ON BILINEAR FORMS OF HEIGHT 2 AND DEGREE 1 OR 2 IN CHARACTERISTIC 2

AHMED LAGHRIBI AND ULF REHMANN

Abstract. Our aim in this paper is to complete some results given in [15] on the classification of symmetric bilinear forms of height 2 and degree d = 1 or 2 over fields of characteristic 2, i.e., those whose anisotropic parts over their own function fields are similar to d-fold bilinear Pfister forms.

1. Introduction and Main results

Let F be a field of characteristic 2. Throughout this paper, the expression “bilinear form” means “finite dimensional regular symmetric bilinear form”.

For a field extension L/F and a bilinear (or quadratic) form B over L, we say that B is definable over F if B is isometric to C_{L} for some bilinear (or quadratic) form C over F. If moreover, C is unique, then we say that B is defined (by C) over F.

To a bilinear form B with underlying vector space V, we associate a unique quadratic form B̃ given on V by: B̃(v) = B(v, v) for v ∈ V. The function field of B, denoted by F (B), is by definition the function field of B̃. The standard splitting tower of a nonzero bilinear form B is the sequence of forms and fields defined as follows:

\begin{cases}F_{0} = F and B_{0} = B_{an} \\ For n ≥ 1 : F_{n} = F_{n−1}(B_{n−1}) and B_{n} = ((B_{n−1})_{F_{n}})_{an}, \end{cases}

where C_{an} denotes the anisotropic part of a bilinear form C. The height h(B) of B is the smallest integer h such that dim B_{h} ≤ 1, where dim C denotes the dimension of a bilinear form C. As was done by the first author in [15], we associate to the form B another numerical invariant deg(B), called the degree of B, as follows: If h = h(B) and (B_{i}, F_{i})_{0≤i≤h} is the standard splitting tower of B, then the form B_{h(B)−1} is of height 1. By the classification of height 1 bilinear forms [15, Th. 4.1], there exists a unique bilinear Pfister form π over F_{h(B)−1} such that B_{h(B)−1} is similar to π or to the pure part of π according as dim B is even or odd. If dim B is even, then we put deg(B) = d where dim π = 2^{d}. Otherwise, we put deg(B) = 0.

Key words and phrases. Bilinear and quadratic forms, function fields of quadrics, standard splitting theory, norm degree.

MSC 2000. 11E04; 11E81.

We call π the leading form of B. The form B is called good if π is definable over F, and in this case, we know from [15, Prop. 5.3] that π is defined over F by a d-fold bilinear Pfister form. For example, if B is of even dimension and nontrivial determinant, then it is good of degree 1 and leading form (⟨1, det B⟩_{b})_{F_{h−1}}.

An important problem considered in [15] is the classification of bilinear forms by height and degree. Bilinear forms of height 1 are completely classified as we said before in the definition of the degree. For good bilinear forms of height 2, the first author gave in [15] a complete classification of those of degree 0, and a partial classification of those of degree ≥ 1. In this paper, we complete the classification of bilinear forms which are good of height 2 and degree 1 or 2, and with [15, Th. 5.10] we get the following theorem:

Theorem 1.1. Let B be an anisotropic bilinear form over F which is good of height 2 and degree d = 1 or 2. Let λ be the unique d-fold bilinear Pfister form over F such that λ_{F (B)} is the leading form of B. Then, we are in one of the following cases:

(1) dim B = 2^{n} with n ≥ d + 1: In this case, there exists α ∈ F^{∗} := F \ {0} and π similar to an n-fold bilinear Pfister form such that B ⊥ αλ ⊥ π is metabolic.

(2) dim B = 2^{m} − 2^{d} with m ≥ d + 2: In this case, B ≃ ρ ⊗ λ such that dim ρ is odd and B ⊥ ⟨det ρ⟩_{b} ⊗ λ is similar to an m-fold bilinear Pfister form. Conversely, any anisotropic bilinear form satisfying the conditions described in (1) or (2) is good of height 2 and degree d.

Moreover, the first author gave a formula on the possible dimensions of bilinear forms of height 2 (good or not) [15, Cor 5.20]. As a consequence of it, we get that the dimension of any anisotropic bilinear form of height and degree 2 (good or not) can be 2^{n}, 2^{n} − 2, or 2^{n} − 4 for some n ≥ 3 [15, Comment after Remark 5.21]. Note that Theorem 1.1 shows that the integers 2^{n} for n ≥ 3, and 2^{n} − 4 for n ≥ 4 occur as dimensions of good anisotropic bilinear forms of height and degree 2. We know by [15] that any Albert bilinear form, i.e., a 6-dimensional bilinear form of trivial determinant, is of height and degree 2 but not good. Before this work and except for the integer 6, we did not know other integers which really do occur as dimensions of anisotropic nongood bilinear forms of height and degree 2. Here, we clarify this point by proving the following theorem:

Theorem 1.2. (1) There are 8-dimensional anisotropic nongood bilinear forms of height and degree 2.

(2) An anisotropic bilinear form B over F is nongood of height and degree 2 iff one of the following conditions holds:

(i) B is an Albert form. (ii) dim B = 8 and there exists an anisotropic Albert bilinear form θ, unique up to similarity, that becomes isotropic over F (B) and satisfies B ⊥ θ ∈ I^{3}F.

The existence of 8-dimensional anisotropic bilinear forms of height and degree 2 which are not good is new in comparison with what is known in characteristic ≠ 2. In fact, in this case, Kahn proved that a nongood anisotropic quadratic form of height and degree 2 is necessarily of dimension 6 and trivial discriminant, i.e., an Albert quadratic form [7]. Kahn’s proof is based on the index reduction theorem of Merkurjev [18], [21]. But we do not have such a theorem for bilinear forms in characteristic 2. In our case, we will be inspired from a descent method due to Kahn [8], and we will use a result of Aravire and Baeza [1] to get the following theorem which is essential for the proofs of Theorems 1.1 and 1.2:

Theorem 1.3. Let B be an anisotropic bilinear form over F of dimension ≥ 3, and τ an anisotropic bilinear form similar to a d-fold bilinear Pfister form over F (B), with d = 1 or 2. Let C be a bilinear form over F.

(1) Suppose that d = 2 and τ ⊥ C_{F (B)} ∈ I^{3}F (B). Then, there exists an Albert bilinear form θ over F such that τ ⊥ θ_{F (B)} ∈ I^{3}F (B). Furthermore, if dim B > 8, then there exists a unique 2-fold bilinear Pfister form λ over F such that τ is similar to λ_{F (B)}.

(2) Suppose that dim B > 2^{d+1} and τ ⊥ C_{F (B)} ∈ I^{d+2}F (B). Let λ be as in (1) if d = 2, or λ = ⟨1, det C⟩_{b} if d = 1. Moreover, suppose that B_{F (λ)} is anisotropic or C_{F (λ)} is metabolic. Then, τ is defined over F by a form similar to λ.

Obviously, statement (1) of Theorem 1.3 implies that an anisotropic nongood bilinear form of height and degree 2 is of dimension 6 or 8. Moreover, as we see in Theorem 1.2, a complete classification of such bilinear forms consists in studying the isotropy of Albert bilinear forms over function fields of quadrics. This is an affair of norm field and norm degree (see subsection 2.3 for the definitions). More precisely, for θ an Albert bilinear form, there exists scalars x, u, v, r, s ∈ F^{∗} such that θ ≃ x ⟨r, s, rs, u, v, uv⟩ (because θ is of trivial determinant). The norm field of θ̃ is F^{2}(r, s, u, v), and if θ is anisotropic then [F^{2}(r, s, u, v) : F^{2}] = 8 or 16, since this degree is a power of 2, and it is at least equal to dim θ. The condition that this degree equals 8 is equivalent to say that the 4-fold bilinear Pfister form ⟨⟨r, s, u, v⟩⟩ is isotropic. In this case, the form θ̃ is a quasi-Pfister neighbor of a quasi-Pfister form π ([5, Def. 8.8], [5, Prop. 8.9(ii)]), and θ becomes isotropic over the function field of an anisotropic bilinear form B iff π is isotropic over F (B) iff B̃ is similar to a subform of π (we use [5, Prop. 8.9(iii)] and [12, Prop. 2.4]). If [F^{2}(r, s, u, v) : F^{2}] = 16 we ask the following question:

Question 1.4. Let θ be an anisotropic Albert bilinear form over F such that the norm field of θ̃ is of degree 16 over F^{2}. Let B be an anisotropic bilinear form of dimension ≥ 2 such that θ_{F (B)} is isotropic and B̃ is not similar to a 2-fold quasi-Pfister form. Is it true that B̃ is similar to a subform of θ̃?

We have a partial answer to this question:

Proposition 1.5. Question 1.4 has a positive answer if dim B = 2 or 3.

For 8-dimensional nongood bilinear forms of height and degree 2, Theorem 1.2 can be refined for a special class of fields as follows:

Proposition 1.6. Let F be a field of characteristic 2 satisfying one of the following conditions: (C1) Any 4-fold bilinear Pfister form over F is isotropic. (C2) Question 1.4 has a positive answer. Then, an 8-dimensional anisotropic bilinear form B over F is of height and degree 2 but not good iff there exists an anisotropic Albert bilinear form C, a 3-fold bilinear Pfister form π, and scalars x, y, z ∈ F^{∗} such that xB ⊥ yC ⊥ zπ is metabolic, and the forms B̃ and C̃ are similar to subforms of π̃.

The rest of this paper is organized as follows. In the next section we recall some definitions, notions and results on bilinear forms and totally singular quadratic forms, like Witt decompositions, the notion of norm degree, and some facts on transfer for bilinear forms. After that, we give the proofs of the results announced in this section. We start with the proof of Theorem 1.3 since we will need it for the proofs of Theorems 1.1 and 1.2.

2. Backgrounds on bilinear forms

The details of the most results that we present in this section can be found in [2], [5] and [20].

2.1. Some definitions. A quadratic form φ is called totally singular if it is isometric to B̃ for some bilinear form B.

A bilinear (or quadratic) form C is called a subform of B, denoted by C ⊂ B, if B ≃ C ⊥ C′ for some bilinear (or quadratic) form C′.

Two forms (bilinear or quadratic) B and C are called similar if B ≃ αC for some scalar α ∈ F^{∗}.

For a_{1}, ⋯, a_{n} ∈ F^{∗}, the diagonal bilinear form B given by the polynomial ∑_{i=1}^{n} a_{i}x_{i}y_{i} will be denoted by ⟨a_{1}, ⋯, a_{n}⟩_{b}, and the quadratic form B̃ will be denoted by ⟨a_{1}, ⋯, a_{n}⟩.

For any integer n ≥ 1 and a_{1}, ⋯, a_{n} ∈ F^{∗}, the form ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n}⟩_{b} is called an n-fold bilinear Pfister form, denoted by ⟨⟨a_{1}, ⋯, a_{n}⟩⟩_{b}. (⊗ means the product of bilinear forms.) The 0-fold bilinear Pfister form is just ⟨1⟩_{b}.

The pure part of a bilinear Pfister B is the unique form B′ satisfying B ≃ ⟨1⟩_{b} ⊥ B′.

Let IF be the ideal of the Witt ring W (F) of bilinear forms of even dimension, and I^{n}F = (IF)^{n} for any n ≥ 0 (with I^{0}F = W (F)). The ideal I^{n}F is additively generated by n-fold bilinear Pfister forms.

A basic result that we will use, called the Arason-Pfister Hauptsatz or simply the Hauptsatz, asserts that any anisotropic bilinear form B in I^{n}F is of dimension ≥ 2^{n}, and if dim B = 2^{n} then B is similar to an n-fold bilinear Pfister form [15, Lem. 4.8].

For any integer n ≥ 0, let \overline{I^{n}}F denote the quotient I^{n}F/I^{n+1}F. A quasi-Pfister form is a totally singular quadratic form φ such that

φ ≃ B̃ for some bilinear Pfister form B. A totally singular form ψ is called a quasi-Pfister neighbor if it is similar to a subform of a quasi-Pfister form φ and 2 dim ψ > dim φ.

2.2. Witt decompositions. A quadratic (or bilinear) form B with underlying vector space V is called isotropic if B(v) = 0 (or B̃(v) = 0) for some nonzero vector v ∈ V, and it is called anisotropic otherwise.

For any scalar a ∈ F, the 2-dimensional bilinear form given by the

matrix (\begin{matrix}a & 1 \\ 1 & 0 \end{matrix}) is called a metabolic plane. Such a bilinear form is

denoted by 𝕄(a). An orthogonal sum of metabolic planes is called a metabolic bilinear form.

Two bilinear forms B and C are called equivalent, denoted by B ∼ C, if B ⊥ M ≃ C ⊥ M′ for some metabolic forms M and M′.

For totally singular quadratic forms, the Witt decomposition states that any such quadratic form φ decomposes as follows: φ ≃ ψ ⊥ i×⟨0⟩ for some integer i ≥ 0, and an anisotropic form ψ which is unique up to isometry [5]. We call ψ (resp. i) the anisotropic part of φ, denoted by φ_{an} (resp. the defect index of φ, denoted by i_{d}(φ)).

The same decomposition is also known for bilinear forms and states that any such a form is isometric to an orthogonal sum of an anisotropic form and a metabolic form [9], [19]. In this paper we need the following refinement of this decomposition:

Proposition 2.1. ([15, Prop. 5.15]) Let B be a bilinear form over F of dimension ≥ 1. Then, there exists a bilinear form C, a unique pair of integers (m, n), and scalars a_{1}, ⋯, a_{m} ∈ F^{∗} such that: B ≃ C ⊥ 𝕄(a_{1}) ⊥ ⋯ ⊥ 𝕄(a_{m}) ⊥ n × 𝕄(0) and C ⊥ ⟨a_{1}, ⋯, a_{m}⟩_{b} is anisotropic. Consequently:

(1) (B̃)_{an} ≃ C̃ ⊥ ⟨a_{1}, ⋯, a_{m}⟩.

(2) m + dim C = dim(B̃)_{an}.

With the same notations and hypotheses as in Proposition 2.1, the bilinear form C is unique, we call it the anisotropic part of B, denoted by B_{an}. The Witt index of B is the integer m + n, denoted by i_{W} (B).

2.3. Norm degree. The norm field of a nonzero totally singular form φ, denoted by N_{F} (φ), is the field F^{2}(αβ | α, β ∈ D_{F} (φ)), where D_{F} (φ) is the set of scalars in F^{∗} represented by φ. We denote by ndeg_{F} (φ) the integer [N_{F} (φ) : F^{2}], and we call it the norm degree of φ. It is clear that N_{F} (φ) = N_{F} (αφ) for any scalar α ∈ F^{∗}. If φ is anisotropic and 2^{n} < dim φ ≤ 2^{n+1}, then ndeg_{F} (φ) ≥ 2^{n+1}, and ndeg_{F} (φ) = 2^{n+1} if and only if φ is a quasi-Pfister neighbor. If φ and φ′ are anisotropic quadratic forms such that φ is totally singular and φ_{F (φ′)} is isotropic, then φ′ is also totally singular and N_{F} (φ′) ⊂ N_{F} (φ). We refer to [5, Section 8] for more details on norm field and some of its applications.

2.4. Transfer. For a finite extension K/F, a bilinear form B over K with underlying vector space V, and a nonzero F -linear map s : K ⟶ F, we get an F -bilinear form s_{∗}(B) : V × V ⟶ F, given by: (v, v′) ↦ s(B(v, v′)). We call it the transfer of B by s. As was proved in [20, Lem. 5.5, Page 47], the bilinear form s_{∗}(B) is also regular. Moreover, the transfer operation is compatible with isometry and orthogonal sum. Since the transfer of a metabolic form is a also a metabolic form, it is clear that we get a group homomorphism s_{∗} : W (K) ⟶ W (F). By the same argument as in [20, Th. 5.6, page 48], we also have the Frobenius reciprocity which means that for a bilinear form B over F and a bilinear form B′ over K, there is an isometry s_{∗}(B_{K} ⊗B′) ≃ B⊗s_{∗}(B′). Finally, let us denote by i_{∗} : W (F) ⟶ W (K) the ring homomorphism induced by the inclusion F ⊂ K.

3. Proof of Theorem 1.3

We start with preliminary results.

Lemma 3.1. Let d ∈ F^{∗} \ F^{∗2} and τ a bilinear form over F (\sqrt{d}) similar to a 2-fold bilinear Pfister form. If τ is definable over F, then there exists θ similar to a 2-fold bilinear Pfister form over F such that τ ≃ θ_{F (\sqrt{d})}.

Proof. Let p, q, r, s ∈ F^{∗} be such that τ ≃ ⟨p, q, r, s⟩_{F (\sqrt{d})}. By comparing determinants, we get s = u^{2}pqr for some u ∈ F (\sqrt{d})^{∗}. Then, τ ≃ ⟨p, q, r, pqr⟩_{F (\sqrt{d})}.

We need a computation due to Aravire and Baeza, and another one due to the first author:

Theorem 3.2. ([1, Cor. 3.3]) Let B = ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ be an anisotropic n-fold bilinear Pfister form. Then, the kernel of the natural homomorphism \overline{I^{m}}F ⟶ \overline{I^{m}}F (B) is trivial if n > m, and it is equal to \{\overline{ψ ⊗ ⟨⟨x_{1}, ⋯, x_{n}⟩⟩} | ψ ∈ I^{m−n}F, and x_{1}, ⋯, x_{n} ∈ F^{2}(a_{1}, ⋯, a_{n})^{∗}\} if n ≤ m.

Proposition 3.3. ([15, Prop. 4.13]) Let n ≥ 1 be an integer, and B be an anisotropic bilinear form of dimension > 2^{n}. Then, the kernel of the natural homomorphism \overline{I^{n}}F ⟶ \overline{I^{n}}F (B) is trivial.

We give two lemmas on transfer which are well-known for quadratic forms in characteristic ≠ 2:

Lemma 3.4. Let L = F (\sqrt{k}) with k ∈ F^{∗} \ F^{∗2}, and s : L ⟶ F the F -linear map given by: 1 ↦ 0 and \sqrt{k} ↦ 1. Then, an anisotropic bilinear form B ∈ W (L) satisfies s_{∗}(B) = 0 if and only if B belongs to the image of the homomorphism i_{∗}.

Proof. We use the same argument as for the proof of [20, Th. 5.10, page 50].

Lemma 3.5. Let L = F (\sqrt{k}) with k ∈ F^{∗} \ F^{∗2}, and s : L ⟶ F a nonzero F -linear map. Then, s_{∗}(I^{n}L) ⊂ I^{n}F for any integer n ≥ 0.

Proof. We use the same proof as for [20, Cor. 14.9] after generalizing without difficulty [20, Lem. 14.8, page 92] to the case of bilinear forms in characteristic 2.

Proof of Theorem 1.3. Let B be an anisotropic bilinear form over F of dimension ≥ 3, τ be an anisotropic bilinear form similar to a d-fold bilinear Pfister form over F (B), with d = 1 or 2. Let C be a bilinear form over F.

(1) Suppose that d = 2 and τ ⊥ C_{F (B)} ∈ I^{3}F (B). Note that C ∈ I^{2}F since C_{F (B)} ∈ I^{2}F (B) and dim B ≥ 3. We may suppose, modulo I^{3}F (B), that τ is isometric to a 2-fold bilinear Pfister form.

We have F (B) = L(\sqrt{k}) for a purely transcendental extension L/F and k ∈ L^{∗} \ L^{∗2}. Let s : F (B) ⟶ L be the L-linear map given by: 1 ↦ 0 and \sqrt{k} ↦ 1. We have τ ⊥ C_{F (B)} ∈ I^{3}F (B). It follows from Lemma 3.5 that s_{∗}(τ ⊥ C) = s_{∗}(τ) ∈ I^{3}L. Since ⟨1⟩_{b} ⊂ τ, we deduce by the Hauptsatz that s_{∗}(τ) = 0. By Lemma 3.4, there exists θ_{1} a bilinear form over L such that τ ∼ (θ_{1})_{F (B)}. By [13, Cor. 3.5], we may suppose that dim θ_{1} = 4. By Lemma 3.1 and the multiplicativity of bilinear Pfister forms, we may suppose that θ_{1} is isometric to a 2-fold bilinear Pfister. Since (θ_{1})_{F (B)} ⊥ C_{F (B)} ∈ I^{3}F (B), we deduce that θ_{1} ⊥ C_{L} belongs to the kernel of the natural homomorphism \overline{I^{2}}L ⟶ \overline{I^{2}}L(\sqrt{k}). By applying Theorem 3.2 in the case m = 2 and n = 1, we conclude that θ_{1} ⊥ C_{L} ⊥ D ⊗ ⟨1, x^{2} + y^{2}k⟩_{b} ∈ I^{3}L, where D ∈ IL and x, y ∈ L such that x^{2} + y^{2}k ≠ 0. Hence, θ_{1} ⊥ C_{L} ⊥ θ_{2} ∈ I^{3}L, where θ_{2} = ⟨1, det D⟩_{b} ⊗ ⟨1, x^{2} + y^{2}k⟩. Now, by specializing the variables defining the field L to suitable scalars in F, we deduce by [10] that

(1)  C ⊥ γ_{1} ⊥ γ_{2} ∈ I^{3}F

where γ_{1} and γ_{2} are 2-fold bilinear Pfister forms over F. If we extend (1) to the field F (B), we get τ ⊥ (γ_{1} ⊥ γ_{2})_{F (B)} ∈ I^{3}F (B). Hence, the Albert bilinear form θ that we need is the orthogonal sum of the pure parts of γ_{1} and γ_{2}.

Suppose that dim B > 8 and τ ⊥ θ_{F (B)} ∈ I^{3}F (B) for an Albert bilinear form θ over F. Let ρ = θ_{an}. Since τ ≁ 0, it follows from the Hauptsatz that dim ρ ∈ {4, 6}. By [6, Th. 1.1] the bilinear form ρ_{F (B)} is anisotropic. By the Hauptsatz, we have ρ_{F (B)(τ)} ∼ 0, and thus, by [13, Th. 1.2] dim ρ_{F (B)} is divisible by 4. Hence, dim ρ_{F (B)} = 4, i.e., dim ρ = 4. Hence, ρ is similar to a 2-fold bilinear Pfister form, denoted by λ. By the Hauptsatz τ ≃ λ_{F (B)}, and thus τ is definable over F. For the uniqueness of λ, let δ be another 2-fold bilinear Pfister form satisfying τ ≃ δ_{F (B)}, then (λ ⊥ δ)_{F (B)} ∼ 0. Since dim(λ ⊥ δ)_{an} < 8, it follows from [13, Prop. 1.1] that λ ≃ δ.

(2) Suppose that dim B > 2^{d+1} and τ ⊥ C_{F (B)} ∈ I^{d+2}F (B). Let λ be as in (1) if d = 2, or λ = ⟨1, det C⟩_{b} if d = 1. Suppose that one of the following conditions holds:

(C1) B_{F (λ)} is anisotropic.

(C2) C_{F (λ)} is metabolic.

The form λ_{F (B)} is similar to τ. Hence, (C ⊥ λ)_{F (B)} ∈ I^{d+1}F (B). By using dim B > 2^{d+1} and Proposition 3.3, we get C ⊥ λ ∈ I^{d+1}F. Moreover,

(C ⊥ λ)_{F (B)(λ)} ≡ (τ ⊥ λ)_{F (B)(λ)} (mod I^{d+2}F (B)(λ)).

Since (τ ⊥ λ)_{F (B)(λ)} is metabolic, it follows that C ⊥ λ + I^{d+2}F (λ) belongs to the kernel of the homomorphism \overline{I^{d+1}}F (λ) ⟶ \overline{I^{d+1}}F (λ)(B).

Now, if the condition (C1) is satisfied, then Proposition 3.3 with the hypothesis dim B > 2^{d+1} implies that (C ⊥ λ)_{F (λ)} ∈ I^{d+2}F (λ). If the condition (C2) is satisfied, then it is clear that (C ⊥ λ)_{F (λ)} ∈ I^{d+2}F (λ). Hence, C ⊥ λ + I^{d+2}F belongs to the kernel of \overline{I^{d+1}}F ⟶ \overline{I^{d+1}}F (λ). By Theorem 3.2, C ⊥ λ ⊥ ν ⊗μ ∈ I^{d+2}F for suitable ν ∈ IF and μ a dfold bilinear Pfister form over F. Hence, C ⊥ λ ⊥ ⟨1, β⟩_{b} ⊗ μ ∈ I^{d+2}F, where β = det ν. The form λ ⊥ ⟨1, β⟩ ⊗ μ is isotropic (because λ and ⟨1, β⟩ ⊗ μ represent 1). Hence, by the Hauptsatz, and after extending scalars to F (B), we get

τ ∼ (λ ⊥ ⟨1, β⟩ ⊗ μ)_{F (B)} ∼ (λ′ ⊥ ⟨β⟩_{b} ⊥ ⟨1, β⟩ ⊗ μ′)_{F (B)},

where λ′ and μ′ denote the pure parts of λ and μ, respectively. Hence, i_{W} ((λ′ ⊥ ⟨β⟩_{b} ⊥ ⟨1, β⟩ ⊗ μ′)_{F (B)}) = 2^{d} − 1, and thus, any subform of λ′ ⊥ ⟨β⟩_{b} ⊥ ⟨1, β⟩ ⊗ μ′ of dimension 2^{d+1} becomes isotropic over F (B) [6, Lem. 2.11]. Since dim B > 2^{d+1}, it follows from [6, Th. 1.1] that any subform of λ′ ⊥ ⟨β⟩_{b} ⊥ ⟨1, β⟩ ⊗ μ′ of dimension 2^{d+1} is isotropic. Hence, dim(λ′ ⊥ ⟨β⟩_{b} ⊥ ⟨1, β⟩ ⊗ μ′)_{an} < 2^{d+1}. Again, by [6, Th. 1.1], we conclude that τ is definable over F. Now, if ν and ν′ are bilinear forms over F such that τ ≃ ν_{F (B)} ≃ ν′_{F (B)}, then (ν ⊥ ν′)_{F (B)} is metabolic. Since dim B > 2^{d+1}, it follows from [13, Prop. 1.1] that ν ⊥ ν′ is metabolic, i.e., ν ≃ ν′. Hence, τ is defined over F. Moreover, since dim B > 2^{d+1}, it follows that the unique bilinear form ν over F satisfying τ ≃ ν_{F (B)} is similar to λ.

4. Proof of Theorem 1.1

We give a lemma:

Lemma 4.1. Let B be an anisotropic bilinear form over F which is good of height 2 and degree > 0, and let λ_{F (B)} be its leading form with λ a bilinear Pfister form over F. If B_{F (λ)} is isotropic, then B_{F (λ)} is metabolic.

Proof. Let α ∈ F (B)^{∗} be such that B_{F (B)} ∼ α(λ_{F (B)}). Let [1, t^{−1}] be the quadratic form given by the polynomial x^{2} + xy + t^{−1}y^{2} over the rational function field F (t) in one variable t. The relation B_{F (B)} ∼ α(λ_{F (B)}) implies that

B ⊗ [1, t^{−1}]_{F (t)(B)} ∼ α(φ_{F (t)(B)}),

where ⊗ means the module action of W (F (t)) on the Witt group W_{q}(F (t)) of nonsingular quadratic forms over F (t) [2], and φ = λ ⊗ [1, t^{−1}]. Hence, B ⊗ [1, t^{−1}] is hyperbolic over F (t)(B)(φ). Since B_{F (λ)} is isotropic, it follows from [13, Prop. 3.9] that B ⊗ [1, t^{−1}] is also hyperbolic over F (t)(λ)(φ). Moreover, the extension F (t)(λ)(φ)/F (t)(λ) is purely transcendental, since φ is isotropic over F (t)(λ). Hence, B ⊗ [1, t^{−1}] is hyperbolic over F (t)(λ). By [15, Lem. 4.6], we conclude that B_{F (λ)} ∼ 0.

Proof of Theorem 1.1. Let B be an anisotropic bilinear form over F, good of height 2 and degree d = 1 or 2. Let λ be the unique d-fold bilinear Pfister form over F such that λ_{F (B)} is the leading form of B.

We know from [15, Th. 5.10] that dim B = 2^{n} for some n ≥ d + 1, or dim B = 2^{m} − 2^{d} for some m ≥ d + 2. Moreover, the classification given in the theorem in the case n = d + 1 or m ≥ d + 2 already exist in [15, Th. 5.10]. So, to complete our proof we have to consider the remaining case n > d + 1.

Suppose that dim B = 2^{n} > 2^{d+1}, and let α ∈ F (B)^{∗} be such that B_{F (B)} ∼ α(λ_{F (B)}). If B_{F (λ)} is isotropic, then it is metabolic by Lemma 4.1. Hence, by applying statement (2) of Theorem 1.3 for the forms λ and C := B, we conclude that α(λ_{F (B)}) is definable over F. Without loss of generality, we may suppose that α ∈ F^{∗}. Hence, (B ⊥ αλ)_{F (B)} ∼ 0. Since dim(B ⊥ αλ)_{an} < 2^{n+1}, it follows from [13, Th. 1.2] that B ⊥ αλ ∼ π, where π is similar to an n-fold bilinear Pfister form.

Conversely, if we proceed as in the proof of [15, Th. 5.10 ], we prove that any anisotropic bilinear form satisfying the conditions given in statement (1) or (2) of Theorem 1.1 is good of height 2 and degree d.

5. Proof of Theorem 1.2

(1) Let F = 𝔽_{2}(x, y, z) be the rational function field in the variables x, y, z over 𝔽_{2}. Let us consider the following forms:

π = ⟨⟨x, y, z⟩⟩ ,
θ = ⟨x, y, xy, 1 + x, z, z(1 + x)⟩ ,
B = ⟨⟨x, y⟩⟩ ⊥ (x + y + z) ⟨⟨1 + x, z⟩⟩ .

We verify without difficult that the quadratic forms B̃ and θ̃ are subforms of the quadratic form π̃ (it suffices to use the well-known isometry ⟨a⟩ ⊥ ⟨b⟩ ≃ ⟨a⟩ ⊥ ⟨a + b⟩ for any scalars a, b). Hence, B̃ and θ̃ are quasi-Pfister neighbors of π̃. By [5, Prop. 8.9] θ̃_{F (B)} is isotropic, i.e., θ_{F (B)} is isotropic. We easily check that B ⊥ θ ∈ I^{3}F, and by statement (2), B is of height and degree 2 but not good.

(2) Let B be an anisotropic bilinear form, and τ its leading form. (a) Suppose that B is of height and degree 2 but not good: We have

B_{F (B)} ∼ ατ for a suitable scalar α ∈ F (B)^{∗}. In particular, B_{F (B)} ⊥ τ ∈ I^{3}F (B). Since τ is not definable over F and dim B > 4, it follows from statement (1) of Theorem 1.3 that dim B ∈ {6, 8}. We discuss the two cases:

– Suppose dim B = 6: Then B is an Albert bilinear form since det B = 1.

– Suppose dim B = 8: By Theorem 1.3, there exists an Albert bilinear form θ such that τ ⊥ θ_{F (B)} ∈ I^{3}F (B). The form θ is anisotropic, otherwise θ_{an} would be similar to a 2-fold bilinear Pfister form, and by the Hauptsatz τ would be definable over F. Since (B ⊥ θ)_{F (B)} ∈ I^{3}F (B) and dim B > 4, it follows from Proposition 3.3 that B ⊥ θ ∈ I^{3}F. By the Hauptsatz, θ_{F (B)(τ)} ∼ 0. By [13, Th. 1.2] the form θ_{F (B)} can not be metabolic, and dim(θ_{F (B)})_{an} is divisible by 4. Hence, dim(θ_{F (B)})_{an} = 4, which means that θ_{F (B)} is isotropic. The uniqueness of θ, up to similarity, is a consequence of [16].

(b) Conversely, if B is an Albert bilinear form, then we know by [15, Th. 5.10] that B is of height and degree 2 but not good. So, suppose that dim B = 8 and B ⊥ θ ∈ I^{3}F for some anisotropic Albert bilinear form θ that becomes isotropic over F (B).

Let λ be a bilinear form over F (B) similar to a 2-fold bilinear Pfister form such that θ_{F (B)} ∼ λ. The bilinear form B is not similar to a 3-fold bilinear Pfister form, otherwise θ ∈ I^{3}F, and by the Hauptsatz θ would be isotropic. Consequently, B_{F (B)} is not metabolic [15, Cor. 5.5], and thus 0 < dim(B_{F (B)})_{an} ≤ 6. Again by the Hauptsatz, the bilinear form (B_{F (B)})_{an} is metabolic over F (B)(λ), and thus (B_{F (B)})_{an} is similar to a 2-fold bilinear Pfister form [13, Th. 1.2]. Hence, B is of height and degree 2. Moreover, if B is good and δ is the unique 2-fold bilinear Pfister form over F such that τ ≃ δ_{F (B)}, then we get B ⊥ δ ∈ I^{3}F [15, Prop. 5.3]. This implies that θ ⊥ δ ∈ I^{3}F, and, again by [15, Prop. 5.3], the bilinear Albert form θ is good, a contradiction. Hence, B is of height and degree 2, but not good.

6. Proof of Proposition 1.5

We start with some preliminary results.

Lemma 6.1. Let L = F (\sqrt{d}) with d ∈ F^{∗}\F^{∗2}. An anisotropic bilinear form B becomes isotropic over L if and only if B contains a subform similar to ⟨1, x^{2} + d⟩_{b} for suitable x ∈ F such that x^{2} + d ≠ 0 (this is equivalent to saying that ⟨1, d⟩ is similar to a subform of B̃).

Proof. It is clear that the condition given in the lemma is sufficient. Conversely, suppose that B_{L} is isotropic. By [13, Lem. 3.4], there exists a ≠ 0, b ∈ F such that the bilinear form B′ given by the matrix

(\begin{matrix}a & b \\ b & ad \end{matrix}) is a subform of B. It is easy to show that B′ is similar to

⟨1, x^{2} + d⟩_{b}, where x = 0 or x = b^{2}a^{−2} according as b = 0 or not. Since B is anisotropic we have x^{2} + d ≠ 0.

Since ⟨1, x^{2} + d⟩ ≃ ⟨1, d⟩, it follows that the isotropy of B_{L} is equivalent to say that ⟨1, d⟩ is similar to a subform of B̃.

Proposition 6.2. ([14, Cor. 2.4]) If an anisotropic totally singular form φ over F represents a nonzero polynomial p(x_{1}, ⋯, x_{n}) over F (x_{1}, ⋯, x_{n}), and if c = (c_{1}, ⋯, c_{n}) ∈ F^{n} satisfies p(c) ≠ 0, then p(c) ∈ D_{F} (φ).

To prove Proposition 1.5 in dimension 3, we will adapt to our case some arguments used by Leep in his complete answer to the isotropy of Albert quadratic forms over function fields of quadrics in characteristic ≠ 2 [17] (cf. PhD Thesis of Hoffmann [3]). Another important ingredient that we will use is the norm theorem for bilinear forms due to Knebusch [10].

Proof of Proposition 1.5. Let θ be an anisotropic Albert bilinear form such that ndeg_{F} (θ̃) = 16. Let B be an anisotropic bilinear form such that dim B ∈ {2, 3}. Obviously, if B̃ is similar to a subform of θ̃, then θ_{F (B)} is isotropic. Conversely, suppose that θ_{F (B)} is isotropic.

(1) The case dim B = 2: By Lemma 6.1 the quadratic form B̃ is similar to a subform of θ̃.

(2) The case dim B = 3: We may suppose that B ≃ ⟨1, α, β⟩_{b}. Let F [t] be the polynomial ring in one variable t, and F (t) its quotient field. It is well-known that the fields F (t)(B) and F (t)(⟨1, α + βt^{2}⟩_{b}) are isomorphic. By Lemma 6.1, there exists polynomials f, g, h, h_{1}, ⋯, h_{4} ∈ F [t] such that:

(2)  θ_{F (t)} ≃ f ⟨1, g^{2} + h^{2}(α + βt^{2})⟩_{b} ⊥ ⟨h_{1}, ⋯, h_{4}⟩_{b} .

This implies the following:

(3)  θ̃_{F (t)} ≃ f ⟨1, α + βt^{2}⟩ ⊥ ⟨h_{1}, ⋯, h_{4}⟩ .

It is clear that we may suppose g and h coprime, and the polynomials f, h_{1}, ⋯, h_{4} are square free.

Our aim is to reduce in equation (3) to the case where the polynomial f is constant, and after that we conclude, by Proposition 6.2, that θ̃ represents the scalars f, αf and βf. Since f ⟨1, α, β⟩ is anisotropic, we deduce that f ⟨1, α, β⟩ = fB̃ is a subform of θ̃.

Suppose that deg f > 0, and let p be a monic irreducible factor of f. Let F_{p} denote the residue field of the p-adic valuation of F (t), and ∂^{1} : W (F (t)) ⟶ W (F_{p}) the first residue homomorphism.

• Suppose that α + βt^{2} is a square in F_{p} ≃ F [t]/(p), then α + βt^{2} = r^{2} + p · s for suitable r, s ∈ F [t]. We may suppose that deg r < deg p. If p is linear, then for a ∈ F such that p(a) = 0, we get α + a^{2}β = r(a)^{2}, which implies that ⟨1, α, β⟩_{b} is isotropic, a contradiction. Hence, deg p ≥ 2. Consequently, deg s ≤ deg p − 2 since deg(p · s) = deg(α + βt^{2} + r^{2}) ≤ 2 deg p − 2. Moreover, p · s is represented by ⟨1, α + βt^{2}⟩_{b}. By the multiplicativity of quasi-Pfister forms, we get f ⟨1, α + βt^{2}⟩ ≃ f_{1} ⟨1, α + βt^{2}⟩, where f_{1} = \frac{s·f}{p} is of degree smaller than deg f.

• Suppose that α + βt^{2} is not a square in F_{p}. This implies that p does not divide g^{2} + h^{2}(α + βt^{2}). In fact, if p divides g^{2} + h^{2}(α + βt^{2}), then p does not divide h since g and h are coprime, which implies that α + βt^{2} is a square in F_{p}, a contradiction.

Since det θ_{F (t)} = (g^{2} + h^{2}(α + βt^{2})) h_{1}h_{2}h_{3}h_{4} ∈ F (t)^{2}, we have three possibilities:

(a) p | h_{i} for 1 ≤ i ≤ 4.

(b) p only divides two polynomials among h_{1}, ⋯, h_{4}, say h_{1} and h_{2}.

(c) p|/h_{i} for 1 ≤ i ≤ 4.

– Case (a) or case (b): By applying the homomorphism ∂^{1} to equation (2), we get

∂^{1}(θ_{F (t)}) ∼ θ_{F_{p}} ∼ 0

or

∂^{1}(θ_{F (t)}) ∼ θ_{F_{p}} ∼ ⟨\overline{h_{3}}, \overline{h_{4}}⟩_{b}

according as we are in case (a) or (b) (here \overline{u} denote the class of u ∈ F [t] in F_{p}). In case (b) we get \overline{h_{3}}\overline{h_{4}} ∈ F_{p}^{2}, and thus ⟨\overline{h_{3}}, \overline{h_{4}}⟩_{b} ∼ 0. Hence, in both cases the form θ becomes metabolic over F_{p}. By the norm theorem, we deduce that p is a norm of θ_{F (t)}. Hence,

θ̃_{F (t)} ≃ f_{2} ⟨1, α + βt^{2}⟩ ⊥ ⟨f · h_{1}, ⋯, f · h_{4}⟩ ,

where f_{2} = \frac{f}{p} is of degree smaller than deg f.

– Case (c): We have ∂^{1}(θ_{F (t)}) ∼ θ_{F_{p}} ∼ ⟨\overline{h_{1}}, ⋯, \overline{h_{4}}⟩_{b}, which implies that D := ⟨\overline{h_{1}}, ⋯, \overline{h_{4}}⟩_{b} is similar to a 2-fold bilinear Pfister form. If D is isotropic, then it is metabolic, and we may conclude as in the previous cases.

So we suppose that D is anisotropic. Then, ndeg_{F_{p}}(D̃) = 4.

Claim: N_{F_{p}}(θ̃_{F_{p}}) ⊂ N_{F_{p}}(D̃), and thus ndeg_{F_{p}}(θ̃_{F_{p}}) ≤ 4.

To prove the claim its suffices to verify that D_{F} (θ) ⊂ D_{F_{p}}(D). In fact, for every a ∈ D_{F} (θ), we get the following by equation (3):

(4)  k^{2}a = f (l^{2} + (α + βt^{2})m^{2}) + ∑_{i=1}^{4} n_{i}^{2}h_{i}

for suitable polynomials k, l, m, n_{1}, ⋯, n_{4} ∈ F [t], which we may suppose coprime. If p divides the polynomials k, n_{1}, ⋯, n_{4}, then p also divides l^{2} + (α + βt^{2})m^{2} since f is square free. Since k, l, m, n_{1}, ⋯, n_{4} are coprime, the polynomial p does not divide m. In particular, α+βt^{2} becomes a square in F_{p}, which is excluded. Hence, at least one polynomial among k, n_{1}, ⋯, n_{4} is not divided by p. Moreover, since

\overline{k}^{2}a = ∑_{i=1}^{4} \overline{n_{i}}^{2}\overline{h_{i}}

and D is anisotropic, we conclude that p does not divide k. Hence, a ∈ D_{F_{p}}(D). This finishes the proof of the claim.

Moreover, since θ_{F_{p}} is isotropic, the polynomial p is inseparable, i.e., \frac{∂p}{∂t} = 0. Hence, we may write F_{p} = S( \sqrt[2^{n}]{d}) for an integer n ≥ 1, a separable extension S/F and d ∈ S. Then, the norm degree of θ̃ decreases to 8 after extending scalars to F_{p}. But by, the claim above, we have ndeg_{F_{p}}(θ̃_{F_{p}}) ≤ 4, which is not possible. Hence, the case (c) does not happen.

Now in equation (3) we may change f by another polynomial f′ such that deg f′ < deg f. Let u_{1}, ⋯, u_{4} ∈ F [t] be such that

θ̃_{F (t)} ≃ f′ ⟨1, α + βt^{2}⟩ ⊥ ⟨u_{1}, ⋯, u_{4}⟩ .

It is clear that this isometry implies the following:

(5)  θ_{F (t)} ≃ f′ ⟨1, (g′)^{2} + (α + βt^{2})(h′)^{2}⟩_{b} ⊥ ⟨v_{1}, ⋯, v_{4}⟩_{b}

for suitable polynomials g′, h′, v_{1}, ⋯, v_{4} ∈ F [X].

If deg f′ = 0 then we are done, if not, we apply to f′ the same argument used for f to change f′ with another polynomial of degree smaller than deg f′. By continuing this process we get the desired conclusion.

7. Proof of Proposition 1.6

Let F be a field satisfying one of the following conditions:

(C1) Any 4-fold bilinear Pfister form over F is isotropic.

(C2) Question 1.4 has a positive answer.

Let B be an anisotropic bilinear form over F of dimension 8, and let τ be its leading form.

(1) ⟹ (2) Suppose that B is of height and degree 2 but not good. By Theorem 1.2 there exists an anisotropic Albert bilinear form θ which becomes isotropic over F (B) and satisfies B ⊥ θ ∈ I^{3}F. Then, by condition (C1) or (C2), the form θ̃ is a quasi-Pfister neighbor since ndeg_{F} (θ̃) = 8.

Moreover, the isotropy of θ_{F (B)} implies that N_{F} (B̃) ⊂ N_{F} (θ̃). Since ndeg_{F} (B̃) ≥ 8, we conclude that N_{F} (B̃) = N_{F} (θ̃). In particular, ndeg_{F} (B̃) = 8 and B̃ is also a quasi-Pfister neighbor. By [5, Th. 8.11], we have dim(B̃_{F (B)})_{an} = dim(θ̃_{F (B)})_{an} = 4. Moreover, θ_{F (B)} can not be metabolic since dim B > dim θ. Then, dim(θ_{F (B)})_{an} = 4. We also have dim(B_{F (B)})_{an} = 4 since h(B) = 2. Now all this data with Proposition 2.1 imply the following:

(6)  \begin{matrix}B_{F (B)} ≃ xθ_{1} ⊥ 2 × 𝕄(0) \\ θ_{F (B)} ≃ yθ_{2} ⊥ 𝕄(0) \end{matrix}

for suitable scalars x, y ∈ F (B)^{∗} and 2-fold bilinear Pfister forms θ_{1} and θ_{2} over F (B).

On the one hand, x̃θ_{1} and ỹθ_{2} are definable over F since, by equation (6), x̃θ_{1} ≃ (B̃_{F (B)})_{an} and ỹθ_{2} ≃ (θ̃_{F (B)})_{an} (recall that the anisotropic part of a totally singular quadratic form over any field extension is definable over the ground field). In particular, xθ_{1} and yθ_{2} represent scalars in F^{∗}, and hence we may suppose that x, y ∈ F^{∗}.

On the other hand, the condition B ⊥ θ ∈ I^{3}F implies that θ_{1} ⊥ θ_{2} ∈ I^{3}F (B), and by the Hauptsatz we have θ_{1} ≃ θ_{2}.

Then we may conclude from equation (6) that (xB ⊥ yθ)_{F (B)} ∼ 0. Since ndeg_{F} (B̃) = 8, and xB ⊥ yθ is not metabolic (because B is anisotropic), it follows from [13, Th. 1.2] that xB ⊥ yθ ∼ zπ for some z ∈ F^{∗} and π a 3-fold bilinear Pfister form such that π̃ is similar to B̃. In particular, N_{F} (B̃) = N_{F} (π̃). Since N_{F} (θ̃) = N_{F} (B̃) = N_{F} (π̃), it follows that θ̃ is similar to a subform of π̃.

(2) ⟹ (1) Suppose we have an anisotropic Albert bilinear form θ, a 3-fold bilinear Pfister form π, and scalars x, y, z ∈ F^{∗} such that xB ⊥ yθ ⊥ zπ is metabolic and the forms B̃ and C̃ are similar to subforms of π̃. Then, xB ⊥ yθ ∈ I^{3}F and the forms B̃ and θ̃ are quasi-Pfister neighbors of π̃. Consequently, θ̃_{F (B)} is isotropic, i.e., θ_{F (B)} is isotropic and B ⊥ θ ∈ I^{3}F (because B ⊥ xB ∈ I^{3}F and θ ⊥ yθ ∈ I^{3}F). By Theorem 1.2 we get that B is of height and degree 2 but not good.

Acknowledgment. This work was completed during a visit of the first author at the Universität Bielefeld for the period April-May 2008. The first author would like to thank the second author for organizing this visit, and he is grateful to the SFB 701 ”Spektrale Strukturen und topologische Methoden in der Mathematik” for this support.

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Faculté Jean Perrin, Laboratoire de mathématiques de Lens EA 2462, rue Jean Souvraz SP-18, F-62307 Lens

  E-mail address: laghribi@euler.univ-artois.fr

Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501, Bielefeld

  E-mail address: rehmann@math.uni-bielefeld.de