Harrison’s criterion, Witt equivalence and reciprocity equivalence

N. Grenier-Boley^{∗}

February 5, 2006

Abstract. Harrison’s criterion characterizes the isomorphy of the Witt rings of two fields in terms of properties of these fields. In this article, we discuss about the existence of such characterizations for the isomorphism of Witt groups of hermitian forms over certain algebras with involution. In the cases where we consider the Witt group of a quadratic extension with its non-trivial automorphism or the Witt group of a quaternion division algebra with its canonical involution, such criteria are proved. In the framework of global fields, the first of these criteria is reformulated in terms of properties involving real places of the considered fields.

Mathematics subject classification (2000): Primary 11E39 ; Secondary 11E81

Keywords: Harrison’s criterion, Witt equivalence, reciprocity equivalence, algebras with involution.

1 Introduction

One of the basic questions in the algebraic theory of quadratic forms is to give necessary and sufficient conditions for two fields K_{1} and K_{2} to have isomorphic Witt rings: in this case, K_{1} and K_{2} are said to be Witt equivalent. In [7], Harrison expresses Witt equivalence in the following terms:

Theorem 1.1 (Harrison). Let K_{1} and K_{2} be two fields of characteristic different from 2. Then the following are equivalent:

(1) K_{1} and K_{2} are Witt equivalent.

(2) There is a group isomorphism t : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2} sending −1 to −1 such that the quadratic form ⟨x, y⟩ represents 1 over K_{1} if and only if the quadratic form ⟨t(x), t(y)⟩ represents 1 over K_{2} for all x, y ∈ K_{1}^{∗}.

In the literature, the previous theorem is known as “Harrison’s criterion”. In [4, Theorem 2.3], Cordes shows that the two conditions of Theorem 1.1 are equivalent to the following one, known as “Harrison-Cordes condition”:

(3) There is a group isomorphism t : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2} sending −1 to −1 such that t(D_{K_{1}}(⟨1, a⟩)) = D_{K_{2}}(⟨1, t(a)⟩) (if K is a field of characteristic different from 2 and q is a nondegenerate quadratic form, recall that D_{K}(q) denotes the set of nonzero square classes represented by q). With this criterion, Cordes classifies Witt rings of order at most 32 up to Witt equivalence by their group structure: see [4, Theorem 7.1].

∗The author was supported by the European research network HPRN-CT-2002-00287 “K-theory, Linear Algebraic Groups and Related Structures”

In [1], Baeza and Moresi study the possibilities to extend Harrison’s criterion to fields K_{1} and K_{2} of characteristic 2. On the one hand, they show that the bilinear Witt rings W (K_{1}) and W (K_{2}) of K_{1} and K_{2} are isomorphic if and only if K_{1} and K_{2} are isomorphic in the case where dim_{K_{1}}_{2} K_{1} = dim_{K_{2}}_{2} K_{2} > 2, and they give a complete treatment of the cases where dim_{K_{1}}_{2} K_{1} = dim_{K_{2}}_{2} K_{2} = 1, 2: see [1, Theorem 2.9, Proposition 2.10]. On the other hand, in [1, Theorem 3.1], they characterize the isomorphy of the quadratic Witt modules W_{q}(K_{1}) et W_{q}(K_{2}) in the following way:

Theorem 1.2 (Baeza-Moresi). Let K_{1} and K_{2} be two fields of characteristic 2. Then the following are equivalent:

(1) There exist a ring isomorphism Φ : W (K_{1}) → W (K_{2}) and a group isomorphism Ψ : W_{q}(K_{1}) → W_{q}(K_{2}) such that

Ψ(b.q) = Φ(b).Ψ(q)

for all b ∈ W (K_{1}) and for all q ∈ W_{q}(K_{1}).

(2) There exist groups isomorphisms

t_{1} : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2}, t_{2} : K_{1}/℘(K_{1}) → K_{2}/℘(K_{2})

such that

t_{1}(D_{K_{1}}(⟨1, a⟩)) = D_{K_{2}}(⟨1, t_{1}(a)⟩), t_{2}(D_{K_{1}}[1, b]) = D_{K_{2}}[1, t_{2}(b)]

for all a ∈ K_{1}^{∗} and for all b ∈ K_{1} (where ℘(K_{i}) = {a + a^{2} | a ∈ K_{i}}, i = 1, 2).

Note that the first condition in (2) is similar to “Harrison-Cordes condition”.

Such criteria are very useful. For example, Harrison’s criterion is used by Mináč and Spira to connect the Witt equivalence of two fields K_{1} and K_{2} to the isomorphy of some groups G_{K_{1}} and G_{K_{2}} (called W-groups), G_{K_{i}} being the Galois group of a certain field extension K_{i}^{(3)} of K_{i} for i = 1, 2: see [14].

In this context, a natural question arises: is it possible to obtain such criteria for the Witt group of a central simple algebra with involution ? After recalling some notations and basic facts in Section 2, we explain how to obtain such criteria in two particular cases in Section 3. First, we obtain a criterion similar to Theorem 1.1 for the Witt group of a quadratic field extension equipped with its nontrivial automorphism:

Theorem 1.3. Let K_{1} and K_{2} be two fields of characteristic different from 2. Let L_{1} = K_{1}(\sqrt{a}) (resp. L_{2} = K_{2}(\sqrt{b})) be a quadratic field extension of K_{1} (resp. K_{2}) equipped with its non trivial automorphism σ_{1} (resp. σ_{2}). Then, the following are equivalent:

(1) W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) as rings.

(2) There is a group isomorphism t : K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) → K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗}) sending −1 to −1 such that the quadratic form ⟨⟨a, x, y⟩⟩ is hyperbolic over K_{1} if and only if the quadratic form ⟨⟨b, t(x), t(y)⟩⟩ is hyperbolic over K_{2} for all x, y ∈ K_{1}^{∗}, where N_{L_{i}/K_{i}}(L_{i}^{∗}) denotes the norm group of the extension L_{i}/K_{i} for i = 1, 2.

Moreover, in Assertion (2), we show that the condition “t(−1) = −1” is not a consequence of the two other conditions: see Example 3.5. Next, in the case of the Witt group of a quaternion division algebra endowed with its canonical involution, two examples point out that its structure of module seems to be the right choice to obtain such a criterion: see Examples 3.6. In this direction, we obtain Theorem 3.9 which is at the same time similar to Theorem 1.1 and to Theorem 1.2 (in some sense). At first sight, Theorem 3.9 seems to be more analogous to Baeza and Moresi’s results, but the similarity with Harrison’s criterion shows up by taking K_{1} = K_{2} = K:

Corollary 1.4. Let Q_{1} (resp. Q_{2}) be quaternion division algebras over K endowed with its canonical involution γ_{1} (resp. γ_{2}). Then, the following are equivalent:

(1) W (Q_{1}, γ_{1}) ≃ W (Q_{2}, γ_{2}) as W (K)-modules.

(2) There is a group isomorphism t̃: K^{∗}/Nrd_{Q_{1}/K}(Q_{1}^{∗}) ≃ K^{∗}/Nrd_{Q_{2}/K}(Q_{2}^{∗}) with t̃(−1) = −1 such that the quadratic form ⟨⟨a, b, u, v⟩⟩ is hyperbolic over K if and only if the quadratic form ⟨⟨c, d, t̃(u), t̃(v)⟩⟩ is hyperbolic over K for all u, v ∈ K^{∗}.

In this framework, another interesting problem is to give necessary and sufficient conditions for two global fields to be Witt equivalent. This problem is now entirely solved. The first step is due to Baeza and Moresi who show that two global fields of characteristic 2 are always Witt equivalent: see [1] or [16, §2]. Besides, if two global fields are Witt equivalent then either char(K_{1}), char(K_{2}) ≠ 2 or char(K_{1}) = char(K_{2}) = 2: see [16, §2]. Lastly, in [16, §3, §4], Perlis, Szymiczek, Conner and Litherland prove that two global fields K_{1} and K_{2} of characteristic different from 2 are Witt equivalent if and only if they are reciprocity equivalent (i.e. if there exist a group isomorphism t between their square class groups and a bijection T between their non trivial places such that the Hilbert symbols (x, y)_{P} and (t(x), t(y))_{T (P)} are equal for any x, y ∈ K_{1}^{∗}/K_{1}^{2} and for any non trivial place P over K_{1}).

In Section 4, we explain how to obtain such a classification for the Witt group of a quadratic field extension endowed with its nontrivial automorphism. For this purpose, we define the notion of (a, b)-quadratic reciprocity equivalence between two global fields of characteristic different from 2. This notion is a natural adaptation of the reciprocity equivalence when considering norm class groups of quadratic extensions instead of square class groups (see Definition 4.7).

Theorem 1.5. Let K_{1} and K_{2} be two global fields of characteristic different from 2. Let L_{1} = K_{1}(\sqrt{a}) (resp. L_{2} = K_{2}(\sqrt{b})) be a quadratic field extension of K_{1} (resp. K_{2}) equipped with its nontrivial automorphism σ_{1} (resp. σ_{2}). Then, the following are equivalent:

(1) W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) as rings.

(2) There is an (a, b)-quadratic reciprocity equivalence between K_{1} and K_{2}.

2 Basic results and notations

In this Section, we fix some notations and recall some basic definitions and results.

2.1 Central simple algebras wth involution

The general reference for the theory of central simple algebras with involution is [10]: see also [17, Chapter 8]

In this Section, K will denote a field of characteristic different from 2, D will denote a finitedimensional division algebra over K. Then dim_{K} D = n^{2} for some n ∈ ℕ, and n = deg D is called the degree of D. Suppose that D is endowed with an involution σ. The map σ restricts to an involution of the center K and we can distinguish two cases: if σ|_{K} is the identity, we say that σ is of the first kind, otherwise σ|_{K} is an automorphism of order two of K and we say that σ is of the second kind.

A central simple algebra D of degree 2 is called a quaternion algebra. As char(K) ≠ 2, every quaternion algebra has a quaternion basis {1, i, j, k}, that is a basis of the K-algebra Q subject to the relations

i^{2} = a ∈ K^{∗}, j^{2} = b ∈ K^{∗}, ij = k = −ji.

This algebra Q is then denoted by Q = (a, b)_{K}. Note also that every quaternion algebra has a canonical involution (usually denoted by γ) which is of the first kind and defined as follows:

γ(i) = −i, γ(j) = −j.

2.2 Hermitian forms

The standard reference for the theory of hermitian forms is [17]. All the vector spaces considered are supposed to be finite dimensional right vector spaces.

A hermitian form over (D, σ) is a pair (V, h) where V is a finite dimensional D-vector space and h is a map h : V × V → D which is σ-sesquilinear in the first argument, K-linear in the second argument and which satisfies

σ(h(x, y)) = h(y, x) for any x, y ∈ V.

If D = K and σ = id_{K} then a hermitian form is a symmetric bilinear form which can be identified with a quadratic form as char(K) ≠ 2. The integer dim_{K} V is called the dimension (or the rank) of the hermitian form. Since char(K) ≠ 2 and D is division, a hermitian form over (D, σ) may be given by a diagonalization ⟨a_{1}, ⋯, a_{n}⟩.

Note that the dual vector space V^{∗} := Hom_{D}(V, D) can be considered as a right D-vector space by twisting the action of D by σ. We define the adjoint map of (V, h) by the following morphism of right D-vector spaces

ĥ : \{ \begin{matrix}V & → & V^{∗} \\ x & ↦ & h_{x} \end{matrix}

where h_{x}(y) = h(x, y) for all y ∈ M. If this map is an isomorphism of right D-vector spaces, h is said to be nondegenerate. We say that h is isotropic if there exists an x ∈ V \ {0} such that h(x, x) = 0, anisotropic otherwise. If y is an element of D such that h(x, x) = y for a certain x ∈ V \ {0}, then we say that h represents y.

Let (V, h) and (V′, h′) be two hermitian forms over (D, σ). If these forms are isometric then we write h ≃ h′ for short. Their orthogonal sum is denoted by h⊥h′.

If (V, h) is a hermitian form over (D, σ), one can attach to it a nondegenerate hermitian form over (D, σ) denoted by (V ⊕ V^{∗}, Ih_{V}) with

Ih_{V} (v ⊕ f, v′ ⊕ g) = f(v′) + σ(g(v))

for all f, g ∈ V^{∗}, v, v′ ∈ V. A hermitian form over (D, σ) which is isometric to such a form is called a hyperbolic form.

Witt’s cancelation theorem says that if (V, h), (V′, h′), (V′′, h′′) are three hermitian forms over (D, σ) and if h⊥h′ ≃ h⊥h′′ then h′ ≃ h′′: see [9, Chapter 5.11].

2.3 The Witt group of a division algebra with involution

We refer to [17] for more details about the Witt group.

Let (V, h) be a nondegenerate hermitian form over (D, σ). The isometry class of this form is denoted by [V, h] or [h] for short and [0] denotes the zero-dimensional hermitian form which is supposed to be nondegenerate by convention. Let S(D, σ) denote the set of isometry classes of nondegenerate hermitian forms over (D, σ). This set is endowed with a structure of commutative monoïd by the law

[h] + [h′] = [h⊥h′].

The Witt group of (D, σ) is the quotient group of the Grothendieck group of S(D, σ) by the subgroup generated by hyperbolic forms and is denoted by W (D, σ). In the case where D = K, Kronecker’s product can be used to define a structure of ring on W (K, σ). If moreover σ = id_{K}, this ring is called the Witt ring of K and is denoted by W (K).

If [V, h] ∈ W (K, σ|_{K}) and [V′, h′] ∈ W (D, σ), we define [V ⊗_{K} V′, h · h′] ∈ W (D, σ) as the Witt class of the following nondegenerate hermitian form

h · h′ : \{ \begin{matrix}(V ⊗_{K} V′) × (V ⊗_{K} V′) & → & D \\ (v_{1} ⊗ v′_{1}, v_{2} ⊗ v′_{2}) & ↦ & h(v_{1}, v_{2})h′(v′_{1}, v′_{2}) \end{matrix} .

It is easy to see that this action endows W (D, σ) with a structure of W (K, σ|_{K})-module. The submodule generated by nondegenerate hermitian forms of even rank is denoted by I_{1}(D, σ) or by I(K) if D = K and σ = id_{K}. We write I^{n}(K) for (I(K))^{n}. It is easy to see that I^{n}(K) is additively generated by the so-called Pfister forms

⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ := ⟨⟨a_{1}, ⋯, a_{n}⟩⟩,

for a_{1}, ⋯, a_{n} ∈ K^{∗}.

2.4 The (refined) discriminant of a hermitian form

We refer to [2, §2] and [3, §3] for more general statements about this invariant.

Let (V, h) be a hermitian form over (D, σ) and suppose first that σ is an involution of the first kind. Let {e_{1}, ⋯, e_{n}} be a D-basis of the right D-vector space V. Let M be the matrix of h with respect to this basis, E = M_{n}(D) and m = n deg(D). We define the signed discriminant of (V, h) by

d_{±}(h) = (−1)^{\frac{m(m−1)}{2}} Nrd_{E/K}(M) ∈ K^{∗},

where Nrd_{E/K} denotes the usual reduced norm map: see [5, §22]. One can show that d_{±} induces a well-defined group homomorphism, again denoted by d_{±}

d_{±} : I_{1}(D, σ) → K^{∗}/K^{∗}^{2}.

If σ is an involution of the second kind and if F is the fixed field of σ in K, one can define the signed discriminant of (V, h) by the formula above and show that it induces a group homomorphism

d_{±} : I_{1}(D, σ) → F^{∗}/N_{K/F} (K^{∗}),

where N_{K/F} (K^{∗}) is the group of norms of K/F. The kernel of the signed discriminant is denoted by I_{2}(D, σ).

In both cases, the signed discriminant can be refined as follows. We will describe it only for σ of the first kind. Denote by Nrd_{D/K}(D^{∗}) the group of reduced norms from D. With the same formula as above, on can define a group homomorphism

Disc : I_{1}(D, σ) → Nrd_{D/K}(D^{∗})/Nrd_{D/K}(D^{∗})^{2},

which is called the refined discriminant.

2.5 Chain equivalence

Throughout this Subsection we will use the notations of [11, Chapter I, §5] and will refer to it for more general statements.

Let D be a division algebra over K endowed with an involution σ (of arbitrary kind). Let h = ⟨a_{1}, ⋯, a_{n}⟩ and h′ = ⟨a′_{1}, ⋯, a′_{n}⟩ be two hermitian forms over (D, σ). They are said to be simply equivalent if there exists indices i, j ∈ {1, ⋯, n} such that ⟨a_{i}, a_{j}⟩ ≃ ⟨b_{i}, b_{j}⟩ and a_{k} = b_{k} for every k different from i and j (note that, if i = j, the expression ⟨a_{i}, a_{j}⟩ is understood to be ⟨a_{i}⟩). Two (diagonalized) hermitian forms h and h′ over (D, σ) are chain equivalent if there is a sequence of diagonalized hermitian forms f_{0}, ⋯, f_{m} over (D, σ) such that h = f_{0}, h′ = f_{m} and such that f_{i} is simply equivalent to f_{i+1} for 0 ≤ i ≤ m − 1. We immediately see that two chain equivalent forms are isometric. In fact, the converse is also true by “Witt’s Chain equivalence Theorem”:

Theorem 2.1 (Witt). If h and h′ are two (diagonalized) hermitian forms over (D, σ) and if h is isometric to h′ then h and h′ are chain equivalent.

Proof. The proof can be easily adapted from [11, Chapter I, Theorem 5.2] by replacing usual squares by hermitian squares (that is, elements of the form σ(x)x).

Remark 2.2. The above proposition is very useful for showing general properties for quadratic (resp. hermitian) forms by reducing it to properties for two-dimensional quadratic (resp. hermitian) forms. See the proofs of Theorems 1.1, 1.3 and 3.9.

2.6 Further results

Here we want to state two important results that are used several times in this paper.

The following result known as “Arason-Pfister Hauptsatz” gives a sufficient condition for a quadratic form to belong to I^{n}(K).

Theorem 2.3 (Arason-Pfister). Let q be a positive-dimensional anisotropic quadratic form over K. If q ∈ I^{n}(K), then dim q ≥ 2^{n}.

Proof. See [11, Chapter X, Hauptsatz 5.1] or [17, Chapter 4, Theorem 5.6].

Let L/K, L = K(\sqrt{a}), be a quadratic field extension endowed with its non trivial automorphism − and D = (a, b)_{K} be a quaternion algebra endowed with its canonical involution γ. We define the following usual transfer maps

π_{L} : \{ \begin{matrix}W (L, −) & → & W (K) \\ [h] & ↦ & [x ↦ h(x, x)] \end{matrix} , π_{D} : \{ \begin{matrix}W (D, γ) & → & W (K) \\ [h] & ↦ & [x ↦ h(x, x)] \end{matrix} .

Theorem 2.4 (Jacobson). With the above notations, the maps π_{L} and π_{D} are injective.

Proof. See [17, Chapter 10, 1.1, 1.2, 1.7] or [8].

Moreover, im(π_{L}) = ⟨⟨a⟩⟩W (K) and for any positive integer n, π_{L}(I_{1}(L, −)^{n}) = ⟨⟨a⟩⟩I^{n}(K).

3 Analogues of Harrison’s criterion

In this Section, we state and prove results for quadratic field extension with their nontrivial automorphism and for quaternion division algebras with their canonical involutions in analogy with Harrison’s criterion 1.1.

3.1 The case of quadratic field extensions

First, we rephrase Theorem 1.1 by introducing another equivalent condition and it is this condition we will then generalize to the setting of hermitian forms.

Lemma 3.1. Let K_{1} and K_{2} be two fields of characterisic different from 2. Then the following are equivalent:

(1) K_{1} and K_{2} are Witt equivalent.

(2) There is a group isomorphism t : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2} sending −1 to −1 and such that the quadratic form ⟨⟨x, y⟩⟩ is hyperbolic over K_{1} if and only if the quadratic form ⟨⟨t(x), t(y)⟩⟩ is hyperbolic over K_{2} for all x, y ∈ K_{1}^{∗}.

Proof. The quadratic form ⟨x, y⟩ represents 1 over K_{1} if and only if the quadratic form ⟨1, −x, −y⟩ is isotropic over K_{1} if and only if the Pfister form ⟨⟨x, y⟩⟩ is hyperbolic over K_{1}. The equivalence then follows from the second statement of Harrison’s criterion 1.1.

We now come to the proof of Theorem 1.3:

Theorem Let K_{1} and K_{2} be two fields of characteristic different from 2. Let L_{1} = K_{1}(\sqrt{a}) (resp. L_{2} = K_{2}(\sqrt{b})) be a quadratic field extension of K_{1} (resp. K_{2}) equipped with its non trivial automorphism σ_{1} (resp. σ_{2}). Then, the following are equivalent:

(1) W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) as rings.

(2) There is a group isomorphism t : K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) → K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗}) sending −1 to −1 such that the quadratic form ⟨⟨a, x, y⟩⟩ is hyperbolic over K_{1} if and only if the quadratic form ⟨⟨b, t(x), t(y)⟩⟩ is hyperbolic over K_{2} for all x, y ∈ K_{1}^{∗}.

In the proof, we will need the following two lemmas.

Lemma 3.2. For i = 1, 2, the signed discriminant induces a group isomorphism

d_{±} : I_{1}(L_{i}, σ_{i})/I_{2}(L_{i}, σ_{i}) ≃ K_{i}^{∗}/N_{L_{i}/K_{i}}(L_{i}^{∗}).

Proof. We know that

d_{±} : I_{1}(L_{i}, σ_{i}) → K_{i}^{∗}/N_{L_{i}/K_{i}}(L_{i}^{∗})

is a group homomorphism whose kernel is I_{2}(L_{i}, σ_{i}). If b ∈ K_{i}^{∗} then

d_{±}(⟨1, −b⟩) = (−1)^{1}Nrd_{M_{2}(L_{i})/L_{i}}(⟨1, −b⟩) = −Nrd_{L_{i}/K_{i}}(−b) = b mod N_{L_{i}/K_{i}}(L_{i}^{∗}),

hence d_{±} is onto.

We consider W (L_{i}, σ_{i}) as a ring, I_{1}(L_{i}, σ_{i}) and I_{2}(L_{i}, σ_{i}) as ideals for i = 1, 2. As in the case of quadratic forms, these two ideals are related in the following way:

Lemma 3.3. For i = 1, 2, (I_{1}(L_{i}, σ_{i}))^{2} = I_{2}(L_{i}, σ_{i}).

Proof. By using the signed discriminant, it is easy to show that (I_{1}(L_{i}, σ_{i}))^{2} ⊆ I_{2}(L_{i}, σ_{i}). Conversely, if ϕ ∈ I_{2}(L_{i}, σ_{i}) with ϕ = ⟨a_{1}, ⋯, a_{2s}⟩, a_{j} ∈ K_{i}^{∗}, j = 1, ⋯, 2s, then

1 = d_{±}(ϕ) = (−1)^{\frac{2s(2s−1)}{2}} ( ∏_{j=1}^{2s} a_{j}) ∈ K_{i}^{∗}/N_{L_{i}/K_{i}}(L_{i}^{∗}).

We proceed by induction on s.

If s = 1 and ϕ ≃ ⟨a, b⟩ then −a = b ∈ K_{i}^{∗}/N_{L_{i}/K_{i}}(L_{i}^{∗}). Now ϕ ≃ ⟨a, −a⟩ and ϕ ∈ (I_{1}(L_{i}, σ_{i}))^{2}.

If s = 2 and ϕ ≃ ⟨a, b, c, d⟩ then d = abc ∈ K_{i}^{∗}/N_{L_{i}/K_{i}}(L_{i}^{∗}) and

ϕ ≃ ⟨a⟩ ⊗ ⟨1, ab, ac, bc⟩ ≃ ⟨a⟩ ⊗ ⟨1, ab⟩ ⊗ ⟨1, ac⟩.

We thus have ϕ ∈ (I_{1}(L_{i}, σ_{i}))^{2}.

Suppose now s ≥ 3. One can write ϕ = ⟨a, b, c⟩⊥ϕ′ and

ϕ = \underbrace{⟨a, b, c, abc⟩}_{α} ⊥ \underbrace{(ϕ′⊥⟨−abc⟩)}_{β} ∈ W (K, σ_{a}).

By assumption, d_{±}(ϕ) = 1 and d_{±}(α) = 1 and this implies that d_{±}(β) = 1. By induction, β ∈ (I_{1}(L_{i}, σ_{i}))^{2} and ϕ ∈ (I_{1}(L_{i}, σ_{i}))^{2}, thus finishing the proof.

Proof of Theorem 1.3: (1) ⇒ (2) : Let Φ : W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) be a ring isomorphism. The rank induces a group homomorphism, rank : W (L_{1}, σ_{1}) → ℤ/2ℤ and this shows that I_{1}(L_{1}, σ_{1}) is a maximal ideal of index 2 in W (L_{1}, σ_{1}). In fact, it can be seen that I_{1}(L_{1}, σ_{1}) is the unique ideal of index 2 in W (L_{1}, σ_{1}). As a consequence, Φ induces the following group isomorphism

Φ|_{I_{1}(L_{1},σ_{1})} : I_{1}(L_{1}, σ_{1}) → I_{1}(L_{2}, σ_{2}).

Hence, by Lemma 3.3, Φ(I_{2}(L_{1}, σ_{1})) = I_{2}(L_{2}, σ_{2}) and we can define the following group isomorphism by means of Lemma 3.2:

t : \{ \begin{matrix}K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) & → & K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗}) \\ c & ↦ & d_{±} (Φ(⟨1, −c⟩)) \end{matrix} .

Now, it is easy to see that t(−1) = −1.

Φ induces also the following group isomorphism

u : I_{1}(L_{1}, σ_{1})^{2}/I_{1}(L_{1}, σ_{1})^{3} ≃ I_{1}(L_{2}, σ_{2})^{2}/I_{1}(L_{2}, σ_{2})^{3},

which can be used to obtain the following commutative diagram:

[figure: (K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗})) × (K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗})); (t, t); θ_{L_{1}}; (I_{1}(L_{1}, σ_{1}))^{2}/(I_{1}(L_{1}, σ_{1}))^{3}; u; (K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗})) × (K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗})); θ_{L_{2}}; (I_{1}(L_{2}, σ_{2}))^{2}/(I_{1}(L_{2}, σ_{2}))^{3}]

where

θ_{L_{i}}(x, y) = ⟨1, −x⟩ ⊗ ⟨1, −y⟩ mod I_{1}(L_{i}, σ_{i})^{3}

for all x, y ∈ K_{i}^{∗} and for i = 1, 2. If the hermitian form ⟨1, −x, −y, xy⟩ is hyperbolic over (L_{1}, σ_{1}), then ⟨1, −x, −y, xy⟩ ∈ I_{1}(L_{1}, σ_{1})^{3}. Conversely, let ⟨1, −x, −y, xy⟩ ∈ I_{1}(L_{1}, σ_{1})^{3}. Using the notations of Subsection 2.6, we know that the map π_{L_{1}} : W (L_{1}, σ_{1}) → ⟨⟨a⟩⟩W (K_{1}) is an additive group isomorphism and that

Ψ(⟨1, −x, −y, xy⟩) ∈ I^{4}(K_{1}).

By Theorem 2.3, the quadratic form Ψ(⟨1, −x, −y, xy⟩) is hyperbolic over K_{1} and it follows that the hermitian form ⟨1, −x, −y, xy⟩ is hyperbolic over (L_{1}, σ_{1}).

Lastly, if the quadratic form ⟨⟨a, x, y⟩⟩ = π_{L_{1}}(⟨1, −x, −y, xy⟩) is hyperbolic over K_{1} then, the hermitian form ⟨1, −x, −y, xy⟩ is hyperbolic over (L_{1}, σ_{1}) and ⟨1, −x, −y, xy⟩ ∈ I_{1}(L_{1}, σ_{1})^{3}. By commutativity of the previous diagram,

0 = u(θ_{L_{1}}(x, y)) = θ_{L_{2}}(t(x), t(y)).

We conclude that the hermitian form ⟨1, −t(x), −t(y), t(xy)⟩ ∈ I_{1}(L_{2}, σ_{2})^{3} which implies that the quadratic form ⟨⟨b, t(x), t(y)⟩⟩ = π_{L_{2}}(⟨1, −t(x), −t(y), t(xy)⟩) is hyperbolic over K_{2}. The converse is similar.

(2) ⇒ (1) : We define a map Φ on diagonal forms by

Φ(⟨a_{1}, ⋯, a_{n}⟩) = ⟨t(a_{1}), ⋯, t(a_{n})⟩.

We have to show that this definition does not depend on the chosen diagonalization. If n = 1, this is clear. If n = 2, suppose that ⟨u, v⟩ ≃ ⟨u′, v′⟩ as hermitian forms over (L_{1}, σ_{1}). By taking the signed discriminant on both sides, we obtain that uv = u′v′ ∈ K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}).

If we let the one-dimensional hermitian form ⟨u⟩ act on both sides, it follows that the hermitian form ⟨1, −uu′, −uv′, u′v′⟩ is hyperbolic over (L_{1}, σ_{1}). As a consequence, the hermitian form ⟨1, −t(u)t(u′), −t(u)t(v′), t(u′)t(v′)⟩ is hyperbolic over (L_{2}, σ_{2}) and, multiplying by the form ⟨t(u)⟩, it follows that the hermitian forms ⟨t(u), t(v)⟩ and ⟨t(u′), t(v′)⟩ are isometric over (L_{2}, σ_{2}). If n > 2, the result comes from Theorem 2.1 and from the fact that the property holds for n = 2. As t(−1) = −1, Φ preserves hyperbolicity and induces a well-defined map between W (L_{1}, σ_{1}) and W (L_{2}, σ_{2}). Besides, Φ is additive and multiplicative (Φ being multiplicative over rank one forms which generate additively W (L_{1}, σ_{1})) and t^{−1} provides an inverse for Φ which is thus a ring isomorphism. □

Remarks 3.4. (1) The previous proof is similar to the proof of Harrison’s criterion: see [7] or [16, §2].

(2) As K_{i}^{∗}^{2} = D_{K_{i}}(⟨1⟩) and N_{L_{i}/K_{i}}(L_{i}^{∗}) = D_{K_{i}}(⟨1, −a⟩) for i = 1, 2, Theorem 1.3 is a quadratic analogue of Theorem 1.1.

In Theorem 1.3 we can show that the condition t(−1) = −1 is not a consequence of the other two conditions of Assertion (2):

Example 3.5. Let K_{1} = ℚ_{3} and K_{2} = ℚ_{5}. Then

K_{1}^{∗}/K_{1}^{∗}^{2} = {1, −1, 3, −3}, K_{2}^{∗}/K_{2}^{∗}^{2} = {1, 2, 5, 10}, u(K_{1}) = u(K_{2}) = 4,

and the unique anisotropic quadratic form of dimension 4 over K_{1} (resp. over K_{2}) is ⟨1, 1, −3, −3⟩ (resp. ⟨1, −2, −5, 10⟩) (see [11, Chapter 6, Theorem 2.2]). Let L_{1} = K_{1}(\sqrt{3}) and L_{2} = K_{2}(\sqrt{2}). It is easy to show that |K_{1}^{∗}/D_{K_{1}}(⟨1, −3⟩)| = 2 = |K_{2}^{∗}/D_{K_{2}}(⟨1, −2⟩)| and that we have a group isomorphism defined by

t : K_{1}^{∗}/D_{K_{1}}(⟨1, −3⟩) → K_{2}^{∗}/D_{K_{2}}(⟨1, −2⟩)
1 ↦ 1
−1 ↦ 5

As u(K_{1}) = u(K_{2}) = 4, the quadratic form ⟨⟨3, x, y⟩⟩ (resp. ⟨⟨2, t(x), t(y)⟩⟩) is hyperbolic over K_{1} (resp. over K_{2}) for all x, y ∈ K_{1}^{∗}. Finally, ⟨1, −2⟩ clearly represents −1 over K_{2} and

t(−1) ≠ −1 = 1 ∈ K_{2}^{∗}/D_{K_{2}}(⟨1, −2⟩).

3.2 The case of quaternion division algebras

In this Subsection, Q_{1} = (a, b)_{K_{1}} (resp. Q_{2} = (c, d)_{K_{2}}) will denote a quaternion division algebra over K_{1} (resp. over K_{2}) with its canonical involution γ_{1} (resp. γ_{2}).

First, we motivate our choice of the module structure in Theorem 3.9 and Corollary 1.4 by giving two examples showing that, for quadratic forms, the group structure of the Witt ring is not sufficient to classify fields up to Witt equivalence as in Theorem 1.1. In the first example, the cardinality of the Witt ring is infinite and in the second, it is finite.

Examples 3.6. (1) One can find this example in [16, §7]. If K_{1} = ℚ(\sqrt[3]{2}) and K_{2} = ℚ, one can show that W (K_{1}) ≃ W (K_{2}) as groups. But, by [16, §4, Corollary 2], W (K_{1}) and W (K_{2}) are not isomorphic as rings.

(2) One can find this example in [4, Example 7.2]. The construction is based on [6, §II.1] which was obtained in 1965 by Gross and Fischer. We choose K_{1} = ℚ_{2}(\sqrt{d}) where d ∈ ℚ_{2}^{∗}\ ± ℚ_{2}^{∗}^{2}. Then K_{1} is a local field and we have

|K_{1}^{∗}/K_{1}^{∗}^{2}| = 16, s(K_{1}) = 2, u(K_{1}) = 4

(see [11, Chapter VI, Corollary 2.24], [11, Chapter XI, Examples 2.4(7), 6.2(4)]). Let F be a field satisfying

|F^{∗}/F^{∗}^{2}| = 4, s(F) = 2, u(F) = 2

(there is such a field by Cordes’ results in [4]). Then K_{2} := F ((X)) is such that

|K_{2}^{∗}/K_{2}^{∗}^{2}| = 8, s(K_{2}) = 2, u(K_{2}) = 4.

By [4, Theorem 4.5],

W (K_{1}) ≃ W (K_{2}) ≃ C_{4} × C_{4} × C_{2} × C_{2}

as groups. But W (K_{1}) and W (K_{2}) are not isomorphic as rings by Harrison’s criterion 1.1 as we have |K_{1}^{∗}/K_{1}^{∗}^{2}| = 16 ≠ 8 = |K_{2}^{∗}/K_{2}^{∗}^{2}|.

In order to simplify the statement of Theorem 3.9, we define:

Definition 3.7. Two fields K_{1} and K_{2} whose characteristic is different from 2 are said to be (Q_{1}, Q_{2})-equivalent if there is a group morphism t : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2} sending −1 to −1 such that, if the quadratic form ⟨⟨x, y⟩⟩ is hyperbolic over K_{1}, then the quadratic form ⟨⟨t(x), t(y)⟩⟩ is hyperbolic over K_{2} for all x, y ∈ K_{1}^{∗} and which induces a group isomorphism t̃: K_{1}^{∗}/D_{K_{1}}(⟨⟨a, b⟩⟩) ≃ K_{2}^{∗}/D_{K_{2}}(⟨⟨c, d⟩⟩). The pair (t, t̃) is called a (Q_{1}, Q_{2})-equivalence.

Remark 3.8. If A is a central simple algebra over a field K, let

SH_{0}(A) = K^{∗}/Nrd_{A/K}(A^{∗}).

In the literature, the group SH_{0}(A) is called the reduced norm residue group of the algebra A. In particular, if Q is a quaternion division algebra with norm form N_{Q}, SH_{0}(Q) = K^{∗}/D_{K}(N_{Q}) which allows us to translate Definition 3.7 in terms of these groups.

Theorem 3.9. The following are equivalent:

(1) There exist a ring homomorphism Φ : W (K_{1}) → W (K_{2}) sending one-dimensional forms to one-dimensional forms and a group isomorphism Ψ : W (Q_{1}, γ_{1}) → W (Q_{2}, γ_{2}) such that Ψ(⟨1⟩) = ⟨1⟩ and

Ψ(q.h) = Φ(q).Ψ(h),

for all q ∈ W (K_{1}), h ∈ W (Q_{1}, γ_{1}).

(2) There is a (Q_{1}, Q_{2})-equivalence (t, t̃) between K_{1} and K_{2} such that the hermitian forms ⟨u, v⟩ and ⟨u′, v′⟩ are isometric over (Q_{1}, γ_{1}) if and only if the hermitian forms ⟨t̃(u), t̃(v)⟩ and ⟨t̃(u′), t̃(v′)⟩ are isometric over (Q_{2}, γ_{2}) for all u, v, u′, v′ ∈ K_{1}^{∗}.

(3) There is a (Q_{1}, Q_{2})-equivalence (t, t̃) between K_{1} and K_{2} such that the quadratic form ⟨⟨a, b, u, v⟩⟩ is hyperbolic over K_{1} if and only if the quadratic form ⟨⟨c, d, t̃(u), t̃(v)⟩⟩ is hyperbolic over K_{2} for all u, v ∈ K_{1}^{∗}.

First, we need to prove the following lemma:

Lemma 3.10. Let Q be a quaternion division algebra over a field K with norm form N_{Q}. Let u, v, u′, v′ ∈ K^{∗}. Suppose that the quadratic form N_{Q} ⊗ ⟨u, v, −u′, −v′⟩ is hyperbolic over K. Then uvu′v′ ∈ D_{K}(N_{Q}).

Proof. As the quadratic forms q = ⟨u, v, −u′, −v′⟩ and q′ = ⟨1, −uu′vv′⟩ have the same signed discriminant, q⊥(−q′) belongs to I^{2}(K). Thus,

N_{Q} ⊗ ⟨u, v, −u′, −v′⟩ ≡ N_{Q} ⊗ ⟨1, −uu′vv′⟩ mod I^{4}(K).

By assumption and by Theorem 2.3, the quadratic form N_{Q} ⊗ ⟨1, −uu′vv′⟩ is hyperbolic over K and it follows that uvu′v′ ∈ D_{K}(N_{Q}).

Proof of Theorem 3.9: (3) ⇒ (2) : Let u, v, u′, v′ ∈ K_{1}^{∗} be such that

⟨u, v⟩ ≃ ⟨u′, v′⟩  (1)

as hermitian forms over (Q_{1}, γ_{1}). We now use the notations of Subsection 2.6. By Theorem 2.4, π_{Q_{1}} : W (Q_{1}, γ_{1}) → W (K_{1}) is injective. The fact (1) is thus equivalent to the fact that

⟨⟨a, b⟩⟩ ⊗ ⟨u, v⟩ ≃ ⟨⟨a, b⟩⟩ ⊗ ⟨u′, v′⟩

as quadratic forms, which in turn is equivalent to the hyperbolicity of the quadratic form ⟨⟨a, b⟩⟩⊗ ⟨u, v, −u′, −v′⟩ over K_{1}. By Lemma 3.10, uvu′v′ ∈ D_{K_{1}}(⟨⟨a, b⟩⟩) and t(uvu′v′) ∈ D_{K_{2}}(⟨⟨c, d⟩⟩)). Now we let the quadratic form ⟨u⟩ act on (1) so we have ⟨1, uv⟩ ≃ ⟨uu′, uv′⟩ as hermitian forms over (Q_{1}, γ_{1}) and, since uvu′v′ ∈ D_{K_{1}}(⟨⟨a, b⟩⟩), it follows that ⟨1, u′v′⟩ ≃ ⟨vv′, vu′⟩ as hermitian forms over (Q_{1}, γ_{1}). This is equivalent to the hyperbolicity of the quadratic form ⟨⟨a, b, vv′, vu′⟩⟩ over K_{1} and by Assertion (3), the quadratic form ⟨⟨c, d, t̃(vv′), t̃(vu′)⟩⟩ is hyperbolic over K_{2}. Using the fact that t(uvu′v′) ∈ D_{K_{2}}(⟨⟨c, d⟩⟩)), we deduce that the hermitian forms ⟨t̃(u), t̃(v)⟩ and ⟨t̃(u′), t̃(v′)⟩ are isometric over (Q_{2}, γ_{2}). The converse is similar.

(2) ⇒ (1) : let (t, t̃) be a (Q_{1}, Q_{2})-equivalence between K_{1} and K_{2} satisfying the conditions of Assertion (2). Mimicking the first part of the proof of Theorem 1.3, one can define a group homomorphism Φ : W (K_{1}) → W (K_{2}) sending a one-dimensional form to a one-dimensional form. We define Ψ in the following way

Ψ : \{ \begin{matrix}W (Q_{1}, γ_{1}) & → & W (Q_{2}, γ_{2}) \\ ⟨a_{1}, ⋯, a_{n}⟩ & ↦ & ⟨t̃(a_{1}), ⋯, t̃(a_{n})⟩ \end{matrix} .

As in the proof of Theorem 1.3, by using Proposition 2.1, we can show that Ψ is a well-defined map which induces a group homomorphism, and that the inverse of t̃ induces an inverse for Ψ. Finally, the compatibility relation between Φ and Ψ is easily proved.

(1) ⇒ (3) : Let us suppose the existence of Φ and Ψ as in Assertion (1). As Φ(I(K_{1})) ⊂ I(K_{2}), Φ induces the following group homomorphism

t : \{ \begin{matrix}K_{1}^{∗}/K_{1}^{∗}^{2} & → & K_{2}^{∗}/K_{2}^{∗}^{2} \\ a & ↦ & d_{±} (Φ(⟨1, −a⟩)) \end{matrix}

and t satisfies the other properties stated in Definition 3.7 by the implication (1) ⇒ (2) in Harrison’s criterion 1.1. We are going to show that

D_{K_{1}}(⟨⟨a, b⟩⟩)/K_{1}^{∗}^{2} = t^{−1}(D_{K_{2}}(⟨⟨c, d⟩⟩)/K_{2}^{∗}^{2}).  (2)

Let \overline{u} ∈ D_{K_{1}}(⟨⟨a, b⟩⟩)/K_{1}^{∗}^{2}. Then Ψ(⟨u⟩) = Ψ(⟨1⟩) = ⟨1⟩ on the one hand, and Ψ(⟨u⟩) = Φ(⟨u⟩).⟨1⟩ on the other hand (note that Φ(⟨u⟩) is a quadratic form over K_{2} whereas Ψ(⟨u⟩) is a hermitian form over (Q_{2}, γ_{2})). Letting Φ(⟨u⟩) = ⟨x⟩, we easily see that t(\overline{u}) = x and that x ∈ D_{K_{2}}(⟨⟨c, d⟩⟩). This implies that t(\overline{u}) ∈ D_{K_{2}}(⟨⟨c, d⟩⟩)/K_{2}^{∗}^{2} and finally D_{K_{1}}(⟨⟨a, b⟩⟩)/K_{1}^{∗}^{2} ⊂ t^{−1}(D_{K_{2}}(⟨⟨c, d⟩⟩)/K_{2}^{∗}^{2}).

Let \overline{v} ∈ D_{K_{2}}(⟨⟨c, d⟩⟩)/K_{2}^{∗}^{2} be such that t(\overline{y}) = \overline{v} for a y ∈ K_{1}^{∗}. As Φ(⟨y⟩) = ⟨v⟩,

Ψ(⟨y⟩) = Φ(⟨y⟩).⟨1⟩ = ⟨v⟩.⟨1⟩ = ⟨1⟩.

By injectivity of Ψ, it follows that ⟨y⟩ ≃ ⟨1⟩ and (2) holds.

Hence, t induces after factorization a unique injective group homomorphism

t̃: \{ \begin{matrix}K_{1}^{∗}/D_{K_{1}}(⟨⟨a, b⟩⟩) & → & K_{2}^{∗}/D_{K_{2}}(⟨⟨c, d⟩⟩) \\ x & ↦ & t(x) \end{matrix}

Denote by s_{1} : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{1}^{∗}/D_{K_{1}}(⟨⟨a, b⟩⟩) and s_{2} : K_{2}^{∗}/K_{2}^{∗}^{2} → K_{2}^{∗}/D_{K_{2}}(⟨⟨c, d⟩⟩) the two canonical surjections. Then, the following diagram is commutative

[figure: K_{1}^{∗}/K_{1}^{∗}^{2}; t; s_{1}; K_{2}^{∗}/K_{2}^{∗}^{2}; s_{2}; K_{2}^{∗}/D_{K_{2}}(⟨⟨c, d⟩⟩); K_{1}^{∗}/D_{K_{1}}(⟨⟨a, b⟩⟩); t̃]

Now, we show that s_{2} ∘ t is onto. Let \overline{w} ∈ K_{2}^{∗}/D_{K_{2}}(⟨⟨c, d⟩⟩). Ψ being surjective, there is a hermitian form h over (Q_{1}, γ_{1}) such that Ψ(h) = ⟨w⟩ = Φ(q).⟨1⟩ where h = q.⟨1⟩ and q is a quadratic form over K. Without loss of generality, one can suppose that h = ⟨a_{1}, ⋯, a_{n}⟩ et Φ(q) = ⟨b_{1}, ⋯, b_{n}⟩ with a_{1}, ⋯, a_{n}, b_{1}, ⋯, b_{n} ∈ K_{1}^{∗} (note that n is odd). By taking the refined signed discriminant on both sides of the previous equality, we obtain

∏_{i=1}^{n} b_{i}^{2} = w^{2} mod D_{K_{2}}(⟨⟨c, d⟩⟩)^{2}.

Consequently, there is a δ ∈ Q_{2}^{∗} such that ( ∏_{i=1}^{n} b_{i}^{2}).Nrd_{Q_{2}/K_{2}}(δ)^{2} = w^{2} and

w = ±( ∏_{i=1}^{n} b_{i}).Nrd_{Q_{2}/K_{2}}(δ).

The element ±( ∏_{i=1}^{n} a_{i}) is an antecedent of w via s_{2} ∘ t. Indeed,

t(± ∏_{i=1}^{n} a_{i}) = ± ∏_{i=1}^{n} t(a_{i})
= ± d_{±} (n⟨1⟩⊥ − Φ(⟨a_{1}, ⋯, a_{n}⟩))
= ± d_{±}(n⟨1⟩⊥⟨−b_{1}, ⋯, −b_{n}⟩)
= ± ∏_{i=1}^{n} b_{i} ∈ K_{2}^{∗}/K_{2}^{∗}^{2}.

So

(s_{2} ∘ t)(± ∏_{i=1}^{n} a_{i}) = ± ∏_{i=1}^{n} b_{i} mod D_{K_{2}}(⟨⟨c, d⟩⟩) = \overline{w}.

It follows that s_{2} ∘ t is surjective and that t̃ is a group isomorphism.

Let u, v ∈ K_{1}^{∗} be such that the quadratic form ⟨⟨a, b, u, v⟩⟩ is hyperbolic over K_{1}. The hermitian form ⟨1, −u, −v, uv⟩ is also hyperbolic over (Q_{1}, γ_{1}) by Theorem 2.4. We obtain (in W (Q_{1}, γ_{1}))

0 = Ψ(⟨1, −u, −v, uv⟩)
= Φ(⟨1, −u, −v, uv⟩).⟨1⟩
= (Φ(⟨1, −u⟩) ⊗ Φ(⟨1, −v⟩)).⟨1⟩.

By definition of t and t̃, we then have

Ψ(⟨1, −u, −v, uv⟩) = ⟨1, −t(u)⟩ ⊗ ⟨1, −t(v)⟩
= ⟨1, −t(u), −t(v), t(u)t(v)⟩
= ⟨1, −t̃(u), −t̃(v), t̃(u)t̃(v)⟩.

It follows that the quadratic form ⟨⟨c, d, t̃(u), t̃(v)⟩⟩ is hyperbolic over K_{2}. Conversely, if the quadratic form ⟨⟨c, d, t̃(u), t̃(v)⟩⟩ is hyperbolic over K_{2} then the quadratic form ⟨⟨a, b, u, v⟩⟩ is hyperbolic over K_{1} by Theorem 2.4 and by injectivity of Ψ. □

In the particular case where K_{1} = K_{2} = K, Theorem 3.9 readily implies:

Corollary Then, the following are equivalent:

(1) W (Q_{1}, γ_{1}) ≃ W (Q_{2}, γ_{2}) as W (K)-modules.

(2) There is a group isomorphism t̃: K^{∗}/Nrd_{Q_{1}/K}(Q_{1}^{∗}) ≃ K^{∗}/Nrd_{Q_{2}/K}(Q_{2}^{∗}) with t̃(−1) = −1 such that the quadratic form ⟨⟨a, b, u, v⟩⟩ is hyperbolic over K if and only if the quadratic form ⟨⟨c, d, t̃(u), t̃(v)⟩⟩ is hyperbolic over K for all u, v ∈ K^{∗}.

Remarks 3.11. (1) The condition on the hyperbolicity of quadratic forms in Theorem 3.9 can be replaced by a condition similar to the “Harrison-Cordes condition” (see Introduction). Theorem 3.9 is thus analogous to Theorem 1.2 proved by Baeza and Moresi. There are also some differences: − we do not suppose Φ to be bijective in Assertion (1) ; − in addition, we suppose that Φ respects one-dimensional forms and that Ψ(⟨1⟩) = ⟨1⟩ in Assertion (1) ;

(2) As K_{1}^{∗}^{2} = D_{K_{1}}(⟨1⟩) and Nrd_{Q_{1}/K_{1}}(Q_{1}^{∗}) = D_{K_{1}}(⟨⟨a, b⟩⟩), Corollary 1.4 is an analogue of Theorem 1.1 for quaternion division algebras (in the case where K = L).

(3) In [12], Leep and Marshall construct a surjective map between Aut (W (K_{1})) and the set of the so-called “Harrison maps” (i.e. satisfying Assertion (2) of Theorem 1.1) and describe the kernel of this map. To prove their results, they used the fact that every ρ ∈ Hom_{ring} (W (K_{1}), W (K_{2})) induces an element \overline{ρ} ∈ Hom_{ring} (W (K_{1}), W (K_{2})) respecting the dimension of quadratic forms and characterized by

\overline{ρ}(q) ≡ ρ(q) mod (I(K_{2}))^{2}

for all q ∈ W (K_{1}). It might be interesting to check if such properties hold for hermitian forms over a quaternion division algebra.

(4) In Assertion (1) of Theorem 3.9, the fact that Ψ(q.h) = Φ(q).Ψ(h), for all q ∈ W (K_{1}), h ∈ W (Q_{1}, γ_{1}) is equivalent to the commutativity of the following diagram (with exact rows)

[figure: W (K_{1}); Φ; ρ_{1}; W (Q_{1}, γ_{1}); Ψ; 0; W (K_{2}); ρ_{2}; W (Q_{2}, γ_{2}); 0]

where ρ_{i} : W (K_{i}) → W (Q_{i}, γ_{i}) is the scalar extension map, for i = 1, 2.

4 Reciprocity equivalence

In this Section we recall some basic results about global fields and reciprocity equivalence: we refer to [15] for more details about quadratic forms over global fields and to [16] for a complete treatment of reciprocity equivalence. Finally, we define the notion of quadratic reciprocity equivalence and prove Theorem 1.5.

4.1 Preliminaries

A global field is either an algebraic number field (i.e. a finite field extension of ℚ) or an algebraic function field (i.e. a finite field extension of a field of the form 𝔽_{q}(X) where 𝔽_{q} is the finite field with q elements for some prime power q and X is an indeterminate).

Let K be a global field, P be a nontrivial place of K and K_{P} be a completion of K at P. If P is non archimedian then P is discrete and K_{P} is a local field whose residue field is finite: we say that P is a finite place. If P is archimedian then there is a topological isomorphism between K_{P} and ℝ or ℂ: the place is called real in the first case, complex in the second case. An archimedian place is also called an infinite place. If K is an algebraic function field, every place over K is finite and if K is an algebraic number field, there is a finite number of archimedian places (more precisely [K : ℚ] = 2r + s where r is the number of complex places and s is the number of real places). We introduce the following notations:

Ω_{K} = {nontrivial places of K},
Ω_{K}^{f} = {finite places of K},
Ω_{K}^{∞} = {infinite places of K},
Ω_{K}^{r} = {real places of K} = {P ∈ Ω_{K}^{∞} | K_{P} ≃ ℝ},
Ω_{K}^{c} = {complex places of K} = {P ∈ Ω_{K}^{∞} | K_{P} ≃ ℂ}.

So we have the following disjoint unions: Ω_{K} = Ω_{K}^{f} ∪ Ω_{K}^{∞} and Ω_{K}^{∞} = Ω_{K}^{r} ∪ Ω_{K}^{c}.

Let K be a global field and suppose P ∈ Ω_{K}^{r}. Then, there is a topological isomorphism ϕ : K_{P} ≃ ℝ. Via ϕ, K_{P} is an ordered field, real closed and euclidian, with unique ordering K_{P}^{2} (see [17, Chapter 3, Theorem 1.1.4]). We thus say that an element a ∈ K^{∗} is positive (resp. negative) at P if a ∈ K_{P}^{∗}^{2} (resp. a ∈ −K_{P}^{∗}^{2}), and we write a >_{P} 0 (resp. a <_{P} 0). If a ∈ K^{∗}, we introduce the notation

Ω_{K}^{a} = {P ∈ Ω_{K}^{r} | a <_{P} 0}

to denote the set of real places at which a is negative.

The following is immediate:

Lemma 4.1. Let P ∈ Ω_{K}^{r} and q be a quadratic form over K. Then q is hyperbolic over K_{P} if and only if sgn_{P} (q) = 0 where sgn_{P} denotes the usual signature at P (see [17, Chapter 2, §4]).

It is easy to calculate the signature of a 2-fold Pfister form ⟨⟨x, y⟩⟩ at a real place P:

Table 4.2. \begin{matrix}x & y & xy & sgn_{P} (⟨⟨x, y⟩⟩) \\ + & + & + & 0 \\ − & + & − & 0 \\ + & − & − & 0 \\ − & − & + & 4 \end{matrix}

Finally, we get the following result which will be useful in Subsection 4.3:

Lemma 4.3. Let K be a global field and P ∈ Ω_{K}^{r}. A n-fold Pfister form q = ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ is anisotropic over K_{P} if and only if a_{i} <_{P} 0 for i = 1, ⋯, n.

Proof. The fact that a_{i} <_{P} 0 for all i = 1, ⋯, n is equivalent to sgn_{P} (q) = 2^{n} by mean of Table 4.2 and an induction on n. This property is also equivalent to the fact that q is anisotropic over K_{P} by Lemma 4.1, a Pfister form being either anisotropic or hyperbolic.

4.2 Reciprocity equivalence

Throughout this Subsection, K_{1} and K_{2} will denote global fields of characteristic different from 2. The notion of reciprocity equivalence between such fields has been defined in [16, §1]:

Definition 4.4. A reciprocity equivalence between K_{1} and K_{2} is a pair of maps (t, T), where t is a group isomorphism t : K_{1}^{∗}/K_{1}^{∗}^{2} → K_{2}^{∗}/K_{2}^{∗}^{2} and T is a bijection T : Ω_{K_{1}} → Ω_{K_{2}} such that (t, T) respects Hilbert symbols, i.e.

(x, y)_{P} = (tx, ty)_{T P}

for all x, y ∈ K_{1}^{∗}/K_{1}^{∗}^{2} and for all P ∈ Ω_{K_{1}}.

Remark 4.5. As (x, y)_{P} = 1 if and only if the 2-fold Pfister form ⟨⟨x, y⟩⟩ is hyperbolic over (K_{1})_{P}, one can replace the condition concerning Hilbert symbols in Definition 4.4 by ⟨⟨x, y⟩⟩ being hyperbolic over (K_{1})_{P} if and only if the quadratic form ⟨⟨t(x), t(y)⟩⟩ is hyperbolic over (K_{2})_{T (P)}.

The main Theorem of [16] says that

Theorem 4.6. K_{1} and K_{2} are Witt equivalent if and only if they are reciprocity equivalent.

The proof of the “if”-part of this Theorem very much relies on Harrison’s criterion 1.1. The proof of the converse is much more difficult and is based on a description of the 2-torsion of the Brauer group by the set of subsets of Ω_{K_{1}}\Ω_{K_{1}}^{c} of even order endowed with symmetric difference. We refer to[16, §3, 4] for more details.

4.3 Quadratic Reciprocity equivalence

The purpose of this Subsection is to give a proof of Theorem 1.5. Throughout, K_{1} (resp. K_{2}) will denote a global field of characteristic different from 2, L_{1} = K_{1}(\sqrt{a}) (resp. L_{2} = K_{2}(\sqrt{b})) will denote a quadratic field extension of K_{1} (resp. K_{2}) and σ_{1} (resp. σ_{2}) will denote the nontrivial automorphism of L_{1} (resp. L_{2}).

Definition 4.7. An (a, b)-quadratic reciprocity equivalence between K_{1} and K_{2} is a pair of maps (t, T) where t is a group isomorphism t : K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) → K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗}) with t(−1) = −1 and where T is a bijection T : Ω_{K_{1}}^{a} → Ω_{K_{2}}^{b} such that the quadratic form ⟨⟨a, x, y⟩⟩ is hyperbolic over (K_{1})_{P} if and only if the quadratic form ⟨⟨b, t(x), t(y)⟩⟩ is hyperbolic over (K_{2})_{T (P)} for all x, y ∈ K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) and for all P ∈ Ω_{K_{1}}^{a}.

We now prove Theorem 1.5:

Theorem The following are equivalent:

(1) W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) as rings.

(2) There is an (a, b)-quadratic reciprocity equivalence between K_{1} and K_{2}.

Proof. (2) ⇒ (1) : by Theorem 1.3, it suffices to show that the quadratic form ⟨⟨a, x, y⟩⟩ is hyperbolic over K_{1} if and only if the quadratic form ⟨⟨b, t(x), t(y)⟩⟩ is hyperbolic over K_{2}. Note that, for all P ∈ Ω_{K_{1}}\Ω_{K_{1}}^{a} (resp. for all Q ∈ Ω_{K_{2}}\Ω_{K_{2}}^{b}) and for all x, y ∈ K_{1}^{∗}, the quadratic form ⟨⟨a, x, y⟩⟩ (resp. ⟨⟨b, t(x), t(y)⟩⟩) is hyperbolic over (K_{1})_{P} (resp. over (K_{2})_{Q}). This fact is obvious if the place is complex or if a >_{P} 0 (resp. if b >_{Q} 0). If P (resp. Q) is finite, it comes

from the fact that (K_{1})_{P} (resp. (K_{2})_{Q}) is a local field with u((K_{1})_{P}) = 4 = u((K_{2})_{Q}) (see [11, Chapter XI, Example 6.2(4)]). Suppose now that the quadratic form ⟨⟨a, x, y⟩⟩ is hyperbolic over K_{1}. Then ϕ = ⟨⟨b, t(x), t(y)⟩⟩ is hyperbolic over (K_{2})_{Q} for all Q ∈ Ω_{K_{2}}^{b} hence ϕ is hyperbolic over K_{2} by the Hasse-Minkowski-Principle (see [11, Chapter VI, Hasse-Minkowski-Principle 3.1]). The converse is similar.

(1) ⇒ (2) : by Theorem 1.3, we only have to find the map T of Definition 4.7.

We first show that K_{1} and K_{2} have the same number of distinct real places for which a and b are negative. First of all, K_{1} has no real place where a is negative if and only if K_{2} has no real place where b is negative (this is a consequence of Theorem 1.3, Lemma 4.3 and the Hasse-Minkowski Principle applied to ⟨⟨a, −1, −1⟩⟩ and to ⟨⟨b, t(−1), t(−1)⟩⟩ = ⟨⟨b, −1, −1⟩⟩). In this case, (for example, if K_{1} and K_{2} are algebraic function fields), there is nothing to prove. Suppose now that Ω_{K_{1}}^{a} = {P_{1}, ⋯, P_{n}}, n ≥ 1. As K_{1} is a global field which has at least one real place, K_{1} is an SAP field (see [13, §2]). As a consequence, there exist x_{1}, ⋯, x_{n} ∈ K_{1}^{∗} such that

x_{i} >_{P_{i}} 0, x_{i} <_{P_{j}} 0 for all j ≠ i,

for i = 1, ⋯, n. On the one hand, it is easy to see that for each i,

(−1)^{n}x_{1} ⋯ x_{i−1}x_{i+1} ⋯ x_{n} <_{P_{i}} 0.

This implies, by Lemma 4.3, that the quadratic form ⟨⟨a, −1, (−1)^{n}x_{1} ⋯ x_{i−1}x_{i+1} ⋯ x_{n}⟩⟩ is anisotropic over K_{1}. Consequently, the quadratic form

⟨⟨b, −1, (−1)^{n}t(x_{1}) ⋯ t(x_{i−1})t(x_{i+1}) ⋯ t(x_{n})⟩⟩

is anisotropic over K_{2} and there is a place Q_{i} (which has to be real) such that the quadratic form ⟨⟨b, −1, (−1)^{n}t(x_{1}) ⋯ t(x_{i−1})t(x_{i+1}) ⋯ t(x_{n})⟩⟩ is anisotropic over (K_{2})_{Q_{i}} by the Hasse-Minkowski-Principle. Applying Lemma 4.3, we see that b <_{Q_{i}} 0 and

(−1)^{n}t(x_{1}) ⋯ t(x_{i−1})t(x_{i+1}) ⋯ t(x_{n}) <_{Q_{i}} 0,  (3)

for all i = 1, ⋯, n. On the other hand, one also has

(−1)^{n}x_{1} ⋯ x_{n} <_{P_{i}} 0

for all i = 1, ⋯, n. This implies that the quadratic form ⟨⟨a, −1, (−1)^{n}x_{1} ⋯ x_{n}⟩⟩ is anisotropic over K_{1} and that ⟨⟨b, −1, (−1)^{n}t(x_{1}) ⋯ t(x_{n})⟩⟩ is anisotropic over K_{2}. By the Hasse-Minkowski-Principle,

(−1)^{n}t(x_{1}) ⋯ t(x_{n}) <_{Q_{i}} 0,  (4)

for all i = 1, ⋯, n. By combining (3) and (4), we get that t(x_{i}) >_{Q_{i}} 0 for all i = 1, ⋯, n. We now

show that the n places Q_{i} are distinct. If conversely, there are i ≠ j such that Q = Q_{i} = Q_{j} then

t(x_{i}) >_{Q} 0, t(x_{j}) >_{Q} 0

and the quadratic form ⟨⟨b, t(−x_{i}), t(−x_{j})⟩⟩ is anisotropic over K_{2}. This implies that the quadratic form ⟨⟨a, −x_{i}, −x_{j}⟩⟩ is anisotropic over K_{1}. By the Hasse-Minkowski-Principle and Lemma 4.3, there is a place P ∈ Ω_{K_{1}}^{a} for which

x_{i} >_{P} 0, x_{j} >_{P} 0.

That is impossible, by construction of the x_{i}’s. We have thus shown that there is an injection T : Ω_{K_{1}}^{a} → Ω_{K_{2}}^{b}. By Symmetry, this is a bijection.

Finally, suppose that ⟨⟨b, t(x), t(y)⟩⟩ is anisotropic over (K_{2})_{Q_{i}}. By Lemma 4.3, t(x) <_{Q_{i}} 0 and

t(y) <_{Q_{i}} 0. As t(x_{i}) >_{Q_{i}} 0 by definition, the quadratic forms ⟨⟨b, t(−x_{i}), t(x)⟩⟩ and ⟨⟨b, t(−x_{i}), t(y)⟩⟩

are anisotropic over (K_{2})_{Q_{i}}, hence over K_{2}. As a consequence, the quadratic forms ⟨⟨a, −x_{i}, x⟩⟩ and ⟨⟨a, −x_{i}, y⟩⟩ are anisotropic over K_{1}. By the Hasse-Minkowski-Principle, there are P, R ∈ Ω_{K_{1}}^{a} such that

x_{i} >_{P} 0, x <_{P} 0 et x_{i} >_{R} 0, y <_{R} 0.

Thus P = R = P_{i} and the quadratic form ⟨⟨a, x, y⟩⟩ is anisotropic over (K_{1})_{P_{i}}. The converse is similar.

Remarks 4.8. (1) By means of Remark 4.5, one can say that Theorem 4.6 and Theorem 1.5 are analogous. There are also some major differences between the two results. In the proof of Theorem 4.6, the most important places are the finite places whereas in Theorem 1.5, they do not play a part, the completion of a field at such places being local with u-invariant 4. Besides, in Theorem 1.5, it seems that we have to suppose that t(−1) = −1 which is not the case in Theorem 4.6. In fact, in this last case, this condition comes from the “Global Square Theorem” (see [11, Chapter VI, Theorem 3.7]) which has no analogue (as far as we know) in our situation.

(2) While proving (1) ⇒ (2) in Theorem 1.5, instead of saying that a certain field is SAP, we could have used the Strong Approximation Theorem for global fields: see [15, Chapter II, Theorem 2.1.2].

(3) In the case of algebraic function fields, the ring isomorphism W (L_{1}, σ_{1}) ≃ W (L_{2}, σ_{2}) is equivalent to having an isomorphism K_{1}^{∗}/N_{L_{1}/K_{1}}(L_{1}^{∗}) ≃ K_{2}^{∗}/N_{L_{2}/K_{2}}(L_{2}^{∗}) sending −1 to −1.

Similarly, we can prove:

Theorem 4.9. Let K be a global field of characteristic different from 2. Let Q_{1} = (a, b)_{K} (resp. Q_{2} = (c, d)_{K}) be a quaternion algebra over K endowed with its canonical involution γ_{1} (resp. γ_{2}). For α, β ∈ K^{∗}, denote by Ω_{K}^{(α,β)} the set of real places at which α and β are negative. Then, the following are equivalent:

(1) W (Q_{1}, γ_{1}) ≃ W (Q_{2}, γ_{2}) as W (K)-modules.

(2) There exists a pair of maps (t, T) where t is a group isomorphism t : K^{∗}/Nrd_{Q_{1}/K}(Q_{1}^{∗}) ≃ K^{∗}/Nrd_{Q_{2}/K}(Q_{2}^{∗}) with t(−1) = −1 and where T is a bijection T : Ω_{K}^{(a,b)} → Ω_{K}^{(c,d)} such that the quadratic form ⟨⟨a, b, x, y⟩⟩ is hyperbolic over K_{P} if and only if the quadratic form ⟨⟨c, d, t(x), t(y)⟩⟩ is hyperbolic over K_{T (P)} for all x, y ∈ K^{∗}/Nrd_{Q_{1}/K}(Q_{1}^{∗}) and for all P ∈ Ω_{K}^{(a,b)}.

References

[1] R. Baeza, R. Moresi: On the Witt-equivalence of fields of characteristic 2, J. Algebra 92 (1985), 446–453.

[2] E. Bayer-Fluckiger, R. Parimala: Galois cohomology of the Classical groups over fields of cohomological dimension ≦ 2, Invent. Math. 122 (1995), 195–229.

[3] E. Bayer-Fluckiger, R. Parimala: Classical groups and Hasse principle, Annals of Math. 147 (1998), 651–693.

[4] C. M. Cordes: The Witt group and the equivalence of fields with respect to quadratic forms, J. Algebra 26 (1973), 400–421.

[5] P. K. Draxl: Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983.

[6] H. Gross, H. R. Fischer: Non real fields and infinite dimensional k-vectorspaces, Math. Ann. 159 (1965), 285–308.

[7] D. K. Harrison: Witt rings, University of Kentucky Notes, Lexington, Kentucky, 1970.

[8] N. Jacobson: A note on hermitian forms, Bull. Amer. Math. Soc. 46 (1940), 264–268.

[9] N. Jacobson: Algebra, volume 2, Van Nostrand, Princeton 1953.

[10] M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol: The book of involutions, Coll. Pub. 44. Providence, RI: Amer. Math. Soc. (1998).

[11] T. Y. Lam: Introduction to quadratic forms over fields, Graduate Studies in Mathematics, 67. American Mathematical Society, Providence, RI, 2005.

[12] D. Leep, M. Marshall: Isomorphisms and automorphisms of Witt rings, Canad. Math. Bull. 31 (2) (1988), 250–256.

[13] D. W. Lewis, C. Scheiderer, T. Unger: A weak Hasse principle for central simple algebras with an involution, Proceedings of the Conference on Quadratic Forms and related topics (Baton-Rouge, LA) (2001), doc. math. 2001, extra vol. 241–251 (electronic).

[14] J. Mináč, M. Spira: Witt rings and Galois groups, Annals of Math. 144 (1996), 35–60.

[15] O. T. O’Meara: Introduction to quadratic forms, Berlin-Göttingen-Heidelberg, Springer-Verlag 1963.

[16] R. Perlis, K. Szymiczek, P. E. Conner, R. Litherland: Matching Witts with global fields, Contemp. Math. 155 (1994), 365–387.

[17] W. Scharlau: Quadratic and hermitian forms, Grundlehren Math. Wiss. 270, Berlin, Springer-Verlag 1985.

Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

E-mail address: grenier@math.univ-fcomte.fr