A SECOND DESCENT PROBLEM FOR QUADRATIC FORMS

BRUNO KAHN AHMED LAGHRIBI

Abstract. Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied in [20] and [31]. More precisely, we conjecture that for any quadratic form q over F and any φ ∈ Im(W (F) ⟶ W (F (q))), there exists a quadratic form ψ ∈ W (F) such that dim ψ ≤ 2 dim φ and φ ∼ ψ_{F (q)}, where F (q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim φ ≤ 3 and any q, and get partial results for dim φ ∈ {4, 5, 6}. We also give other related results.

Contents

1. Introduction 2 2. Notation and rappels 5 3. Cohomological kernels 7 4. Conjugate forms 9 5. Lower bounds 11 6. The homomorphism W (F) → W_{nr}(K/F) 12 7. Detailed statements of results 13 8. A rough outline of the proofs 18 9. More lemmas 19 10. Proof of Theorem 7.3: the cases n = 1, 2, 3 22 11. Proof of Theorem 7.3: the case of a form similar to a 2-fold

Pfister form 22 12. Proof of Theorem 7.3: the case of an Albert form 25 13. Proof of Theorem 7.3: the case of a 5-dimensional form 29 14. Proof of Theorem 7.3: the case of a 4-dimensional form

which is not a neighbour 30 References 34

2000 Mathematics Subject Classification. 11E04, 12G05, 14F42. Keywords: quadratic form, Galois cohomology, rationality.

1. Introduction

Let F be a field of characteristic different from 2 and K/F an extension. Let φ be a quadratic form over K. We say that φ is defined over F if there exists a quadratic form ψ over F such that φ ≃ ψ_{K}, where ≃ denotes isometry of quadratic forms.

Problem 1. Under which conditions is φ defined over F?

This problem is studied in [20] and [31] when K is the function field of a quadric: Conjecture 1 below predicts sufficient conditions for a positive answer in that case. In the present paper, our aim is to study a complementary problem, which we now explain.

If we replace ≃ by ∼ (Witt equivalence), we say that φ is defined over F up to Witt equivalence: this means that φ ∈ Im(W (F) ⟶ W (K)). Clearly, if φ is defined over F, it is defined over F up to Witt equivalence. We then may ask:

Problem 2. Suppose that φ is defined over F up to Witt equivalence.

(1) What is the smallest dimension of an F -quadratic form ψ such that φ ∼ ψ_{K}?

(2) Can one describe those ψ which have this smallest dimension?

Conjecture 2 below tackles this issue, still when K is the function field of a quadric; Theorem 7.3 will provide a very detailed answer to Problem 2 for low-dimensional φs in this case, and prove part of Conjecture 2 as a consequence.

More generally, suppose K/F finitely generated and regular: this means that K is the function field of a geometrically irreducible F - variety. The condition that φ ∈ Im(W (F) ⟶ W (K)) implies that φ belongs to the unramified Witt group of K/F

W_{nr}(K/F) = Ker (W (K) ⟶^{(∂_{v}^{2})} ⨁_{v} W (K_{v}))

where v runs through all discrete valuations of K which are trivial on F, K_{v} and ∂_{v}^{2} denote respectively the residue field of v and the second residue homomorphism at v (associated to a local uniformiser). This condition is sufficient in the following cases, in which φ is even defined over F:

• K/F is purely transcendental.

• K is the function field of a conic or a quadric defined by a 2-fold

Pfister form ([4, Lem. 3.1], [5, Appendix], [38], [41]). In general, however, the homomorphism W (F) → W_{nr}(K/F) is not

surjective (cf. Theorem 6.4 d)). When K = F (q) is the function field of the projective quadric with equation q = 0, one has the following conjecture [20]:

Conjecture 1. Let q be a quadratic form over F, K = F (q) and φ a quadratic form over K such that:

(1) φ ∈ W_{nr}(K/F),

(2) dim φ < \frac{1}{2} dim q. Then φ is defined over F.

This conjecture was proved in certain cases where dim φ is small ([20], [31] and [17, Th. 3.9]).

In this paper, we study the following complementary conjecture to Conjecture 1, and prove it in certain cases:

Conjecture 2. Let q be a quadratic form over F, K = F (q) and φ an anisotropic quadratic form over K. Suppose that φ ∈ Im(W (F) → W (K)). Then there exists a quadratic form ψ over F such that ψ_{K} ∼ φ and dim ψ ≤ 2 dim φ.

In order to state our results, it is convenient to introduce some notation. For (q, φ) as in Conjecture 2, define

C_{F} (q, φ) = inf{dim ψ | ψ ∈ W (F) and φ ∼ ψ_{F (q)}}.

Note that C_{F} (q, φ) and dim φ have the same parity. Moreover, define:

C_{F} (q, n) = sup{C_{F} (q, φ) | dim φ = n} ≤ +∞
C(m, n) = sup{C_{F} (q, n) | F a field and dim q = m} ≤ +∞
C(n) = sup{C(m, n) | m ≥ 2} ≤ +∞.

A reformulation of Conjecture 2 in terms of these constants is as follows:

Reformulation. For any integer n ≥ 1, one has C(n) ≤ 2n. In particular, C(n) ≤ 2n − 1 for n odd.

(When n is odd, the inequality C(n) ≤ 2n−1 follows from C(n) ≤ 2n and the fact that C(n) has the same parity as n.)

These bounds are best possible: indeed, we shall show in Section 5:

Theorem 1. For any n ≥ 1, one has

C(n) ≥ C(4, n) ≥ \begin{cases}2n & if n is even \\ 2n − 1 & if n is odd. \end{cases}

The appearance of C(4, n) in the formulation of this theorem suggests that the case dim q = 4 is the most difficult to study: this is amply vindicated by our computations below.

Here are now the main results of this paper.

Theorem 2. a) Conjecture 2 holds if

(i) dim φ ≤ 3.

(ii) φ is similar to a 2-fold Pfister form.

(iii) φ is a Pfister neighbour of dimension 5.

b) For all m ≠ 4, C(m, 4) ≤ 8 and C(m, 5) ≤ 9.

c) C(4, 4) ≤ 10. In particular, C(4) ≤ 10.

For n = 5 or 6, we also get the following partial results, which overlap with Theorem 2 b):

Theorem 3. Let q, K, φ be as in Conjecture 2; let D be a central simple F -algebra such that [D_{K}] = c(φ).

a) If dim φ = 5 and φ is not a Pfister neighbour, Conjecture 2 holds, except perhaps when dim q = 4, d_{±}q ≠ 1 and ind(D) = ind(D ⊗ C_{0}(q)) = 4.

b) If dim φ = 6 and φ is an Albert form, Conjecture 2 holds and one even has C_{F} (q, φ) ≤ 8, except perhaps in the same exceptional case as in a).

Observe that in Theorem 2 and Theorem 3, the only instances where we cannot fully conclude are when dim q = 4 (and d_{±}q ≠ 1), confirming that this case is particularly difficult.

Theorems 2 and 3 follow from more precise results, which will be stated in Theorem 7.3. Roughly, we are able not only to prove the existence of the forms ψ appearing in Conjecture 2, but also to determine exactly those of smallest dimension, provided φ is not too complicated. (When φ is defined over F, ψ is usually unique, cf. [20, Lem. 3]. This is not the case in general, but all ψ of minimal dimension are at least of the same shape). It turns out that one is often in a “standard” situation, of the same type as the one in Proposition 4.5 below. Conversely, the fact that the exceptional case of Theorem 3 is the same as the one showing up in [15] is not a coincidence.

See Section 8 for the general strategy of the proofs. Unfortunately they are not as simple as one might hope, and we are forced to go through tedious discussions involving cases and subcases. As in [20] for Conjecture 1, we hope that a more geometric understanding of Conjecture 2 will eventually lead to a direct and general proof.

2. Notation and rappels

Most of the notations and definitions that we use are well-established (cf. inter alia [32], [43], [26, 27]). Let us only specify those which may not be standard:

(1) For i ≥ 0, we denote by H^{i}F the Galois cohomology group H^{i}(F, ℤ/2).

(2) We denote by e^{1}(φ) = (d_{±}φ) ∈ H^{1}F where d_{±}φ is the signed discriminant of a quadratic form φ, and by e^{2}(φ) = c(φ) ∈ H^{2}F where c(φ) is its Clifford invariant.

(3) For φ ∈ I^{3}F, we denote by e^{3}(φ) ∈ H^{3}F its Arason invariant [2].

(4) We denote by D(φ) (resp. G(φ)) the set of values (resp. the group of similarity factors) of an anisotropic form φ.

(5) For two quadratic forms φ and ψ, we write φ ∼ ψ if φ ⊥ −ψ is hyperbolic (Witt equivalence), and ψ ≤ φ if ψ is isometric to a subform of φ.

(6) We denote by φ_{an} the anisotropic part of a quadratic form φ.

(7) For a_{1}, …, a_{n} ∈ F^{∗}, we denote by ⟨⟨a_{1}, …, a_{n}⟩⟩ the n-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩.

(8) We write P_{n}(F) for the set of n-fold Pfister forms, GP_{n}(F) = F^{∗}P_{n}(F) and GP (F) = ⋃_{n≥1} GP_{n}(F).

(9) We shall usually abbreviate the expression Pfister neighbour [27, Def. 7.4] into neighbour.

(10) An Albert form is a 6-dimensional quadratic form with trivial signed discriminant. An Albert form may be isotropic.

(11) A virtual Albert form is an anisotropic 6-dimensional quadratic form which remains anisotropic on the quadratic extension given by its signed discriminant.

(12) For any quadratic form φ of dimension ≥ 2, we denote by X_{φ} the projective quadric of equation φ = 0 (dim X_{φ} = dim φ − 2) and by F (φ) the function field of X_{φ} (if dim φ = 2 and φ is isotropic, one therefore has F (φ) = F × F).

(13) For any finite-dimensional central simple F -algebra A, we write ind(A) for the Schur index of A; [A] for the class of A in H^{2}F; SB(A) for the Severi-Brauer variety of A and F (A) for the function field of SB(A).

(14) If φ and φ′ are two quadratic forms, we denote by F (φ, φ′) the function field of the product variety X_{φ}×_{F} X_{φ′}. If A is a central simple F -algebra and φ is a quadratic form, we write F (A, φ) for the function field of SB(A) ×_{F} X_{φ}. We have F (φ, φ′) = F (φ)(φ′) = F (φ′)(φ), F (A, φ) = F (A)(φ) = F (φ)(A). Similarly for more quadratic forms or more algebras.

(15) If φ and ψ are two anisotropic F -quadratic forms, we say that φ is dominated by ψ if ψ_{F (φ)} is isotropic. Notation: φ ≼ ψ or ψ ≽ φ. This is a preorder relation. One has: (φ ≤ aψ for a scalar a ∈ F^{∗}) ⇒ (φ ≼ ψ).

(16) If φ ≼ ψ and ψ ≼ φ, we say that φ and ψ are stably birationally equivalent^{1}. Notation: φ ≍ ψ. This is an equivalence relation.

(17) If φ is an F -quadratic form, we denote by (φ_{0}, φ_{1}, …, φ_{h}) the sequence of its “higher kernel forms” in the sense of Knebusch [26, 27]. In particuliar, φ_{0} = φ_{an} and (if dim φ_{0} ≥ 2) φ_{1} = (φ_{F (φ_{0})})_{an}. We write (i_{0}(φ), i_{1}(φ), …, i_{h}(φ)) for the sequence of the “higher Witt indices” of φ (this is sometimes called the splitting pattern of φ): in particular, i_{0}(φ) = i_{W} (φ) (the classical Witt index) and i_{1}(φ) = i_{W} (φ_{F (φ_{0})}). The integer h is the height of φ; if φ ∈ IF, the unique integer d such that φ_{h−1} ∈ GP_{d}(F_{h−1}) is the degree of φ, and the unique Pfister form which is determined by φ_{h−1} is called the leading form of φ. The height and degree are respectively denoted by h(φ) and deg(φ).

(18) [11, Def. 3.4] Let 1 ≤ m < n. We say that an anisotropic form φ of dimension 2^{n} is a twisted (n, m)-fold Pfister form if there exists σ ∈ P_{n}(F) − {0} and π ∈ P_{m}(F) − {0} such that φ ∼ σ ⊥ −π. The form π is called the twist of φ. Notation: φ ∈ P_{n,m}(F). We write GP_{n,m}(F) = F^{∗}P_{n,m}(F).

The Arason-Pfister Hauptsatz will be of constant use in the proofs. Recall its content:

2.1. Theorem (Arason-Pfister [3]). If φ ∈ I^{n}F is anisotropic, then dim φ ≥ 2^{n}. Moreover, if dim φ = 2^{n}, then φ ∈ GP_{n}(F).

Recall also that, for n ≤ 3, the cohomological invariant e^{n} induces an isomorphism

(2.1)  I^{n}F/I^{n+1}F ⟶^{∼} H^{n}F

(Kummer for n = 1, Merkurjev [34] for n = 2, Rost [42] and Merkurjev-Suslin [36] for n = 3). We shall not need the fact that this result extends to all n in characteristic zero (Orlov-Vishik-Voevodsky [37]).

Finally, let us recall the statement of the index reduction theorems of Merkurjev [35], [46] and Schofield-van den Bergh [44]:

2.2. Theorem. Let D be a division algebra with center F.

1Strictly speaking, this terminology should rather apply to X_{φ} and X_{ψ}.

(1) For q an F -quadratic form, we have ind(D_{F (q)}) < ind(D) if and only if there exists a homomorphism C_{0}(q) → D.

(2) For a central simple F -algebra A and K = F (SB(A)), we have

ind(D_{K}) = inf{ind(D ⊗_{F} A^{⊗i}) | i ∈ ℤ}.

3. Cohomological kernels

Certain cohomological results will play a central rôle in the proofs. For any extension K/F and any integer n ≥ 0, define

H^{n}(K/F) = Ker(H^{n}F → H^{n}K).

In degree 2, one has: 3.1. Theorem.

(1) (Amitsur [1]) For any central simple F -algebra A of exponent 2, H^{2}(F (A)/F) = {0, [A]}.

(2) (Witt, Arason [2]) For any F -quadratic form q of dimension ≥ 4, one has H^{2}(F (q)/F) = 0, except if q ∈ GP_{2}(F); in this case the kernel is generated by e^{2}(q).

In degree 3, there is a general theorem due to Peyre [40] concerning the function fields of projective homogeneous varieties. We shall only need special cases of this theorem, first of all for quadrics and Severi-Brauer varieties. In the first case, there is a previous, more precise result, due to Arason: 3.2. Theorem (Arason [2, Satz 5.6]). Let q be an F -quadratic form of dimension ≥ 3, and α ∈ H^{3}(F (q)/F) − {0}. Then there exists τ ∈ GP_{3}(F) such that

(i) e^{3}(τ) = α, (ii) q ≤ τ. One deduces from this:

3.3. Lemma. Let q be an anisotropic F -quadratic form of dimension 4 and discriminant d. Write E = F (\sqrt{d}) and let N : E^{∗} → F^{∗} be the norm map. Then

H^{3}(F (q)/F) = {e^{3}(⟨⟨ N(x) ⟩⟩ ⊗q) | x ∈ E^{∗}}.

Proof. For x ∈ E^{∗}, the quadratic form ⟨⟨ N(x) ⟩⟩ ⊗q has trivial Clifford invariant, hence belongs to GP_{3}(F). So {e^{3}(⟨⟨ N(x) ⟩⟩ ⊗q) | x ∈ E^{∗}} ⊂ H^{3}(F (q)/F). Conversely, let α ∈ H^{3}(F (q)/F) − {0} and τ be as in the conclusion of Theorem 3.2. If one writes τ ≃ q ⊥ ψ, ψ_{E} is similar to q_{E}, hence ψ ≃ −aq for a suitable a ∈ F^{∗} by a theorem of Wadsworth [48]. But the quadratic form q ⊥ −aq is in I^{3}F if and only if c(q ⊥ −aq) = (a, d) = 0, that is, a ∈ N(E^{∗}).

3.4. Lemma. Keep the same notation as in Lemma 3.3, and suppose q ∉ GP_{2}(F). Let ρ be a quadratic form such that ρ_{F (q)} ∼ q_{F (q)}. Then ρ ≡ N(x)q (mod I^{4}F) for some x ∈ E^{∗}.

Proof. Since H^{i}(F (q)/F) = {0} for i = 1, 2, one has q ⊥ −ρ ∈ I^{3}F and therefore e^{3}(q ⊥ −ρ) ∈ H^{3}(F (q)/F). By Lemma 3.3, there exists x ∈ E^{∗} such that

q ⊥ −ρ ≡ ⟨⟨ N(x) ⟩⟩ ⊗q (mod I^{4}F).

Hence the result.

3.5. Theorem (Peyre [39]). Let A be a central simple F -algebra of exponent 2. Then

(1) There is an injection \frac{H^{3}(F (A)/F)}{[A] · H^{1}F} ↪ _{2}CH^{2}(SB(A)), where

CH^{2}(X) is the codimension 2 Chow group of a smooth variety X.

(2) For ind(A) ≤ 4, _{2}CH^{2}(SB(A)) = 0.

3.6. Theorem (Karpenko [24]). One also has _{2}CH^{2}(SB(A)) = 0 if A is a tensor product of three quaternion algebras.

3.7. Corollary. H^{3}(F (A)/F) = [A] · H^{1}F if A is a tensor product of at most three quaternion algebras.

We shall also need the case of a product of a quadric by a Severi-Brauer variety:

3.8. Theorem (Izhboldin-Karpenko [16, Prop. 2.1]). Let A be a central simple F -algebra of exponent 2 and q an F -quadratic form of dimension ≥ 3. Then there is an isomorphism

\frac{H^{3}(F (q, A)/F)}{H^{3}(F (q)/F) + H^{3}(F (A)/F)} ≃ \frac{CH^{2}(X_{q} × SB(A))_{tors}}{CH^{2}(X_{q})_{tors} + CH^{2}(SB(A))_{tors}} .

Unfortunately, the results of the next theorem cannot always be derived directly from this general statement.

3.9. Theorem. Let A and q be as in Theorem 3.8.

(1) (Izhboldin-Karpenko [14, Th. 4]) Suppose that A is a division algebra of degree 8, and that dim q ≥ 5. If ind(A_{F (q)}) < ind(A), then

H^{3}(F (q, A)/F) = [A] · H^{1}F.

(2) (Izhboldin-Karpenko [16, Th. 4.1 and Prop. 6.8]) Suppose ind(A) ≤ 4 and dim q = 4, d = d_{±}q ≠ 1. Let E = F (\sqrt{d}). Then

CH^{2}(X_{q} × SB(A))_{tors} = 0

and thus

H^{3}(F (q, A)/F) = H^{3}(F (q)/F) + H^{3}(F (A)/F)

except perhaps if ind(C(q_{E}) ⊗ A_{E}) = 2 or 4. In the first case,

the quotient \frac{H^{3}(F (q, A)/F)}{H^{3}(F (q)/F) + H^{3}(F (A)/F)} is generated by e^{3}(ρ)

for any quadratic form ρ ∈ I^{3}F of type

ρ = kq + q′′ + lγ

where γ is an Albert form verifying c(γ) = [A], k, l ∈ F^{∗} and q′′ is a 4-dimensional form verifying d_{±}q′′ = d, c(q′′_{E}) = c(q_{E}) + [A_{E}].

(3) (Izhboldin-Karpenko [15, Th. 3.1]) Suppose that q = ⟨−a, −b, ab, d⟩ with d ≠ 1 and that A = (a, b) ⊗_{F} (u, v) ⊗_{F} (d, s) is a division algebra. Then

H^{3}(F (q, A)/F) = H^{3}(F (q)/F) + [A] · H^{1}F.

4. Conjugate forms

4.1. Definition ([27, Def. 8.7]). Two anisotropic F -quadratic forms φ and φ′ are conjugate if dim φ = dim φ′ and φ ⊥ −φ′ ∈ GP (F). We denote this relation by φ ≈ φ′.

If two quadratic forms are conjugate, their common dimension is a power of 2.

4.2. Lemma. Let φ, ψ be two anisotropic F -quadratic forms, with φ ≄ ψ and dim ψ ≤ dim φ. Suppose that (ψ ⊥ −φ)_{F (φ)} ∼ 0. Then, either φ is a Pfister neighbour with complementary form −ψ, or φ and ψ are conjugate.

Proof. We have the inequality dim(ψ ⊥ −φ) − 2^{deg(ψ⊥−φ)} < 2 dim φ. Therefore, we are in the position to apply Fitzgerald’s theorem [8, Th. 1.6], which states that ψ ⊥ −φ ∈ GP (F)−{0}. The conclusion follows immediately.

4.3. Proposition. a) For two anisotropic F -quadratic forms φ, φ′ of dimension 2^{n}, the following conditions are equivalent:

(i) φ ≈ φ′.

(ii) φ ⊥ −φ′ ∈ I^{n+1}F.

(iii) (φ ⊥ −φ′)_{F (φ)} ∼ 0.

(iv) φ ≍ φ′ (Section 2, notation 16) and φ_{K} ∼ φ′_{K} for any extension

K/F such that φ_{K} and φ′_{K} are isotropic.

b) The relation of conjugation is an equivalence relation.

Proof. a) (i) ⇒ (ii) is clear. (ii) ⇒ (iii) follows from the Hauptsatz. (iii) ⇒ (i) follows from Lemma 4.2. (i) ⟺ (iv) is [27, Th. 8.8].

b) follows from the equivalence between (i) and (ii) in a).

4.4. Proposition. Let φ ∈ GP_{n,n−1}(F), of twist π. For another F - quadratic form φ′, the following conditions are equivalent:

(i) φ ≈ φ′.

(ii) φ′ ≃ aφ with a ∈ G(π).

Proof. One has φ ≡ −π (mod I^{n}F). (ii) ⇒ (i) Since ⟨⟨ a ⟩⟩ ⊗π ∼ 0, one has ⟨⟨ a ⟩⟩ ⊗φ ∈ I^{n+1}F, that is, φ ⊥ −φ′ ∈ I^{n+1}F. By Proposition 4.3, one has φ ≈ φ′. In the other direction, one has φ′ ≃ aφ by [12, Cor. 3.4, Th. 3.4] (see [28, Th. 3] for n = 3). Hence

φ ⊥ −φ′ ≡ ⟨⟨ a ⟩⟩ ⊗φ ≡ − ⟨⟨ a ⟩⟩ ⊗π (mod I^{n+1}F).

By the Hauptsatz, ⟨⟨ a ⟩⟩ ⊗π ∼ 0.

4.5. Proposition. Let q be an anisotropic F -quadratic form, K = F (q) and q_{1} = (q_{F (q)})_{an}. Then,

C_{F} (q, q_{1}) = \begin{cases}dim q_{1} & if q is a neighbour \\ dim q & otherwise. \end{cases}

In the first case, the complementary form of q is the unique F -quadratic form ψ of dimension < dim q such that ψ_{K} ≃ −q_{1}. In the second case, the set of those F -quadratic forms ψ such that dim ψ ≤ dim q and ψ_{K} ∼ q_{1} is reduced to {q} if dim q is not a power of 2, and equals the set of F -quadratic forms conjugate to q otherwise.

Proof. If q is a neighbour of complementary form ψ, then q_{1} is defined over F by the form −ψ by a theorem of Hoffmann [10, Th. 1]. Moreover, let ψ′ be another F -quadratic form such that ψ′_{K} ≃ q_{1}. Then (ψ ⊥ ψ′)_{K} ∼ 0; if ψ ⊥ ψ′ ≁ 0, one has dim(ψ ⊥ ψ′) ≥ dim q + dim q_{1} by [20, Lem. 2]. This implies dim ψ′ ≥ dim q.

Suppose now that q is not a neighbour. Let ψ be an anisotropic F -quadratic form of minimal dimension such that ψ_{K} ∼ q_{1}. One has dim ψ ≤ dim q and (ψ ⊥ −q)_{K} ∼ 0. If dim q is not a power of 2, then q ≃ ψ by Lemma 4.2. Otherwise, one reapplies Lemma 4.2 to deduce that ψ is conjugate to q; conversely, any F -quadratic form conjugate to q is a solution, thanks to the equivalence between (i) and (iii) in Proposition 4.3 a).

5. Lower bounds

5.1. Lemma. a) For any m ≥ 2 and n ≥ 1, C(m, n) ≥ n.

b) For any n ≥ 1, C(2, n) = C(3, n) = n. If q ∈ GP_{2}(F), then C_{F} (q, n) = n.

c) One has C(m, n) = n for any m > 2n as soon as Conjecture 1 is verified.

Proof. a) and c) are obvious; b) follows from the excellence of the function field of a quadratic form of dimension ≤ 3 or of a 2-fold Pfister form.

5.2. Lemma. We have

a) C(m, 1) = 1 for any m ≥ 2.

b) C(4, 2) ≥ 4.

Proof. a) follows from [20, Th. 2 (a)] for m > 2 and from Lemma 5.1 b) for m = 2. b) Let F, q be such that q is anisotropic over F, dim q = 4 and d_{±}q ≠ 1. Then q is not a neighbour: the statement therefore follows from Proposition 4.5.

5.3. Proposition. Let m ≥ 2 and k ≥ 1 be integers. Then, C(m, k + 1) ≥ C(m, k) + 1 and C(m, 2k) ≥ 2C(m, k).

Proof. Let q be an F -quadratic form of dimension m, K = F (q) and φ′, φ′′ ∈ Im(W (F) ⟶ W (K)) be anisotropic, with dim φ′ = k. Let L = F ((t)) be the field of formal power series in t over F.

Set φ = φ′ ⊥ tφ′′. By Springer’s theorem, φ is anisotropic. The statements will therefore follow from the inequality

(5.1)  C_{L}(q_{L}, φ) ≥ C_{F} (q, φ′) + C_{F} (q, φ′′).

applied respectively with dim φ′′ = 1 (cf. Lemma 5.2 a)) and with φ′′ = φ′.

We have φ ∈ Im(W (L) ⟶ W (K ·L)). Let η ∈ W (L) be anisotropic and such that φ ∼ η_{K·L}. Write η ≃ η′ ⊥ tη′′ with η′, η′′ ∈ W (F). Taking residue forms, we get:

φ′ ∼ η′_{K} and φ′′ ∼ η′′_{K}.

The forms η′ and η′′ are anisotropic. Therefore

dim η′ ≥ C_{F} (q, φ′)
dim η′′ ≥ C_{F} (q, φ′′)

and

dim η ≥ C_{F} (q, φ′) + C_{F} (q, φ′′).

But this is true for any form η verifying φ ∼ η_{K·L}, hence (5.1).

Proof of Theorem 1. For n = 1 the statement is trivial. Suppose n ≥ 2. If n is even, write it n = 2k; the statement then follows from Lemme 5.2 iterated k times and Proposition 5.3. If n is odd, write it n = n′ + 1; the statement follows from the even case and Proposition 5.3.

6. The homomorphism W (F) → W_{nr}(K/F)

6.1. Proposition. Let q, q′ be two anisotropic F -quadratic forms of dimension ≥ 3 such that q ≼ q′, and let K = F (q), K′ = F (q′).

a) There is a commutative diagram

W_{nr}(K/F)
↗
W (F)  ↑
↘
W_{nr}(K′/F).

If moreover q′ ≼ q, the vertical arrow in this diagram is an isomorphism.

b) Let φ′ ∈ W_{nr}(K′/F), and let φ be an anisotropic representative of the image of φ′ in W_{nr}(K/F) by the map of a). Let ψ be an F -quadratic form such that φ ∼ ψ_{K}. Then φ′ ∼ ψ_{K′} in the following cases:

(i) q′ ≍ q (cf. Notation 2.16).

(ii) The following inequality holds:

dim φ′ + dim ψ < 2(dim q − i_{1}(q)).

Proof. a) can be proven as [21, Prop. 2.5 c)] by noting that the extension K · K′/K is purely transcendental.

b) Observe that dim φ ≤ dim φ′. If φ ∼ ψ_{K}, then (ψ_{K′} ⊥ −φ′)_{K·K′} is hyperbolic. In the first case, the extension K · K′/K′ is purely transcendental, hence ψ_{K′} ⊥ −φ′ is hyperbolic. In the second case this form is still hyperbolic, as one sees by applying [20, Lem. 2] and using the fact that i_{1}(q_{K′}) = i_{1}(q) since K · K′/K is purely transcendental.

6.2. Corollary. Conjectures 1 and 2 only depend on the stable birational equivalence class of q.

6.3. Remark. The integer dim q − i_{1}(q) = \frac{1}{2}(dim q + dim q_{1}) is a stable birational invariant of q, by a result of Vishik and Karpenko [47, Cor. A.18], [25, Th. 8.1].

6.4. Theorem. Let q be an F -quadratic form and K = F (q).

a) If dim q = 3, the homomorphism W (F) → W_{nr}(K/F) is surjective. This is also true if q is similar to a 2-fold Pfister form.

b) If F is of characteristic 0 and if q is a neighbour, the homomorphism W (F) → W_{nr}(K/F) is surjective.

c) Suppose that q is an Albert form: (q_{K})_{an} is therefore of the form ατ, with α ∈ K^{∗} and τ ∈ P_{2}(K). Then τ ∈ W_{nr}(K/F), but τ ∉ Im(W (F) → W_{nr}(K/F)).

Proof. The first statement of a) is due to Parimala (cf. [4, Lem. 3.1]); the second one follows from Proposition 6.1 a). b) is [23, Th. 4]. Finally, c) is stated in [19, Remark 3 p. 249]: since this reference is sparing with details, let us give a proof for the reader’s convenience.

Since c(τ) = c(q)_{K}, τ is unramified (its residues are binary forms with trivial discriminants). It is now sufficient to show that φ = ⟨⟨ α ⟩⟩ ⊗τ ∉ Im(W (F) → W_{nr}(K/F)), since ατ is in this image. Suppose that φ ∼ ψ_{K} for some F -quadratic form ψ. As φ ∈ I^{3}K and H^{i}(K/F) = {0} for i = 1, 2 (since dim q > 4), necessarily ψ ∈ I^{3}F. Then e^{3}(φ) = e^{3}(ψ)_{K}. But this contradicts the fact that e^{3}(φ) ∈ H^{3}K is not defined on F (cf. [19, proof of Th. 2 d), middle p. 249]).

6.5. Question. For K = F (q), is it true that W (F) → W_{nr}(K/F) is surjective if dim q ∈ {4, 5}?

At least W (F) → \frac{W_{nr}(K/F)}{I^{5}K ∩ W_{nr}(K/F)} is surjective in these cases: this

follows from [21, Th. 4, 5 and 9] and [22, Th. 3].

7. Detailed statements of results

7.1. Definition. Let q, K, φ be as in Conjecture 2. We write

Desc(φ) = {ψ ∈ W (F) | ψ_{K} ∼ φ}
Desc_{0}(φ) = {ψ ∈ Desc(φ) | dim ψ = dim φ}.

An element of Desc(φ) is called a Witt descent of φ; an element of Desc_{0}(φ) is called a descent of φ.

By definition, φ is defined over F if and only if Desc_{0}(φ) ≠ ∅. Any element of Desc(φ) of minimal dimension is automatically anisotropic.

By Lemma 5.1 b), φ is always defined over F if dim q ≤ 3 or q ∈ GP_{2}(F). This allows us to limit ourselves to dim q ≥ 4, q ∉ GP_{2}(F) to prove Theorems 2 and 3. We shall make this assumption henceforth without further comments.

7.2. Lemma. Let ψ, ψ′ ∈ Desc(φ). Then d_{±}ψ = d_{±}ψ′ and c(ψ) = c(ψ′).

Proof. The form ψ ⊥ −ψ′ belongs to W (K/F). Since the extension K/F is regular, the first claim is obvious. The assumption on q implies that H^{2}(K/F) = 0 (Theorem 3.1), hence the second claim.

Let q, K, φ be as in Conjecture 2. We choose ψ ∈ Desc(φ) of minimal dimension. We denote by D the central division F -algebra such that [D_{K}] = c(φ). Moreover, we set

L = F (D), n = dim φ, d = d_{±}ψ, d′ = d_{±}q,

q_{1} = (q_{F (q)})_{an}, E = F (\sqrt{d}) and E′ = F (\sqrt{d′}).

By Lemma 7.2, d and D are independent of the choice of ψ.

When q ∈ GP_{2,1}(F) or GP_{3,2}(F), we denote its twist by π. In the tableaux below, the expression “other ψ” means the other members of Desc(φ) of minimal dimension.

7.3. Theorem. Suppose that φ is not defined over F. Then

a) One has n > 1.

b) For n = 2, we are in the following situation:

0  1  2  3  4  5
dim q conditions on q conditions on φ value of ψ  other ψ  C_{F} (q, φ)
4  q ∈ GP_{2,1}(F) φ ≃ aq_{1}, a ∈ F^{∗}  aq  abq, b ∈ G(π)  4

c) For n = 3, we are in one of the following situations:

0  1  2  3  4  5
dim q conditions on q  conditions on φ  value of ψ other ψ C_{F} (q, φ)
4  q ∈ GP_{2,1}(F) φ ≃ aq_{1} ⊥ ⟨c⟩, a, c ∈ F^{∗}, aq ⊥ ⟨c⟩ abq ⊥ ⟨c⟩,  5
−ac ∉ G(π)D(q)  b ∈ G(π)
5  not a neighbour  φ ≃ aq_{1}, a ∈ F^{∗}  aq  no  5

d) For n = 4, φ ∈ GP_{2}(K), we are in one of the following situations:

A SECOND DESCENT PROBLEM 15

0  1  2  3  4  5
dim q  conditions on q  conditions on φ value of ψ  other ψ  C_{F} (q, φ)
4  q ∈ GP_{2,1}(F)  φ ∼ aq_{1} ⊥ θ_{K},  aq ⊥ θ  conjugate or  8 if −a
a ∈ F^{∗}, dim θ = 4  Witt-equivalent to ∉ G(π)D(q)D(θ)
abq ⊥ θ, b ∈ G(π)  4 or 6 else
4 or 5  q ≤ γ, γ Albert  φ ∼ aγ_{K},  aγ  abγ,  6
anisotropic  a ∈ F^{∗}  b ∈ G(φ)
6  Albert  φ ≃ aq_{1},  aq  no  6
a ∈ F^{∗}
5, 7 or 8 not a neighbour, q ≤ q′, φ ∼ aq′_{K}, a ∈ F^{∗},  aq′  abq′,  8
q′ ∈ GP_{3,2}(F)  ψ_{L} ∉ I^{4}L if  b ∈ G(π)
anisotropic  dim q = 5

e) For n = 4 and φ ∉ GP_{2}(K), we have ind(D_{E}) ∈ {2, 4}, and we are in one of the following situations:

0  1  2  3  4  5
dim q conditions on q conditions on φ  value of ψ  other ψ C_{F} (q, φ)
4  q ∈ G_{2,1}F,  ind(D_{E}) = 2  a ⟨1, −d⟩ ⊥ τ, or  ?  ≤ 10
a ⟨1, −d⟩ ⊥ bq ⊥ δ,
a, b ∈ F^{∗}, τ ∈ GP_{2}(F)
dim δ = 4
4  q ∈ GP_{2,1}(F),  ind(D_{E}) = 4,  aγ = aq ⊥ θ,  abq ⊥ θ,  6
q ≤ γ,  φ ∼ aγ_{K}, a ∈ F^{∗}  dim θ = 2  b ∈ G(π)

γ virtual Albert

5 not a neighbour φ ≃ aq_{1} ⊥ ⟨b⟩, aq ⊥ ⟨b⟩ no 6
a, b ∈ F^{∗},
−ab ∉ D(q)
6 neither Albert φ ≃ aq_{1}, aq no 6
nor neighbour  a ∈ F^{∗}

f) For n = 5, φ a neighbour, we have φ ≃ τ ⊥ ⟨d⟩ with τ ∈ GP_{2}(K). Then we are in one of the following situations^{2}:

2In column 4 of rows 1 and 4, we only give those other forms ψ representing d_{±}φ.

0  1  2  3  4  5
dim q  conditions on q  conditions on φ  value of ψ  other ψ  C_{F} (q, φ)
4  q ∈ GP_{2,1}(F)  τ ∼ aq_{1} ⊥ θ_{K},  aq ⊥ θ ⊥ ⟨d⟩  q′ ⊥ ⟨d⟩,  7 or 9
a ∈ F^{∗}, dim θ = 4,  q′ ≈ abq ⊥ θ
−a ∉ G(π)D(q)D(θ)
4 or 5  γ := q ⊥ ξ  τ ∼ aγ_{K}, a ∈ F^{∗}  aγ ⊥ ⟨d⟩  abq ⊥ aξ ⊥ ⟨d⟩,  7
anisotropic Albert  −ad ∉ D(γ) if  b ∈ G(π) if
dim q = 5; −ad ∉  dim q = 4; no
D(bq ⊥ ξ) ∀ b ∈ G(π)  if dim q = 5
if dim q = 4
6 Albert τ ≃ aq_{1}, aq ⊥ ⟨d⟩ no 7
−ad ∉ D(q)
5, 7 or 8 not a neighbour, q ≤ q′,  τ ∼ aq′_{K},  aq′ ⊥ ⟨d⟩  q′′ ⊥ ⟨d⟩,  7 or 9
q′ ∈ GP_{3,2}(F)  a ∈ F^{∗},  q′′ ≈ abq′,
anisotropic  (ψ ⊥ ⟨−d⟩)_{L} ∉ I^{4}L  b ∈ G(π)
if dim q = 5

g) For n = 5, φ not a Pfister neigbour, γ = φ ⊥ ⟨−d⟩ is an anisotropic Albert form. If ind(D) = 4, let γ′ be an Albert form such that c(γ′) = [D].

g.1) Suppose that γ is defined over F, and let γ_{0} ∈ Desc_{0}(γ). Then:

(1) −d ∉ D(γ_{0}), hence γ_{0} ⊥ ⟨d⟩ is a descent of φ of minimal dimension and C_{F} (q, φ) = 7.

(2) If dim q ≥ 5, then γ_{0} ⊥ ⟨d⟩ is the unique descent of φ of minimal dimension.

(3) If dim q = 4 and d_{±}q ≠ 1, then a descent ψ of φ of minimal dimension satisfies γ_{0} ⊥ ⟨d⟩ ≃ ψ or there exists ρ ∈ GP_{3}(F) − {0} such that q ≼ ρ, ρ ≃ ρ_{1} ⊥ ρ_{2} and γ_{0} ⊥ ⟨d⟩ ≃ ρ_{1} ⊥ ρ_{3} with dim ρ_{1} = 4, and ψ ≃ ρ_{3} ⊥ −ρ_{2}. In the case d ∈ D(ψ), one has ψ ≃ aγ_{0} ⊥ ⟨d⟩ for some a ∈ F^{∗}.

g.2) Suppose that γ is not defined over F. Then, we are in one of the following situations ^{3}:

3In column 4 we only give those other forms ψ representing d_{±}φ.

A SECOND DESCENT PROBLEM 17

0  1  2  3  4  5
dim q  conditions on q  conditions on φ  value of ψ  other ψ  C_{F} (q, φ)
4  not a neighbour  ind(C_{0}(q) ⊗ D) =  ?  ?  ?
ind(D) = 4
4 q ∈ GP_{2,1}F γ ∼ aq_{1} ⊥ θ_{K}, aq ⊥ θ ⊥ ⟨d⟩ q′ ⊥ ⟨d⟩, 7 or 9
a ∈ F^{∗}, ind(D) = 4,  q′ ≈ acq ⊥ θ,
ind(C_{0}(q) ⊗ D) = 2,  c ∈ G(π)
c(θ_{E′}) = c(q_{E′}) + [D_{E′}],
dim θ = 4,
−a ∉ G(π)D(q)D(θ)
4 q ∈ G_{2,1}(F) γ ≃ aq_{1} ⊥ θ_{K}, aq ⊥ θ ⊥ ⟨d⟩ q′ ⊥ ⟨d⟩, 7 or 9
a ∈ F^{∗}, dim θ = 4  q′ ≈ acq ⊥ θ,
ind(C(aq ⊥ θ)) = 8  c ∈ G(π)
5 to 8 not a neighbour, q ≤ q′,  γ ∼ aq′_{K}, a ∈ F^{∗}  aq′ ⊥ ⟨d⟩,  q′′ ⊥ ⟨d⟩,  7 or 9
q′ ∈ I^{2}F, dim q′ = 8,  ind(D) = 4,  q′′ ≈ aq′
q_{L} anisotropic  e^{3}(ψ ⊥ ⟨−d⟩ ⊥ −γ)_{L} ≠ 0
ind C(q′) = 4
5 to 8 not a neighbour, q ≤ q′, γ ∼ aq′_{K} , a ∈ F^{∗} aq′ ⊥ ⟨d⟩, q′′ ⊥ ⟨d⟩, 7 or 9
q′ ∈ I^{2}F, dim q′ = 8,  q′′ ≈ aq′
ind C(q′) = 8

h) For n = 6, φ an Albert form, we have ind(D) ∈ {4, 8}. If ind(D) = 4, let γ be an Albert form such that D ∼ C(γ). Then we are in one of the following situations:

0  1  2  3  4  5
dim q  conditions on q  conditions on φ  value of ψ  other ψ  C_{F} (q, φ)
4  not a neighbour  ind(C_{0}(q) ⊗ D) =  ?  ?  ?
ind(D) = 4
4 q ∈ GP_{2,1}F φ ∼ aq_{1} ⊥ bq′′_{K} aq ⊥ bq′′ conjugate to 8
a, b ∈ F^{∗}, ind(D) = 4,  acq ⊥ bq′′,
ind(C(q_{E′}) ⊗ D_{E′}) = 2,  c ∈ G(π)
c(q′′_{E′}) = c(q_{E′}) + [D_{E′}],
dim q′′ = 4,
−ab ∉ G(π)D(q)D(q′′)
4 q ∈ G_{2,1}(F) φ ≃ aq_{1} ⊥ θ_{K}, aq ⊥ θ conjugate to 8
a ∈ F^{∗}, dim θ = 4,  abq ⊥ θ,
ind(C(aq ⊥ θ)) = 8  b ∈ G(π)
5 to 8 not a neighbour, q ≤ q′,  φ ∼ aq′_{K}, a ∈ F^{∗}  aq′  conjugate to  8
q′ ∈ I^{2}F, dim q′ = 8,  e^{3}(ψ ⊥ −γ)_{L} ≠ 0  aq′

q_{L} anisotropic, ind(C(q′)) = 4

5 to 8 not a neighbour, q ≤ q′, φ ∼ aq′_{K}, a ∈ F^{∗} aq′ conjugate 8
q′ ∈ I^{2}F, dim q′ = 8,  to aq′

ind C(q′) = 8

7.4. Remarks. a) In column 2 of row 4 in the tableau of g.2), the condition given on ψ amounts to saying that ψ ≢ bγ′ (mod I^{4}F) for any b ∈ F^{∗}, thanks to Corollary 3.7. Moreover, this condition is independent of the choice of ψ, because for ψ′ ∈ Desc(φ) we get ψ ≡ ψ′ (mod I^{4}F) since q is not a Pfister neighbour of dimension ≥ 5 and hence H^{i}(K/F) = {0} for i ≤ 3.

b) The same justification as in a) shows that the condition ψ_{L} ∉ I^{4}L in column 2 of row 4 in the tableau of d) is also independent of the choice of ψ.

8. A rough outline of the proofs

For the proof of Theorem 7.3, we first establish a few more technical results in the next section. We then prove Theorem 7.3 in the following order, relatively to the type of φ: forms of dimension ≤ 3; forms in GP_{2}(K); Albert forms; 5-dimensional forms; and finally 4-dimensional forms not in GP_{2}(K). Each case uses the results established for the previous cases.

Since the proof of Theorem 7.3 is an extremely involved case-by-case analysis, we thought it would be helpful to give the reader a global idea of what is going on and what is the general strategy. The following sketch should only be taken as a guideline, however. For simplicity we only discuss the strategy for bounding the minimal dimension of forms in Desc(φ), not the one for finding all members of Desc(φ) of minimal dimension.

We keep the notation of the previous section.

(1) Find a form ψ_{0} over F with e^{i}(ψ_{0})_{K} = e^{i}(φ) for i ≤ 2, with small dimension. Then (ψ_{0})_{K} ≡ φ (mod I^{3}K).

(2) By a cohomological kernel argument in degree 3, modify ψ_{0} into a form ψ (often ψ = aψ_{0} for some scalar a) such that ψ_{K} ≡ φ (mod I^{4}K).

(3) If lucky, dim ψ + dim φ < 16 and we can conclude that ψ_{K} ∼ φ by the Arason-Pfister Hauptsatz (Theorem 2.1).

9. More lemmas

The following lemma will be needed to justify the “uniqueness” statements (column 4) in the proof of Theorem 7.3.

9.1. Lemma. Let q, K, φ be as in Conjecture 2, and let ψ, ψ′ ∈ Desc(φ) anisotropic. Then

(i) If dim q ≥ 5 and q is not a neighbour, then ψ ≡ ψ′ (mod I^{4}F). If furthermore, dim ψ′ ≤ dim ψ ≤ 8, then ψ ≈ ψ′ or ψ ≃ ψ′ according as dim ψ = 8 or not.

(ii) If dim q = 4 and d_{±}q ≠ 1, then ψ ≡ ψ′ (mod I^{3}F).

Proof. From Theorems 3.1 and 3.2 we deduce that H^{i}(K/F) = {0} for i ≤ 2 (resp. for i ≤ 3) when dim q = 4 (resp. dim q ≥ 5). We finish the proof by using the bijectivity of e^{2} and e^{3}, respectively.

The following proposition is essentially contained in [28].

9.2. Proposition. Let γ be a possibly isotropic Albert form over F, and let L denote the field F (C(γ)) (L is a generic splitting field for γ). Let ψ be an F -quadratic form such that ψ ≡ γ (mod I^{3}F). Suppose that ψ_{L} ∈ I^{4}L. Then there exists a scalar a ∈ F^{∗} such that ψ ≡ aγ (mod I^{4}F). If dim ψ < 10, one has ψ ∼ aγ.

Proof. Consider the Arason invariant e = e^{3}(ψ ⊥ −γ) ∈ H^{3}F. By [28, Cor. 6], one has e = (a) · c(γ) = e^{3}(⟨a, −1⟩ ⊗ γ) for some a ∈ F^{∗}. Hence, by (2.1)

ψ ⊥ −γ ≡ aγ ⊥ −γ (mod I^{4}F)

and the claim follows. The last assertion follows from the Hauptsatz.

9.3. Proposition. Let K/F be a finitely generated extension and φ ∈ Im(W (F) → W (K)) + I^{n}K be a form of dimension < 2^{n−1}. Then φ ∈ W_{nr}(K/F).

Proof. Same as [20, Prop. 1 (a)].

The following technical corollary will be needed to handle the case where dim φ = 4 in Theorem 7.3.

9.4. Corollary. Let φ ∈ I^{3}F, q be a 4-dimensional form and τ ∈ GP_{2}(F). We assume φ_{F (τ,q)} ∼ 0. Then there exists a form θ of dimension 4 such that d_{±}q = d_{±}θ, c(q) ∈ {c(θ), c(τ) + c(θ)} and

(i) If c(q) = c(θ), then π := −q ⊥ θ ∈ GP_{3}(F) and φ ⊥ π ⊥ τ ⊥ ατ ∈ I^{4}F for some α ∈ F^{∗}. In this case, π_{F (q)} ∼ 0.

(ii) If c(q) = c(τ) + c(θ), then φ ⊥ β(−q ⊥ θ) ⊥ τ ∈ I^{4}F for some β ∈ F^{∗}.

Proof. By assumption one has e^{3}(φ)_{F (τ)} ∈ H^{3}(F (τ, q)/F (τ)). By Lemma 3.3, e^{3}(φ)_{F (τ)} = e^{3}(−q ⊥ aq) for some a ∈ F (τ)^{∗}. By the bijectivity of e^{3}, one has φ_{F (τ)} ⊥ −q_{F (τ)} ⊥ aq ∈ I^{4}F (τ). Proposition 9.3 implies that aq ∈ W_{nr}(F (τ)/F). By Theorem 6.4 b), W (F) → W_{nr}(F (τ)/F) is surjective; hence, by the excellence of F (τ)/F, there exists θ ∈ W (F) of dimension 4 such that aq ≃ θ_{F (τ)}. Since −q ⊥ aq ∈ GP_{3}(F (τ)) one gets c(q)_{F (τ)} = c(aq) = c(θ)_{F (τ)}, hence c(q) ∈ {c(θ), c(τ) + c(θ)}. We distinguish two cases:

(i) If c(q) = c(θ), then π := −q ⊥ θ ∈ GP_{3}(F) and π_{F (q)} ∼ 0. Also, e^{3}(φ) + e^{3}(−q ⊥ θ) ∈ H^{3}(F (τ)/F). By Theorem 3.2, there exists α ∈ F^{∗} such that e^{3}(φ) + e^{3}(−q ⊥ θ) = e^{3}(τ ⊥ ατ). So φ ⊥ π ⊥ τ ⊥ ατ ∈ I^{4}F.

(ii) If c(q) = c(τ) + c(θ). Then −q ⊥ θ ⊥ τ ∈ I^{3}F, hence e^{3}(φ) + e^{3}(−q ⊥ θ ⊥ τ) ∈ H^{3}(F (τ)/F). One reapplies Theorem 3.2 to find β ∈ F^{∗} such that e^{3}(φ) + e^{3}(−q ⊥ θ ⊥ τ) = c(−q ⊥ θ) · (β) (because c(τ) = c(−q ⊥ θ)). So φ ⊥ β(−q ⊥ θ) ⊥ τ ∈ I^{4}F.

The following proposition reinforces [20, Lem. 5].

9.5. Proposition. Let D be a central division F -algebra. Let q be an F -quadratic form such that 5 ≤ dim q ≤ 8, and let K = F (D). Consider the following conditions:

(i) q_{K} is an anisotropic neighbour, but q is not a neighbour.

(ii) There exists q′ ∈ I^{2}F of dimension 8 such that a) q ≤ q′

b) c(q′) = [D]. Then (i) ⇒ (ii), and (ii) ⇒ (i) if ind(D) ≥ 4 and dim q ≥ 6.

Proof. (i) ⇒ (ii): set d = d_{±}q and L = F (\sqrt{d_{±}q}). We distinguish three cases:

• dim q = 5. Since q is not a neighbour and q_{K} is a neighbour, we have ind(C_{0}(q)) = 4 and ind(C_{0}(q)_{K}) ≤ 2 [27, Page 10]. By Theorem 2.2 (2), we have ind(C_{0}(q) ⊗ D) = 2, hence there exists τ = ⟨1⟩ ⊥ τ′ ∈ P_{2}(F) such that [D] = c(q) + c(τ). We take q′ = q ⊥ dτ′.

• dim q = 6. By assumption q_{K} is anisotropic, hence d ≠ 1. Moreover, q_{K·L} ∼ 0, hence c(q)_{L} ∈ H^{2}(K · L/L) = {0, [D_{L}]}. Since q_{L} ≁ 0 (because q is not a neighbour [27, Page 10]), we have c(q)_{L} ≠ 0. So c(q)_{L} = [D_{L}], and thus c(q) = [D] + (f, d) for some f ∈ F^{∗}. We take q′ = q ⊥ −f⟨1, −d⟩.

• dim q = 7 or 8. By [27, Page 11], c(q) ∈ H^{2}(K/F) = {0, [D]}. Since q is not a neighbour, we have c(q) = [D]. We take q′ = q ⊥ ⟨−d⟩ or q according as dim q = 7 or 8.

(ii) ⇒ (i): note that q′_{K} ∈ I^{3}K. By [28, Th. 4] (for ind(D) = 4) or [6, Cor. 9.2] (for ind(D) = 8), q′_{K} is anisotropic, so q_{K} is an anisotropic neighbour. If q were a neighbour of a 3-fold Pfister form π, then dim(q′ ⊥ −aπ)_{an} would be ≤ 4 for some a ∈ F^{∗}. This is impossible since c(q′ ⊥ −aπ) = c(q′) has index ≥ 4.

We shall also need the following proposition.

9.6. Proposition. Let D be a central division F -algebra of exponent 2 and index 8, and let q be an F -quadratic form. Consider the following conditions:

(i) ind(D_{F (q)}) < ind(D).

(ii) There exists a form q′ ∈ I^{2}F of dimension 8 such that c(q′) = [D] and q ≤ q′.

Then (ii) ⇒ (i), and (i) ⇒ (ii) if q ∉ GP_{2}(F) and q is not an Albert form.

Proof. (ii) ⇒ (i) is obvious. For (i) ⇒ (ii), use Theorem 2.2 (1) and proceed as in the proof of [29, Cor. 3].

9.7. Remark. In Proposition 9.6, condition (ii) cannot hold in the exceptional cases where q ∈ GP_{2}(F) or q is an Albert form: otherwise, we would get a contradiction with the complementary form which would be in I^{2}F. In these cases however, one may find an q′ as in (ii) which contains a codimension 1 subform of q.

10. Proof of Theorem 7.3: the cases n = 1, 2, 3

From now on, we keep the same notation as at the beginning of Section 7.

10.A. The cases n = 1, 2. The case n = 1 has been seen previously (Lemma 5.2 a)). For n = 2, by [20, Th. 2 (b)], one has necessarily dim q = 4, d = d_{±}q = d_{±}φ. Moreover, [20, Th. 6] shows that one then has φ ≃ aq_{1} for some a ∈ F^{∗}, so we may choose ψ = aq. The uniqueness assertion follows from Propositions 4.5 and 4.4.

10.B. The case n = 3. According to [20, Th. 2 (c)], we are in one of the following three cases:

(1) q is an Albert form.

(2) dim q = 5, q not a neighbour.

(3) dim q = 4, d′ ≠ 1.

In case (1), [20, Th. 6] shows that φ ≃ −dτ′, where τ = ⟨1⟩ ⊥ τ′ is the leading form of q. Then φ ⊥ ⟨−d⟩ ≃ −dτ. But τ ∉ Im(W (F) → W_{nr}(K/F)) by Theorem 6.4 c), a contradiction. This case is therefore impossible.

In case (2), we argue as in [20, Proof of Theorem 6]: the invariant c(φ) is of the form c′_{F (q)}, where c′ is of index 4. Merkurjev’s index reduction theorem implies that c′ = c(q). But then c(φ) = c(q_{1}), hence φ is similar to q_{1} and φ ≃ dd′q_{1}. We may therefore choose ψ = dd′q. The uniqueness assertion follows from Proposition 4.5.

In case (3), the same argument gives that c(q)_{E′} = c′_{E′}. Then c(φ)_{E′(q)} = c′_{E′(q)} = 0 and φ_{E′(q)} is isotropic. Therefore φ is of the form λ⟨1, −d′⟩ ⊥ ⟨dd′⟩ for some λ ∈ F (q)^{∗}. But then, λ⟨1, −d′⟩ ∈ Im(W (F) → W (F (q))). By the case n = 2, we therefore may choose ψ = aq ⊥ ⟨dd′⟩ for some a ∈ F^{∗}. For uniqueness, let ψ′ be of dimension 5 such that φ ∼ ψ′_{K}. By Lemma 3.4 we get

ψ′ ≡ aN(x)q ⊥ ⟨dd′⟩ (mod I^{4}F)

for some x ∈ E^{′∗}, hence ψ′ ≃ aN(x)q ⊥ ⟨dd′⟩ by the Hauptsatz.

It follows that, in row 1 of the tableau, C_{F} (q, φ) > 3 if and only if the condition in column 2 is satisfied.

11. Proof of Theorem 7.3: the case of a form similar to a 2-fold Pfister form

By [20, Th. 2 and 6], φ is defined over F except perhaps in the following cases:

(1) dim q ≤ 5.

(2) q is an Albert form.

(3) dim q = 7, c(q)_{K} = c(φ).

(4) dim q = 8, q ∈ I^{2}F, c(q)_{K} = c(φ).

The cases (2) and (4) are handled in [20, Th. 6]: in each of them, φ is necessarily of the form aq_{1} for some a ∈ F^{∗}. This yields row 4 of the tableau in dimension 8 as well as row 3, the uniqueness claim on ψ coming from Proposition 4.5 (and Proposition 4.4 in case (4)).

Observe that cases (3) and (4) are equivalent: indeed, in these two cases, we have ind c(q) = 2 by [20, Lem. 4]. But a subform of dimension 7 in a form of dimension 8 of that type is stably birationally equivalent to it. Conversely, if q is anisotropic with dim q = 7 and ind c(q) = 2, then q cannot represent its discriminant d_{±}q, hence q ⊥ ⟨−d_{±}q⟩ is anisotropic of the above type. The equivalence between (3) and (4) now follows from Proposition 6.1 b).

From this, one derives easily row 4 of the tableau in dimension 7. It remains to deal with the cases where dim q ≤ 5. If q is a Pfister

neighbour, φ is defined over F by reduction to a 3-fold Pfister form (Proposition 6.1 b)). There are therefore two cases to examine:

(1a) dim q = 5, q not a neighbour.

(1b) dim q = 4, d_{±}q ≠ 1.

In both cases, observe that ind(C(ψ)) ≤ 4. Since ψ_{L} ∈ I^{3}L, we have by the Hauptsatz

(11.1)  ψ_{K·L} ∼ φ_{K·L} ∼ 0

In particular,

(11.2)  e^{3}(ψ_{L}) ∈ H^{3}(K · L/L)

We have the following diagram of field extensions:

\begin{matrix}L & K · L \\ | & | \\ F & K \end{matrix}

11.A. The case ind(C(ψ)) = 4. Let γ be an anisotropic Albert form such that c(ψ) = c(γ). Then ψ ≡ γ (mod I^{3}F). The form γ_{K} is isotropic. It follows from [33] (cf. also Hoffmann’s unpublished thesis) that q is similar to a subform of γ. Therefore q_{L} is isotropic since γ_{L} ∼ 0. The extension L · K/L is therefore purely transcendental and ψ_{L} ∼ 0 by (11.1). By Proposition 9.2, there exists a ∈ F^{∗} such that ψ ≡ aγ (mod I^{4}F). Extending scalars to K, we conclude by the Hauptsatz that φ ∼ aγ_{K} since dim φ < 10.

This justifies row 2 of the tableau, except for the uniqueness claim on ψ and the exact value of C_{F} (q, φ). For another anisotropic Witt descent ψ′ of φ, of dimension ≤ 6, we deduce by Lemma 9.1 that ψ′ ∈ I^{2}F and c(ψ′) = c(γ). Hence dim ψ′ = 6 since ind(C(γ)) = 4. Moreover, ψ′ is similar to γ. If ψ′ = abγ, one has (⟨⟨ b ⟩⟩ ⊗aγ)_{K} ∼ 0, which is equivalent to b ∈ G(φ). Conversely, one does have abγ_{K} ∼ φ for any b verifying this condition.

11.1. Remark. In case (1a), the map H^{3}F → H^{3}K is injective and one gets something slightly better: b ∈ G(φ) ⟺ b ∈ G(γ).

11.B. The case ind(C(ψ)) = 2. The reasoning is similar but more complicated. Of course, we assume that q is of type (1a) or (1b).

Let τ ∈ P_{2}(F) be such that c(ψ) = c(τ). Then ψ ≡ τ (mod I^{3}F). We shall distinguish the following cases:

(i) q_{L} is isotropic.

(ii) dim q = 5 and q_{L} is not a neighbour.

(iii) dim q = 5 and q_{L} is an anisotropic neighbour.

(iv) dim q = 4.

11.B.1. Suppose that q is as in (i) or (ii). Then

H^{3}(K · L/L) = 0.

From equation (11.2) and the bijectivity of e^{3}, we deduce that

ψ_{L} ∈ I^{4}L.

By Proposition 9.2, there exists a ∈ F^{∗} such that ψ ≡ aτ (mod I^{4}F), hence φ ≡ aτ_{K} (mod I^{4}K). By the Hauptsatz, we therefore have φ ≃ aτ_{K} and φ is defined over F.

11.B.2. Suppose that q is as in (iii). By Proposition 9.5 there exists a form q′ ∈ I^{2}F of dimension 8 such that q ≤ q′ and c(q′) = c(τ). Set

ξ = q′ ⊥ −τ.

Then ξ ∈ I^{3}F. In particular, ψ ⊥ −ξ ≡ τ (mod I^{3}F).

Note that q_{L} and q′_{L} are neighbours of the same 3-fold Pfister form, and e^{3}(q′_{L}) = e^{3}(ξ)_{L}. Hence, by Theorem 3.2 we get that H^{3}(K·L/L) = {0, e^{3}(ξ)_{L}}.

• If e^{3}(ψ_{L}) = 0, then ψ_{L} ∈ I^{4}L and we conclude as in case 11.B.1 that φ is defined over F.

• If e^{3}(ψ_{L}) = e^{3}(ξ)_{L}, then (ψ ⊥ −ξ)_{L} ∈ I^{4}L. Applying Proposition 9.2, we can find a scalar a ∈ F^{∗} such that ψ ⊥ −ξ ≡ aτ (mod I^{4}F), or

ψ ≡ a(ξ ⊥ τ) ≡ aq′ (mod I^{4}F)

since ξ ∈ I^{3}F. The Hauptsatz then implies that

φ ∼ aq′_{K}.

This justifies row 4 of the tableau in dimension 5, except for the uniqueness claim on ψ and the exact value of C_{F} (q, φ).

Observe first that q′ is anisotropic since q_{L} is anisotropic and q_{L}, q′_{L} are neighbours of the same 3-fold Pfister from. If ψ′ ∈ Desc(φ) of dimension ≤ 8, then we deduce by Lemma 9.1 that ψ′ ≡ aq′ (mod I^{4}F). Hence we necessarily have dim ψ′ = 8 and ψ′ ≈ aq′. By Proposition 4.4, ψ′ ≃ baq′ where b ∈ G(π) and π is the twist of q′.

11.B.3. Suppose that q is as in (iv). We apply Corollary 9.4 to the form ψ ⊥ −τ to get the existence of a form θ of dimension 4 and π ∈ GP_{3}(F) hyperbolic over K such that

(ψ ⊥ −τ) ⊥ π ⊥ τ ⊥ −ατ ∈ I^{4}F

or

(ψ ⊥ −τ) ⊥ −aq ⊥ −θ ⊥ τ ∈ I^{4}F

for suitable scalars α, a ∈ F^{∗}. In both cases and after simplification, we extend scalars to K and apply the Hauptsatz to get ατ_{K} ∼ φ or (aq ⊥ θ)_{K} ∼ φ. In the first case, φ is defined over F; in the second case, C_{F} (q, φ) ≤ 8.

To finish the justification of row 1 of the tableau, we prove the uniqueness claim on ψ and the exact value of C_{F} (q, φ). Let ψ′ ∈ Desc(φ) of dimension ≤ 8. Then (ψ′ ⊥ −aq ⊥ −θ)_{K} ∼ 0. By Lemma 3.4, we have

ψ′ ≡ aN(x)q ⊥ θ (mod I^{4}F)

for some x ∈ E^{′∗}. So ψ′ is conjugate or Witt-equivalent to aN(x)q ⊥ θ, according as dim ψ′ = 8 or not. Moreover, by the Hauptsatz C_{F} (q, φ) ≤ 6 if and only if −a ∈ G(π)D(q)D(θ).

12. Proof of Theorem 7.3: the case of an Albert form

The method is parallel to that in the previous section. By [20, Th. 2], φ is defined over F except perhaps in the following cases:

(1) dim q ≤ 7.

(2) dim q = 8, q ∈ I^{2}F, c(q)_{K} = c(φ). From these cases we exclude the following, after Proposition 6.1 b):

(3) q is a neighbour of a 3-fold Pfister form

(4) dim q = 7, ind(C_{0}(q)) = 2.

Indeed, in case (3) (resp. (4)), q is stably birationally equivalent to a 3-fold Pfister form (resp. to an 8-dimensonal form in I^{2}F with Clifford invariant of index 2).

Moreover, in case (2), φ is necessarily of the form aq_{1} with a ∈ F^{∗} by [20, Th. 6] (see also loc. cit., note bottom p. 153).

It therefore remains to handle the cases dim q ∈ {4, 5, 6, 7}, q not a neighbour and not of type (4).

We have ind(C(ψ)) = 4 or 8 since ind(C(ψ)_{K}) = 4.

12.A. The case ind(C(ψ)) = 8. We first note that in this case, we have automatically dim ψ ≥ 8 since ψ ∈ I^{2}F.

12.1. Lemma. If ind(C(ψ)) = 8, then C_{F} (q, φ) = 8 and q cannot be an Albert form.

Proof. The index of C(ψ) gets reduced by extension of scalars to K. Let δ be a form defined as follows: δ is an arbitrary 5-dimensional subform of q if q is an Albert form; δ = q otherwise. Since F (q, δ)/F (δ) is purely transcendental, we have ind(C(ψ)_{F (δ)}) ≤ 4. By Proposition 9.6, there exists q′ ∈ I^{2}F of dimension 8 such that δ ≤ q′ and c(ψ) = c(q′). So, ψ ⊥ −q′ ∈ I^{3}F. Now, on the one hand

q′_{L(δ)} ∼ 0

because q′_{L} ∈ GP_{3}(L) and q′_{F (δ)} is isotropic. On the other hand, by the Hauptsatz,

ψ_{L(q)} ∼ 0

because φ ∼ ψ_{F (q)} and ψ_{L} ∈ I^{3}L. Therefore, e^{3}(ψ ⊥ −q′) ∈ H^{3}(L(q, δ)/F). Note that

H^{3}(L(q, δ)/F) ⊆ H^{3}(L(δ)/F)

because L(δ, q)/L(δ) is purely transcendental.

(A) Suppose that dim q ∈ {5, 6, 7}. By Theorem 3.9 (1), we have e^{3}(ψ ⊥ −q′) = c(−q′)(r) for some r ∈ F^{∗}, hence ψ ⊥ −rq′ ∈ I^{4}F. To finish, extend scalars to K and apply the Hauptsatz to get φ ∼ rq′_{K}.

(B) Suppose dim q = 4. Up to a scalar, we have q = ⟨−a, −b, ab, d_{±}q⟩. By the index reduction theorem of Merkurjev (Theorem 2.2 (1)), we have [C(ψ)] = (a, b) + (d_{±}q, u) + (r, s) for suitable scalars r, s, u ∈ F^{∗}. By Theorem 3.9 (3), there exists x, y ∈ F^{∗} such that e^{3}(ψ ⊥ −q′) = c(q′)(x) + e^{3}(q ⊗ ⟨1, −y⟩) with q ⊗ ⟨1, −y⟩ ∈ GP_{3}(F). So, ψ ⊥ −xq′ ⊥ q ⊗ ⟨1, −y⟩ ∈ I^{4}F. Extend scalars to K and apply the Hauptsatz and the fact that (q ⊗ ⟨1, −y⟩)_{K} ∼ 0 to get φ ∼ xq′_{K}.

In both cases, we get an 8-dimensional form q′ ∈ I^{2}F such that δ ≤ q′, ind(C(q′)) = 8 and φ ∼ aq′_{K} for some scalar a ∈ F^{∗}. Hence C_{F} (q, φ) = 8.

Since φ ∼ aq′_{K}, the form q′_{K} is isotropic, hence q cannot be an Albert form otherwise q′ would be isotropic by [29], which is impossible because ind(C(q′)) = 8.

Let us justify the uniqueness claims on ψ. Let ψ′ ∈ Desc(φ) of dimension 8. It follows from Lemma 9.1 that c(q′) = c(ψ′).

• If dim q ≥ 5, then by Lemma 9.1 again aq′ ≈ ψ′.

• If dim q = 4, write aq′ = aq ⊥ θ with dim θ = 4. By Lemma 3.4 (applied to ρ = a(ψ′ ⊥ −θ)), we then have aN(y)q ⊥ θ ≡ ψ′ (mod I^{4}F) for some y ∈ E^{′∗}, hence aN(y)q ⊥ θ ≈ ψ′ by Proposition 4.3 a).

This justifies rows 3 and 5 of the tableau.

12.B. The case ind(C(ψ)) = 4.

12.2. Lemma. Suppose ind(C(ψ)) = 4.

a) If dim q ≥ 5, φ is defined over F unless q_{L} is an anisotropic neighbour. In the latter case, C_{F} (q, φ) ≤ 8.

b) If dim q = 4, then φ is defined over F unless ind(C(q_{E′})⊗C(ψ)_{E′}) = 2 or 4. In the first case, C_{F} (q, φ) ≤ 8. (In the second case, we cannot bound C_{F} (q, φ).)

Proof. Let γ be an anisotropic Albert form such that c(γ) = c(ψ). In particular, ψ ⊥ −γ ∈ I^{3}F. We distinguish the following cases:

(i) dim q ≥ 5, q_{L} is not a neighbour or q_{L} is isotropic.

(ii) dim q ≥ 5, q_{L} is an anisotropic neighbour.

(iii) dim q = 4.

By [18], φ ≃ xγ_{K} for some x ∈ K^{∗}. We have φ_{K·L} ∼ ψ_{K·L} ∼ 0 since γ_{L} ∼ 0. Then

(12.1)  e^{3}(ψ ⊥ −γ) ∈ H^{3}(K · L/F).

(A) Suppose that q satisfies (i). Then

H^{3}(K · L/F) ⊂ H^{3}(L/F).

By Corollary 3.7, we have e^{3}(ψ ⊥ −γ) = c(γ)(u) for some u ∈ F^{∗}. So, ψ ⊥ −uγ ∈ I^{4}F. Extend scalars to K and apply the Hauptsatz to get φ ∼ uγ_{K}. In this case, φ is defined over F.

(B) Suppose that q satisfies (ii). In particular, q is not an Albert form. By Proposition 9.5, there exists q′ ∈ I^{2}F of dimension 8 such that q ≤ q′ and c(q′) = c(γ).

By equation (12.1), e^{3}(ψ ⊥ −γ)_{L} ∈ {0, e^{3}(q′_{L})} because q_{L} and q′_{L} are neighbours of the same 3-fold Pfister form.

• Suppose e^{3}(ψ ⊥ −γ)_{L} = 0. By Corollary 3.7, there exists v ∈ F^{∗} such that e^{3}(ψ ⊥ −γ) = c(γ)(v). So ψ ⊥ −vγ ∈ I^{4}F. Extend scalars to K and apply the Hauptsatz to get φ ∼ vγ_{K}. In this case, φ is defined over F.

• Suppose e^{3}(ψ ⊥ −γ)_{L} = e^{3}(q′_{L}). We obviously have e^{3}(q′_{L}) = e^{3}(q′ ⊥ γ)_{L}. By Corollary 3.7, we have e^{3}(ψ ⊥ −γ) + e^{3}(q′ ⊥ γ) = c(q′)(−a) for some a ∈ F^{∗}. So ψ ⊥ −aq′ ∈ I^{4}F. Extend scalars to K and apply the Hauptsatz to get φ ∼ aq′_{K}.

(C) Suppose that q verifies (iii). If we are not in the exceptional cases of the lemma, we have (use Corollary 3.7, Theorem 3.9 (2), and Lemma 3.3):

e^{3}(ψ ⊥ −γ) = e^{3}(⟨⟨ N(x) ⟩⟩ ⊗q) + e^{3}(⟨⟨ a ⟩⟩ ⊗γ)

for some x ∈ E^{′∗} and a ∈ F^{∗}, hence

ψ ⊥ −γ ≡ ⟨⟨ N(x) ⟩⟩ ⊗q ⊥ − ⟨⟨ a ⟩⟩ ⊗γ (mod I^{4}F)

or

ψ ≡ ⟨⟨ N(x) ⟩⟩ ⊗q ⊥ aγ (mod I^{4}F).

Extending scalars to K and applying the Hauptsatz, we find that φ ≃ aγ_{K}. In this case, φ is defined over F.

If we are in case ind(C(q_{E′}) ⊗ C(ψ)_{E′}) = 2, the same references give this time:

e^{3}(ψ ⊥ −γ) = e^{3}(⟨⟨ N(x) ⟩⟩ ⊗q) + e^{3}(⟨⟨ f ⟩⟩ ⊗γ) + e^{3}(bq ⊥ q′′ ⊥ cγ)

for q′′ as in Theorem 3.9 (2) and some x, f, b, c ∈ F^{∗}. Clearly e^{3}(bq ⊥ q′′ ⊥ cγ) = e^{3}(−fc(bq ⊥ q′′ ⊥ cγ)). Hence, by the bijectivity of e^{3} and after simplification, we get

ψ ≡ ⟨⟨ N(x) ⟩⟩ ⊗q ⊥ −fc(bq ⊥ q′′) (mod I^{4}F)

and, thanks to the Hauptsatz, φ ∼ −fc(bq ⊥ q′′)_{K}. So C_{F} (q, φ) ≤ 8. Set α = −fcb, β = −fc and δ = αq ⊥ βq′′.

It remains to justify the uniqueness claim on ψ and the exact value of C_{F} (q, φ). We keep the case distinction in the proof of Lemma 12.2.

(B) Let ψ′ ∈ Desc(φ), with dim ψ′ ≤ 8. It follows from Lemma 9.1 that ψ′ ≡ aq′ (mod I^{4}F). We have C_{F} (q, φ) = 8, otherwise by the Hauptstaz aq′ would be isotropic, hence also q_{L}, a contradiction. This justifies row 4 of the tableau.

(C) Let ψ′ ∈ Desc(φ) of dimension ≤ 8. Then (δ ⊥ −ψ′)_{K} ∼ 0. By Lemma 3.4, we have

αN(y)q ⊥ βq′′ ≡ ψ′ (mod I^{4}F)

for some y ∈ E^{′∗}. By the Hauptsatz, we have dim ψ′ = 6 if and only if αN(y)q ⊥ βq′′ is isotropic, that is, −αβ ∈ G(π)D(q)D(q′′), where π is the twist of q. This justifies row 2 of the tableau.

13. Proof of Theorem 7.3: the case of a 5-dimensional form

13.1. Proposition. Let q, K, φ be as in Conjecture 2, with φ of dimension 5, and let ψ ∈ Desc(φ). If ind(C_{0}(ψ)) = ind(C_{0}(q) ⊗ C_{0}(ψ)) = 4, we assume that dim q ≠ 4. Then C_{F} (q, φ) ≤ 9. In particular, C(m, 5) ≤ 9 for all m ≠ 4.

Proof. It will be divided into several cases.

13.A. φ is of one of the two following forms.

(1) φ is not a neighbour and γ = φ ⊥ ⟨−d⟩ is not defined over F.

(2) φ is a neighbour.

In case (1) and by the case of an Albert form, we deduce the existence of a form q′ of dimension ≤ 8 such that q ≤ q′ and γ ∼ aq′_{K} for some scalar a ∈ F^{∗}. Then φ ∼ (aq′ ⊥ ⟨d⟩)_{K} and C_{F} (q, φ) ≤ 9. In particular, C(m, 5) ≤ 9 for all m ≠ 4.

In case (2) write φ = τ ⊥ ⟨d⟩ with τ ∈ GP_{2}(K) and get back to the case of τ by applying Theorem 2.

Assume that φ is not defined over F. Notice that in case (2) τ is not defined over F.

Let us give a few justifications for column 4 of the tableaux in f) and g.2). First, observe that the rows of these tableaux, except for column 4, can be deduced respectively from those of the tableaux in d) and h).

13.A.1. Case of the tableau in g.2). Since γ is not defined over F, it follows from the tableau of an Albert form that q′ is anisotropic of dimension 8. Let ψ′ ∈ Desc(φ) of dimension ≤ 9.

(i) dim q ≥ 5: then φ is not defined over F, otherwise there would be ν of dimension 5 such that φ ≃ ν_{K}. By Lemma 9.1, we would get aq′ ⊥ ⟨d⟩ ⊥ −ν ∈ I^{4}F. By the Hauptsatz, q′ would be isotropic, a contradiction. So C_{F} (q, φ) = 7 or 9. For the uniqueness claims, one reapplies Lemma 9.1 to get aq′ ⊥ ⟨d⟩ ⊥ −ψ′ ∈ I^{4}F. If furthermore d ∈ D(ψ′) (as was supposed in the tableau in g.2)), then ψ′ = q′′ ⊥ ⟨d⟩ with dim q′′ ≤ 8, and thus aq′ ⊥ −q′′ ∈ I^{4}F. In particular, q′′ is anisotropic of dimension 8. Hence the uniqueness claim.

(ii) dim q = 4: Set aq′ = aq ⊥ θ with dim θ = 4. By Lemma 3.4, we have

aN(x)q ⊥ θ ⊥ ⟨d⟩ ⊥ −ψ′ ∈ I^{4}F

for some x ∈ E^{′∗}. The condition of column 2 implies that φ is not defined over F. Hence, C_{F} (q, φ) = 7 or 9. If furthermore d ∈ D(ψ′), then ψ′ = q′′ ⊥ ⟨d⟩ with dim q′′ ≤ 8, and

aN(x)q ⊥ θ ⊥ −q′′ ∈ I^{4}F

It is clear that the condition of column 2 implies that q′′ is anisotropic of dimension 8. Hence, the uniqueness claim.

This justifies column 4 in the tableau in g.2).

13.A.2. Case of the tableau in f). We have H^{i}(K/F) = {0} for i = 1, 2 (resp. for i ≤ 3) when dim q = 4 (resp. dim q > 4). Let ψ′ ∈ Desc(φ) anisotropic of dimension ≤ 9.

(i) Case of row 1: we get by Lemma 3.4 that abq ⊥ θ ⊥ ⟨d⟩ ⊥ −ψ′ ∈ I^{4}F for some b ∈ G(π). The condition of column 2 implies that C_{F} (q, φ) = 7 or 9. If furthermore ψ′ = ⟨d⟩ ⊥ q′, then by the same condition q′ is anisotropic of dimension 8, and thus abq ⊥ θ ≈ q′.

(ii) Case of row 4: suppose that ψ′ = q′′ ⊥ ⟨d⟩. Since H^{i}(K/F) = {0} for i ≤ 3, we get aq′ ≡ q′′ (mod I^{4}F). By the tableau in d) the form q′ is anisotropic of dimension 8. Then, q′′ is also anisotropic of dimension 8. In particular, C_{F} (q, φ) = 7 or 9. By Proposition 4.4 we get that abq′ ≈ q′′ with b ∈ G(π).

(iii) Case of row 3 and row 2 in dimension 5: we argue as in (ii). In this case, C_{F} (q, φ) = 7 by the condition in column 2.

(iv) Case of row 2 in dimension 4: write γ ≃ q ⊥ ξ. In this case one has C_{F} (q, φ) ≤ 7. We get by Lemma 3.4

aN(y)q ⊥ aξ ⊥ ⟨d⟩ ⊥ −ψ′ ∈ I^{4}F

for some y ∈ E^{′∗}. By the Hauptsatz, we get aN(y)q ⊥ aξ ⊥ ⟨d⟩ ≃ ψ′. Now it is clear that the condition in the column 2 implies that C_{F} (q, φ) = 7.

13.B. φ is not a neighbour and Desc_{0}(γ) ≠ ∅. Let γ_{0} ∈ Desc_{0}(γ). The justifications of statements (1) and (2) in g.2) are easy to do. For statement (3), let ψ′ ∈ Desc(φ) be anisotropic of dimension 7. If γ_{0} ⊥ ⟨d⟩ ⊥ −ψ′ ≁ 0, then by a result of Fitzgerald there exists ρ ∈ GP_{3}(F) such that (γ_{0} ⊥ ⟨d⟩ ⊥ −ψ′)_{an} ≃ ρ and q ≼ ρ. Hence the claim since i_{W} (γ_{0} ⊥ ⟨d⟩ ⊥ −ρ) = 4. If d ∈ D(ψ′), then ψ′ ≃ δ ⊥ ⟨d⟩ for some Albert form δ. Since c(δ) = c(γ), we get by [18] that δ ≃ aγ for some a ∈ F^{∗}.

14. Proof of Theorem 7.3: the case of a 4-dimensional form which is not a neighbour

By [20, Th. 2], φ is defined over F except perhaps in the following cases:

(1) dim q = 4 and d_{±}q ≠ 1.

(2) dim q = 5.

(3) dim q = 6 and d_{±}q ∈ {1, d_{±}φ}.

Moreover, thanks to Proposition 6.1 b), φ is also defined over F if q is a neighbour of a 3-fold Pfister form.

Since φ_{K·E} ∈ GP_{2}(K·E), we have ind(C(ψ)_{K·E}) ≤ 2, hence ind(C(ψ)_{E}) ≤ 4. So there exists an Albert form γ and some r ∈ F^{∗} such that

(14.1)  c(ψ) = c(γ) + (r, d).

Let F̃ = F ((t)) be the field of formal power series in the variable t over F, and K̃ = F̃(q). Let D′ = C(γ)F̃ ⊗F̃ (rt, d), M = F̃(D′) and φ′ = (φ ⊥ −t⟨1, −d⟩)K̃, which is an anisotropic Albert form. We have the following diagram of field extensions:

\begin{matrix}M & K · M & K̃ · M \\ | & | & | \\ F & K & K̃ \end{matrix}

14.1. Lemma. If dim q ≠ 4, then C_{F} (q, φ) ≤ 6.

Proof. We have φ′ ∈ Im(W (F̃) ⟶ W (K̃)). By Theorem 3, there exists δ ∈ W (F̃) anisotropic such that dim δ ≤ 8 and φ′ ∼ δK̃. Write δ = δ_{1} ⊥ tδ_{2} with δ_{1}, δ_{2} ∈ W (F) anisotropic. By Springer’s Theorem, we get:

• φ ∼ (δ_{1})_{K},

• dim δ_{2} ≥ 2 because d ≠ 1, hence dim δ_{1} ≤ 6. Since φ is not defined over F, the form δ_{1} is of dimension 6. Hence

(δ_{1})_{K} is isotropic. By the isotropy results in dimension 6 [9], [30], the form q cannot be an Albert form. Again by [9], [30] and as q is not a neighbour, we get δ_{1} ≃ aq when dim q = 6 (resp. δ_{1} ≃ aq ⊥ ⟨b⟩ when dim q = 5) for some a, b ∈ F^{∗}.

Uniqueness in dimension 6 follows from Propsoition 4.5. Concerning dimension 5, if ψ′ is another anisotropic Witt descent of φ of dimension 6, then Lemma 9.1 and the Hauptsatz implie that ψ′ ≃ aq ⊥ ⟨b⟩. Clearly, −ab ∉ D(q), because δ_{1} is anisotropic.

This justifies rows 3 and 4 of the tableau.

14.2. Lemma. If dim q = 4, then C_{F} (q, φ) ≤ 10. More precisely, we have:

(1) If ind(C(ψ)_{E}) = 4, then C_{F} (q, φ) = 6.

(2) If ind(C(ψ)_{E}) = 2, then ψ is of one of the two following types: • ψ ≃ a ⟨1, −d⟩ ⊥ τ with a ∈ F^{∗} and τ ∈ GP_{2}(F),

• ψ ≃ b ⟨1, −d⟩ ⊥ cq ⊥ θ for some b, c ∈ F^{∗} and dim θ = 4.

Proof. Up to scaling we may write q = ⟨−α, −β, αβ, d_{±}q⟩. We shall distinguish two cases:

(A) Suppose that ind(C(ψ)_{E}) = 4. This amounts to say that γ_{E} is anisotropic. The algebra D′ is then a triquaternion division algebra [45, Prop. 2.4]. The index of D′ gets reduced by extension of scalars to K̃, since c(φ′) = [D′ ; K̃]. By Proposition 9.6, there exists η ∈ I^{2}F̃ of dimension 8 such that qF̃ ≤ η and

(14.2)  c(η) = [D′].

By equations (14.1) and (14.2), we have

ψF̃ ⊥ −t⟨1, −d⟩ ⊥ η ∈ I^{3}F̃

hence

φK̃ ⊥ (−t⟨1, −d⟩ ⊥ η)K̃ ∈ I^{3}K̃.

We have η_{M(q)} ∼ 0, because η_{M} ∈ GP_{3}(M) and η_{F̃(q)} is isotropic. By the Hauptsatz, we get

(14.3)  (φ ⊥ −t⟨1, −d⟩ ⊥ η)_{M(q)} ∼ 0.

So e^{3}(ψF̃ ⊥ −t⟨1, −d⟩ ⊥ η) ∈ H^{3}(M(q)/F̃). Since qF̃ ≤ η, We get by equation (14.2) that [D′] = (d_{±}q, u) + (α, β) + (r, s) for some r, s, u ∈ F̃^{∗}. By Theorem 3.9 (3), the kernel

H^{3}(M(q)/F̃)

equals

c(η) · H^{1}F̃ + {e^{3}(q ⊗ ⟨1, −y⟩) | y ∈ F̃^{∗}, q ⊗ ⟨1, −y⟩ ∈ GP_{3}(F̃)}.

Let a, b ∈ F̃^{∗} be such that

e^{3}(ψF̃ ⊥ −t⟨1, −d⟩ ⊥ η) = e^{3}(η ⊥ aη) + e^{3}(q ⊗ ⟨1, −b⟩)

with q ⊗ ⟨1, −b⟩ ∈ GP_{3}(F̃). Then

(14.4)  ψF̃ ⊥ −t⟨1, −d⟩ ⊥ −aη ⊥ q ⊗ ⟨1, −b⟩ ∈ I^{4}F̃

We have (q ⊗ ⟨1, −b⟩)K̃ ∼ 0. In (14.4), we extend scalars to K̃ and apply the Hauptsatz to get

(14.5)  φK̃ ∼ (aη ⊥ t⟨1, −d⟩)K̃

Set aη = q′ ⊥ tq′′ with q′, q′′ ∈ W (F) anisotropic. We apply Springer’s Theorem to (14.5) to deduce:

• (q′′ ⊥ ⟨1, −d⟩)_{K} ∼ 0, hence necessarily dim q′′ ≥ 2 since d ≠ 1.

• φ ∼ q′_{K} and dim q′ ≤ 6 by the first assertion.

As φ is not defined over F, necessarily dim q′ = 6, hence dim q′′ = 2. In particular, C_{F} (q, φ) = 6 and d_{±}q′ = d.

Since aq ≤ q′ ⊥ tq′′ and dim q′′ < dim q, we deduce again from Springer’s Theorem that, up to a square, a ∈ F^{∗} and aq ≤ q′.

By Lemma 7.2 c(q′) = c(ψ). So ind(C(q′)_{E}) = 4, that is, q′ is a virtual Albert form.

It remains to justify the uniqueness claims in this case. Let ψ′ ∈ Desc(φ) of dimension 6. By Lemma 3.4

aN(x)q ⊥ θ ≡ ψ′ (mod I^{4}F)

for some x ∈ E^{′∗} and some form θ satisfying q′ ≃ aq ⊥ θ. By the Hauptsatz, ψ′ ≃ aN(x)q ⊥ θ. This justifies row 2 of the tableau.

(B) Suppose that ind C(ψ)_{E} = 2. There exists τ = ⟨1⟩ ⊥ τ′ ∈ P_{2}(F) and s ∈ F^{∗} such that c(ψ) = c(τ) + (s, d). Consider the form μ = −s(⟨d⟩ ⊥ τ′).

We have d_{±}μ = d = d_{±}φ and c(μ) = c(ψ). In particular, c(μ)_{K} = c(φ). By [48], the form φ is similar to μ_{K}. Hence K(φ) ≃ K(μ). We now repeat an argument from [19, Page 149]. Indeed, we have ψ ⊥ −μ ∈ I^{3}F, hence

(ψ_{K} ⊥ −μ_{K})_{K(μ)} ∼ (φ ⊥ −μ_{K})_{K(φ)} ∼ (φ_{K(φ)} ⊥ −μ_{K(μ)}) ∈ I^{3}K(μ).

Since dim(φ_{K(φ)})_{an} and dim(μ_{K(μ)})_{an} ≤ 2, we have dim(φ_{K(φ)} ⊥ −μ_{K(μ)})_{an} ≤ 4. By the Hauptsatz (ψ ⊥ −μ)_{K(μ)} ∼ 0. Since μ_{F (τ)} is isotropic, we get (ψ ⊥ −μ)_{K(τ)} ∼ 0. By Corollary 9.4, there exists a 4-dimensional form ζ and π ∈ GP_{3}(F) hyperbolic over K such that

(ψ ⊥ sτ ⊥ −s⟨1, −d⟩) ⊥ π ⊥ τ ⊥ uτ ∈ I^{4}F

or

(ψ ⊥ sτ ⊥ −s⟨1, −d⟩) ⊥ v(−q ⊥ ζ) ⊥ τ ∈ I^{4}F

for some u, v ∈ F^{∗}. Observe that modulo I^{4}F, the two relations become

(ψ ⊥ sτ ⊥ −s⟨1, −d⟩) ⊥ π ⊥ −s(τ ⊥ uτ) ∈ I^{4}F

or

(ψ ⊥ sτ ⊥ −s⟨1, −d⟩) ⊥ −sv(−q ⊥ ζ) ⊥ −sτ ∈ I^{4}F.

In these last two relations and after simplification, we extend scalars to K and apply the Hauptsatz to get C_{F} (q, φ) ≤ 6 in the first case (because π_{K} ∼ 0), and C_{F} (q, φ) ≤ 10 in the second case. This justifies row 1 in the tableau.

Still concerning row 1 of the tableau, we finally give certain cases where the form φ is defined over F.

14.3. Proposition. Keep the same notation and hypotheses as in case (B) in the proof of Lemma 14.2. Then φ is defined over F when one of the following two conditions is satisfied:

(1) d_{±}q = d_{±}φ

(2) ind(C_{0}(μ) ⊗ C_{0}(q)) = 4.

Proof. By assumption, we have (use [14, Cor. 2.13, Theorem 5.1, Theorem 5.8, Theorem 5.9])

H^{3}(K(μ)/F) = H^{3}(K/F) + H^{3}(F (μ)/F).

By Lemma 3.3, we have e^{3}(ψ ⊥ −μ) = e^{3}(μ ⊥ −aμ) + e^{3}(q ⊥ bq) for some a, b ∈ F^{∗} such that μ ⊥ −aμ, q ⊥ bq ∈ GP_{3}(F). So ψ ⊥ −aμ ⊥ q ⊥ bq ∈ I^{4}F. Since (q ⊥ bq)_{K} ∼ 0, we extend scalars to K and apply the Hauptsatz to get φ ≃ aμ_{K}.

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Institut de Mathématiques de Jussieu, 175–179 rue du Chevaleret, 75013 Paris, France

  E-mail address: kahn@math.jussieu.fr

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

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