadratic forms cα 2 + dβ 2 and ecα 2 + eαβ + edβ 2 over F will be denoted by [c] ⊥ [d] and e [c, d], respectively.For a quadratic form q : V → F (V an F -vector space), D F q = {q(u) ∈ F, u ∈ V \ {0}} denotes the subset of elements of F represented by q.For instance, The quadratic form q is universal if q represents all α ∈ Ḟ .We refer to [1] for general facts about quadratic forms in characteristic two.
The element x ∈ Q is said to be symmetric if σ(x) = x, and we denote by

Preliminaries
Let Q be a quaternion division algebra over a field F. An hermitian form on a finite dimensional Q-right vector space V is a map h : V × V → Q which satisfies the following conditions: and σ(h(u, v)) = h(v, u), for all u, v, w ∈ V and all α, β ∈ Q.
We will refer to h as being an hermitian form over Q and V as its underlying vector space.The pair (V, h) is called an hermitian space.The Q-dimension of V is said to be the dimension of h over Q; dim Q h, and also the dimension of the hermitian space (V, h) over Q.
The hermitian form h over Q (or hermitian space (V, h) is said to be regular Otherwise, h or (V, h) is said to be singular or degenerate.
We say that an hermitian form h, or hermitian space (V, h) is isotropic if there exists a vector u ∈ V \ {0} such that h(u, u) = 0, and h or (V, h) is anisotropic in otherwise.
We say that the hermitian form h represents the element z ∈ Q if there exists u ∈ V \ {0} such that h(u, u) = z.Denote by Dh the subset of elements of Q represented by h.Thus 0 ∈ Dh if and only if h is isotropic.Of course, Dh ⊂ Sym(Q) and we say that h is universal if h represents all z ∈ Sym(Q).
An isometry between two hermitian spaces (V 1 , h 1 ) and (V 2 , h 2 ), or between h 1 and h 2 is an isomorphism of Q-vector spaces τ : Given two hermitian spaces An hermitian form h with underlying vector space V is said to be diagonalizable if there exists a basis {e 1 , e 2 , . .
We denote h by a 1 , a 2 . . ., a n , or also by n a , if a i = a for all i = 1, 2, . . ., n.It follows that h is regular if and only if a i = 0, for all a i ∈ Q.
Two elements a, b ∈ Sym(Q) are congruent if there exists c ∈ Q such that b = σ(c)ac, which is equivalent to saying that a ≃ b over Q.
For an element a ∈ Ḟ we define the scaled hermitian form ah by (ah)(u, v) = a.h(u, v), for all u, v belonging to underlying vector space of h.In particular a a 1 , a 2 , . . ., a n = aa 1 , aa 2 , . . ., aa n .Two hermitian forms h and h 1 over Q are said to be similar if h 1 ≃ ah, for some a ∈ Ḟ .
The Grothendieck group and the Witt group of the regular hermitian forms over Q are denoted, respectively, by W (Q) and by W (Q). A regular hermitian space (V, h) such that there exists a decomposition is called metabolic hermitian space.We denote the twodimensional metabolic hermitian space h = 0 1 1 a by IM(a).The following lemma is due to knebusch and can be seen in ( [4], Chapter I, Proposition 3.7.6)or ( [7], Chapter 7, Lemma 3.7).
Lemma 2.1.Let (V, h) be a metabolic hermitian space.Then (V, h) Proposition 2.2.(Chapter I; 6.1.1 and 6.1.4 of [4]) Let (V, h) be a regular hermitian space over Q.There exists an orthogonal decomposition ), with h an anisotropic or zero and r ≥ 0. Furthermore, (V ′ , h an ) is uniquely determined up to isometry by (V, h).In particular, (V, h), (or h) is isotropic if and only if r ≥ 1.
We write h ≃ h an ⊥ h IM , where h IM is a metabolic hermitian space.
As in [8] and also [7] the discriminant of an hermitian form h (or hermitian space (V, h)) over Q will be denoted by disc(h) and it is defined as follows: Let {e 1 , e 2 , . . ., e n } be an Q-basis of V. Denoting by N rd the reduced norm from M n (Q) • to Ḟ , we put disc(h) = (−1) n N rd (h(e i , e j )) mod Ḟ 2 .It is known that disc(h) is independent of the choice of the basis of V and is also independent of the choice of the splitting field of Q,( see Chapter 8, Lemma 5.7 of [7], 16.1 of [5] or §22 of [3]).In particular, given the quaternion algebra (b, a)/F , if we take the algebraic closure F of F, we have an F -algebra homomorphism ϕ : (b, a)/F → M 2 (F ) given by ϕ(i) = i 0 and ϕ(j) = j 0 , where i 0 = 0 α α 0 and j 0 = Since reduced norm is multiplicative ( [3], §22 (7) and §20, Theorem 1 and §22, Theorem 1) or ( [5], §16.5, Corollary b), it follows that disc(h ) is an isometry and B 1 , B 2 are Fbasis of V 1 and V 2 respectively, take (h 1 ) B1 , (h 2 ) B2 , (α ij ) = (τ ) B1B2 the matrices of h 1 , h 2 and τ , with respect to the given basis.Then (h where σ : Q → Q, is the standard involution.From Lemma 5 of ( [3], §22) we get Thus N rd and disc does not depend of the isometry class of (V 1 , h 1 ).Furthermore, as hyperbolic space and metabolic hermitian space has Dieudonné determinant 1. Ḟ 2 , (see [3] §19 Example 1, §20 Definitions 1 and 3) the Proposition 5 of [8] holds for any characteristic: The mapping W (Q) → Ḟ / Ḟ 2 will also be denoted by disc.

Main Results
The field F in question is local field of characteristic two, that is, F = K((t)) (the field of Laurent's power series of K), where K is a finite field of characteristic two.Every element f ∈ F is of the form f = t m (1+a f can be written in the form f = g 2 + th 2 , for some g, h ∈ F. Thus {1, t} is a basis for the F 2 -vector space F and the quadratic form (b) A two-dimensional regular hermitian form over Q is isotropic if and only if has trivial discriminant.
(c) Any regular hermitian form even, for some metabolic hermitian space h IM and h an = 0 or 1, z for some z ∈ Sym(Q).
(d) Let h 1 and h 2 be regular hermitian forms of equal dimension over Thus the hermitian forms z 0 and 1, . . ., z 0 have discriminant α.Now, we show (d) for 1-dimensional forms.Let z 1 , z 2 ∈ Sym(Q) and assume that hermitian forms z 1 and z 2 over Q have the same discriminant.According to Proposition (2.4), by Lemma (2.5) we obtain cz 2 ≃ z 2 and so z 1 ≃ z 2 .
(b) Let z 1 , z 2 ∈ Sym(Q) be such that the form z 1 , z 2 has discriminant 1.Then N rd( z 1 ) and N rd( z 2 ) represent the same element in Ḟ / Ḟ 2 .This means that z 1 ≃ z 2 by what we showed above.It follows that z 1 , z 2 is isotropic.
Conversely, if h is an 2-dimensional regular hermitian form over Q and h is isotropic then there is a basis B = {u, v} such that h(u, u) = 0, h(u, v) = h(v, u) = 1.Thus h ≃ IM(h(v, v)) and disc(h) = 1.
(c) First we give Tsukamoto's argument to show that every 3-dimensional regular hermitian form over Q is isotropic.Suppose that h is anisotropic.Since h can be diagonalized ( [4], Chapter I, Lemma 6.2.1) we may assume that h = z 1 , z 2 , z 2 , with z 1 , z 2 , z 3 ∈ Sym(Q)\{0}.From (a) there exists z 0 ∈ Sym(Q) such that N rd( z 0 ) = disc(h).As Sym(Q) has F -dimension 3, there exist c 0 , c 1 , c 2 , c 3 ∈ F, not all zero, such that c 0 z 0 + c 1 z 1 + c 2 z 2 + c 3 z 3 = 0.For c i = 0, the Proposition (2.4) implies that c i z i and z i are similar, that is, c i z i = σ(d i )z i d i , for some d i ∈ Ḟ , i = 0, 1, 2, 3.If we take d i = 0 for c i = 0, we obtain h is isotropic, absurd.This concludes the first part.From Proposition (2.2) and the first part every regular hermitian form h with dim Q h ≥ 2 is the form h ≃ z ⊥ h IM , for some z ∈ Sym(Q), if dim Q h is odd, and h ≃ h an ⊥ h IM , for some metabolic hermitian space h IM and h an = 0 or for some two-dimensional regular hermitian form h 1 and a ∈ Sym(Q).As before disc(h 1 ) = 1 and by part (b) and Proposition (2.2) we obtain 1, z 0 ⊥ h an = 0 in W (Q). Thus h an ≃ 1, z 0 .

[ 1 ]
⊥ [t] is universal over F. The unique quaternion division algebra over F, up to isomorphism, is Q = (b, t)/F , for some b ∈ F and their norm form is N = [1, b] ⊥ t [1, b] up to isometry, (see, for instance, ( [1], Chapter II, Proposition 1.19 and [6], Lemma 1.7) Theorem 3.1.Let F = K((t)) be a local field of characteristic two and Q = (b, t)/F be the unique quaternion division algebra over F, up to isomorphism.Then (a) For any dimension ≥ 1 there are regular hermitian forms of any discriminant.
Proposition 2.4.Two one-dimensional hermitian forms over Q are similar if and only if their discriminants are the same in Ḟ / Ḟ 2 .Let λ ∈ Sym(Q)\{0} and c ∈ Ḟ .If λ / ∈ F, then the hermitian forms λ and cλ are isometric over Q if and only if c is represented over F by the quadratic form [1] ⊥ [a].If λ ∈ F, then the hermitian form λ and cλ are isometric over Q if and only if c is a norm in F.