FIELDS OF U-INVARIANT 9

OLEG T. IZHBOLDIN

Abstract. Let F be a field of characteristic ≠ 2. The u-invariant of the field F is defined as the maximal dimension of anisotropic quadratic forms over F. It is well known that the u-invariant cannot be equal to 3, 5, or 7. We construct a field F with u-invariant 9. It is the first example of a field with odd u-invariant > 1. The proof uses the computation of the third Chow group of projective quadrics X_{ϕ} corresponding to quadratic forms ϕ. We compute CH^{3}(X_{ϕ}) completely except for the case dim ϕ = 8. In our computation we use the results of B. Kahn, M. Rost, and R. Sujatha on the unramified cohomology and the third Chow group of quadrics ([KRS1]). We compute the unramified cohomology H_{nr}^{4}(F (ϕ)/F) for all forms of dimension ≥ 9. We apply our results to prove several conjectures. In particular, we prove a conjecture of Bruno Kahn on the classification of forms of height 2 and degree 3 for all fields of characteristic zero.

Contents

0. Introduction 2 1. Quadratic forms 6 2. Graded Grothendieck groups of quadrics 12 3. Chow groups of quadrics 15 4. Galois cohomology 18 5. Unramified cohomology of quadrics 19 6. Pfister neighbors over function fields 21 7. Construction of a field with u-invariant 9 25 8. Special pair of forms: definition and basic properties 29 9. Special pairs of degree 4 and unramified cohomology 35 10. Proof of the Conjectures 0.8 and 0.9 38 11. The group Tors CH^{3}(X_{ϕ}) for forms ϕ ∈ I^{2}(F) 40 12. Proof of Theorems 0.5 and 0.6 42 13. Forms of height 2 and degree 3 46 References 48

Key words and phrases. Quadratic form, Chow groups, unramified cohomology. This research was supported by the Alexander von Humboldt-Stiftung.

0. Introduction

Let F be a field of characteristic ≠ 2, and let ϕ = a_{1}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2} be a quadratic form over F. We always assume that the forms under consideration are nondegenerate. The form ϕ is called isotropic if the homogeneous equation a_{1}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2} = 0 has a nontrivial solution. Otherwise, the form ϕ is called anisotropic. The u-invariant of the field F is defined as the maximal dimension of anisotropic quadratic forms over F:

u(F) = sup{dim ϕ | ϕ is an anisotropic form over F }.

Since many questions about a quadratic form can be reduced to that about its anisotropic part, u(F) is a fundamental measure of the complexity of quadratic form theory over F. However, u(F) is often very difficult to determine, and its value is unknown for many fields. Nevertheless, in the following cases the u-invariant is known:

• If F is a quadratically closed field, then u(F) = 1. In particular, u(ℂ) = 1.

• If F is a formally real field, then u(F) = ∞. In particular, u(ℚ) = ∞ and u(ℝ) = ∞.

• If F is a finite field, then u(F) = 2.

• If F is a local field, then u(F) = 4.

• If F is a nonreal global field, then u(F) = 4. In particular, u(ℚ(i)) = 4.

• For any field F, we have u(F ((t_{1}))((t_{2})) … ((t_{n}))) = 2^{n}u(F). In particular, u(ℂ((t_{1}))((t_{2})) … ((t_{n}))) = 2^{n}.

• If F = ℂ(t_{1}, …, t_{n}) denotes the field of rational functions in n variables

over ℂ, then u(F) = 2^{n}. Note that in all examples listed above the value of the u-invariant is always a power of 2 or infinite. This observation was probably a reason for the wellknown conjecture of Kaplansky (1953) that only powers of 2 are possible for finite values of the u-invariant. It is known that Kaplansky’s conjecture is true for finitely generated fields over algebraically closed or finite fields (see [Kah3]). More precisely, if F is such a field, then it is proved that u(F) = 2^{cd(F)}, where cd(F) is the cohomological dimension of F. The problem concerning the uinvariant of finitely generated fields of transcendence degree ≠ 0 over number or local fields seems to be very difficult. For example, the finiteness of the u-invariant is known only in the case when F is of transcendence degree 1 over a local nondyadic field (1998, [HvG] and [PS]). In a series of papers, R. Elam and T. Y. Lam studied the u-invariant for fields satisfying some additional hypotheses ([Elm, EL1, EL2]). In particular, Kaplansky’s conjecture was proved for all linked fields.

In ([M2], 1989), A. Merkurev disproved this conjecture by constructing a field F with u(F) = 6 (see also [Lam2]). Later, he proved that the u-invariant of a field can be any even number ([M3], 1991). It should be noted that the basic idea of his proof (index reduction formula) cannot be used to construct a field with odd u-invariant. It was still an open problem whether the u-invariant can take odd values other than 1. However, it is known that the u-invariant never equals 3, 5, or 7 (see e.g., [Lam1, Prop. 4.8]). The principal result of this paper is the following

Theorem 0.1. There exists a field F with u(F) = 9.

In fact, we prove a more precise result. To formulate it, we introduce the following notion:

Definition 0.2. We say that ϕ is an essential 9-dimensional form if the following conditions hold:

• ϕ is an anisotropic 9-dimensional form;

• ϕ is not a Pfister neighbor;

• ind C_{0}(ϕ) ≥ 4.

The role of this notion is illuminated by the following version of Theorem 0.1.

Theorem 0.3. Let ϕ be a 9-dimensional quadratic form over a field F. Then the following conditions are equivalent:

(1) there exists a field extension E/F such that u(E) = 9 and the form ϕ_{E} is anisotropic,

(2) ϕ is an essential form. Moreover, if these conditions hold, we can construct a field E with the following additional properties:

• all anisotropic 9-dimensional forms over E are similar to ϕ_{E}.

• E has no nontrivial odd extensions and cd_{2}(E) = 3.

The basic tool for the proof of Theorem 0.1 is the following

Proposition 0.4. (cf. Theorem 7.3). Let ϕ be an essential 9-dimensional quadratic form and ψ be a form of dimension ≥ 10. Then ϕ_{F (ψ)} is an essential form.

Using this proposition we can construct a field with u-invariant 9 (cf. [M3, §3] and/or [M2]). Indeed, we start with an arbitrary field k_{0} and consider the form ϕ = ⟨t_{1}, …, t_{9}⟩ over the rational function field k = k_{0}(t_{1}, …, t_{9}). It is easy to show that the k-form ϕ is essential. By iterated passages to function fields of 10-dimensional quadratic forms, one can obtain a field F with no anisotropic 10dimensional quadratic forms. By Proposition 0.4, the form ϕ_{F} is still essential. In particular, ϕ_{F} is an anisotropic 9-dimensional form. Hence, u(F) = 9.

Our proof of Proposition 0.4 (and Theorem 0.3) uses the following recent results of B. Kahn, M. Rost, R. Sujatha, and N. Karpenko:

• the results of B. Kahn, M. Rost, and R. Sujatha, concerning the unramified cohomology of quadrics [KRS1].

• a new result of N. Karpenko related to isotropy of 9-dimensional essential forms over the function fields of 9-dimensional forms (see Theorem 1.13).

An essential part of the proof of Proposition 0.4 is based on the computation of the third Chow group and the fourth unramified cohomology group of quadrics. We compute these groups completely for all quadrics of dimension ≥ 7 (i. e., in the case where the corresponding quadratic forms are of dimension ≥ 9).

Theorem 0.5. Let ϕ be an F -form of dimension ≥ 9. Let X = X_{ϕ} be the projective quadric corresponding to the form ϕ. Then Tors CH^{3}(X) = 0, with the following exceptions, where Tors CH^{3}(X) ≃ ℤ/2ℤ : (9-a) ϕ = π ⊥ ⟨d⟩, where π is similar to an anisotropic 3-fold Pfister form (in

this case, ind C_{0}(ϕ) = 1). (9-b) ϕ is an anisotropic 9-dimensional form with the following properties:

ind C_{0}(ϕ) = 2, det ϕ ∉ D_{F} (ϕ), and ϕ contains no 7-dimensional Pfister neighbors.

(10-a) ϕ = π ⊥ ℍ, where π is similar to an anisotropic 3-fold Pfister form (in

this case, ϕ ∈ I^{2}(F) and ind C(ϕ) = 1). (10-b) ϕ is an anisotropic 10-dimensional form such that ϕ ∈ I^{2}(F) and

ind C(ϕ) = 2. (10-c) ϕ is an anisotropic 10-dimensional form with nontrivial discriminant

d = d_{±}ϕ ∉ F^{∗2} which is similar to a subform of an anisotropic 12dimensional form τ ∈ I^{3}(F) and such that ϕ_{F (\sqrt{d})} is not hyperbolic (in this case, ind C_{0}(ϕ) = 1).

(11-a) ϕ is an anisotropic 11-dimensional form with ind C_{0}(ϕ) = 1. (12-a) ϕ is an anisotropic 12-dimensional form belonging to I^{3}(F) (in particular,

ind C(ϕ) = 1).

Remarks. 1) The statement that the group Tors CH^{3}(X) is either zero or isomorphic to ℤ/2ℤ is due to N. Karpenko. He also proved that this group is trivial for all forms of dimension > 12 ([Kar2]).

2) In many cases the triviality of the group Tors CH^{2}(X_{ϕ}) was proved by B. Kahn and R. Sujatha in [KS3].

3) Theorem 0.5 together with the results of N. Karpenko [Kar1] complete the computation of the group CH^{3}(X_{ϕ}) for all forms except for the case when ϕ is an 8-dimensional form with nontrivial discriminant.

Theorem 0.5 is closely related to the computation of the fourth unramified cohomology of quadrics. Let H^{n}(F) be the Galois cohomology group of F with ℤ/2ℤ-coeficients. For a form ϕ, we denote by H̃_{nr}^{4}(F (ϕ)/F) the homology group of the complex

H^{4}(F) → H^{4}(F (X_{ϕ})) → ⊕_{x∈X_{ϕ}^{(1)}}H^{3}(F (x)),

where X_{ϕ} is the projective quadric corresponding to ϕ. This group was studied in detail in [KRS1, KS2, KS3]. Among many other results, it was proved that if ϕ has dimension ≥ 9 and ϕ is not a 4-fold neighbor, then there exists an injective homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) → Tors CH^{3}(X_{ϕ}) (see [KRS1, Th. 6(1) and Prop.3]). The construction of this homomorphism fundamentally uses mixing the Hochschild-Serre and Bloch-Ogus spectral sequences. Another tool, originally due to Bloch, is an exact sequence that relates H-cohomology and K-cohomology (cf. [CT2, 3.6]). It is a very important question to ask when the homomorphism ϵ is an isomorphism.

Theorem 0.6. Let ϕ be an anisotropic form of dimension ≥ 9 that is not a 4-fold Pfister neighbor. Then the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) → Tors CH^{3}(X_{ϕ}) is an isomorphism.

Remarks. 1) Under the hypothesis of the theorem, B. Kahn and R. Sujatha proved that the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F, ℚ/ℤ(3)) → Tors CH^{3}(X_{ϕ}) is an isomorphism ([KS3, Th. 1(b)]). Moreover, they proved Theorem 0.6 for fields containing all 2-primary roots of unity (see Corollary 1 in [KS3]).

2) Theorem 0.6 together with Theorem 0.5, complete the computation of the groups H̃_{nr}^{4}(F (ϕ)/F) for all forms ϕ of dimension ≥ 9. Here we note that in the case of 4-fold neighbors (which is excluded from the formulation of Theorem 0.6) the group H̃_{nr}^{4}(F (ϕ)/F) was computed in [KRS1, Cor. 8(3)].

The following easy corollary of Theorem 0.6 gives a partial answer to the question stated at the beginning of Subsection 1.1 of [KS2].

Corollary 0.7. (cf. [KS2, Theorem 1]). Let ϕ be a form of dimension ≥ 9. Then the natural homomorphism H̃_{nr}^{4}(F (ϕ)/F) → H̃_{nr}^{4}(F (ϕ)/F, ℚ/ℤ(3)) is surjective.

In this paper we also prove the following conjectures.

Conjecture 0.8. ([Kah1, p.154]). Let ϕ ∈ I^{3}(F) be an anisotropic form of dimension 12. Let ψ be a form of dimension > 12. Then ϕ_{F (ψ)} is anisotropic.

Conjecture 0.9. ([Lag, Conj. 3]). Let ϕ ∈ I^{2}(F) be an anisotropic form of dimension 10 with ind C(ϕ) = 2. Let ψ be a form of dimension > 10. Then ϕ_{F (ψ)} is anisotropic.

Conjecture 0.10. ([Izh1, Conj. 2.1]). Let ϕ be a 10-dimensional anisotropic form. Then ϕ has maximal splitting if and only if at least one of the following conditions holds:

• ϕ is a Pfister neighbor,

• ϕ ≃ ⟨ ; ⟨a⟩ ; ⟩ ⊗ τ with dim τ = 5.

The proofs of all results mentioned above depend only on published papers or papers accepted for publication. However, our following result depends on the theory of Voevodsky related to the proof of Milnor’s conjecture. Using some recent results announced by Alexander Vishik [Vi3, Vi4], we prove the following conjecture of Bruno Kahn in the case when n = 3 and char F = 0:

Conjecture 0.11. ([Kah2, Conj. 7]). Let ϕ be a nongood form of height 2 and degree n. Then ϕ is of the form τ ⊗ q, where τ is an (n − 2)-fold Pfister form and q is an Albert form.

Acknowledgments. The work on this paper was done while the author was a visiting researcher at the Bielefeld University supported by the Alexander von Humboldt Stiftung. I would like to thank this university for the excellent working conditions. I am grateful to Nikita Karpenko for numerous helpful discussions concerning Chow groups. I also would like to thank Boris Bekker, Ivan Panin, and Alexander Vishik for many useful suggestions. The author is greatly indebted to Ulf Rehmann, Bruno Kahn, and Ramdorai Sujatha for their comments and suggestions which lead to the final form of the paper.

1. Quadratic forms

This section contains some preliminary results concerning quadratic forms. The basic notations are the same as in the books of T. Y. Lam and W. Scharlau ([Lam1], [Sch]). We recall some of them and introduce several additional notations.

Let ϕ be a quadratic form over F. We denote the Witt index of ϕ by i_{W} (ϕ). The anisotropic part of ϕ is denoted by ϕ_{an}. We say that ϕ is Witt equivalent to ψ if ϕ_{an} is isometric to ψ_{an}. We call two forms ϕ and ψ stably birationally equivalent if the corresponding quadrics X_{ϕ} and X_{ψ} are stably birationally equivalent. It is well known that the forms ϕ and ψ are stably birationally equivalent if and only if ϕ_{F (ψ)} and ψ_{F (ϕ)} are both isotropic (see, e.g., [Ohm, Sect. 3]). In this paper we deal with the following four equivalence relations on the set of quadratic forms:

ϕ ≃ ψ - isometry of the forms ϕ and ψ,

ϕ ∼ ψ - similarity of the forms ϕ and ψ (i.e., ϕ ≃ kψ for a suitable k ∈ F^{∗}),

ϕ ∼^{st} ψ - stable rational equivalence of the forms ϕ and ψ,

ϕ = ψ - Witt equivalence of the forms ϕ and ψ.

The Clifford algebra (resp., the even part of the Clifford algebra) of ϕ will be denoted by C(ϕ) (resp., C_{0}(ϕ)). We recall that if ϕ ∈ I^{2}(F), then C_{0}(ϕ) is a semisimple algebra of the form A × A, where A is a central simple F -algebra. Moreover, in this case we have C(ϕ) = M_{2}(A).

For any form ϕ, we define F -algebra C′_{0}(ϕ) as follows:

• if dim ϕ is odd, then C′_{0}(ϕ) = C_{0}(ϕ). In this case, C′_{0}(ϕ) is a central simple F -algebra.

• if dim ϕ is even and d = d_{±}ϕ ∉ F^{∗2}, then C′_{0}(ϕ) = C_{0}(ϕ). In this case, C′_{0}(ϕ) is a central simple L-algebra, where L/F is the discriminant extension (i.e., L = F (\sqrt{d})).

• if dim ϕ is even and d = d_{±}ϕ ∈ F^{∗2}, then C′_{0}(ϕ) = A, where A is a central simple F -algebra such that C_{0}(A) = A × A.

By definition, C′_{0}(ϕ) is a simple algebra of degree 2^{[(dim ϕ−1)/2]}. By Wedderburn’s theorem, there exists a division algebra D and an integer s ≥ 0 such that C′_{0}(ϕ) = M_{2^{s}}(D). We define Schur invariants ind ϕ and i_{S}(ϕ) as follows ^{1}:

ind ϕ = ind C′_{0}(ϕ) = deg D, i_{S}(ϕ) = s

We say that ind ϕ is the Schur index of ϕ. The integer i_{S}(ϕ) will be called the Schur splitting index of ϕ. Many properties of the Schur splitting index i_{S}(ϕ) are the same as the properties of the Witt index i_{W} (ϕ). For example, i_{S}(ϕ ⊥ mℍ) = i_{S}(ϕ) + m and i_{S}(kϕ) = i_{S}(ϕ). Moreover, i_{S}(ϕ) ≥ i_{W} (ϕ).

The following lemma is trivial

Lemma 1.1. Let ϕ be a form over F.

• if dim ϕ = 2n + 1, then deg C′_{0}(ϕ) = 2^{i_{S}(ϕ)} · ind ϕ = 2^{n},

• if dim ϕ = 2n, then deg C′_{0}(ϕ) = 2^{i_{S}(ϕ)} · ind ϕ = 2^{n−1}.

The following statement is well known:

Lemma 1.2. Let E/F be either an odd extension or a unirational extension (i.e., a subfield of a purely transcendental extension). Then for any F -form ϕ, we have

i_{W} (ϕ_{E}) = i_{W} (ϕ), dim(ϕ_{E})_{an} = dim ϕ_{an},
i_{S}(ϕ_{E}) = i_{S}(ϕ), ind(ϕ_{E}) = ind(ϕ).

In particular, the homomorphism W (F) → E(E) is injective. Moreover, the homomorphism W (F)/I^{n}(F) → W (E)/I^{n}(E) is injective for all n.

Theorem 1.3 ([Ti]). Let F be a field and F (t) be the field of rational functions in one variable over F.

• if A is a simple algebra over F, then ind A_{F (t)} = ind A,

• if A is a central simple algebra over F of exponent 2, and d is an element of F^{∗} such that d ∉ F^{∗2}, then ind(A_{F (t)} ⊗ (d, t)) = 2 ind A_{F (\sqrt{d})}.

Corollary 1.4. Let F be a field and F (t) be the field of rational functions in one variable over F. Let ϕ be a quadratic form over F, and let ϕ̃ = ϕ_{F (t)} ⊥ ⟨kt⟩ with k ∈ F^{∗}.

• if dim ϕ is odd or ϕ ∈ I^{2}(F), then ind ϕ̃ = ind ϕ,

• if dim ϕ is even and ϕ ∉ I^{2}(F), then ind ϕ̃ = 2 ind ϕ.

Sketch of the proof. Using the formulas for Clifford algebras given in [Lam1], one can compute C_{0}(ϕ̃) in terms of C(ϕ) or C_{0}(ϕ). After this, the required result follows easily from Tignol’s Theorem 1.3.

The following lemma is a trivial consequence of Merkurev’s index reduction formula ([M3]).

Lemma 1.5. Let ϕ and ψ be forms over F.

• If ψ ∈ I^{3}(F), then ind ϕ_{F (ψ)} = ind ϕ.

• If ind ϕ ≤ 8 and dim ψ ≥ 9, then ind ϕ_{F (ψ)} = ind ϕ.

• If ind ϕ ≥ 4 and dim ψ ≥ 9, then ind ϕ_{F (ψ)} ≥ 4.

1In [H2] the author used the notation i_{S}(ϕ) for the index of c(ϕ) ∈ Br(F).

We recall Cassels–Pfister subform theorem (which will be used in the sequel without any reference).

Theorem 1.6. [Sch, Ch.4, Th. 5.4(ii)] Let ϕ be a nonhyperbolic form, and let ψ be a form such that ϕ_{F (ψ)} is hyperbolic. Then for any k ∈ D_{F} (ϕ) · D_{F} (ψ), we have kψ ⊂ ϕ. In particular, dim ψ ≤ dim ϕ.

In what follows, we use the following very specific consequence of this theorem.

Corollary 1.7. Let ψ be a form over F, and let d be an element of F such that d ∉ F^{∗2}. Let K be

• either an odd extension of F,

• or a unirational extension of F,

• or K = F (ψ), where ψ is a form with dim ψ > dim ϕ. Then the form ϕ_{K(\sqrt{d})} is hyperbolic if and only if the form ϕ_{F (\sqrt{d})} is hyperbolic.

Proof. If K/F is odd or unirational, then the extension K(\sqrt{d})/F (\sqrt{d}) is also odd or unirational. In these cases, the result follows immediately from Lemma 1.2. Now, we assume that K = F (ψ) with dim ψ > dim ϕ. Let L = F (\sqrt{d}). By Theorem 1.6, the form ϕ_{K(\sqrt{d})} ≃ ϕ_{L(ψ)} is hyperbolic if and only if the form ϕ_{F (\sqrt{d})} ≃ ϕ_{L} is hyperbolic.

The following theorem we will call Pfister’s theorem on 10-dimensional forms in I^{3} (or simply Pfister’s theorem).

Theorem 1.8. (see [Pf2, Satz 14 and Zautsats], also [Sch, Ch.2, Th.14.4] or [H3, Th. 2.9]). Let ϕ ∈ I^{3}(F) and dim ϕ = 10. Then ϕ is isotropic and can be written in the form ϕ ≃ π ⊥ ℍ for some π ∈ GP_{3}(F).

Corollary 1.9. Let ϕ ∈ I^{3}(F). We have

(a) if dim ϕ_{an} < 12, then there exists π ∈ GP_{3}(F) such that ϕ and π are Witt equivalent,

(b) if dim ϕ = 12, then i_{W} (ϕ_{F (ϕ)}) ≥ 2,

(c) if dim ϕ = 12 and ϕ is nonhyperbolic, then (ϕ_{F (ϕ)})_{an} ∈ GP_{3}(F (ϕ)).

The following theorems concern the so-called forms with maximal splitting.

Theorem 1.10 ([H1, Kah1, H2, Izh1]). Let ϕ be an anisotropic form of dimension 2^{n} + m, where 0 < m ≤ 2^{n}. Then

• 1 ≤ i_{1}(ϕ) ≤ m (in the case where i_{1}(ϕ) = m, we say that ϕ has maximal splitting),

• if ϕ is a Pfister neighbor, then i_{1}(ϕ) = m (i.e., ϕ has maximal splitting),

• if n ≥ 3, m ≥ 2^{n} −5, and ϕ has maximal splitting, then ϕ is an (n+1)-fold Pfister neighbor,

• if n ≥ 3, m = 2^{n} − 6, ϕ ∈ I^{2}(F), and ϕ has maximal splitting, then ϕ is an (n + 1)-fold Pfister neighbor.

The first two statements of this theorem (as well as the notion of forms with maximal splitting) are due to Detlev Hoffmann [H1]. The proofs of the third and fourth statements follow easily from the results of Bruno Kahn [Kah1, Remark after Th.4]. Besides, an elementary proof of the third statement can be found in [Izh1], and an elementary proof of the fourth statement is given in [H2] and [Izh1].

Theorem 1.11 ([H1, Izh4]). Let ϕ be an anisotropic form with 2^{n} < dim ϕ ≤ 2^{n+1}. Suppose that ϕ has maximal splitting (e.g., dim ϕ = 2^{n} + 1). Let ψ be a form of dimension ≥ 2^{n} + 1 such that ϕ_{F (ψ)} is isotropic. Then

• ψ has maximal splitting and 2^{n} < dim ψ ≤ 2^{n+1},

• ψ_{F (ϕ)} is isotropic (and hence ϕ ∼^{st} ψ),

• if ψ is a Pfister neighbor, then ϕ is also a Pfister neighbor.

The first and third statements of this theorem are due to D. Hoffmann [H1]. The second statement is proved in [Izh4]. Setting n = 3 in Theorems 1.10 and 1.11, we get the following corollary.

Corollary 1.12. Let ϕ be a form 9 ≤ dim ϕ ≤ 16. Suppose that ϕ has maximal splitting (e.g., dim ϕ = 9), and ϕ is not a Pfister neighbor. Let ψ be a form of dimension ≥ 9 such that ϕ_{F (ψ)} is isotropic. Then

• 9 ≤ dim ϕ, dim ψ ≤ 10,

• ψ is not a Pfister neighbor and ψ ∉ I^{2}(F),

• ψ ∼^{st} ϕ.

The following result concerning the isotropy of 9-dimensional forms over function fields of quadrics has recently been proved by Nikita Karpenko.

Theorem 1.13. (N. Karpenko [Kar3]). Let ϕ be a 9-dimensional essential form (see Definition 0.2). Let ψ be a form of dimension 9 such that ϕ_{F (ψ)} is isotropic. Then ψ is similar to ϕ.

It should be pointed out that Theorem 1.13 plays a key role in the proof of our main Theorem 0.1. We also need the following statement from Karpenko’s paper [Kar3].

Lemma 1.14. Let ϕ be a 9-dimensional form which is not a Pfister neighbor. Then ϕ_{L} is not a Pfister neighbor for all odd extensions L/F.

The following theorem was proved by A. Vishik by using calculations in Voevodsky’s motivic category [Vi2]. In [Kar4], N. Karpenko simplified Vishik’s proofs by using arguments in the framework of the classical category of Grothendieck Chow-motives and presented the current formulation of the theorem:

Theorem 1.15. Let ϕ and ψ be anisotropic forms such that ϕ ∼^{st} ψ. Then dim ϕ − i_{1}(ϕ) = dim ψ − i_{1}(ψ).

The rest of this section contains several lemmas concerning quadratic forms. These results will be used in other sections.

Lemma 1.16. Let F be a field which has no nontrivial odd extensions. Let ϕ be an F -form with ind ϕ ≥ 2^{s}. Then there exists a finite field extension E/F such that

• ind ϕ_{E} = 2^{s},

• if dim ϕ is even, then ϕ_{E} ∈ I^{2}(E),

• the homomorphism N_{E/F} : K(C_{0}(ϕ_{E})) → K(C_{0}(ϕ)) is surjective. ^{2}

Proof. Let D be the simple division algebra corresponding to the algebra C′_{0}(ϕ). Let L be the center of D (in the cases when dim ϕ is odd or ϕ ∈ I^{2}(F) we obviously have L = F). Let M/L be a maximal subfield of D. Clearly, [M : L] = 2^{r} with r = ind ϕ. By our assumption, [M : L] = 2^{r} ≥ 2^{s}. Since F has no odd extensions, Galois theory shows that there exists a tower of fields

M = M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{r} = L

such that [M_{i} : M_{i+1}] = 2 for all i. Setting E = M_{s}, we get the field with the required properties.

Lemma 1.17. Let L/F be a field extension, A be a central simple F -algebra of exponent 2, and m be an integer. Suppose that one of the following conditions holds:

(i) [L : F] ≤ 2, ind A_{L} = 1, and m = 2; (ii) L = F, ind A ≤ 2, and m = 3; (iii) [L : F] ≤ 2, ind A_{L} ≤ 2, and m = 4; (iv) L = F, ind A ≤ 4, and m = 5. Then there exists an m-dimensional form μ over F such that the algebra C′_{0}(μ) is Brauer equivalent to A_{L}.

Proof. (i) If L = F, we set μ = ℍ, if L = F (\sqrt{d}), we set μ = ⟨ ; ⟨d⟩ ; ⟩.

(ii) Since ind A ≤ 2, it follows that A is Brauer equivalent to a quaternion algebra (a, b). Now, it suffices to set μ = ⟨1, −a, −b⟩.

(iii) If L = F, then A_{L} = A is Brauer equivalent to a biquaternion algebra (a, b). In this case we set μ = ⟨ ; ⟨a, b⟩ ; ⟩. If [L : F] = 2, the condition ind A_{L} ≤ 2 implies that ind A ≤ 4. Therefore A is Brauer equivalent to a biquaternion algebra. Let q be an Albert form corresponding to A. Since ind A_{L} ≤ 2, the form q_{L} is isotropic. Hence q can be written in the form q ≃ c ⟨ ; ⟨d⟩ ; ⟩ ⊥ q_{0}, where d is such that L = F (\sqrt{d}). Now, it suffices to set μ = q_{0}.

(iv) Let q be an Albert form corresponding to A. We define μ as an arbitrary 5-dimensional subform of q.

Lemma 1.18. Let ϕ and ψ be F -forms such that dim ϕ ≡ dim ψ (mod 2ℤ) and C′_{0}(ϕ) is Brauer equivalent to C′_{0}(ψ). Then there exists k ∈ F^{∗} such that ϕ ≡ kψ (mod I^{3}(F)).

Proof. If dim ϕ and dim ψ are odd, it suffices to set k = d_{±}ϕ/d_{±}ψ. Then γ = ϕ ⊥ −kψ ∈ I^{2}(F) and c(γ) = c(ϕ) + c(ψ) = 0. Therefore, γ ∈ I^{3}(F). In the case when dim ϕ and dim ψ are even, the lemma is proved in [Izh2, Lemma 4.3]

2Here K(A) denotes the Grothendieck group of the ring A.

Lemma 1.19. Let ϕ be a form satisfying one of the following conditions:

(i) dim ϕ = 2n, and ind ϕ = 1, (ii) dim ϕ = 2n − 1 and ind ϕ ≤ 2, (iii) dim ϕ = 2n − 2 and ind ϕ ≤ 2, (iv) dim ϕ = 2n − 3 and ind ϕ ≤ 4. Then there exists a (2n + 1)-dimensional form ϕ̃ and a (2n + 2)-dimensional form γ ∈ I^{3}(F) such that ϕ ⊂ ϕ̃ ⊂ γ and ind ϕ̃ = 1.

Proof. Let m = 2n + 2 − dim ϕ. By Lemma 1.17, there exists an m-dimensional form μ such that C′_{0}(ϕ) is Brauer equivalent to C′_{0}(μ). Then there exists k ∈ F^{∗} such that ϕ ≡ kμ (mod I^{3}(F)) (Lemma 1.18). We set γ = ϕ ⊥ −kμ. Now it suffices to set ϕ̃ = ϕ ⊥ −kμ_{0}, where μ_{0} is a subform of μ of codimension 1.

Lemma 1.20. (cf. [H1, Lemma 3]). Let ψ be a form over F, and let ϕ ⊂ ψ. If dim ϕ ≥ dim ψ − i_{W} (ψ_{F (ψ)}) + 1, then ϕ ∼^{st} ψ.

Proof. It suffices to verify that ϕ_{F (ψ)} is isotropic, but this is an obvious consequence of [H1, Lemma 3].

Lemma 1.21. (1) Let γ be a 12-dimensional form from I^{3}(F), and let ψ be an 11-dimensional subform of γ. Then ψ ∼^{st} γ.

(2). Let ψ be an 11-dimensional form with ind ψ = 1. Then there exists a 12-dimensional form γ ∈ I^{3}(F) such that ψ ∼^{st} γ.

Proof. (1). Obvious in view of Lemma 1.20 and Corollary 1.9.

(2). We set γ = ψ ⊥ ⟨det ψ⟩. Clearly γ ∈ I^{2}(F) and ind γ = ind ψ = 1. Hence, γ ∈ I^{3}(F). Item (1) shows that ψ ∼^{st} γ.

Lemma 1.22. (see, e.g., [Izh4, Lemma 2.11]). Let μ and ν be F -forms of dimension ≥ 1, and let ϕ = μ ⊥ −tν be a form over F̃ = F (t). Then the extension F̃(ϕ)/F is purely transcendental.

Lemma 1.23. Let ϕ be an F -form and E/F be a unirational field extension. Then the form ϕ_{E} is a Pfister neighbor if and only if the form ϕ is a Pfister neighbor.

Proof. It suffices to consider the case where E/F is purely transcendental. In this case lemma is proved in [H1, Prop.7].

Lemma 1.24. Let ϕ be a 9-dimensional essential form over F (see Definition 0.2). Let E/F be either an odd extension or a unirational extension. Then ϕ_{E} is an essential form.

Proof. By Springer’s theorems, ϕ_{E} is anisotropic. By Lemma 1.2, ind ϕ_{E} = ind ϕ ≥ 4. By Lemmas 1.14 and 1.23, ϕ_{E} is not a Pfister neighbor.

At the end of this section, we prove the implication (1)⇒(2) in Theorem 0.3 (this is the simplest part of the theorem).

Proof. Let E be as in Item (1) of Theorem 0.3. We must prove that ϕ_{E} is essential. By our assumption, ϕ_{E} is anisotropic. Since u(E) = 9, it follows that all 4-fold Pfister forms over E are isotropic. Hence all 4-fold Pfister neighbors are also isotropic. Hence, ϕ_{E} is not a 4-fold Pfister neighbor. Now, it suffices to verify that ind ϕ_{E} ≥ 4. Since all 10-dimensional forms over E are isotropic, it follows that all 9-dimensional E-form are universal (i.e., D_{E}(ϕ) = E^{∗}). In particular, d = det ϕ ∈ D_{E}(ϕ). Hence, ϕ_{E} can be written in the form ϕ_{E} = τ ⊥ ⟨d⟩, where τ is an 8-dimensional form over E with trivial discriminant. Since ind τ = ind ϕ_{E}, it suffices to prove that ind τ ≥ 4. Assume the converse, ind τ ≤ 2. Then there exists a quaternion E-algebra (a, b) such that c(τ) = (a, b). We have τ ≡ ⟨ ; ⟨a, b⟩ ; ⟩ (mod I^{3}(F)). Let γ = dτ ⊥ ⟨ ; ⟨a, b⟩ ; ⟩. Since dim γ = 12 > 9 = u(E), the form γ is isotropic. Therefore, the forms (−d)τ and ⟨ ; ⟨a, b⟩ ; ⟩ have a nontrivial common value. Let k ∈ E^{∗} be such that k ∈ D_{E}(−dτ) and

k ∈ D_{E}(⟨ ; ⟨a, b⟩ ; ⟩). Set π = ⟨ ; ⟨k⟩ ; ⟩ ⊗ τ = τ ⊥ −kτ. Since k ∈ D_{E}(−dτ), it follows

that that d ∈ D_{E}(−kτ), and hence ϕ_{E} = τ ⊥ ⟨d⟩ ⊂ τ ⊥ −kτ = π.

Since k ∈ D_{E}(⟨ ; ⟨a, b⟩ ; ⟩), it follows that ⟨ ; ⟨a, b, k⟩ ; ⟩ = 0. Therefore π = ⟨ ; ⟨k⟩ ; ⟩ ⊗ τ ≡ ⟨ ; ⟨k⟩ ; ⟩ ⊗ ⟨ ; ⟨a, b⟩ ; ⟩ = 0 (mod I^{4}(F)). Since dim π = 16, we have π ∈ GP_{4}(E). Since ϕ_{E} ⊂ π, we see that ϕ_{E} is a Pfister neighbor. However, we have proved earlier that ϕ_{E} is not a Pfister neighbor, a contradiction.

2. Graded Grothendieck groups of quadrics

For a smooth variety X we will denote by K(X) the Grothendieck ring of X. This ring is supplied with the “topological” filtration K(X)^{(i)} (which respects multiplication). The factor group K(X)^{(i)}/K(X)^{(i+1)} is denoted by G^{i}K(X). Thus, we get the adjoint graded ring G^{∗}K(X) = ⊕_{i}G^{i}K(X). There exists a canonical surjective homomorphism of the graded Chow ring CH^{∗}(X) onto G^{∗}K(X).

Now, let X = X_{ϕ} be the projective quadric corresponding to the ndimensional form ϕ. We consider X as a subvariety of codimension 1 in ℙ^{n−1} (in particular, dim X_{ϕ} = n − 2). The embedding X_{ϕ} ⊂ ℙ^{n−1} determines the local free sheaf O_{X}(−1) on X. The group K(X)^{(1)} contains the element h := 1 − [O_{X}(−1)]. By h̄, we denote the class of the element h in the group G^{1}K(X). It is well known that h̄ ∈ G^{1}K(X) corresponds the “hyperplane” class in CH^{1}(X) via the natural isomorphism CH^{1}(X) ≃ G^{1}K(X). For all i = 0, …, dim X = dim ϕ − 2 the homomorphism ℤ → G^{i}K(X), 1 → h̄^{i} is injective. The image of this homomorphism will be denoted by h̄^{i}ℤ.

Let us recall some basic properties of the group K(X_{ϕ}) (see [Kar1]). Let s = i_{S}(ϕ) (i.e., C′_{0}(ϕ) has the form M_{2^{s}}(D), where D is a division algebra). Clearly, s ≤ \frac{1}{2} dim X. If s = 0, then the group K(X) is generated (as a free Abelian group) by the elements h^{i}, where i = 0, …, dim X. If s > 0, the group K(X) contains elements l_{0}, …, l_{s−1} such that 2^{i+1}l_{i} = h^{d} + 2h^{d−1} + ⋯ + 2^{i}h^{d−i}, where d = dim X. Since the group K(X_{ϕ}) is torsion free, the elements l_{0}, …, l_{s−1} are uniquely defined. There exists a convenient geometric characterization of the elements l_{i}. Namely, l_{i} coincides with the “class of an i-dimensional line”. More formally, this means that the image of the element l_{i} under the pushforward homomorphism K(X) → K(ℙ^{n−1}) coincides with the image of the unit 1 = [O_{ℙ^{i}}] under the push-forward homomorphism K(ℙ^{i}) → K(ℙ^{n−1}) induced by the natural embedding ℙ^{i} ⊂ ℙ^{n−1}.

Theorem 2.1. (N. Karpenko, [Kar1, Th. 3.8 and 3.10]^{3} ). Let ϕ be a quadratic form over F. Let X be the quadric corresponding to ϕ. Let i be an arbitrary integer such that 0 ≤ i < dim X = dim ϕ − 2.

(1) if i < \frac{1}{2} dim X = \frac{1}{2} dim ϕ − 1, then G^{i}K(X) = Tors G^{i}K(X) ⊕ h̄^{i}ℤ,

(2) if ϕ ∉ I^{2}(F) and l_{i_{S}(ϕ)−1} ∈ K(X)^{(i+1)}, then

Tors G^{0}K(X) = ⋯ = Tors G^{i}K(X) = 0.

(3) if i < \frac{1}{2} dim ϕ − 1 and Tors G^{0}K(X) = ⋯ = Tors G^{i}K(X) = 0, then l_{i_{S}(ϕ)−1} ∈ K(X)^{(i+1)}.

Lemma 2.2. Let ϕ ⊂ ϕ̃ be quadratic forms such that ϕ, ϕ̃ ∉ I^{2}(F). Let m = dim ϕ̃ − dim ϕ. Let p be an integer such that 0 ≤ p < \frac{1}{2} dim ϕ − 1.

(1) if i_{S}(ϕ̃) = i_{S}(ϕ) and Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p, then Tors G^{i}K(Xϕ̃) = 0 for all i ≤ p + m.

(2) if i_{S}(ϕ̃) = i_{S}(ϕ) + m and Tors G^{i}K(Xϕ̃) = 0 for all i ≤ p, then Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p.

Proof. Let s = i_{S}(ϕ) and s̃ = i_{S}(ϕ̃).

(1) Theorem 2.1(3) shows that l_{s−1} ∈ K(X_{ϕ})^{(p+1)}. Let us consider the pushforward homomorphism K(X_{ϕ}) → K(Xϕ̃). This homomorphism maps K(X_{ϕ})^{(i)} to K(Xϕ̃)^{(i+m)} for all i, and maps l_{j} to l_{j} for all j ≤ s − 1. Taking the image of l_{s−1} under this homomorphism, we get l_{s−1} ∈ K(Xϕ̃)^{(p+1+m)}. Since s̃ = s, Theorem 2.1(2) shows that Tors G^{i}K(Xϕ̃) = 0 for all i ≤ p + m.

(2) Theorem 2.1(3) shows that l_{s̃−1} ∈ K(Xϕ̃)^{(p+1)}. Let us consider the pullback homomorphism K(Xϕ̃) → K(X_{ϕ}). This homomorphism maps K(Xϕ̃)^{(i)} to K(X_{ϕ})^{(i)} for all i, and maps l_{j} to l_{j−m} for all j = m, …, s̃ − 1. Taking the image of l_{s̃−1} under this homomorphism, we get l_{s̃−1−m} ∈ K(X_{ϕ})^{(p+1)}. Since s̃ − m = s, Theorem 2.1(2) shows that Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p.

Corollary 2.3. Let ϕ ⊂ ϕ̃ be odd-dimensional forms. Let k be an integer such that dim ϕ̃ = dim ϕ + 2k and ind ϕ̃ = \frac{1}{2^{k}} ind ϕ. Let p be an integer such that 0 ≤ p < \frac{1}{2} dim ϕ − 1. Suppose also that Tors G^{i}K(Xϕ̃) = 0 for all i ≤ p. Then Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p.

3Actually, Theorems 3.8 and 3.10 from [Kar1] are proved only for anisotropic quadrics.

However, the proof of three statements included in the formulation of Theorem 2.1 does not use anything specific from the anisotropic case.

Proof. By Lemma 2.2(2), it suffices to prove that i_{S}(ϕ̃) = i_{S}(ϕ) + 2k. Applying Lemma 1.1, we get 2^{(dim ϕ̃−dim ϕ)/2} = 2^{i_{S}(ϕ̃)−i_{S}(ϕ)} · ind ϕ̃/ ind ϕ. Since dim ϕ̃ − dim ϕ = 2k, and 2^{k} ind ϕ̃ = ind ϕ, we have i_{S}(ϕ̃) − i_{S}(ϕ) = 2k.

Corollary 2.4. Let ϕ be an even-dimensional form with nontrivial discriminant and ϕ̃ be an odd-dimensional form such that ϕ ⊂ ϕ̃. Let k be an integer such that dim ϕ̃ = dim ϕ + (2k + 1) and ind ϕ̃ = \frac{1}{2^{k}} ind ϕ. Let p be an integer such that 0 ≤ p < \frac{1}{2} dim ϕ − 1. Suppose that Tors G^{i}K(Xϕ̃) = 0 for all i ≤ p. Then Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p.

Proof. By Lemma 2.2(2), it suffices to prove that i_{S}(ϕ̃) = i_{S}(ϕ) + 2k + 1. Let dim ϕ = 2n and dim ϕ̃ = 2ñ + 1. Since dim ϕ̃ − dim ϕ = 2k + 1, we have ñ − n = k. Applying Lemma 1.1, we get 2^{ñ−(n−1)} = 2^{i_{S}(ϕ̃)−i_{S}(ϕ)} · ind ϕ̃/ ind ϕ. Since ñ − n = k and 2^{k} ind ϕ̃ = ind ϕ, we have i_{S}(ϕ̃) − i_{S}(ϕ) = 2k + 1.

Corollary 2.5. Let ϕ = ϕ_{0} ⊥ ⟨a⟩ be an even-dimensional form with nontrivial discriminant such that that ind ϕ = ind ϕ_{0}. Let p be an integer such that 1 ≤ p < \frac{1}{2} dim ϕ − 1. Suppose that Tors G^{i}K(X_{ϕ_{0}}) = 0 for i ≤ p − 1. Then Tors G^{i}K(X_{ϕ}) = 0 for i ≤ p.

Proof. Since ind ϕ = ind ϕ_{0}, we have i_{S}(ϕ) = i_{S}(ϕ_{0}). Clearly, 0 ≤ p − 1 < \frac{1}{2} dim ϕ_{0} − 1. Now, Lemma 2.2(1) completes the proof.

Proposition 2.6 (Karpenko). Let ϕ be an arbitrary quadratic form over F, and let E/F be a finite extension such that the norm map

N_{E/F} : K(C_{0}(ϕ_{E})) → K(C_{0}(ϕ))

is surjective (e.g., E is a subfield of the division algebra corresponding to the simple algebra C′_{0}(ϕ)). Let p ≥ 0 be such that G^{i}K(X_{ϕ_{E}}) = h̄^{i}ℤ for all i < p. Then

• G^{i}K(X_{ϕ}) = h̄^{i}ℤ for all i < p,

• the norm map N_{E/F} : G^{p}K(C_{0}(ϕ_{E}))/h̄_{E}^{p}ℤ → G^{p}K(C_{0}(ϕ))/h̄^{p}ℤ is surjective.

Proof. The first statement coincides with Corollary 4.9 in [Kar2]. The proof of the second statement is the same as that of Corollary 4.9 in [Kar2].

Corollary 2.7. Let m, s, and p be integers such that p < m/2 − 1. Suppose that for any field F and any F -form ρ satisfying the following conditions:

• ρ has dimension m,

• ind ρ = s,

• if m is even, then ρ ∈ I^{2}(F), we necessarily have Tors G^{i}K(X_{ρ}) = 0 for all i ≤ p.

Then for any field F and any F -form ϕ satisfying two conditions:

• ϕ has dimension m,

• ind ϕ ≥ s, we necessarily have Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ p.

Proof. Standard transfer arguments reduce the general case to the case where F has no nontrivial odd extensions. Let E/F be the extension constructed in Lemma 1.16. Applying Theorem 2.1(1) to the form ϕ_{E}, we have G^{i}K(X_{ϕ_{E}}) = Tors G^{i}K(X_{ϕ_{E}}) ⊕ h̄^{i}ℤ for all i ≤ p. Applying the hypothesis of the corollary to the form ρ = ϕ_{E}, we have Tors G^{i}K(X_{ϕ_{E}}) = 0 for i ≤ p. Hence, G^{i}K(X_{ϕ_{E}}) = h̄^{i}ℤ for all i ≤ p. By Proposition 2.6, we have G^{i}K(X_{ϕ}) = h̄^{i}ℤ for all i ≤ p. Therefore, Tors G^{i}K(X_{ϕ}) = 0 for i ≤ p.

Proposition 2.8. Let n ≥ 3 and p ≥ 0 be integers satisfying the following condition: for any (2n + 1)-dimensional form τ over an arbitrary field F, the group Tors G^{i}K(X_{τ}) is zero for all i ≤ p.

Now, let ϕ be a form such that p < \frac{1}{2} dim ϕ − 1. Suppose also that ϕ satisfies one of the following conditions:

(i) dim ϕ = 2n, d_{±}ϕ ∉ F^{∗2}, and ind ϕ = 1, (ii) dim ϕ = 2n − 1 and ind ϕ ≥ 2, (iii) dim ϕ = 2n − 2, d_{±}ϕ ∉ F^{∗2}, and ind ϕ = 2, (vi) dim ϕ = 2n − 3 and ind ϕ ≥ 4. Then Tors G^{i}K(X_{ϕ}) is zero for all i ≤ p.

Proof. Corollary 2.7 shows that, instead of the cases (ii) and (iv), it suffices to consider their subcases in which we have (ii’) dim ϕ = 2n − 1 and ind ϕ = 2, (iv’) dim ϕ = 2n − 3 and ind ϕ = 4. After this, Lemma 1.19 and Corollaries 2.3 and 2.4 complete the proof.

3. Chow groups of quadrics

Our computation of the third Chow group of quadrics is based on the following assertion.

Theorem 3.1 (Karpenko, [Kar1, Kar2]). Let ϕ be a quadratic form and X_{ϕ} be the projective quadric corresponding to ϕ. Then

• the homomorphism CH^{i}(X_{ϕ}) → G^{i}K(X_{ϕ}) is an isomorphism for i ≤ 3.

• Tors CH^{i}(X_{ϕ}) = Tors G^{i}K(X_{ϕ}) = 0 for i ≤ 1.

• Tors CH^{2}(X_{ϕ}) = Tors G^{2}K(X_{ϕ}) = 0 except for the case when ϕ is an anisotropic 3-fold Pfister neighbor. If ϕ is an anisotropic 3-fold Pfister neighbor, then Tors CH^{2}(X_{ϕ}) = Tors G^{2}K(X_{ϕ}) = ℤ/2ℤ.

• Tors CH^{3}(X_{ϕ}) is either 0 or ℤ/2ℤ. If dim ϕ > 12, then Tors CH^{3}(X_{ϕ}) = 0.

We will also use the following statement concerning the Chow groups of isotropic quadrics.

Lemma 3.2 ([Kar1]). If ϕ = ψ ⊥ ℍ, then Tors CH^{i}(X_{ϕ}) = Tors CH^{i−1}(X_{ψ}) for all i. If ϕ splits (i.e., dim ϕ_{an} ≤ 1), then Tors CH^{i}(X_{ϕ}) = 0 for all i.

Corollary 3.3. Let ϕ be an isotropic form. Then the group Tors CH^{3}(X_{ϕ}) is nonzero only in the case when

ϕ ≃ (anisotropic 3-fold neighbor) ⊥ ℍ.

In particular, this implies that 7 ≤ dim ϕ ≤ 10.

Proof. Since ϕ is isotropic, we can write ϕ in the form ϕ ≃ μ ⊥ ℍ. By Lemma 3.2, Tors CH^{3}(X_{ϕ}) ≃ Tors CH^{2}(X_{μ}). Theorem 3.1 completes the proof.

Corollary 3.4. Let ϕ be an isotropic form of dimension ≥ 9. Then the group Tors CH^{3}(X_{ϕ}) is nonzero only in the following two cases:

• dim ϕ = 10 and ϕ ≃ π ⊥ ℍ, where π is similar to an anisotropic 3-fold Pfister form,

• dim ϕ = 9 and ϕ ≃ μ ⊥ ℍ, where μ is an anisotropic 7-dimensional Pfister neighbor.

Corollary 3.5. Let ϕ be a quadratic form of dimension ≥ 9. Then • Tors G^{i}K(X_{ϕ}) = Tors CH^{i}(X_{ϕ}) = 0 for i ≤ 2,

• Tors G^{3}K(X_{ϕ}) = Tors CH^{3}(X_{ϕ});

• If dim ϕ > 12, then Tors G^{3}K(X_{ϕ}) = 0;

Remark 3.6. In the remaining part of this section we mostly work with forms of dimension ≥ 9. On the other hand, we are interested in the group Tors CH^{i}(X_{ϕ}) only in the case when i ≤ 3. In this case, the condition i < \frac{1}{2} dim ϕ − 1 obviously holds. This shows that we can use all results of the previous section.

Now, we can prove our first result concerning the third Chow group of quadrics.

Proposition 3.7. Let ϕ be a form satisfying one of the following conditions:

(i) dim ϕ = 12, d_{±}ϕ ∉ F^{∗2}, and ind ϕ = 1 (ii) dim ϕ = 11 and ind ϕ ≥ 2, (iii) dim ϕ = 10, d_{±}ϕ ∉ F^{∗2}, and ind ϕ = 2, (vi) dim ϕ = 9 and ind ϕ ≥ 4. Then Tors CH^{3}(X_{ϕ}) = 0.

Proof. Let n = 6 and p = 3. By Corollary 3.5, we have Tors G^{i}K(X_{τ}) = 0 for i ≤ p = 3 and all forms τ of dimension 13 = 2n + 1. Applying Proposition 2.8, we see that Tors G^{3}K(X_{ϕ}) = 0 for all quadratic forms satisfying conditions (i)–(iv). Since CH^{3}(X_{ϕ}) ≃ G^{3}K(X_{ϕ}), the proof is complete.

Lemma 3.8 (Karpenko). Let ϕ be an arbitrary quadratic form over F of dimension ≥ 9, and let E/F be a finite extension such that the norm map

N_{E/F} : K(C_{0}(ϕ_{E})) → K(C_{0}(ϕ))

is surjective (e.g., E may be any subfield of the division algebra corresponding to C′_{0}(ϕ)). Then the homomorphism N_{E/F} : Tors CH^{3}(X_{ϕ_{E}}) → Tors CH^{3}(X_{ϕ}) is surjective.

Proof. Theorem 2.1 shows that Tors G^{i}K(X_{ϕ}) ≃ G^{i}K(X_{ϕ})/h̄^{i}ℤ for all i ≤ 3. By Corollary 3.5, we have Tors G^{i}K(X_{ϕ}) = 0 for i ≤ 2. Finally, Proposition 2.6 together with the second item of Corollary 3.5 complete the proof.

Corollary 3.9. Let ϕ be an even-dimensional form of dimension > 8. Suppose that d = d_{±}ϕ ∉ F^{∗2} and put L = F (\sqrt{d}). Then the norm homomorphism N_{L/F} : Tors CH^{3}(X_{ϕ_{L}}) → Tors CH^{3}(X_{ϕ}) is surjective.

Corollary 3.10. Let ϕ be a form of even dimension > 8. Suppose that d = d_{±}ϕ ∉ F^{∗2} and ϕ_{F (\sqrt{d})} is hyperbolic. Then Tors CH^{3}(X_{ϕ}) = 0.

Proof. Let L = F (\sqrt{d}). By Lemma 3.2, we have Tors CH^{3}(X_{ϕ_{L}}) = 0. Corollary 3.9 implies that Tors CH^{3}(X_{ϕ}) = 0.

Lemma 3.11. Let d ∉ F^{∗2} and ϕ be a 10-dimensional form with discriminant d. Suppose that ϕ has the form ϕ = τ ⊥ c ⟨ ; ⟨d⟩ ; ⟩, where τ ∈ I^{2}(F) (i.e., ϕ_{F (\sqrt{d})} is isotropic). Then Tors CH^{3}(X_{ϕ}) = 0 except possibly when

ind ϕ = ind τ_{F (\sqrt{d})} = 1,  ind τ = 2,  ϕ_{F (\sqrt{d})} is not hyperbolic.

Proof. Suppose that Tors CH^{3}(X_{ϕ}) ≠ 0. Let L = F (\sqrt{d}). By Lemma 3.9, we have Tors CH^{3}(X_{ϕ_{L}}) ≠ 0. Since ϕ_{L} = τ_{L} ⊥ ℍ, Corollary 3.3 implies that τ_{L} is an anisotropic 8-dimensional Pfister neighbor. In particular, this means that ϕ_{L} is not hyperbolic and ind ϕ = ind τ_{L} = 1.

Now, it suffices to verify that ind τ = 2. Since ind τ_{L} = 1, we obviously have ind τ ≤ 2. Assume that ind τ = 1 (i.e., τ ∈ GP_{3}(F)). Let ϕ_{0} = τ ⊥ ⟨c⟩ ⊂ ϕ. Clearly, ind ϕ_{0} = ind τ = 1 = ind ϕ. Since dim ϕ_{0} = 9 > 8, it follows that Tors G^{i}K(X_{ϕ_{0}}) = 0 for i ≤ 2. By Corollary 2.5, we get Tors G^{i}K(X_{ϕ}) = 0 for i ≤ 3. Hence, Tors CH^{3}(X_{ϕ}) = Tors G^{3}K(X_{ϕ}) = 0. We get a contradiction to our assumption. Hence ind τ = 2.

Lemma 3.12. Let ϕ be a 9-dimensional form such that ind ϕ ≠ 1. Suppose that ϕ has one of the following forms:

(i) ϕ = γ ⊥ ⟨u, v⟩, where γ is a 7-dimensional Pfister neighbor, (ii) ϕ = τ ⊥ ⟨d⟩, where τ ⊂ I^{2}(F). Then Tors CH^{3}(X_{ϕ}) = 0.

Proof. The case ind ϕ ≥ 4 was considered in Proposition 3.7. Thus, we can assume that ind ϕ = 2.

(i) Since ind ϕ = 2 and dim ϕ = 9, Lemma 1.1 shows that i_{S}(ϕ) = 3. Since γ is a 7-dimensional Pfister neighbor, we have ind γ = 1. By Lemma 1.1, we get i_{S}(γ) = 3. By Theorem 3.1, we have Tors G^{i}K(X_{γ}) = 0 for i ≤ 1. Lemma 2.2(1) shows that Tors G^{i}K(X_{ϕ}) = 0 for i ≤ 3. Thus, Tors CH^{3}(X_{ϕ}) = Tors G^{3}K(X_{ϕ}) = 0.

(ii) Since τ ∈ I^{2}(F) and ind τ = ind ϕ = 2, we can write τ in the form ϕ = ⟨ ; ⟨a⟩ ; ⟩ ⊗ ρ with dim ρ = 4. Let L = F (\sqrt{a}). Since ϕ_{L} splits, we have Tors CH^{i}(X_{ϕ_{L}}) = 0 for all i. By Lemma 3.8, the homomorphism N_{L/F} : Tors CH^{3}(X_{ϕ_{L}}) → Tors CH^{3}(X_{ϕ}) is surjective. Hence Tors CH^{3}(X_{ϕ}) = 0.

Lemma 3.13. Let F be a field such that all 14-dimensional forms from I^{3}(F) are isotropic. Let ϕ be a form over F satisfying one of the following conditions:

• dim ϕ = 10, ϕ ∈ I^{2}(F) and ind ϕ = 4,

• dim ϕ = 12, ϕ ∈ I^{2}(F) and ind ϕ = 2. Then Tors CH^{3}(X_{ϕ}) = 0

Proof. Let us define the form τ as follows: if dim ϕ = 10, then τ is an Albert form corresponding to c(ϕ); if dim ϕ = 12, then τ is the 2-fold Pfister form corresponding to c(ϕ). Let k ∈ F^{∗} be such that the form γ := ϕ ⊥ −kτ is isotropic. By definition, we have dim γ = 16 and γ ∈ I^{3}(F). Since γ is isotropic, we obtain dim γ_{an} ≤ 14. By the hypothesis of the lemma, we have dim γ_{an} ≤ 12. Therefore ϕ and kτ contain a common subform of dimension 2. Hence, there exists a quadratic extension L/F such that ϕ_{L} and τ_{L} are isotropic. By Corollary 3.3, we have Tors CH^{3}(X_{ϕ_{L}}) = 0. Since τ_{L} is isotropic, it follows that L is a subfield of the division algebra corresponding to C′_{0}(ϕ). Then Lemma 3.8 shows that the homomorphism N_{L/F} : Tors CH^{3}(X_{ϕ_{L}}) → Tors CH^{3}(X_{ϕ}) is surjective. Hence Tors CH^{3}(X_{ϕ}) = 0.

4. Galois cohomology

Throughout the paper we use the notation H^{n}(F) for the Galois cohomology

H^{n}(Gal(F^{sep}/F), ℤ/2ℤ)

of the field F. It is well known that H^{1}(F) ≃ F^{∗}/F^{∗2}. Thus, any element a ∈ F^{∗} determines an element of H^{1}(F), which is denoted by (a). The cupproduct (a_{1}) ∪ ⋯ ∪ (a_{n}) ∈ H^{n}(F) is denoted by (a_{1}, …, a_{n}). It is well known that there exists a well-defined map

P_{n}(F) → H^{n}(F),  ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ ↦ (a_{1}, …, a_{n}).

If n ≤ 4, this map yields the homomorphism ([Ara, JR, Szy]):

e^{n} : I^{n}(F)/I^{n+1}(F) → H^{n}(F).

Recently, Orlov-Vishik-Voevodsky have announced deep results concerning existence and bijectivity of e^{n} for arbitrary n, but in our paper, we need only old (already published) results concerning e^{n}. We list these results in the following theorem ([Ara, AEJ2, M1, MS, R1, Szy, JR]).

Theorem 4.1. Let F be a field, and π, π_{1}, π_{2} ∈ P_{n}(F), where n ≤ 4. • The form π is isotropic (and hence hyperbolic) if and only if e^{n}(π) = 0.

• If e^{n}(π_{1}) = e^{n}(π_{2}), then π_{1} ≃ π_{2}.

• If n ≤ 3, then the homomorphism e^{n} : I^{n}(F)/I^{n+1}(F) → H^{n}(F) is an isomorphism.

Corollary 4.2. Let n ≤ 3, and let E/F be a field extension such that the homomorphism H^{i}(F) → H^{i}(E) is injective for i = 0, …, n. Then the homomorphism

W (F)/I^{n+1}(F) → W (E)/I^{n+1}(E)

is injective.

Definition 4.3. Let E/F be a field extension. By H^{n}(E/F) we denote the kernel of the homomorphism H^{n}(F) → H^{n}(E).

The following theorem was proved by J. K. Arason in the case when n ≤ 3 and proved by B. Kahn, M. Rost, and R. Sujatha, in the case n = 4.

Theorem 4.4 ([Ara, KRS1]). Let ϕ be a form of dimension ≥ 9. Then

• for n ≤ 3, the homomorphism H^{n}(F) → H^{n}(F (ϕ)) is injective (i.e., H^{n}(F (ϕ)/F) = 0),

• if ϕ is not an anisotropic 4-fold Pfister neighbor, then the homomorphism H^{4}(F) → H^{4}F (ϕ)) is injective (i.e., H^{4}(F (ϕ)/F) = 0).

• If ϕ is a Pfister neighbor of a Pfister form π = ⟨ ; ⟨a, b, c, d⟩ ; ⟩, then the group H^{4}(F (ϕ)/F) is generated by the element e^{4}(π) = (a, b, c, d).

Corollary 4.5. (cf. [KRS1, Cor.10]). Let ϕ be a form of dimension ≥ 9. Then the homomorphism

W (F)/I^{n}(F) → W (F (ϕ))/I^{n}(F (ϕ))

is injective for all n ≤ 4.

Proof. Obvious in view of Corollary 4.2 and Theorem 4.4.

Proposition 4.6. ([AEJ1, AEJ2]). Let F be a field satisfying two conditions: F has no nontrivial odd extensions and I^{4}(F) = 0. Then H^{4}(F) = 0 and cd_{2}(F) ≤ 3.

Lemma 4.7. Let E/F be an arbitrary field extension and ψ be a 4-fold Pfister neighbor over F. Then H^{4}(E(ψ)/F) = H^{4}(E/F) + H^{4}(F (ψ)/F).

Proof. Clearly, H^{4}(E(ψ)/F) ⊃ H^{4}(E/F) + H^{4}(F (ψ)/F). It suffices to verify that any element u ∈ H^{4}(E(ψ)/F) belongs to H^{4}(E/F) + H^{4}(F (ψ)/F). Let π = ⟨ ; ⟨a, b, c, d⟩ ; ⟩ be the Pfister form associated with ψ. Since u ∈ H^{4}(E(ψ)/F), we have u_{E} ∈ H^{4}(E(ψ)/E). By Theorem 4.4, the group H^{4}(E(ψ)/E) is generated by the element (a, b, c, d)_{E} ∈ H^{4}(E). Let m ∈ ℤ be such that u_{E} = m · (a, b, c, d)_{E}. We have u − m · (a, b, c, d) ∈ H^{4}(E/F). Since m · (a, b, c, d) ∈ H^{4}(F (ψ)/F), we get u ∈ H^{4}(E/F) + H^{n}(F (ψ)/F).

Corollary 4.8. Let ψ be a 4-fold Pfister neighbor over F. Then for any form ϕ, we have H^{4}(F (ϕ, ψ)/F) = H^{4}(F (ϕ)/F) + H^{4}(F (ψ)/F).

Proof. It suffices to set E = F (ϕ) in the Lemma 4.7.

Corollary 4.9. Let ϕ and ψ be forms of dimension ≥ 9. Suppose that ϕ is not a 4-fold neighbor and ψ is a 4-fold Pfister neighbor. Then H^{4}(F (ϕ, ψ)/F) = H^{4}(F (ψ)/F).

Proof. By Theorem 4.4, the group H^{4}(F (ϕ)/F) is zero. Hence, H^{4}(F (ϕ, ψ)/F) = H^{4}(F (ϕ)/F) + H^{4}(F (ψ)/F) = H^{4}(F (ψ)/F).

5. Unramified cohomology of quadrics

In this section we use the terminology and notation of [CT1] and [KRS1, KS2]. For a smooth variety X we define the unramified cohomology H_{nr}^{n}(F (X)/F) and the unramified Witt ring W_{nr}(F (X)/F) as follows:

H_{nr}^{n}(F (X)/F) : = ker(H^{n}(F (X)) → ∐_{x∈X^{(1)}} H^{n−1}(F (x))),
W_{nr}(F (X)/F) : = ker(W (F (X)) → ∐_{x∈X^{(1)}} W (F (x))).

Besides that, we set

I_{nr}^{n}(F (X)/F) := ker(I^{n}(F (X)) → ∐_{x∈X^{(1)}} I^{n−1}(F (x))).

Clearly, I_{nr}^{n}(F (X)/F) = I^{n}(F (X)) ∩ W_{nr}(F (X)/F). For n ≤ 4, the homomorphism e^{n} : I^{n}(F (X)) → H^{n}(F (X)) determines the homomorphism e^{n} : I_{nr}^{n}(F (X)/F) → H_{nr}^{n}(F (X)/F).

Since the image of the homomorphism H^{n}(F) → H^{n}(F (X)) belongs to H_{nr}^{n}(F (X)/F), we get the homomorphism η_{2,X}^{n} : H^{n}(F) → H_{nr}^{n}(F (X)/F). The cokernel of this homomorphism will be denoted by H̃_{nr}^{n}(F (X)/F):

H̃_{nr}^{n}(F (X)/F) := coker(η_{2,X}^{n} : H^{n}(F) → H_{nr}^{n}(F (X)/F)).

For a form ϕ, we set H̃_{nr}^{n}(F (ϕ)/F) := H̃_{nr}^{n}(F (X_{ϕ})/F), where X_{ϕ} is the projective quadric corresponding to the form ϕ.

The following statement is well known (see, e.g., [KRS1, Prop.2.5]).

Lemma 5.1. If ϕ is an isotropic form over F, then H̃_{nr}^{n}(F (ϕ)/F) = 0. If two forms ϕ_{1} and ϕ_{2} are stable rational equivalent (i.e., ϕ_{1} ∼^{st} ϕ_{2}), then

H̃_{nr}^{n}(F (ϕ_{1})/F) ≃ H̃_{nr}^{n}(F (ϕ_{2})/F).

To state the following theorem, we need one more notation:

H^{n}(F (X)/F)_{0} = {α ∈ H^{n}(F (X)/F) | (−1) ∪ α = 0 ∈ H^{n+1}(F)}.

We note, that H^{n}(F (X)/F)_{0} is a subgroup of H^{n}(F).

The essential part of the following theorem is contained in the paper of B. Kahn, M. Rost, and R. Sujatha [KRS1].

Theorem 5.2. Let X = X_{ϕ} be the projective quadric corresponding to the quadratic form ϕ of dimension ≥ 9. Then

(1) There exists a natural exact sequence

0 ⟶ H^{4}(F (X)/F)_{0} ⟶^{δ} H̃_{nr}^{4}(F (X)/F) ⟶^{ϵ} Tors CH^{3} X.

(2) The orders ^{4} of the groups H^{4}(F (X)/F)_{0}, H̃_{nr}^{4}(F (X)/F), and Tors CH^{3} X are at most 2.

(3) If ϕ is not a 4-fold Pfister neighbor, then H^{4}(F (X)/F)_{0} = 0. This (in particular) means that the homomorphism

ϵ : H̃_{nr}^{4}(F (X)/F) → Tors CH^{3} X

is injective and |H̃_{nr}^{4}(F (X)/F)| ≤ | Tors CH^{3} X| ≤ 2.

4Below we will use the notation |A| for the order of a set A.

(4) If ϕ is a 4-fold neighbor, then the homomorphism ϵ is zero and the homomorphism δ is an isomorphism

δ : H^{4}(F (X)/F)_{0} ≃ H̃_{nr}^{4}(F (X)/F).

Proof. (1) This is a formal consequence of [KRS1, Prop.3 and Th.6(1)].

(3) The group H^{4}(F (ϕ)/F) is zero in view of Theorem 4.4. Hence, its subgroup H^{4}(F (ϕ)/F)_{0} is also zero. The rest of the statement is a formal consequence of the first item of this theorem and Theorem 3.1.

(4) By Theorem 4.4, the group H^{4}(F (X)/F)_{0} is finite. By Item (1), it suffices to prove that |H̃_{nr}^{4}(F (X)/F)| = |H^{4}(F (X)/F)_{0}|. Since the groups H^{4}(F (X)/F)_{0} and H̃_{nr}^{4}(F (X)/F) are stably birational invariants of X, we can change the 4-fold Pfister neighbor ϕ by its associated 4-fold Pfister form. Thus, we can assume that dim ϕ = 16. In this case, we have Tors CH^{3}(X_{ϕ}) = 0 in view of Theorem 3.1. By Item (1) of the theorem, we obtain that δ is an isomorphism, and hence |H̃_{nr}^{4}(F (X)/F)| = |H^{4}(F (X)/F)_{0}|.

(2) The groups H^{4}(F (X)/F)_{0} and Tors CH^{4}(X_{ϕ}) have orders at most 2 in view of Theorems 4.4 and 3.1. After this, the statement concerning the group H̃_{nr}^{4}(F (X)/F) follows readily from Items (3) and (4).

Corollary 5.3. Let ϕ be a form of dimension ≥ 9. Suppose that ϕ is not a 4-fold Pfister neighbor and H̃_{nr}^{4}(F (ϕ)/F) ≠ 0. Then

H̃_{nr}^{4}(F (ϕ)/F) ≃ Tors CH^{3} X_{ϕ} ≃ ℤ/2ℤ

and the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) → Tors CH^{3} X_{ϕ} is an isomorphism.

Proof. Obvious in view of Item (3) of Theorem 5.2.

Corollary 5.4. ([KRS1, Cor. 8(3)(a)]). If dim ϕ > 12 and ϕ is not a 4-fold neighbor, then H̃_{nr}^{4}(F (ϕ)/F) = 0.

Proof. Follows from Theorems 5.2(3) and Theorem 3.1.

Corollary 5.5. Let ψ be a 4-fold Pfister neighbor. Let E/F be an extension such that H^{4}(E/F) = 0 (for example, E = F (ϕ), where ϕ is a form of dimension ≥ 9 which is not a 4-fold neighbor).

Then the homomorphism H̃_{nr}^{4}(F (X_{ψ})/F) → H̃_{nr}^{4}(E(X_{ψ})/E) is injective.

Proof. Since H^{4}(F) → H^{4}(E) is injective, it follows that the homomorphism H^{4}(F (X_{ψ})/F)_{0} → H^{4}(E(X_{ψ})/E)_{0} is also injective (because H^{4}(F (X_{ψ})/F)_{0} is a subgroup of H^{4}(F)). Now the corollary follows from Theorem 5.2(4).

6. Pfister neighbors over function fields

Definition 6.1. Let ϕ be a quadratic form over F. By Pf(ϕ) we denote the form defined as follows:

• if ϕ is not a Pfister neighbor, then Pf(ϕ) = 0,

• if ϕ is a Pfister neighbor of a Pfister form π, we set Pf(ϕ) = π.

Lemma 6.2. Let X be a smooth F -variety, and let ϕ be a quadratic form over F. Then Pf(ϕ_{F (X)}) ∈ W_{nr}(F (X)/F).

Proof. We can assume that 1 ∈ D_{F} (ϕ). Let π = Pf(ϕ_{F (X)}). If π = 0, the statement is trivial. Hence we can assume that ϕ_{F (X)} is a Pfister neighbor of a Pfister form π ∈ P_{n}(F (X)). We must prove that δ_{x}(π) = 0 for any x ∈ X^{(1)}. Since π ∈ I^{n}(F (X)), it follows that δ_{x}(π) ∈ I^{n−1}(F (x)). Since 1 ∈ D_{F} (ϕ), we have ϕ_{F (X)} ⊂ π. Let ξ be an F (X)-form such that ϕ_{F (X)} ⊥ ξ = π. Clearly, dim ξ < 2^{n−1}. Since ϕ is defined over F, it follows that δ_{x}(ϕ_{F (X)}) = 0. Therefore, δ_{x}(π) = δ_{x}(ξ). Since dim ξ < 2^{n−1} and δ_{x}(ξ) ∈ I^{n−1}(F (x)), the Arason–Pfister Hauptsatz shows that δ_{x}(π) = 0. Therefore, π ∈ W_{nr}(F (X)/F).

Corollary 6.3. Let X be a smooth F -variety and ϕ be a quadratic form over F. Suppose that ϕ_{F (X)} is a Pfister neighbor of π ∈ P_{n}(F (X)), where n ≤ 4. Then e^{n}(π) ∈ H_{nr}^{n}(F (X)/F).

Lemma 6.4. Let ϕ be a quadratic form over F, and let E = F (ϕ). Let X be a smooth F -variety. Suppose that ϕ_{F (X)} is an anisotropic n-fold Pfister neighbor (n ≤ 4). Then one of the following conditions holds:

• the kernel of the natural homomorphism

i_{L/F} : H̃_{nr}^{n}(F (X)/F) → H̃_{nr}^{n}(E(X)/E)

contains the nonzero element ẽ^{n}(Pf(ϕ_{F (X)})). In particular, i_{L/F} is not injective and H̃_{nr}^{n}(F (X)/F) ≠ 0.

• H^{n}(F (ϕ, X)/F) ⊋ H^{n}(F (X)/F). In particular, H^{n}(F (ϕ, X)/F) ≠ 0 and H^{n}(F) ≠ 0.

Proof. Let π = Pf(ϕ_{F (X)}). By Corollary 6.3, we have e^{n}(π) ∈ H_{nr}^{n}(F (X)/F).

Case 1: the element e^{n}(π) is not defined over F. In this case, the element e^{n}(π) determines the nonzero element ẽ^{n}(π) ∈ H̃_{nr}^{n}(F (X)/F). In particular, H̃_{nr}^{n}(F (X)/F) ≠ 0. Since ϕ_{F (X)} is a Pfister neighbor of π, we conclude that π is hyperbolic over the function field of ϕ_{F (X)}. Since the function field of ϕ_{F (X)} coincides with E(X), we see that the element ẽ^{n}(π) maps to zero under the homomorphism i_{L/F} : H̃_{nr}^{n}(F (X)/F) → H̃_{nr}^{n}(E(X)/E).

Case 2: the element e^{n}(π) ∈ H^{n}(F (X)) is defined over F. Let λ ∈ H^{n}(F) be an element such that e^{n}(π) = λ_{F (X)}. Since π is anisotropic, it follows that e^{n}(π) ≠ 0 (see Theorem 4.1). Hence λ ∉ H^{n}(F (X)/F). Since ϕ_{F (X)} is a subform of π, it follows that π_{F (ϕ,X)} is hyperbolic. Hence, λ_{F (ϕ,X)} = e^{n}(π_{F (ϕ,X)}) = 0. Therefore, λ ∈ H^{n}(F (ϕ, X)/F). Thus, we have proved that λ belongs to the group H^{n}(F (ϕ, X)/F) but does not belong to H^{n}(F (X)/F). Since H^{n}(F (ϕ, X)/F) ⊃ H^{n}(F (X)/F), we obtain that H^{n}(F (ϕ, X)/F) ⊋ H^{n}(F (X)/F).

Corollary 6.5. Let F be a field such that H^{4}(F) = 0. Let ψ be a form over F such that H̃_{nr}^{4}(F (ψ)/F) = 0. Then for any form ϕ over F, the form ϕ_{F (ψ)} is not an anisotropic 4-fold neighbor.

Corollary 6.6. Let F be a field such that H^{4}(F) = 0, and let ψ be a form of dimension ≥ 9 over F. Suppose that Tors CH^{3}(X_{ψ}) = 0 (for example, dim ψ > 12). Then for any form ϕ over F, the form ϕ_{F (ψ)} is not an anisotropic 4-fold neighbor.

Proof. Since H^{4}(F) = 0, the form ψ is not an anisotropic 4-fold neighbor. By Theorem 5.2, we have |H̃_{nr}^{4}(F (ψ)/F)| ≤ | Tors CH^{3}(X_{ψ})| = 0. Hence, H̃_{nr}^{4}(F (ψ)/F) = 0. Now, the required result follows from Corollary 6.5.

Lemma 6.7. Let F be a field such that H^{4}(F) = 0. Let ϕ_{1}, ϕ_{2}, and ψ be forms of dimension ≥ 9 such that (ϕ_{1})_{F (ψ)} and (ϕ_{2})_{F (ψ)} are anisotropic 4-fold Pfister neighbors. Then (ϕ_{1})_{F (ψ)} ∼^{st} (ϕ_{2})_{F (ψ)}.

Proof. Let π_{i} = Pf((ϕ_{i})_{F (ψ)}) for i = 1, 2. By our assumption, the forms π_{1} and π_{2} are anisotropic. Hence, e^{4}(π_{1}) and e^{4}(π_{2}) are nonzero elements of the group H_{nr}^{4}(F (ψ)/F). Since |H̃_{nr}^{4}(F (ψ)/F)| ≤ 2 and H^{4}(F) = 0, we have |H_{nr}^{4}(F (ψ)/F)| ≤ 2. Therefore, e^{4}(π_{1}) = e^{4}(π_{2}). Thus π_{1} = π_{2}, and so (ϕ_{1})_{F (ψ)} and (ϕ_{2})_{F (ψ)} are Pfister neighbors in the same Pfister form. Hence

(ϕ_{1})_{F (ψ)} ∼^{st} (ϕ_{2})_{F (ψ)}.

Lemma 6.8. Let ϕ be an anisotropic form of dimension 9 which is not a Pfister neighbor. Let ψ be a 4-fold Pfister neighbor. Then ϕ_{F (ψ)} is anisotropic form which is not a Pfister neighbor.

Proof. The form ϕ_{F (ψ)} is anisotropic in view of Corollary 1.12. Suppose that ϕ_{F (ψ)} is a Pfister neighbor. Let X = X_{ψ} and E = F (ϕ). By Corollary 4.9, we have H^{4}(F (ϕ, X)/F) = H^{4}(F (X)/F). By Corollary 5.5, the homomorphism H̃_{nr}^{4}(F (X)/F) → H̃_{nr}^{4}(E(X)/E) is injective. We get a contradiction to the statement of Lemma 6.4.

Corollary 6.9. Let ϕ be an essential 9-dimensional form, and ψ be a 4-fold Pfister neighbor. Then ϕ_{F (ψ)} is essential.

Proof. Follows from Lemma 6.8 and Lemma 1.5.

Proposition 6.10. For any field F there exists a field extension E/F with the following properties:

(i) E has no nontrivial odd extensions, I^{4}(F) = 0, and H^{4}(F) = 0 (in particular, cd_{2}(F) ≤ 3),

(ii) for any F -form τ, we have ind τ_{E} = ind τ, (iii) for any anisotropic 9-dimensional form ϕ which is not a Pfister neighbor,

the form ϕ_{E} is also anisotropic and is not a Pfister neighbor. (iv) for any essential 9-dimensional form ϕ over F, the form ϕ_{E} is also essen-

tial,

(v) the homomorphism W (F)/I^{n}(F) → W (E)/I^{n}(E) is injective for all n ≤ 4,

(vi) for any 10-dimensional form ϕ with nontrivial discriminant d the form ϕ_{E(\sqrt{d})} is hyperbolic only if ϕ_{F (\sqrt{d})} is hyperbolic.

Proof. Let us construct the fields

F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{2i} ⊂ F_{2i+1} ⊂ …

as follows. First, we set F_{−1} = F.

If n = 2i, we define F_{n} as the maximal odd extension of F_{n−1}.

If n = 2i + 1, we define F_{n} as the free composite of all fields F_{n−1}(ψ), where ψ runs over all 4-fold Pfister forms over F_{n−1}.

Now, we set E = ∪_{n≥0}F_{n}. We claim that E satisfies all needed properties.

(i) By definition, E has no odd extensions. Clearly all 4-fold Pfister forms over E are isotropic. Hence I^{4}(E) = 0. Therefore H^{4}(E) = 0 and cd_{2}(E) ≤ 3 (see Proposition 4.6.)

(ii) Follows from Lemmas 1.2 and 1.5;

(iii) Follows from Lemmas 1.14 and 6.8;

(iv) Follows from Lemma 1.24 and Corollary 6.9;

(v) Follows from Lemma 1.2 and Corollary 4.5;

(vi) Follows from Corollary 1.7. Remark 6.11. It is possible to include many additional properties of the field extension E/F in the formulation of Proposition 6.10. Here we point out only the following modification of property (iii) (which has the same proof): Let ϕ be a 9-dimensional form, and let X be an F -variety such that ϕ_{F (X)} is anisotropic and is not a Pfister neighbor. Then ϕ_{E(X)} is also anisotropic and is not a Pfister neighbor. Lemma 6.12. Let E/F be a field extension constructed ^{5} in Theorem 6.10. Let ϕ be a form with maximal splitting satisfying the condition 9 ≤ dim ϕ ≤ 16. Then

(1) If ϕ is anisotropic and is not a Pfister neighbor, then ϕ_{E} is also anisotropic and is not a Pfister neighbor.

(2) If ψ is an F -form such that ϕ_{F (ψ)} is anisotropic and is not a Pfister neighbor, then ϕ_{E(ψ)} is also anisotropic and is not a Pfister neighbor.

Proof. (1) Let ϕ_{0} be a 9-dimensional subform of ϕ. By Theorem 1.11, ϕ_{0} ∼^{st} ϕ and hence (ϕ_{0})_{E} ∼^{st} ϕ_{E}. Replacing ϕ by ϕ_{0}, we can assume that dim ϕ = 9. In this case the required statement coincides with Item (iii) of Theorem 6.10.

(2). Taking into account Remark 6.11, one can can give the same proof as for Item (1). Proposition 6.13. Let ϕ be an anisotropic form of dimension 9 that is not a Pfister neighbor. Let ψ be a form of dimension > 12. Then ϕ_{F (ψ)} is an anisotropic form that is not a Pfister neighbor. Proof. In view of Proposition 6.10, we can assume that H^{4}(F) = 0. By Corollary 1.12, the form ϕ_{F (ψ)} is anisotropic. Now, the required result follows immediately from Corollary 6.6.

5Here we mean not only the formulation of Theorem 6.10, but also the proof of this theorem. In all other statements we will refer the reader only to the formulation of Theorem 6.10.

Corollary 6.14. Let ϕ be an anisotropic form and ψ be a form of dimension > 12. Suppose that ϕ_{F (ψ)} is an anisotropic 4-fold Pfister neighbor. Then ϕ is a 4-fold Pfister neighbor.

Proof. Since ϕ_{F (ψ)} is an anisotropic 4-fold Pfister neighbor, it follows that ϕ_{F (ψ)} has maximal splitting and 9 ≤ dim ϕ ≤ 16. By [H1, Lemma 5], the form ϕ also has maximal splitting. Let ϕ_{0} be an arbitrary 9-dimensional subform of ϕ. Since ϕ_{F (ψ)} is a 4-fold Pfister neighbor, it follows that (ϕ_{0})_{F (ψ)} is a 4-fold Pfister neighbor. By Proposition 6.13, ϕ_{0} is a 4-fold neighbor. Since ϕ has maximal splitting, it follows that (ϕ_{0})_{F (ϕ)} is isotropic. Hence ϕ is a 4-fold neighbor.

Remark 6.15. As it will be shown in the following sections, Proposition 6.13 and Corollary 6.14 cannot be generalized to the case of 12-dimensional forms ψ.

7. Construction of a field with u-invariant 9

In this section, we prove Theorems 0.1 and 0.3, and Conjecture 0.10. We start the proofs with two easy lemmas.

Lemma 7.1. Let ϕ, ψ, and ψ_{0} be forms over F such that ψ_{F (ψ_{0})} is isotropic (for example, ψ_{0} ⊂ ψ). Suppose that ϕ_{F (ψ_{0})} is an essential 9-dimensional form. Then ϕ_{F (ψ)} is also essential.

Proof. Let E = F (ψ_{0}). Since ψ_{E} = ψ_{F (ψ_{0})} is isotropic, it follows that the extension E(ψ)/E is purely transcendental. Since ϕ_{E} = ϕ_{F (ψ_{0})} is essential and E(ψ)/E is purely transcendental, Lemma 1.24 implies that ϕ_{E(ψ)} is essential. Since F (ψ) ⊂ E(ψ), it follows that ϕ_{F (ψ)} is also essential.

Lemma 7.2. Let F be a field such that H^{4}(F) = 0, and let ϕ be an essential 9-dimensional F -form. Let ψ be a form of dimension ≥ 9 such that ψ ≁ ϕ and Tors CH^{3}(X_{ψ}) = 0. Then ϕ_{F (ψ)} is an essential form.

Proof. By Theorem 1.13, the form ϕ_{F (ψ)} is anisotropic. Lemma 1.5 shows that ind ϕ_{F (ψ)} ≥ 4. By Corollary 6.6, the form ϕ_{F (ψ)} is not a Pfister neighbor. Therefore, ϕ_{F (ψ)} is an essential form.

The following theorem is a basic tool in the proof of our main results concerning the u-invariant.

Theorem 7.3. Let ϕ be an essential 9-dimensional form over F. Then for an F -form ψ of dimension ≥ 9, the following conditions are equivalent:

(1) dim ψ = 9 and there exists k ∈ F^{∗} such that ψ ≡ kϕ (mod I^{4}(F)),

(2) the form ϕ_{F (ψ)} is not essential. In particular, ϕ_{F (ψ)} is always essential if dim ψ ≥ 10.

Proof. (1)⇒(2). Let ψ and k be as in (1). Let ψ_{0} be a 7-dimensional F (ψ)-form such that ψ_{F (ψ)} = ψ_{0} ⊥ ℍ. Put π = ϕ_{F (ψ)} ⊥ −kψ_{0}. Clearly, dim π = 9+7 = 16.

In the Witt ring W (F (ψ)), we have π = ϕ_{F (ψ)} − kψ_{0} = (ϕ − kψ)_{F (ψ)} ∈ I^{4}(F (ψ)). By the Arason-Pfister Hauptsatz, π ∈ GP_{4}(F (ψ)). Since ϕ_{F (ψ)} ⊂ π, the form ϕ_{F (ψ)} is a Pfister neighbor. Therefore, ϕ_{F (ψ)} is not an essential form.

(2)⇒(1). First of all, we introduce the following notation:

Definition 7.4. Let ϕ and ψ be forms over F. We will write ϕ ∼ ψ (mod I^{n}(F)) if there exists k ∈ F^{∗} such that ϕ ≡ kψ (mod I^{n}(F)). Otherwise, we write ϕ ≁ ϕ (mod I^{n}(F)).

Remark 7.5. In Definition 7.4 we do not assume that dim ϕ = dim ψ.

Lemma 7.6. Let ϕ be an odd-dimensional form and n ≥ 2. Then for a form ψ the following conditions are equivalent:

(i) ψ ∼ ϕ (mod I^{n}(F)), (ii) ψ ≡ kϕ (mod I^{n}(F)), where k = d_{±}ψ/d_{±}ϕ.

Proof. (i)⇒(ii). Let x ∈ F^{∗} be such that ψ ≡ xϕ (mod I^{n}(F)). It suffices to verify that x ≡ d_{±}ψ/d_{±}ϕ ∈ F^{∗}/F^{∗2}. Since n ≥ 1 and dim ϕ is odd, it follows that dim ψ is odd. Since n ≥ 2, it follows that d_{±}ψ ≡ d_{±}(xϕ) ∈ F^{∗}/F^{∗2}. Hence, x ≡ d_{±}ψ/d_{±}ϕ ∈ F^{∗}/F^{∗2}.

(ii)⇒(i). Obvious.

Corollary 7.7. Let ϕ be an odd-dimensional form and let n ≥ 2. Let E/F be an extension such that the homomorphism W (F)/I^{n}(F) → W (E)/I^{n}(E) is injective. Then for any F -form ψ, the condition ψ ≁ ϕ (mod I^{n}(F)) implies that ψ_{E} ≁ ϕ_{E} (mod I^{n}(E)).

Proof. Suppose that ψ_{E} ∼ ϕ_{E} (mod I^{n}(E)). By Lemma 7.6, ψ_{E} ≡ kϕ_{E} (mod I^{n}(E)), where k = d_{±}ψ/d_{±}ϕ. Since the homomorphism W (F)/I^{n}(F) → W (E)/I^{n}(E) is injective, it follows that ψ ≡ kϕ (mod I^{n}(F)). Hence, ψ ∼ ϕ (mod I^{n}(F)), a contradiction.

Corollary 7.8. Let ϕ and ψ be odd-dimensional forms such that ϕ ≁ ψ (mod I^{4}(F)). Then

(a) if E/F is an extension such that W (F)/I^{4}(F) → W (E)/I^{4}(E) is injective, then ϕ_{E} ≁ ψ_{E} (mod I^{4}(E)),

(b) if τ_{1}, …, τ_{m} are forms of dimension ≥ 9, then ϕ_{F (τ_{1},...,τ_{m})} ≁ ψ_{F (τ_{1},...,τ_{m})} (mod I^{4}(F (τ))).

Proof. Statement (a) follows from Corollary 7.7; Statement (b) follows from Statement (a) and Corollary 4.5.

Lemma 7.9. Let ψ be an anisotropic form of dimension 10. Then there exists an extension E/F and a 9-dimensional form ψ_{0} ⊂ ψ such that:

(1) for any 9-dimensional form ϕ over F we have ϕ_{E} ≁ ψ_{0} (mod I^{4}(E)).

(2) E/F is a purely transcendental field extension.

(3) ind ψ_{0} ≥ 4 except for the cases where – either ψ ∈ I^{2}(F) and ind ψ ≤ 2,

– or ψ ∉ I^{2}(F) and ind ψ = 1.

Proof. We define ψ_{0} as the “generic subform of ψ of codimension 1”. Let us give the explicit definition of ψ_{0} and E. Let F̃ = F (t) and ψ̃ = ψF̃ ⊥ ⟨−t⟩. We define E as F̃(ψ̃). Since ψ̃_{E} is isotropic, there exists a E-form ψ_{0} such that ψ̃_{E} = ψ_{0} ⊥ ℍ. We have

ψ_{0} ⊥ ⟨t⟩ ⊥ ℍ = ψ̃_{E} ⊥ ⟨t⟩ = ψ_{E} ⊥ ⟨−t⟩ ⊥ ⟨t⟩ = ψ_{E} ⊥ ℍ.

Hence, ψ_{0} ⊥ ⟨t⟩ = ψ_{E}. Therefore, ψ_{0} is a 9-dimensional subform of ψ_{E}.

(1) Let ϕ be an arbitrary 9-dimensional form over F. We set ϕ̃ = ϕF̃ and k̃ = d_{±}ψ̃/d_{±}ϕ̃. Clearly, k̃ = kt, where k = d_{±}ψ/d_{±}ϕ.

We claim that ψ̃ ≁ ϕ̃ (mod I^{4}(F̃)). Indeed, assuming the contrary, we have ψ̃ ≡ k̃ϕ̃ (mod I^{4}(F̃)). Then ψF̃ + ⟨−t⟩ ≡ tkϕF̃ (mod I^{4}(F̃)). Computing the homomorphism δ_{t}^{1} (see e.g., [Lam1, Ch.6. Cor.1.6]), we get ψ ≡ 0 (mod I^{3}(F)). Since dim ψ = 10, Pfister’s theorem shows that ψ is isotropic, a contradiction.

Since E = F̃(ψ̃) and dim ψ̃ = 11 ≥ 9, Corollary 7.8(2), shows that ϕ_{E} ≁ ψ̃_{E} (mod I^{4}(E)). Since ψ̃_{E} coincides with ψ_{0} in the Witt ring W (E), we have ϕ_{E} ≁ ψ_{0} (mod I^{4}(E)).

(2) Obvious in view of Lemma 1.22.

(3) Suppose that ψ is not “exceptional”. By Lemma 1.4, ind ψ̃ ≥ 4. By Lemma 1.5, we have ind ψ_{0} = ind(ψ̃_{F̃(ψ̃)}) ≥ 4.

Now, we return to the proof of the implication (2)⇒(1) in Theorem 7.3. We start with the following case: Case 1. dim ψ = 10, ψ ∈ I^{2}(F), and ind ψ ≥ 4. Since dim ψ ≠ 9, we must prove that ϕ_{F (ψ)} is an essential form. Let E/F

and ψ_{0} be as in Lemma 7.9. Since E/F is purely transcendental, it follows that ϕ_{E} is essential and ind ψ_{E} = ind ψ ≥ 4 (see Lemmas 1.2 and 1.24). Replacing the field F by E, we can assume that the 9-dimensional form ψ_{0} is defined over the ground field and ϕ ≁ ψ_{0} (mod I^{4}(F)). After this, applying Proposition 6.10 together with Corollary 7.8(a), we can assume that H^{4}(F) = 0. Since ind ψ_{0} = ind ψ ≥ 4, Proposition 3.7(iv) shows that Tors CH^{3}(X_{ψ_{0}}) = 0. Since ϕ ≁ ψ_{0} (mod I^{4}(F)), we have ϕ ≁ ψ_{0}. By Lemma 7.2, the form ϕ_{F (ψ_{0})} is essential. By Lemma 7.1, the form ϕ_{F (ψ)} is also essential.

Case 2. dim ψ = 10, ψ ∈ I^{2}(F), and ind ψ ≤ 2.

Since dim ψ ≠ 9, we must prove that ϕ_{F (ψ)} is an essential form. By Proposition 6.10, we can assume that H^{4}(F) = 0. Our assumption concerning the form ψ shows that ψ (up to similarity) can be written in the form π′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′, where π′ is the pure subform of a 3-fold Pfister form π and ⟨ ; ⟨u, v⟩ ; ⟩′ is the pure subform of ⟨ ; ⟨u, v⟩ ; ⟩ (see e.g, [H2, Th. 5.1]). Consider the subform ψ_{0} = π′ ⊥ ⟨u, v⟩ ⊂ ψ. Since ind ψ_{0} = ind ψ ≤ 2 < ind ϕ, we have ψ_{0} ≁ ϕ. By Lemma 3.12(i), we have Tors CH^{3}(X_{ψ_{0}}) = 0. By Lemma 7.2, the form ϕ_{F (ψ_{0})} is essential. Lemma 7.1 shows the form ϕ_{F (ψ)} is also essential.

Case 3. dim ψ = 9. Suppose that ϕ ≁ ψ (mod I^{4}(F)). We must verify that ϕ_{F (ψ)} is essential.

Let τ = ψ ⊥ ⟨−d⟩, where d = det ψ. Since τ ∈ I^{2}(F) and dim τ = 10, the results of Cases 1 and 2 show that ϕ_{F (τ)} is an essential form. By Corollary 7.8, we have ϕ_{F (τ)} ≁ ψ_{F (τ)} (mod I^{4}(F (τ))). Hence, replacing F by F (τ), we can assume that τ is isotropic. Then ψ has the form ψ = ψ_{0} ⊥ ⟨d⟩, where ψ_{0} is a 8-dimensional form from I^{2}(F). By Proposition 6.10 and Corollary 7.8(a), we can assume that H^{4}(F) = 0. All needed properties of ϕ and ψ are preserved. In particular, we still have ϕ ≁ ψ (mod I^{4}(F)). Since H^{4}(F) = 0, all 4-fold Pfister neighbors are isotropic. Hence, we can assume that ψ is not a Pfister neighbor. Then ψ_{0} ∉ GP_{3}(F) and therefore, ind ψ = ind ψ_{0} ≥ 2. By Lemma 3.12(ii), we have Tors CH^{3}(X_{ψ}) = 0. By Lemma 7.2, the form ϕ_{F (ψ)} is essential.

Case 4. dim ψ ≥ 10. Changing ψ by a 10-dimensional subform, we can assume that dim ψ = 10 (Lemma 7.1). As in Case 1, we can assume that there exists a 9-dimensional subform ψ_{0} ⊂ ψ such that ψ_{0} ≁ ϕ (mod I^{4}(F)). By Case 3, the form ϕ_{F (ψ_{0})} is essential. By Lemma 7.1, the form ϕ_{F (ψ)} is also essential.

The proof of Theorem 7.3 is complete.

Corollary 7.10. Let ϕ be an essential 9-dimensional form, and let ψ_{1}, …, ψ_{m} be forms of dimension ≥ 9. Then the following conditions are equivalent:

(1) there exists i such that dim ψ_{i} = 9 and ψ_{i} ∼ ϕ (mod I^{4}(F)),

(2) the form ϕ_{F (ψ_{1},...ψ_{m})} is not essential. In particular, the form ϕ_{F (ψ_{1},...,ψ_{m})} is always essential if dim ψ_{i} ≥ 10 for all i = 1, …, m.

Proof. (1)⇒(2). Obvious in view of Theorem 7.3.

(2)⇒(1). Suppose that ϕ_{F (ψ_{1},...ψ_{m})} is not an essential form. Set F_{0} = F, F_{1} = F (ψ_{1}), F_{2} = F (ψ_{1}, ψ_{2}), …, F_{m} = F (ψ_{1}, … ψ_{m}). By our assumption, ϕ_{F_{0}} is essential and ϕ_{F_{m}} is not. Hence, there exists i ≥ 1 such that ϕ_{F_{i−1}} is essential and ϕ_{F_{i}} is not essential. Since F_{i} = F_{i−1}(ψ_{i}), Theorem 7.3 shows that dim ψ_{i} = 9 and (ψ_{i})_{F_{i−1}} ∼ ϕ_{F_{i−1}} (mod I^{4}(F_{i−1})). Finally, Corollary 7.8(b) shows that ψ_{i} ∼ ϕ (mod I^{4}(F)).

Proof of Theorem 0.3. Implication (1)⇒(2) was proved in Section 1. (2)⇒(1). Let us construct the fields

F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{3i} ⊂ F_{3i+1} ⊂ F_{3i+1} ⊂ …

as follows. First, we set F_{−1} = F.

If n = 3i, we define F_{n} as the maximal odd extension of F_{n−1}.

If n = 3i + 1, we define F_{n} as the free composite of the fields F_{n−1}(ψ), where ψ runs over all F_{n−1}-forms of dimension ≥ 10.

If n = 3i + 2, we define F_{n} as the free composite of the fields F_{n−1}(ψ), where ψ runs over all 9-dimensional F_{n−1}-forms satisfying the condition ψ ≁ ϕ_{F_{n−1}} (mod I^{4}(F_{n−1})).

Induction on n, Lemma 1.14, and Corollary 7.10 show that ϕ_{F_{n}} is an essential form for all n. Now, we set E = ∪_{n≥0}F_{n}. Clearly, ϕ_{E} is an essential form. In particular, ϕ_{E} is anisotropic. Hence u(E) ≥ dim ϕ = 9. By the definition of F_{3i+1}, all 10-dimensional forms over E are isotropic. Hence, u(E) = 9.

Therefore I^{4}(E) = 0. By the definition of F_{3i+2}, each anisotropic 9-dimensional form ψ over E satisfies the condition ψ ∼ ϕ_{E} (mod I^{4}(E)). Since I^{4}(E) = 0, we have ψ ∼ ϕ_{E}. Clearly, E has no odd extensions. Taking into account the equation I^{4}(E) = 0, we conclude that cd_{2}(E) ≤ 3 (see Proposition 4.6). Now, it suffices to verify that cd_{2}(E) ≰ 2. Indeed, assuming the contrary, we obtain I^{3}(E) = 0. In this case, [Lam1, Ch. 11, Lemma 4.9] claims that u(E) is even. We get a contradiction.

Proof of Theorem 0.1. By Theorem 0.3, it suffices to construct at least one example of a 9-dimension essential form. We present here the following example: Let F_{0} be an arbitrary field and F = F_{0}((t_{1})) … ((t_{9})). We define ϕ as ⟨t_{1}, …, t_{9}⟩. An easy computation using Tignol’s Theorem (see Corollary 1.4) shows that ind ϕ = 16. By Springer’s theorem, ϕ is anisotropic. Now, it suffices to prove that ϕ is not a Pfister neighbor. Assume the contrary. Let μ be the complementary 7-dimensional form. Since ϕ ⊥ μ ∈ GP_{4}(F), it follows that ϕ ≡ −μ (mod I^{3}(F)) and hence ind ϕ = ind μ. Since dim μ = 7, we have ind μ ≤ 8, a contradiction.

At the end of this section we present a proof of Conjecture 0.10. ^{6}

Proof of Conjecture 0.10. Let ϕ be an anisotropic 10-dimensional form with maximal splitting. We can assume that ϕ is not a Pfister neighbor. We must prove that ϕ ≃ ⟨ ; ⟨d⟩ ; ⟩ ⊗ τ for a suitable d ∈ F^{∗} and a 5-dimensional form τ. Let us consider the following three cases.

Case 1: ϕ ∈ I^{2}(F). Since all 10-dimensional forms from I^{2}(F) with maximal splitting are necessarily Pfister neighbors (see the last Item of Theorem 1.10), we get a contradiction to our assumption.

Case 2: ϕ ∉ I^{2}(F) and ind ϕ = 1. Let d = det ϕ and L = F (\sqrt{d}). Since ϕ_{L} ∈ I^{2}(L) and ind ϕ_{L} = ind ϕ = 1, it follows that ϕ_{L} ∈ I^{3}(L). By Pfister’s theorem the form ϕ_{L} is isotropic. Since ϕ has maximal splitting, we have dim(ϕ_{L})_{an} ≤ 6. Then the Arason–Pfister Hauptsatz implies that ϕ_{L} is hyperbolic. Therefore, ϕ is divisible by ⟨ ; ⟨d⟩ ; ⟩. Hence ϕ has the form ⟨ ; ⟨d⟩ ; ⟩ ⊗ τ with dim τ = 5.

Case 3: ϕ ∉ I^{2}(F) and ind ϕ ≥ 2. By Lemma 7.9, there exists a purely transcendental extension E/F and a 9-dimensional subform ϕ_{0} ⊂ ϕ_{E} with ind ϕ_{0} ≥ 4. Since ϕ has maximal splitting, the form (ϕ_{0})_{E(ϕ)} is isotropic. Assume that ϕ_{0} is a Pfister neighbor. Since (ϕ_{0})_{E(ϕ)} is isotropic, it follows that ϕ_{E} is a Pfister neighbor. Since E/F is unirational, ϕ is also a Pfister neighbor (Lemma 1.23). We get a contradiction. Hence ϕ_{0} is not a Pfister neighbor. Then ϕ_{0} is an essential E-form. Since dim ϕ_{E} = 10, it follows from Theorem 7.3 that (ϕ_{0})_{E(ϕ)} is anisotropic. We get a contradiction, and the proof is complete.

8. Special pair of forms: definition and basic properties

The main goal of this section is to study the properties of some specific class of pairs of forms. We will call these pairs special (see Definition 8.3 below). Since many basic properties of special pairs are closely related to linkage properties of Pfister forms, we recall some results of R. Elman and T. Y. Lam on this subject.

6This conjecture was proved in [Izh3] in the case when char F = 0 and \sqrt{−1} ∈ F^{∗}.

Theorem 8.1. ([EL3, §4]) ^{7} Let τ_{1} ∈ P_{n_{1}}(F) and τ_{2} ∈ P_{n_{2}}(F).

(1) Let k ∈ F^{∗} be such that the form τ_{1} ⊥ −kτ_{2} is isotropic. Then the forms (τ_{1} ⊥ −kτ_{2})_{an} and (τ_{1} ⊥ −τ_{2})_{an} are similar. In particular,

dim(τ_{1} ⊥ −kτ_{2})_{an} = dim(τ_{1} ⊥ −τ_{2})_{an}.

(2) Let ρ ∈ P_{n}(F), μ ∈ P_{m}(F), and ν ∈ P_{k}(F) be such that τ_{1} ≃ ρ ⊗ μ, τ_{2} ≃ ρ ⊗ ν, and m, k > 0. Let μ′ and ν′ be the pure subforms of μ and ν. Then

– either (τ_{1} ⊥ −τ_{2})_{an} ≃ ρ ⊗ (μ′ ⊥ −ν′) (in particular, this means that the form ρ ⊗ (μ′ ⊥ −ν′) is anisotropic),

– or there exist μ_{0} ∈ P_{m−1}(F), ν_{0} ∈ P_{k−1}(F), and d ∈ F^{∗} such that

τ_{1} ≃ ρ ⊗ ⟨ ; ⟨d⟩ ; ⟩ ⊗ μ_{0} and τ_{2} ≃ ρ ⊗ ⟨ ; ⟨d⟩ ; ⟩ ⊗ ν_{0}.

Lemma 8.2. Let ρ = τ ⊗q, where q is an Albert form and τ ∈ P_{n}(F). Suppose that there exists a form ρ̃ such that dim ρ̃ < dim ρ and ρ ≡ ρ̃ (mod I^{n+3}(F)). Then the form ρ is isotropic.

Proof. Assume that ρ is anisotropic. Then τ is anisotropic. We obviously have ρ ∈ I^{n+2}(F). Since ρ̃ ≡ ρ (mod I^{n+3}(F)), we also have ρ̃ ∈ I^{n+2}(F). Clearly, dim ρ̃ < dim ρ = 6 · 2^{n} < 2^{n+3}.

If we assume that ρ̃ is hyperbolic, we get ρ ≡ ρ̃ ≡ 0 (mod I^{n+3}(F)). Since dim ρ < 2^{n+3}, the Arason–Pfister Hauptsatz shows that ρ is hyperbolic. We get a contradiction. Hence, ρ̃ is not hyperbolic. Changing ρ̃ by its anisotropic part, we can assume that ρ̃ is non-zero and anisotropic.

We have ρ̃_{F (τ)} ≡ ρ_{F (τ)} ≡ 0 (mod I^{n+3}(F)). Since dim ρ̃ < 2^{n+3}, the Arason– Pfister Hauptsatz shows that ρ̃_{F (τ)} is hyperbolic. Hence there exists a form λ such that ρ̃ = τ ⊗ λ. Since dim τ = 2^{n} and dim ρ̃ < dim ρ = 6 · 2^{n}, it follows that dim λ < 6. First, consider the case when dim λ is odd. Then ⟨1⟩ ≡ λ (mod I(F)) and we have τ ≡ τ ⊗ λ ≡ ρ̃ = 0 (mod I^{n+1}(F)). Since dim τ = 2^{n}, the Arason-Pfister Hauptsatz shows that τ is hyperbolic, a contradiction. Now, we can assume that dim λ is even. Since dim λ < 6, we have dim λ ≤ 4. Hence, dim ρ̃ = dim(τ ⊗ λ) ≤ 2^{n} · 4 = 2^{n+2}. Since ρ̃ ∈ I^{n+2}(F), the Arason-Pfister Hauptsatz implies that ρ̃ ∈ GP_{n+2}(F). Therefore, dim ρ̃ = 2^{n+2} and the form ρ̃_{F (ρ̃)} is hyperbolic. Hence ρ_{F (ρ̃)} ≡ ρ̃_{F (ρ̃)} = 0 (mod I^{n+3}(F (ρ̃))). Since dim ρ < 2^{n+3}, the Arason–Pfister Hauptsatz shows that the form ρ_{F (ρ̃)} is hyperbolic. Hence, the form ρ is divisible by ρ̃. On the other hand, dim ρ = 6 · 2^{n} is not divisible by dim ρ̃ = 2^{n+2}, a contradiction.

7For the proof of the first statement, see the proof of Theorem 4.5 in [EL3] or [H4, Lemma 3.2]. The proof of the second statement is the same as Step 2 of the proof of Proposition 4.4 in [EL3].

Definition 8.3. Let n and m be integers such that n ≥ 0 and m ≥ 2. We say that (ρ, ρ_{0}) is an (n, m)-special pair of forms if there exist u, v ∈ F^{∗} and τ ∈ P_{n}(F), μ ∈ P_{m}(F) such that

ρ ≃ τ ⊗ (μ′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′)  and  ρ_{0} ≃ τ ⊗ (μ′ ⊥ ⟨u, v⟩),

where μ′ and ⟨ ; ⟨u, v⟩ ; ⟩′ are the pure subforms of μ and ⟨ ; ⟨u, v⟩ ; ⟩.

Our interest in special pairs is motivated by the following conjecture.

Conjecture 8.4. Let s be a positive integer and ϕ be an F -form such that: • ϕ is not an s-fold Pfister neighbor,

• dim ϕ > 2^{s−1} and H̃_{nr}^{s}(F (ϕ)/F) ≠ 0. Then H̃_{nr}^{s}(F (ϕ)/F) ≃ ℤ/2ℤ and there exists an (n, m)-special pair (ρ, ρ_{0}) with the following properties:

(i) n ≥ 0, m ≥ 2, and m + n + 1 = s, (ii) ρ_{F (ϕ)} is isotropic and (ρ_{0})_{F (ϕ)} is anisotropic, (iii) the group H̃_{nr}^{s}(F (ϕ)/F) is generated by ẽ^{s}(Pf((ρ_{0})_{F (ϕ)})).

Remark 8.5. (1) If s ≤ 2, then the group H̃_{nr}^{s}(F (ϕ)/F) is zero for all non Pfister neighbors ϕ (see [KRS1, Th. 4 and Prop. 3]). Hence, in this case the conjecture is obvious. We also note that for s ≤ 2 there are no integers n and m satisfying condition (i).

(2) In the case s = 3, the conjecture follows easily from the results of [Kah3] and [KRS1]. See also the following section: Example 9.2 and the proof of Lemma 9.5.

(3) In the case s = 4, the conjecture will be proved in Section 12.

(4) In this section, we show (Theorem 8.6(2)) that condition (ii) of Conjecture 8.4 implies that (ρ_{0})_{F (ϕ)} is an anisotropic s-fold Pfister neighbor. Hence Pf((ρ_{0})_{F (ϕ)}) is an anisotropic s-fold Pfister form.

In this section, we study the properties of (n, m)-special pairs. Let us start with the following obvious observation: if (ρ, ρ_{0}) is an (n, m)-special pair over F, then (ρ_{L}, (ρ_{0})_{L}) is an (n, m)-special pair over L for any field extension L/F.

All basic properties of (n, m)-special pair of forms are collected in the following theorem.

Theorem 8.6. Let n and m be integers such that n ≥ 0 and m ≥ 2. Let (ρ, ρ_{0}) be an (n, m)-special pair of forms. Then

(1) dim ρ_{0} = 2^{n}(2^{m} + 1), dim ρ = 2^{n}(2^{m} + 2), and ρ_{0} is a subform of ρ. In

particular,

2^{n+m} < dim ρ_{0} < dim ρ < 2^{n+m+1}.

(2) The following conditions are equivalent:

(a) the form ρ_{0} is a Pfister neighbor,

(b) the form ρ contains an (n + m + 1)-fold Pfister neighbor,

(c) there exists a form ρ̃ such that dim ρ̃ < dim ρ and ρ̃ ≡ ρ (mod I^{n+m+1}(F)),

(d) the form ρ is isotropic.

(3) If ρ_{0} is an anisotropic form, then – ρ_{0} has maximal splitting,

– ρ is not a Pfister neighbor,

– (ρ_{0})_{F (ρ)} is an anisotropic Pfister neighbor.

(4) If ρ is anisotropic, then i_{1}(ρ) = 2^{n} and dim(ρ_{F (ρ)})_{an} = 2^{n+m}.

Proof. Item (1) is obvious.

Let us prove Statement (2).

(a)⇒(b). Obvious.

(b)⇒(c). Suppose that ρ contains an (n+m+1)-fold Pfister neighbor. Then we can write ρ in the form ρ = λ ⊥ μ, where λ is an (n + m + 1)-fold Pfister neighbor. Then there exists a form λ̃ such that λ ⊥ −λ̃ ∈ GP_{n+m+1}(F). In particular, λ̃ ≡ λ (mod I^{n+m+1}(F)). By the definition of Pfister neighbors, we have dim λ̃ < dim λ. To complete the proof, it suffices to set ρ̃ = λ̃ ⊥ μ.

(c)⇒(d). Let ρ̃ be such that dim ρ̃ < dim ρ and ρ ≡ ρ̃ (mod I^{n+m+1}(F)). Set π = (ρ ⊥ −ρ̃)_{an} and λ = τ⊗μ. Clearly, π ∈ I^{n+m+1}(F) and λ ∈ P_{n+m}(F).

Since m ≥ 2, we have

dim π ≤ dim ρ + dim ρ̃ < 2 dim ρ = 2^{n+1}(2^{m} + 2) ≤ 2^{n+1}(2^{m} + 2^{m−1}) = 3 · 2^{n+m}.

We can assume that π is a nonhyperbolic form (otherwise, ρ_{an} = ρ̃_{an} and the proof is obvious). Since π ∈ I^{n+m+1}(F), it follows that dim π ≥ 2^{n+m+1}. We have proved 2 · 2^{n+m} ≤ dim π < 3 · 2^{n+m}.

We consider the cases m > 2 and m = 2 separately.

Case 1: m > 2. In the Witt ring, we have ρ = τ(μ − ⟨ ; ⟨u, v⟩ ; ⟩) = λ − τ ⟨ ; ⟨u, v⟩ ; ⟩. Hence ρ_{F (λ)} = −τ ⟨ ; ⟨u, v⟩ ; ⟩_{F (λ)}. Therefore, dim(ρ_{F (λ)})_{an} ≤ 2^{n+2}. Taking into account the inequality 6 < 2^{m}, we have

dim(π_{F (λ)})_{an} ≤ dim(ρ_{F (λ)})_{an} + dim ρ̃ < 2^{n+2} + dim ρ
≤ 2^{n+2} + 2^{n}(2^{m} + 2) = 2^{n}(6 + 2^{m}) < 2^{n+m+1}.

Since π ∈ I^{n+m+1}(F), the Arason–Pfister Hauptsatz shows that π_{F (λ)} is hyperbolic. Hence, there exists a form ξ such that π = λ ⊗ ξ. Since dim λ = 2^{n+m} and 2 · 2^{n+m} ≤ dim π < 3 · 2^{n+m}, we conclude that dim ξ = 2. Let us write ξ in the form ξ = k ⟨ ; ⟨d⟩ ; ⟩. Then π = λ ⊗ ξ = kτ ⊗ μ ⊗ ⟨ ; ⟨d⟩ ; ⟩. In the Witt ring W (F) we have

ρ̃ = ρ − π = τ(μ − ⟨ ; ⟨u, v⟩ ; ⟩) − kτμ ⟨ ; ⟨d⟩ ; ⟩ =

= τμ ⟨1, −k, kd⟩ − τ ⟨ ; ⟨u, v⟩ ; ⟩ =

= τμ(⟨ ; ⟨k, d⟩ ; ⟩ + ⟨d⟩) − τ ⟨ ; ⟨u, v⟩ ; ⟩ =

= τ(dμ − ⟨ ; ⟨u, v⟩ ; ⟩) + τμ ⟨ ; ⟨k, d⟩ ; ⟩ .

Let ρ̂ = τ ⊗ (dμ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩). Since τμ ⟨ ; ⟨k, d⟩ ; ⟩ ∈ GP_{n+m+2}(F), we have ρ̃ ≡ ρ̂

(mod I^{n+m+2}(F)). On the other hand, dim ρ̃+dim ρ̂ < dim ρ+dim ρ̂ ≤ 2^{n}(2^{m}+ 2)+2^{n}(2^{m} +4) = 2^{n}(2^{m+1} +6) < 2^{n+m+2}. The Arason–Pfister Hauptsatz shows that ρ̃_{an} ≃ ρ̂_{an}. Since dim ρ̃ < dim ρ < dim ρ̂, it follows that ρ̂ = τ ⊗ (dμ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩) is isotropic. Applying Theorem 8.1(1) to the forms τ_{1} = τ ⊗ μ and τ_{2} = τ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩, we see that ρ̂_{an} is similar to ρ_{an}. Hence, ρ_{an} ∼ ρ̂_{an} ≃ ρ̃_{an}. Since dim ρ̃ < dim ρ, the form ρ is isotropic.

Case 2: m = 2. Since μ ∈ P_{m}(F) = P_{2}(F), there exist a, b ∈ F^{∗}

such that μ = ⟨ ; ⟨a, b⟩ ; ⟩. Then ρ = τ ⊗ (μ′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′) = τ ⊗ q, where

q = ⟨−a, −b, ab, u, v, −uv⟩. Since q is an Albert form, the required statement follows readily from Lemma 8.2.

(d)⇒(a). Let us assume that ρ is isotropic. Applying Theorem 8.1(2) to the forms τ_{1} = τ ⊗ μ and τ_{2} = τ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩, we see that there exists d ∈ F^{∗} such that τ ⊗ ⟨ ; ⟨d⟩ ; ⟩ divides the both forms τ ⊗ μ and τ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩. Thus, there exists

μ_{0} ∈ P_{m−1}(F) and k ∈ F^{∗} such that τ ⊗ ⟨ ; ⟨d⟩ ; ⟩ ⊗ μ_{0} = τ ⊗ μ and τ ⊗ ⟨ ; ⟨d, k⟩ ; ⟩ =

τ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩.

We claim that ρ_{0} is a subform of the Pfister form

π := τ ⊗ μ ⊗ ⟨ ; ⟨−uvk⟩ ; ⟩ .

Besides, we claim that the complementary form is equal to

ρ′_{0} := uvτ ⊗ (kμ′_{0} ⊗ ⟨ ; ⟨d⟩ ; ⟩ ⊥ ⟨−d⟩).

Since dim ρ_{0} = 2^{n}(2^{m} + 1), dim ρ′_{0} = 2^{n}(2^{m} − 1), and π ∈ P_{n+m+1}(F), it suffices to verify the equation ρ_{0} + ρ′_{0} = π in the Witt ring. We have

π − ρ_{0} = τ(μ + uvkμ) − τ(μ + ⟨u, v, −1⟩)

= τ(uvkμ + ⟨1, −u, −v⟩) = uvτ(kμ + ⟨uv, −v, −u⟩)

= uvτ(kμ + ⟨ ; ⟨u, v⟩ ; ⟩ − ⟨1⟩)

= uvτ(kμ_{0} ⟨ ; ⟨d⟩ ; ⟩ + ⟨ ; ⟨d, k⟩ ; ⟩ − ⟨1⟩)

= uvτ(kμ′_{0} ⟨ ; ⟨d⟩ ; ⟩ + ⟨−d⟩)

= ρ′_{0}

The proof of Item (2) is complete. To prove Item (3), we need the following lemma.

Lemma 8.7. Let ϕ = τ ⊗ ν, where τ is an n-fold Pfister form. Let L/F be an arbitrary extension such that τ_{L} is anisotropic. Then i_{W} (ϕ_{L}) is divisible by 2^{n}.

Proof. Since ϕ_{L(τ)} is hyperbolic, there exists an L-form γ such that (ϕ_{L})_{an} = τ_{L} ⊗γ. In the Witt ring W (L), we have τ_{L} ·(ν_{L} −γ) = 0. Since τ_{L} ≠ 0 ∈ W (L), (ν_{L}−γ) is a zero-divisor in W (L). By [Lam1, Ch. VIII, Cor. 6.7], dim ν −dim γ is even. Hence dim ϕ − dim(ϕ_{L})_{an} = 2^{n}(dim ν − dim γ) is divisible by 2^{n+1}. Therefore, i_{W} (ϕ_{L}) is divisible by 2^{n}.

Corollary 8.8. Let ϕ = τ ⊗ ν be an anisotropic form, where τ is an n-fold Pfister form and dim ν ≥ 2. Then i_{1}(ϕ) is divisible by 2^{n}.

Proof. Since ϕ is anisotropic, τ is also anisotropic. Since dim ν ≥ 2, we have dim ϕ > dim τ. The Cassels–Pfister subform theorem shows that τ_{F (ϕ)} is anisotropic. Now, Lemma 8.7 shows that i_{1}(ϕ) = i_{W} (ϕ_{F (ϕ)}) is divisible by 2^{n}.

Corollary 8.9. Let ϕ = τ ⊗ ν be an anisotropic form, where τ is an n-fold Pfister form and dim ν = 2^{m} + 1. Then ϕ has maximal splitting (i.e., i_{1}(ϕ) = 2^{n}).

Proof. Since dim ϕ = 2^{n+m}+2^{n}, Theorem 1.10 shows that 1 ≤ i_{1}(ϕ) ≤ 2^{n}. From Corollary 8.8 it follows that i_{1}(ϕ) is divisible by 2^{n}. Hence i_{1}(ϕ) = 2^{n}.

Corollary 8.10. Let ϕ = τ ⊗ ν be an anisotropic form, where τ is an n-fold Pfister form and dim ν = 2^{m} + 2 with m ≥ 1. Then

• either i_{1}(ϕ) = 2^{n+1} (in this case ϕ has maximal splitting),

• or i_{1}(ϕ) = 2^{n} (in this case dim(ϕ_{F (ϕ)})_{an} = 2^{n+m}).

Proof. Since dim ϕ = 2^{n+m}+2^{n+1}, Theorem 1.10 shows that i_{1}(ϕ) ≤ 2^{n+1}. From Corollary 8.8 it follows that i_{1}(ϕ) is divisible by 2^{n}. Hence i_{1}(ϕ) = 2^{n+1} or 2^{n}. The rest of the proof is obvious.

Now, we return to the proof of Item (3) of Theorem 8.6. Here we can assume that ρ_{0} is anisotropic. Corollary 8.9 shows that ρ_{0} has maximal splitting. Now, we must verify that ρ is not a Pfister neighbor. Suppose at the moment that ρ is a Pfister neighbor of π. Then ρ_{0} is also a Pfister neighbor of π. Item (2) of Theorem 8.6 shows that ρ is isotropic. Then π is isotropic. Since ρ_{0} is a neighbor of π, the form ρ_{0} is also isotropic. This contradicts our assumption. Hence ρ is not a Pfister neighbor.

To complete the proof of Item (3) it suffices to verify that (ρ_{0})_{F (ρ)} is an anisotropic Pfister neighbor. Suppose that (ρ_{0})_{F (ρ)} is isotropic. Since ρ_{0} has maximal splitting, ρ also has maximal splitting (Theorem 1.11). By [H1, Prop. 6] there exists an extension K/F such that ρ_{K} is an anisotropic Pfister neighbor. Changing the field F by K, we can assume that ρ is an anisotropic Pfister neighbor. However, we have proved above that ρ is not a Pfister neighbor, a contradiction. Hence (ρ_{0})_{F (ρ)} is anisotropic. To prove that (ρ_{0})_{F (ρ)} is a Pfister neighbor, we consider the special pair (ρ_{F (ρ)}, (ρ_{0})_{F (ρ)}). Since ρ_{F (ρ)} is isotropic, Item (2) of the theorem shows that (ρ_{0})_{F (ρ)} is a Pfister neighbor. The proof of Item (3) is complete.

In the proof of Item (4) of Theorem 8.6, we can assume that ρ is anisotropic. By Corollary 8.10, the form ρ has maximal splitting or i_{1}(ρ) = 2^{n}. If ρ has maximal splitting, then there exists an extension K/F such that ρ_{K} is an anisotropic Pfister neighbor ([H1, Prop. 6]). This contradicts Item (3) of the theorem. By Corollary 8.10, we have i_{1}(ρ) = 2^{n} and dim(ρ_{F (ρ)})_{an} = 2^{n+m}. The proof of Theorem 8.6 is complete.

Definition 8.11. We say that a special pair (ρ, ρ_{0}) is anisotropic, if ρ (and so ρ_{0}) is anisotropic.

Lemma 8.12. Let (ρ, ρ_{0}) be an anisotropic (n, m)-special pair. Then • ρ contains no (n + m + 1)-fold Pfister neighbors, • ρ_{0} and ρ are not Pfister neighbors,

• ρ_{0} is a form with maximal splitting,

• (ρ_{0})_{F (ρ)} is an anisotropic Pfister neighbor.

Proof. Obvious in view of Theorem 8.6.

Proposition 8.13. Let (ρ, ρ_{0}) be an anisotropic (n, m)-special pair, and (ρ̃, ρ̃_{0}) be an anisotropic (ñ, m̃)-special pair with n + m = ñ + m̃. Suppose that ρ_{0} is isotropic over the function field of ρ̃_{0}. Then n = ñ, m = m̃, dim ρ = dim ρ̃,

dim ρ_{0} = dim ρ̃_{0}, ρ ∼^{st} ρ̃, and ρ_{0} ∼^{st} ρ̃_{0}.

Proof. Since n + m = ñ + m̃, we have 2^{n+m} < dim ρ_{0}, dim ρ̃_{0} < 2^{n+m+1}. Since ρ_{0} and ρ̃_{0} have maximal splitting and ρ_{0} is isotropic over the function field of ρ̃_{0}, Theorem 1.11 shows that ρ_{0} ∼^{st} ρ̃_{0}.

By Lemma 8.12, the form (ρ_{0})_{F (ρ)} is a Pfister neighbor. Since ρ_{0} ∼^{st} ρ̃_{0}, it follows that (ρ̃_{0})_{F (ρ)} is also a Pfister neighbor. Applying Theorem 8.6(2) to the special pair (ρ̃_{F (ρ)}, (ρ̃_{0})_{F (ρ)}), we see that the form ρ̃_{F (ρ)} is isotropic. Analogously, ρ_{F (ρ̃)} is isotropic. Hence ρ ∼^{st} ρ̃. By Theorem 1.15, dim ρ−i_{1}(ρ) = dim ρ̃−i_{1}(ρ̃). Taking into account Items (1) and (4) of Theorem 8.6, we have 2^{n}(2^{m}+2)−2^{n} = 2ñ(2m̃ + 2) − 2ñ. Since n + m = ñ + m̃, we obviously get n = ñ and m = m̃.

9. Special pairs of degree 4 and unramified cohomology

We recall that if (ρ, ρ_{0}) is an (n, m)-special pair, then 2^{n+m} < dim ρ_{0} < dim ρ < 2^{n+m+1}. When we say that a pair is “(n, m)-special”, we always assume that n ≥ 0 and m ≥ 2.

Definition 9.1. Let (ρ, ρ_{0}) be an (n, m)-special pair. We define the degree of (ρ, ρ_{0}) by the formula deg(ρ, ρ_{0}) = n + m + 1.

Since n ≥ 0 and m ≥ 0, the degree d of any special pair satisfies the condition d = n + m + 1 ≥ 3.

Example 9.2. Let us consider special pairs of degree 3. Since n ≥ 0, m ≥ 2, and n+m+1 = 3, we obviously have n = 0 and m = 2. In this case, τ ∈ P_{0}(F) and μ ∈ P_{2}(F). Hence τ = ⟨1⟩ and μ is of the form μ = ⟨ ; ⟨s, r⟩ ; ⟩. Then we get

ρ = ⟨ ; ⟨s, r⟩ ; ⟩′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′ = ⟨−s, −r, sr, u, v, −uv⟩ ,
ρ_{0} = ⟨ ; ⟨s, r⟩ ; ⟩′ ⊥ ⟨u, v⟩ . = ⟨−s, −r, sr, u, v⟩

Thus, ρ is an Albert form, and ρ_{0} is a 5-dimensional subform of ρ.

In what follows, we are interested in the case when the degree is equal to 4. Since m ≥ 2 and n ≥ 0, we have exactly two possibilities for special pairs of degree 4, namely:

(i) n = 1 and m = 2, (ii) n = 0 and m = 3. Let us consider consider these cases separately.

Example 9.3. Let n = 1 and m = 2. In this case, τ ∈ P_{1}(F) and μ ∈ P_{2}(F). Then we can write τ and μ in the forms τ = ⟨ ; ⟨a⟩ ; ⟩ and μ = ⟨ ; ⟨s, r⟩ ; ⟩. We obtain

ρ = ⟨ ; ⟨a⟩ ; ⟩ ⊗ (⟨ ; ⟨s, r⟩ ; ⟩′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′) = ⟨ ; ⟨a⟩ ; ⟩ ⊗ ⟨−s, −r, sr, u, v, −uv⟩ ,
ρ_{0} = ⟨ ; ⟨a⟩ ; ⟩ ⊗ (⟨ ; ⟨s, r⟩ ; ⟩′ ⊥ ⟨u, v⟩) = ⟨ ; ⟨a⟩ ; ⟩ ⊗ ⟨−s, −r, sr, u, v⟩ .

We will say that ρ is a special 12-dimensional form and ρ_{0} is a special subform of ρ.

Example 9.4. Let n = 0 and m = 3. In this case, τ ∈ P_{0}(F) and μ ∈ P_{3}(F). Then τ = ⟨1⟩ and μ can be written in the form μ = ⟨ ; ⟨a, b, c⟩ ; ⟩. We get

ρ = ⟨ ; ⟨a, b, c⟩ ; ⟩′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′ ,
ρ_{0} = ⟨ ; ⟨a, b, c⟩ ; ⟩′ ⊥ ⟨u, v⟩ .

We will say that ρ is a special 10-dimensional form and ρ_{0} is a special 9dimensional subform of ρ.

Lemma 9.5. Let (ρ, ρ_{0}) be an anisotropic special pair of degree 4 (in particular, 9 ≤ dim ρ_{0} < dim ρ < 16). Then H̃_{nr}^{4}(F (ρ)/F) = ℤ/2ℤ.

Proof. By Lemma 8.12, the form (ρ_{0})_{F (ρ)} is an anisotropic 4-fold neighbor. Now, we apply Lemma 6.4. This lemma (in particular) shows that either H̃_{nr}^{4}(F (ρ)/F) ≠ 0 or H^{4}(F (ρ_{0}, ρ)/F) ≠ 0.

We claim that H^{4}(F (ρ_{0}, ρ)/F) = 0. To prove this, we note that ρ_{0} ⊂ ρ, and hence H^{4}(F (ρ_{0}, ρ)/F) = H^{4}(F (ρ_{0})/F). Since ρ is anisotropic, it follows that ρ_{0} is not a Pfister neighbor (Lemma 8.12). Hence H^{4}(F (ρ_{0})/F) = 0 and therefore H^{4}(F (ρ_{0}, ρ)/F) = 0.

Since H^{4}(F (ρ_{0}, ρ)/F) = 0, it follows that H̃_{nr}^{4}(F (ρ)/F) ≠ 0. Now, the lemma follows from Theorem 5.2.

Now, we need the following well-known statement.

Lemma 9.6. • (Pfister, [Pf2, Satz 14]). Let ρ be a 12-dimensional form from I^{3}(F). Then there exist a, s, r, u, v ∈ F^{∗} such that ρ is similar to the form

⟨ ; ⟨a⟩ ; ⟩ ⊗ ⟨−s, −r, sr, u, v, −uv⟩ .

In other words, the form ρ up to similarity coincides with the “special 12-dimensional form” defined in Example 9.3.

• (see e.g., [H2, Th.5.1]). Let ρ be a 10-dimensional form from I^{2}(F) with ind C(ϕ) ≤ 2. Then there exist a, b, c, u, v ∈ F^{∗} such that ρ is similar to the form

⟨ ; ⟨a, b, c⟩ ; ⟩′ ⊥ − ⟨ ; ⟨u, v⟩ ; ⟩′ .

In other words, the form ρ up to similarity coincides with the “special 12-dimensional form” defined in Example 9.4.

Corollary 9.7. Let ρ ∈ I^{2}(F) be either an anisotropic 12-dimensional form with ind ρ = 1, or an anisotropic 10-dimensional form with ind ρ = 2. Then ρ does not contain 4-fold Pfister neighbors.

Proof. By Lemma 9.6, we can assume that ρ is either a special 12-dimensional form, or a special 10-dimensional form. In any case, Lemma 8.12 shows that ρ contains no 4-fold Pfister neighbors.

Lemma 9.8. Let ρ be an anisotropic form from I^{2}(F). Suppose also that ρ is either a 10-dimensional form with ind ρ = 2, or a 12-dimensional form with ind ρ = 1. Then ρ is not a Pfister neighbor and H̃_{nr}^{4}(F (ρ)/F) = ℤ/2ℤ.

Proof. Corollary 9.7 shows that ρ is not a Pfister neighbor. The isomorphism H̃_{nr}^{4}(F (ρ)/F) = ℤ/2ℤ follows from Lemmas 9.6 and 9.5.

Corollary 9.9. Let ρ be an anisotropic 11-dimensional form with ind ρ = 1. Then ρ is not a Pfister neighbor and H̃_{nr}^{4}(F (ρ)/F) = ℤ/2ℤ.

Proof. Since c(ρ) = 1, we obtain from Lemma 1.21(2) that there exists a 12dimensional form γ ∈ I^{3}(F) such that ρ ∼^{st} γ. Since ρ is anisotropic, γ is also anisotropic. By Lemma 9.8, γ is not a Pfister neighbor and H̃_{nr}^{4}(F (γ)/F) = ℤ/2ℤ. Since γ ∼^{st} ρ, the proof is complete (see Lemma 5.1).

Proposition 9.10. Let ρ be an F -form satisfying one of the following conditions:

(a) dim ρ = 12 and ρ ∈ I^{3}(F),

(b) dim ρ = 11 and c(ρ) = 1,

(c) dim ρ = 10, ρ ∈ I^{2}(F), and ind C(ρ) = 2. Then

• if ρ is isotropic, then H̃_{nr}^{4}(F (ρ)/F) ≃ Tors CH^{3}(X_{ρ}) ≃ 0,

• if ρ is anisotropic, then H̃_{nr}^{4}(F (ρ)/F) ≃ Tors CH^{3}(X_{ρ}) ≃ ℤ/2ℤ. In any case, the homomorphism ϵ : H̃_{nr}^{4}(F (ρ)/F) → Tors CH^{3}(X_{ρ}) is an isomorphism.

Proof. If ρ is isotropic, then the proposition is trivial in view of Corollary 3.3 and Lemma 5.1. Hence, we can assume that ρ is anisotropic. By Lemma 9.8 and Corollary 9.9, the form ρ is not a Pfister neighbor and H̃_{nr}^{4}(F (ρ)/F) = ℤ/2ℤ. The claim follows now from Corollary 5.3.

Corollary 9.11. Let ϕ be a quadratic form of dimension ≥ 10. If dim ϕ = 10 we suppose in addition that ind ϕ ≠ 1. Then

(i) the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) → Tors CH^{3}(X_{ϕ}) is surjective, (ii) if ϕ is not a Pfister neighbor, then the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) →

Tors CH^{3}(X_{ϕ}) is an isomorphism, (iii) If ϕ is a Pfister neighbor, then Tors CH^{3}(X_{ϕ}) = 0.

Proof. (i) If dim ϕ > 12, then Tors CH^{3}(X_{ϕ}) = 0 by Theorem 3.1. Hence we can assume that dim ϕ = 10, 11, or 12. The standard transfer arguments reduce the general case to the case where F has no nontrivial odd extensions. By Lemma 1.16, there exists an extension E/F such that the form ρ = ϕ_{E} satisfies one of the conditions (a)–(c) of Proposition 9.10, and the homomorphism N_{E/F} :

K(C_{0}(ϕ_{E})) → K(C_{0}(ϕ)) is surjective. By Proposition 9.10, the homomorphism ϵ : H̃_{nr}^{4}(E(ϕ)/E) → Tors CH^{3}(X_{ϕ_{E}}) is an isomorphism. By Theorem 3.8, the homomorphism N_{E/F} : Tors CH^{3}(X_{ϕ_{E}}) → Tors CH^{3}(X_{ϕ}) is surjective.

Clearly, the diagram

H̃_{nr}^{4}(E(ϕ)/E) ϵ_{E} ; ⟶ ; isomorphism Tors CH^{3}(X_{ϕ_{E}})
N_{E/F} ↓  ^{N_{E/F}} ↓surjective
H̃_{nr}^{4}(F (ϕ)/F)  ⟶^{ϵ}  Tors CH^{3}(X_{ϕ})

is commutative. Hence, the homomorphism ϵ is surjective.

(ii) is obvious in view of Item (i) and Theorem 5.2.

(iii) Since ϕ is a Pfister neighbor, Theorem 5.2(4) shows that the homomorphism ϵ is zero. Since ϵ is surjective (Item (i)), we have Tors CH^{3}(X_{ϕ}) = 0.

10. Proof of the Conjectures 0.8 and 0.9

In this section we prove the conjectures 0.8 and 0.9 (see Theorems 10.5 and 10.6). We start with the following lemma.

Lemma 10.1. Let (ρ, ρ_{0}) be an anisotropic special pair of degree 4. Let E/F be the extension constructed in Proposition 6.10 and ψ be an F -form. Then (1) the special pair (ρ_{E}, (ρ_{0})_{E}) is anisotropic,

(2) if ρ_{F (ψ)} is anisotropic, then ρ_{E(ψ)} is also anisotropic,

(3) if dim ψ ≥ 11, then (ρ_{0})_{E(ψ)} is anisotropic.

Proof. (1) Since (ρ, ρ_{0}) is a special pair of degree 4, it follows that ρ_{0} is a form with maximal splitting and 9 ≤ dim ρ_{0} ≤ 16. By Lemma 8.12, ρ_{0} is anisotropic and is not a Pfister neighbor. By Lemma 6.12, (ρ_{0})_{E} is also anisotropic and is not a Pfister neighbor. Then Theorem 8.6(2) shows that ρ_{E} is anisotropic.

(2) The proof is the same as for Item (1).

(3) The form (ρ_{0})_{E} is anisotropic and is not a Pfister neighbor (see the proof of Item (1)). Since ρ_{0} has maximal splitting, it follows that (ρ_{0})_{E} has maximal splitting. Since dim ψ ≥ 11, Corollary 1.12 shows that (ρ_{0})_{E(ψ)} is anisotropic.

Lemma 10.2. Let (ρ, ρ_{0}) be an anisotropic special pair of degree 4, and let ψ be a form such that ρ_{F (ψ)} is isotropic. Then dim ψ ≤ 12. Moreover,

• if dim ψ = 12, then ψ ∈ I^{3}(F),

• if dim ψ = 11, then c(ψ) = 1.

Proof. Since ρ_{F (ψ)} is isotropic, Theorem 8.6 shows that (ρ_{0})_{F (ψ)} is a Pfister neighbor. Let E/F be the extension constructed in Proposition 6.10. Obviously, (ρ_{0})_{E(ψ)} is a Pfister neighbor. By Lemma 10.1(3), the form (ρ_{0})_{E(ψ)} is anisotropic. Hence ϕ_{E(ψ)} is an anisotropic Pfister neighbor, where ϕ is an arbitrary 9-dimensional subform of ρ_{0}. Since H^{4}(E) = 0, Corollary 6.6 shows that Tors CH^{3}(X_{ψ_{E}}) ≠ 0. By Theorem 3.1, we have dim ψ ≤ 12.

Now we consider the case dim ψ = 11. Since Tors CH^{3}(X_{ψ_{E}}) ≠ 0, Proposition 3.7(ii) shows that ind ψ_{E} = 1. Hence ind ψ = 1 (see Theorem 6.10(ii)).

Now, it suffices to consider the case dim ψ = 12. Let ψ_{0} be an 11-dimensional subform of ψ. Since ρ_{F (ψ)} is isotropic, it follows that ρ_{F (ψ_{0})} is also isotropic. We have proved above that ind ψ_{0} = 1. Hence ind ψ = 1 and ind ψ_{E} = 1. Since Tors CH^{3}(X_{ψ_{E}}) ≠ 0, Proposition 3.7(i) shows that d_{±}ψ_{E} = 1. Since ind ψ_{E} = 1, we have ψ_{E} ∈ I^{3}(E). Now, Item (v) of Theorem 6.10 shows that ψ ∈ I^{3}(F). The proof is complete.

Lemma 10.3. Let (ρ, ρ_{0}) be an anisotropic special pair of degree 4, and let ψ be a 12-dimensional form such that ρ_{F (ψ)} is isotropic. Then ψ ∈ I^{3}(F), dim ρ = dim ψ = 12, and ρ ∼^{st} ψ.

Proof. By Lemma 10.2, we have ψ ∈ I^{3}(F). Since dim ψ = 12, we can assume that ψ is a special 12-dimensional form containing a special 10-dimensional subform ψ_{0} (see Lemma 9.6 and Example 9.3). Clearly, the special pair (ψ, ψ_{0}) is anisotropic (since the form ρ_{F (ψ)} is isotropic). Now, let E/F be the field extension constructed in Proposition 6.10. By Lemma 10.1(1), the pairs (ρ_{E}, (ρ_{0})_{E}) and (ψ_{E}, (ψ_{0})_{E}) are anisotropic.

Since ρ_{F (ψ)} is isotropic, (ρ_{0})_{F (ψ)} is a 4-fold neighbor (Theorem 8.6). By Lemma 10.1(3), the Pfister neighbor (ρ_{0})_{E(ψ)} is anisotropic. By Theorem 8.6(3), the form (ψ_{0})_{E(ψ)} is also an anisotropic 4-fold Pfister neighbor. By Lemma 6.7, we have (ψ_{0})_{E(ψ)} ∼^{st} (ρ_{0})_{E(ψ)}. Hence, (ρ_{0})_{E(ψ,ψ_{0})} is isotropic. Since ψ_{0} ⊂ ψ, the form (ρ_{0})_{E(ψ_{0})} is also isotropic. By Lemma 8.13, ρ_{E} ∼^{st} ψ_{E} and dim ρ = dim ψ. Hence the form ψ_{E(ρ)} is isotropic. By Lemma 10.1(2) ^{8}, the form ψ_{F (ρ)} is also isotropic. Since ρ_{F (ψ)} and ψ_{F (ρ)} are both isotropic, we conclude that ρ ∼^{st} ψ.

Corollary 10.4. Let (ρ, ρ_{0}) be an anisotropic special pair of degree 4, and let ψ be an 11-dimensional form such that ρ_{F (ψ)} is isotropic. Then ind ψ = 1,

dim ρ = 12, and ρ ∼^{st} ψ.

Proof. By Lemma 10.2, we have ind ψ = 1. By Lemma 1.21(2), there is a 12-dimensional form γ ∈ I^{3}(F) such that ψ ∼^{st} γ. Since ρ_{F (ψ)} is isotropic and ψ ∼^{st} γ, it follows that ρ_{F (γ)} is isotropic. Lemma 10.3 shows that dim ρ = 12

and ρ ∼^{st} γ. Therefore ρ ∼^{st} γ ∼^{st} ψ.

Theorem 10.5. Let ρ ∈ I^{3}(F) be an anisotropic 12-dimensional form. Let ψ be a form such that ϕ_{F (ψ)} is isotropic. Then dim ψ ≤ 12. Moreover,

(1) if dim ψ = 12, then ψ ∈ I^{3}(F) and ψ ∼^{st} ρ,

(2) if dim ψ = 11, then ind ψ = 1 and ψ ∼^{st} ρ.

Proof. We can assume that ρ is a special 12-dimensional form containing a special 10-dimensional subform ρ_{0}. By our assumption, the special pair (ρ, ρ_{0}) is anisotropic. Now, the results follow readily from Lemmas 10.2 and 10.3 and Corollary 10.4

8Here we apply Lemma 10.1(2) for the special pair (ψ, ψ_{0}) over the function field of the form ρ.

Theorem 10.6. Let ρ ∈ I^{2}(F) be an anisotropic 10-dimensional form with ind C(ϕ) = 2. Let ψ be a form of dimension > 10. Then the form ρ_{F (ψ)} is anisotropic.

Proof. We can assume that ρ is a special 10-dimensional form containing a special 9-dimensional subform ρ_{0}. By Lemma 10.2, dim ψ ≤ 12. Hence dim ψ = 11 or 12. Then Lemma 10.3 and Corollary 10.4 show that dim ρ = 12, a contradiction.

11. The group Tors CH^{3}(X_{ϕ}) for forms ϕ ∈ I^{2}(F)

The main goal of this section is the computation of the group Tors CH^{3}(X_{ϕ}) for all forms ϕ ∈ I^{2}(F). We start the proof with the following easy lemma.

Lemma 11.1. Let ψ ∈ I^{2}(F) be a 14-dimensional Pfister neighbor. Then ψ is hyperbolic.

Proof. By our assumption, there exists a 2-dimensional form μ such that ψ+μ ∈ GP_{4}(F) ⊂ I^{2}(F). Since ψ ∈ I^{2}(F), it follows that μ ∈ I^{2}(F). Since dim μ = 2, the form μ is hyperbolic. Hence π is hyperbolic. Therefore ψ = π − μ is also hyperbolic.

Corollary 11.2. For any 14-dimensional form ϕ ∈ I^{2}(F), we have H̃_{nr}^{4}(F (ψ)/F) = 0.

Proof. The case when ψ is isotropic is obvious (Lemma 5.1). Now we assume that ψ is anisotropic. Then Lemma 11.1 shows that ψ is not a Pfister neighbor. In this case the statement is proved in Corollary 5.4

Lemma 11.3. Let ψ be a 14-dimensional form from I^{2}(F) and let E = F (ψ). Then for any form ϕ of dimension ≥ 9, the homomorphism H̃_{nr}^{4}(F (ϕ)/F) → H̃_{nr}^{4}(E(ϕ)/E) is injective.

Proof. Let X = X_{ϕ} and Y = Y_{ψ} be the projective quadrics corresponding to the forms ϕ and ψ. Let us consider the commutative diagram

H^{4}(F)  ⟶  H^{4}(F (X))  ⟶ ∐_{x∈X^{(1)}} H^{3}(F (x))
↓  ^{α}↓  ↓
H^{4}(F (Y)) ⟶  H^{4}(F (X × Y))  ⟶ ∐_{x∈X_{F (Y )}^{(1)}} H^{3}(F (Y)(x))
↓  ↓
∐_{y∈Y^{(1)}} H^{3}(F (y)) ⟶^{β} ∐_{y∈Y_{F (X)}^{(1)}} H^{3}(F (X)(y))

Clearly, all columns and rows of this diagram are zero sequences. The homology group of the first and the second rows coincide with H̃_{nr}^{4}(F (ϕ)/F) and H̃_{nr}^{4}(E(ϕ)/E) respectively. Thus, we must verify that the homology group of the first row maps injectively to the homology group of the second row. This is a formal consequence of the following three properties of the diagram:

(i) the homomorphism α is injective, (ii) the homomorphism β is injective, (iii) the first column of the diagram is an exact sequence. Let us verify these properties.

(i) By Theorem 4.4, the homomorphism α is not injective only in the case when ψ_{F (X_{ϕ})} is an anisotropic 4-fold Pfister neighbor, which is impossible in view of Lemma 11.1.

(ii) The homomorphism β is injective in view of Theorem 4.4.

(iii) The homology group of the first column equals H̃_{nr}^{4}(F (ψ)/F). By Corollary 11.2, this group is zero. Hence, the first column is exact.

Corollary 11.4. For any field F there exists an extension E/F with the following properties:

(i) all 14-dimensional forms from I^{3}(E) are isotropic; (ii) for all quadratic forms ϕ over F, we have ind ϕ_{E} = ind ϕ, (iii) for all F -forms ϕ of dimension ≥ 9, the homomorphism H̃_{nr}^{4}(F (ϕ)/F) →

H_{nr}^{4}(E(ϕ)/E) is injective. (iv) for any F -form ϕ of dimension ≥ 9 which is not a 4-fold Pfister neighbor,

the form ϕ_{E} is not an anisotropic 4-fold Pfister neighbor.

Proof. Let us construct the fields

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} ⊂ …

as follows. First, we set F_{0} = F. For n ≥ 1, we define F_{n} as the free composite of all fields F_{n−1}(ψ), where ψ runs over all 14-dimensional forms from I^{3}(F). We set E = ∪_{n≥0}F_{n}. We claim that the field E satisfies all required properties. Indeed, Item (i) follows from the definition of E; Item (ii) follows from Lemma 1.5; Item (iii) follows from Lemma 11.3 and Item (iv) follows from Corollary 6.14.

Lemma 11.5. Let ϕ be an F -form which is not an anisotropic 4-fold Pfister neighbor. Suppose also that ϕ satisfies one of the following conditions:

• dim ϕ = 10, ϕ ∈ I^{2}(F), and ind ϕ = 4,

• dim ϕ = 12, ϕ ∈ I^{2}(F), and ind ϕ = 2. Then H̃_{nr}^{4}(F (ϕ)/F) = 0.

Proof. Corollary 11.4 reduces the proof to the case where all 14-dimensional forms from I^{3}(F) are isotropic. Then Lemma 3.13 shows that Tors CH^{3}(X_{ϕ}) = 0. By Theorem 5.2(3), we have H̃_{nr}^{4}(F (ϕ)/F) = 0.

Corollary 11.6. Let ϕ be an F -form satisfying one of the following conditions:

• dim ϕ = 10, ϕ ∈ I^{2}(F), and ind ϕ = 4,

• dim ϕ = 12, ϕ ∈ I^{2}(F), and ind ϕ = 2. Then Tors CH^{3}(X_{ϕ}) = 0. Moreover, Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ 3.

Proof. The first statement follows from Lemma 11.5 and items (ii) and (iii) of Corollary 9.11. The second statement is obvious in view of Theorem 3.1.

Corollary 11.7. Let ϕ be a form satisfying one of the following conditions: • dim ϕ = 10 and ind ϕ ≥ 4,

• dim ϕ = 12 and ind ϕ ≥ 2. Then Tors CH^{3}(X_{ϕ}) = 0.

Proof of Corollary 11.7. Corollaries 2.7 and 11.6 show that Tors G^{i}K(X_{ϕ}) = 0 for all i ≤ 3. Hence Tors CH^{3}(X_{ϕ}) = Tors G^{3}K(X_{ϕ}) = 0.

Proposition 11.8. Let ϕ be an anisotropic quadratic form from I^{2}(F). Then the group Tors CH^{3}(X_{ϕ}) is nonzero only in the following cases:

• dim ϕ = 8 and ϕ is similar to a 3-fold Pfister form,

• dim ϕ = 10 and ind ϕ = 2,

• dim ϕ = 12 and ind ϕ = 1 (i.e., ϕ ∈ I^{3}(F)). In all cases listed above, the group Tors CH^{3}(X_{ϕ}) is isomorphic to ℤ/2ℤ.

Proof. In case dim ϕ ≤ 8, the proposition is proved in [Kar1]. Suppose that dim ϕ > 8. By Theorem 3.1, we can assume that dim ϕ = 10 or 12. If dim ϕ = 10, we necessarily have ind ϕ ≥ 2 (otherwise, the form ϕ is isotropic by Pfister’s theorem). Now, the required result follows readily from Corollary 11.7 and Proposition 9.10.

12. Proof of Theorems 0.5 and 0.6

In this section we complete the computation of the third Chow group of quadrics X_{ϕ} for all forms of dimension ≥ 9 (Theorem 0.5). Besides, we prove our main results concerning unramified cohomology (Theorem 0.6 and Corollary 0.7). In the proofs we will use the following terminology: we say that ϕ is of type (9-a), or (10-a), etc., if the form ϕ satisfies the corresponding conditions given in the formulation of Theorem 0.5.

Lemma 12.1. Let ϕ be a form of dimension ≥ 9 such that Tors CH^{3}(X_{ϕ}) ≠ 0. Then ϕ belongs to the list of forms given in Theorem 0.6.

Proof. First, we consider the case when ϕ is isotropic. Corollary 3.4 shows that

• either dim ϕ = 10 and ϕ = π ⊥ ℍ, where π is similar to an anisotropic 3-fold Pfister form,

• or dim ϕ = 9 and ϕ = μ ⊥ ℍ, where μ is an anisotropic 7-dimensional

Pfister neighbor. Obviously, in the case dim ϕ = 10, the form ϕ has type (10-a). Let us consider the case dim ϕ = 9. Let d ∈ F^{∗} be such that π = μ ⊥ ⟨−d⟩ ∈ GP_{3}(F). Then we have ϕ = μ ⊥ ℍ = μ ⊥ ⟨−d, d⟩ = π ⊥ ⟨d⟩. Hence ϕ is of type (9-a).

In the following, we can assume that ϕ is an anisotropic form. By Theorem 3.1, it suffices to consider the following four cases separately: dim ϕ = 9, 10, 11, or 12.

\underline{dim ϕ = 12}. Since Tors CH^{3}(X_{ϕ}) ≠ 0, Corollary 11.7 shows that ind ϕ = 1. From Proposition 3.7(i) it follows that ϕ ∈ I^{2}(F). Since ind ϕ = 1 and ϕ ∈ I^{2}(F), we have ϕ ∈ I^{3}(F). Therefore ϕ is of type (12-a).

\underline{dim ϕ = 11}. Since Tors CH^{3}(X_{ϕ}) ≠ 0, Proposition 3.7(ii) shows that ind ϕ = 1 and hence ϕ has type (11-a).

\underline{dim ϕ = 10}. If ϕ ∈ I^{2}(F), then Proposition 11.8 shows that ind ϕ = 2. We get a form of type (10-b). Now, we can assume that ϕ ∉ I^{2}(F). Then d = d_{±}ϕ ∉ F^{∗2}. By Corollary 3.10, ϕ_{F (\sqrt{d})} is not hyperbolic. Proposition 3.7(iii) and Corollary 11.7 show that ind ϕ = 1. Hence, there exists a c ∈ F^{∗} such that ϕ ⊥ −c ⟨ ; ⟨d⟩ ; ⟩ ∈ I^{3}(F) (see, e.g., Lemma 1.19(i)). Now, it suffices to verify that the 12-dimensional form γ = ϕ ⊥ −c ⟨ ; ⟨d⟩ ; ⟩ is anisotropic (see Item (10-c) of Theorem 0.5). Assume the contrary. Then dim γ_{an} ≤ 10 and Corollary 1.9 shows that γ is Witt equivalent to some τ ∈ GP_{3}(F). In the Witt ring W (F) we have ϕ = γ + c ⟨ ; ⟨d⟩ ; ⟩ = τ + c ⟨ ; ⟨d⟩ ; ⟩. Since dim ϕ = 10 = 8 + 2 = dim(τ ⊥ c ⟨ ; ⟨d⟩ ; ⟩), it follows that ϕ ≃ τ ⊥ c ⟨ ; ⟨d⟩ ; ⟩. Since ind τ = 1, Lemma 3.11 shows that Tors CH^{3}(X_{ϕ}) = 0. We get a contradiction.

\underline{dim ϕ = 9}. Proposition 3.7(iv) shows that ind ϕ ≤ 2. First, we suppose that d = det ϕ ∈ D_{F} (ϕ). Then ϕ has the form ϕ = τ ⊥ ⟨d⟩, where τ is an 8dimensional form from I^{2}(F). Then Lemma 3.12(ii) shows that ind τ = ind ϕ = 1. Then τ ∈ GP_{3}(F). We get a form of type (9-a).

Now, we can assume that d = det ϕ ∉ D_{F} (ϕ). In this case, the 10-dimensional form μ = ϕ ⊥ ⟨−d⟩ ∈ I^{2}(F) is anisotropic. Pfister’s theorem shows that ind ϕ = ind μ ≠ 1. Since ind ϕ ≤ 2, we have ind ϕ = 2. We claim that ϕ has the type (9-b). We have already proved that ind ϕ = 2 and d ∉ D_{F} (ϕ). Thus, it suffices to verify that ϕ contains no 7-dimensional Pfister neighbors. This has been proved in Lemma 3.12.

Lemma 12.2. Let ϕ be one of the forms listed in Theorem 0.5. Suppose also that the form ϕ is not of type (9-a) or (10-a). Then the homomorphism ϵ : H̃_{nr}^{4}(F (ϕ)/F) → Tors CH^{3}(X_{ϕ}) is an isomorphism and H̃_{nr}^{4}(F (ϕ)/F) ≃ Tors CH^{3}(X_{ϕ}) ≃ ℤ/2ℤ

Proof. If dim ϕ = 11 or 12 the statement is obvious in view of Proposition 9.10. If dim ϕ = 9 or 10, it suffices to verify the following two properties of the form ϕ:

(i) ϕ is not a Pfister neighbor, (ii) H̃_{nr}^{4}(F (ϕ)/F) ≠ 0.

After this, the lemma will be obvious in view of Corollary 5.3.

Now, let us study the forms case by case.

(9-b). Since d = det ϕ ∉ D_{F} (ϕ), the form ρ := ϕ ⊥ ⟨−d⟩ is anisotropic. Clearly, dim ρ = 10, ρ ∈ I^{2}(F) and ind ρ = ind ϕ = 2. In view of Lemma 9.6, we can assume that ρ is a “special 10-dimensional form” containing a special 9-dimensional subform ρ_{0} (Example 9.4).

Since ind ϕ = 2, there exists a 3-dimensional form μ such that ϕ ⊥ μ ∈ I^{3}(F) (see e.g, Lemma 1.19(ii)). Set ρ̃ = ϕ ⊥ μ. Clearly, dim ρ̃ = 12. First, we suppose that ρ̃ is isotropic. Then ρ̃ is Witt equivalent to some form π ∈ GP_{3}(F) (Corollary 1.9). In the Witt ring, we have ϕ + μ = ρ̃ = π. Then π − ϕ = μ. Hence, ϕ and π contain a common subform of dimension \frac{1}{2}(dim π + dim ϕ − dim μ) = \frac{1}{2}(9 + 8 − 3) = 7. Therefore, ϕ contains a 7dimensional Pfister neighbor of π. This contradicts condition (9-b). Thus, we have proved that ρ̃ is anisotropic. In view of Lemma 9.6, we can assume that ρ̃ is a “special 12-dimensional form” which contains a special 10-dimensional subform ρ̃_{0} (Example 9.3).

We have constructed two anisotropic special pairs (ρ, ρ_{0}) and (ρ̃, ρ̃_{0}) such that ϕ ⊂ ρ and ϕ ⊂ ρ̃. Corollary 9.7 shows that ϕ is not a Pfister neighbor. We have realized Item (i) of our plan. Since ϕ ⊂ ρ and ϕ ⊂ ρ̃, the forms ρ_{F (ϕ)} and ρ̃_{F (ϕ)} are isotropic. Theorem 8.6 shows that (ρ_{0})_{F (ϕ)} and (ρ̃_{0})_{F (ϕ)} are Pfister neighbors. We get two elements ẽ^{4}(Pf((ρ_{0})_{F (ϕ)})) and ẽ^{4}(Pf((ρ̃_{0})_{F (ϕ)})) of the group H̃_{nr}^{4}(F (ϕ)/F). If at least one of these elements is nonzero, the proof is complete (see item (ii) of our plan of the proof).

Thus, we can assume that ẽ^{4}(Pf((ρ_{0})_{F (ϕ)})) = ẽ^{4}(Pf((ρ̃_{0})_{F (ϕ)})) = 0.

Let E/F be the field extension constructed in Proposition 6.10. By Lemma 10.1(1), the forms ρ_{E} and ρ̃_{E} are anisotropic. Hence, (ρ_{0})_{E} and (ρ̃_{0})_{E} are also anisotropic. Since H^{4}(E) = 0, we have e^{4}(Pf((ρ_{0})_{E(ϕ)})) = e^{4}(Pf((ρ̃_{0})_{E(ϕ)})) = 0 in the group H_{nr}^{4}(E(ϕ)/E) ⊂ H^{4}(E(ϕ)) (without “tilde” !). By Theorem 4.1, Pf((ρ_{0})_{E(ϕ)})) = Pf((ρ̃_{0})_{E(ϕ)}) = 0. This means that both Pfister neighbors (ρ_{0})_{E(ϕ)} and (ρ̃_{0})_{E(ϕ)} are isotropic. Since (ρ_{0})_{E} and (ρ̃_{0})_{E} are anisotropic forms with maximal splitting (Theorem 8.6), it follows that (ρ_{0})_{E} ∼^{st} ϕ_{E} and (ρ̃_{0})_{E} ∼^{st} ϕ_{E} (see Theorem 1.11). Hence, (ρ_{0})_{E} ∼^{st} (ρ̃_{0})_{E}. By Proposition 8.13, we get dim ρ = dim ρ̃, which contradicts the equations dim ρ = 10 and dim ρ̃ = 12. This completes the proof in case (9-b).

(10-b). In this case the result of the lemma is covered by Proposition 9.10. (10-c). Let ϕ, d, and τ be as in (10-c). Since dim τ = 12 and τ ∈ I^{3}(F),

we can assume that τ is a “special 12-dimensional form” containing a 10dimensional special subform τ_{0} (Example 9.3). Since ϕ ⊂ τ, Corollary 9.7 shows that ϕ is not a Pfister neighbor. This completes the proof of Item (i) of our plan. To prove Item (ii), it suffices to verify that the element ẽ^{4}(Pf((τ_{0})_{F (ϕ)})) is non-zero in the group H̃_{nr}^{4}(F (ϕ)/F). Assume the contrary, ẽ^{4}(Pf((τ_{0})_{F (ϕ)})) = 0. Let E/F be the extension constructed in Proposition 6.10. By Lemma 10.1(1), the form τ_{E} is anisotropic. Hence (τ_{0})_{E} is an anisotropic form which is not a Pfister neighbor. On the other hand, the form (τ_{0})_{E(ψ)} is a Pfister neighbor because the form τ_{E(ψ)} is isotropic. Since H^{4}(E) = 0, we get e^{4}(Pf((τ_{0})_{E(ϕ)})) = 0 ∈ H^{4}(E(ϕ)). Hence, the Pfister neighbor (τ_{0})_{E(ϕ)} is isotropic. Since (τ_{0})_{E} is an anisotropic form with maximal splitting, Theorem 1.11 shows that ϕ_{E} also has maximal splitting. Since ind ϕ = 1, it follows that ϕ_{E(\sqrt{d})} ∈ I^{3}(F). Since dim ϕ = 10, Pfister’s theorem shows that ϕ_{E(\sqrt{d})} is isotropic. Taking into account that ϕ_{E} has maximal splitting, we have dim(ϕ_{E(\sqrt{d})})_{an} ≤ 6. Now, the Arason–Pfister Hauptsatz implies that ϕ_{E(\sqrt{d})} is hyperbolic. By Item (vi) of Proposition 6.10, we conclude that ϕ_{F (\sqrt{d})} is hyperbolic. This contradicts condition (10-c). The proof is complete.

Proof of Theorem 0.6. If Tors CH^{3}(X_{ϕ}) = 0, then ϵ is an isomorphism by Theorem 5.2(3). Now, we can assume that Tors CH^{3}(X_{ϕ}) ≠ 0. By Lemma 12.1, the form ϕ belongs to the list of forms given in Theorem 0.5. Since ϕ is anisotropic and is not a Pfister neighbor, it follows that ϕ is not of type (9-a) or (10-a). Then Lemma 12.2 completes the proof.

Lemma 12.3. If ϕ has type (9-a) or (10-a), then Tors CH^{3}(X_{ϕ}) = ℤ/2ℤ.

Proof. (9-a). If ϕ = π ⊥ ⟨d⟩ is isotropic, then π can be written as μ ⊥ ⟨−d⟩, where μ is a 7-dimensional anisotropic Pfister neighbor. Then ϕ = μ ⊥ ⟨−d, d⟩ = μ ⊥ ℍ. By Corollary 3.3, we have Tors CH^{3}(X_{ϕ}) = ℤ/2ℤ. Now we can assume that ϕ is anisotropic. Since ind ϕ = 1, we have s = i_{S}(ϕ) = 4 (Lemma 1.1). By [Kar1, Th. 3.8], the set

U = {i | Tors G^{i}K(X_{ϕ}) ≠ 0}

consists exactly of s = 4 elements. Since dim X_{ϕ} = dim ϕ − 2 = 7, we have U ⊂ {0, 1, …, 7}. By Theorem 3.1, Tors G^{i}K(X_{ϕ}) = 0 for i ≤ 2. Hence, U ⊂ {3, 4, 5, 6, 7}. By [Sw], we have Tors G^{7}K(X_{ϕ}) = Tors G_{0}K(X_{ϕ}) = 0. Hence, U ⊂ {3, 4, 5, 6}. Since U consists of 4 elements, we get U = {3, 4, 5, 6}. In particular, 3 ∈ U. Hence, Tors CH^{3} X_{ϕ} ≃ Tors G^{3}K(X_{ϕ}) = ℤ/2ℤ.

(10-a). Let ϕ = π ⊥ ℍ, where π is similar to an anisotropic Pfister form. In this case Tors CH^{3}(X_{ϕ}) = ℤ/2ℤ in view of Corollary 3.3.

Proof of Theorem 0.5. The theorem is a formal consequence of Lemmas 12.1 12.2, and 12.3.

Proof of Corollary 0.7. Let us consider the homomorphisms

H̃_{nr}^{4}(F (ϕ)/F) →^{α} H̃_{nr}^{4}(F (ϕ)/F, ℚ/ℤ(3)) →^{β} Tors CH^{3}(X_{ϕ}),

where α is the homomorphism induced by the natural homomorphism

H^{4}(F (ϕ)) → H^{4}(F (ϕ), ℚ/ℤ(3))

and β is the homomorphism defined in [KRS1, Th.6(1)]. We must prove that α is surjective. If ϕ is a 4-fold Pfister neighbor or isotropic, then the group H̃_{nr}^{4}(F (ϕ)/F, ℚ/ℤ(3)) is zero by [KRS1, Th.3(1) and Prop.2.5(a)]. Therefore, α is surjective. If ϕ is anisotropic and is not a 4-fold Pfister neighbor, then the composition β ∘ α is surjective by Theorem 0.6. By [KRS1, Th.6(1)], the homomorphism β is injective. Hence α is surjective.

13. Forms of height 2 and degree 3

.

The main goal of this section is to prove Conjecture 0.11 in the case when n = 3 and char F = 0. We will use the following theorem of A. Vishik.

Theorem 13.1. (A. Vishik, [Vi3, Vi4]). Let F be a field of characteristic zero. Let ϕ be an even-dimensional F -form of height 2 and degree d. Then

dim ϕ = 3 · 2^{d−1}, dim ϕ = 2^{d+1}, or dim ϕ = 2^{N} − 2^{d} for some N ≥ d + 1.

The proof of this theorem depends on Voevodsky’s announced theorem^{9} that Milnor’s conjecture is valid ([Vo]). We need below only the following special case of Theorem 13.1: If ϕ is a form of height 2 and degree 3, then dim ϕ ≠ 14. This special case was proved by Vishik in his thesis [Vi1, Statement 1.2.1] under the additional hypothesis \sqrt{−1} ∈ F^{∗}.

Corollary 13.2. Let F be a field of characteristic 0, and let ϕ ∈ I^{3}(F) be an anisotropic 14-dimensional form. Then dim(ϕ_{F (ϕ)})_{an} = 12.

Proof. The form ϕ_{F (ϕ)} is nonhyperbolic because ϕ is not similar to a Pfister form. If dim(ϕ_{F (ϕ)})_{an} ≠ 12, then dim(ϕ_{F (ϕ)})_{an} ≤ 8 by Pfister’s theorem. Since ϕ ∈ I^{3}(F), we get dim(ϕ_{F (ϕ)})_{an} ∈ GP_{3}(F (ϕ)). Hence ht(ϕ) = 2 and deg ϕ = 3. However, Theorem 13.1 implies that dim ϕ ≠ 14 for all forms of height 2 and degree 3, a contradiction.

Corollary 13.3. Let F be a field of characteristic zero, and τ ∈ I^{3}(F) be an anisotropic form of dimension 12 or 14. Let ψ be a form of dimension > 12. Then dim(τ_{F (ψ)})_{an} ≥ 12.

Proof. If dim τ = 12, the statement follows from Theorem 10.5. Hence we can assume that dim τ = 14. By Corollary 13.2, we have dim(τ_{F (τ)})_{an} = 12. Applying Theorem 10.5 once again, we see that (τ_{F (τ)})_{an} is anisotropic over the function field of the form ψ_{F (τ)}. Hence dim(τ_{F (τ,ψ)})_{an} = 12. Therefore dim(τ_{F (ψ)})_{an} ≥ dim(τ_{F (τ,ψ)})_{an} = 12.

Below we will use the following terminology: we say that an element u ∈ H^{n}(F) is a symbol, if there exist a_{1}, …, a_{n} ∈ F^{∗} such that u = (a_{1}, …, a_{n}).

The following lemma is well known (it is an easy consequence of the isomorphism H^{3}(F) ≃ I^{3}(F)/I^{4}(F) and the “linkage properties” of Pfister forms [EL3]).

Lemma 13.4. Let u = (a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3}) ∈ H^{3}(F). The the following conditions are equivalent:

• the element u is a symbol,

• the form (⟨ ; ⟨a_{1}, a_{2}, a_{3}⟩ ; ⟩ ⊥ − ⟨ ; ⟨b_{1}, b_{2}, b_{3}⟩ ; ⟩)_{an} is zero or belongs to GP_{3}(F).

9More precisely, the proof of Theorem 13.1 is based on the whole technique developed by Voevodsky in his proof of Milnor conjecture. We also note that, for good forms over a field of characteristic not 2, this result was proved earlier (see [H5] and [HR, §3]).

Proposition 13.5. (cf. [Kah1, Prop.5]). Let F be a field of characteristic zero. Let ψ be an F -form of dimension > 12. Let u ∈ H^{3}(F) be an element such that u_{F (ψ)} is a symbol. Then u is also a symbol.

Proof. Suppose that u is not a symbol. By [Kah1, Prop.5], u is a sum of at most two symbols. Thus, u = (a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3}) for suitable a_{i}, b_{i} ∈ F^{∗}.

Let q = (⟨ ; ⟨a_{1}, a_{2}, a_{3}⟩ ; ⟩ ⊥ − ⟨ ; ⟨b_{1}, b_{2}, b_{3}⟩ ; ⟩)_{an}. Clearly, dim q ≤ 14. Since u is not a

symbol, Lemma 13.4 shows that dim q > 8. By Pfister’s theorem, dim q ≥ 12. Hence dim q = 12 or 14. By Lemma 13.3, dim(q_{F (ψ)})_{an} ≥ 12. By Lemma 13.4, the element u_{F (ψ)} is not a symbol, which contradicts the hypothesis of the proposition.

Proposition 13.6. Let F be a field of characteristic zero, and ψ be an F -form of dimension > 12. Let π be a 3-fold Pfister form over F (ψ). Suppose also that xπ is unramified for some element x ∈ F (ψ)^{∗} (i.e., xπ ∈ W_{nr}(F (ψ)/F)). Then π is defined over F by a 3-fold Pfister form (i.e., there exists τ ∈ P_{3}(F) such that π ≃ τ_{F (ψ)}).

Proof. Since xπ ∈ W_{nr}(F (ψ)/F), we have e^{3}(π) = e^{3}(xπ) ∈ H_{nr}^{3}(F (ψ)/F). Since dim ψ > 8, the homomorphism H^{3}(F) → H_{nr}^{3}(F (ψ)/F) is surjective (see [KRS1, Cor.8(2a)]). Hence, there exists an element u ∈ H^{3}(F) such that u_{F (ψ)} = e^{3}(π). Since e^{3}(π) is a symbol, Proposition 13.5 shows that u is also a symbol. Let a_{1}, a_{2}, a_{3} ∈ F^{∗} be such that u = (a_{1}, a_{2}, a_{3}), and let τ = ⟨ ; ⟨a_{1}, a_{2}, a_{3}⟩ ; ⟩. We have e^{3}(τ_{F (ψ)}) = u_{F (ψ)} = e^{3}(π). By Theorem 4.1, we have π = τ_{F (ψ)}. The proof is complete.

Corollary 13.7. Let F be a field of characteristic zero and ϕ be a nongood F -form of degree 3 and height 2. Then dim ϕ ≤ 12.

Proof. Suppose that dim ϕ > 12. Since ϕ has degree 3 and height 2, it follows that (ϕ_{F (ϕ)})_{an} ∈ GP_{3}(F (ϕ)). Hence, there exists π ∈ P_{3}(F (ϕ)) and x ∈ F (ϕ)^{∗} such that (ϕ_{F (ϕ)})_{an} = xπ. By definition, xπ ∈ Im(W (F) → W (F (ϕ))). Hence xπ ∈ W_{nr}(F (ϕ)/F). By Proposition 13.6, π is defined over F by a Pfister form. This contradicts the definition of nongood forms.

We recall the conjecture of Bruno Kahn concerning the classifications of forms of height 2.

Conjecture 13.8. (see [Kah2, Conjecture 7]). Let ϕ be a quadratic form of height 2 and degree n ≥ 2. Then at least one of the following conditions holds:

• (excellent forms) ϕ ≃ aρ ⊗ τ′, where a ∈ F^{∗}, ρ ∈ P_{n}(F) and τ′ is the pure subform of τ ∈ P_{m}(F) with m ≥ 2.

• (good nonexcellent forms) ϕ ≃ ρ ⊗ ψ, where ρ ∈ P_{n−1}(F) and ψ is a 4-dimensional form.

• (nongood forms) ϕ ≃ ρ ⊗ γ, where ρ ∈ P_{n−2}(F) and γ is an Albert form.

Theorem 13.9. Let F be a field of characteristic zero. Then Conjecture 13.8 is true for n ≤ 3.

Proof. M. Knebusch proved Conjecture 13.8 for all excellent forms. B. Kahn proved Conjecture 13.8 for n = 2 ([Kah2]). Moreover, he proved the second item of this conjecture (i.e., for good forms) for n = 3 (see [Kah2, Th.2.12]). To complete the proof of the theorem it suffices to classify nongood forms of height 2 and degree 3.

Since deg ϕ = 3 and ht(ϕ) = 2 > 1, we have ϕ ∈ I^{3}(F) and dim ϕ > 8. By Pfister’s theorem, we have dim ϕ ≥ 12. On the other hand, Corollary 13.7 shows that dim ϕ ≤ 12. Hence, ϕ is a 12-dimensional form from I^{3}(F). Therefore, ϕ = ⟨ ; ⟨a⟩ ; ⟩ ⊗ γ, for suitable a ∈ F^{∗} and an Albert form γ (Lemma 9.6).

Corollary 13.10. Let F be a field of characteristic zero, and let ϕ be an F - form of degree 2 and height 3. ^{10} Then dim ϕ ≠ 14. If dim ϕ = 16, then ϕ ∈ GP_{4,2}F (i.e. ϕ ≃ k ⟨ ; ⟨a⟩ ; ⟩ ⊗ (π′ ⊥ ⟨b⟩), where π′ is the pure subform of some 3-fold Pfister form).

Proof. This follows from Theorem 13.9 and [Lag, Th.6].

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Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, GERMANY

E-mail address: oleg@mathematik.uni-bielefeld.de