ON THE ISOTROPY OF LOW DIMENSIONAL FORMS

OVER THE FUNCTION FIELD OF A QUADRIC

O. T. Izhboldin

Abstract. Let ϕ be a field of characteristic different from 2 and let ϕ and ψ be two anisotropic quadratic form over F. An important problem in the algebraic theory of quadratic forms is to find conditions on ϕ and ψ so that ϕ_{F (ψ)} is isotropic. We prove, that under certain conditions on the quadratic form ϕ (dim ϕ ⩽ 8, and ϕ contains no a quadratic form with trivial determinant) the isotropy of ϕ_{F (ψ)} implies that there exists a homomorphism of even parts of theirs Clifford algebras, C_{0}(ψ) → C_{0}(ϕ).

Let F be a field of characteristic different from 2 and let ϕ be an anisotropic quadratic form over F. For any field extension L/F, we can consider the quadratic form ϕ_{L} = ϕ ⊗_{F} L over L. Of special interest here is the case L = F (ψ), where F (ψ) is the field of rational functions on the quadric hypersurface defined by the equation ψ = 0, ψ being an anisotropic quadratic form over F.

An important problem in the algebraic theory of quadratic forms is to find conditions on ϕ and ψ so that ϕ_{F (ψ)} is isotropic. The fundamental Cassels–Pfister subform theorem states that if ϕ is a Pfister neighbour of an anisotropic Pfister form π, then ϕ_{F (ψ)} is isotropic if and only if ψ is similar to a subform in π (see [Lam], [Sch]). In particular, this completely solves the problem of isotropy of ϕ_{F (ψ)} in the cases where dim(ϕ) ⩽ 3, or dim(ϕ) = 4 and det(ϕ) = 1.

In the case dim(ϕ) = 4, det(ϕ) ≠ 1 the problem was solved by Shapiro [Sh], and in the case dim(ϕ) = 5 by Hoffmann [H1]. The case dim(ϕ) = 6, dim(ψ) ≠ 4 was completely investigated by Hoffmann, Leep, Merkurev and Laghribi (see [H2, Le, Me, La1]), and in the case dim(ϕ) = 8, det(ϕ) = 1, a partial answer has recently been obtained by Laghribi [La2, La3].

The proofs of the mentioned results are based on a combination on the two following methods:

(A) Construct a field extension L/F such that ϕ_{L} is an anisotropic Pfister neighbour of a certain Pfister form π over L. Then the isotropy of ϕ_{F (ψ)} implies that ψ_{L} is similar to a subform of π (by the Cassels–Pfister subform theorem cited above). After that a descent to the ground field is conducted in this or that way.

(B) Use the following corollary of the index reduction formula (Tignole): if C_{0}(ϕ) is a division algebra or has the form D × D, where D is a division algebra, then the

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 11E81, 11E88.

Key words and phrases. Quadratic forms over field, Clifford algebra, semisimple algebra, Brauer group, Pfister form.

date: December 16, 1996.

isotropy of ϕ_{F (ψ)} implies that there is a homomorphism of algebras, C_{0}(ψ) → C_{0}(ϕ) (see [Ti]).

Already in dimension seven there are forms to which neither of these methods can be applied. Indeed, let ϕ = q ⊥ ⟨a⟩, where q is an Albert form. Then there exists no field extension L such that ϕ_{L} is an anisotropic Pfister neighbour. Moreover, C_{0}(ϕ) ≅ M_{2}(C_{0}(q_{0})), where q_{0} is a 5-dimensional subform of q, and hence we cannot use the above-mentioned corollary of the index reduction formula.

It seems a natural idea to try to generalize the second method (B) by eliminating the restriction on the algebra C_{0}(ϕ), namely, to try to prove the following conjecture: if ϕ and ψ are two anisotropic forms such that ϕ_{F (ψ)} is isotropic, then there is a homomorphism of algebras, C_{0}(ψ) → C_{0}(ϕ). This conjecture is true if dim ϕ ⩽ 4. However, it is not true in the full generality. The simplest counterexample can be given by the 3-fold Pfister form ψ = ⟨⟨a, b, c⟩⟩ and 5-dimensional form ϕ = ⟨⟨a, b⟩⟩ ⊥ ⟨−c⟩ (or more generally (n + 5)-dimensional form ψ = ⟨⟨a, b⟩⟩ ⊥ ⟨−c, t_{1}, …, t_{n}⟩ over the field F (t_{1}, …, t_{n})). Note that in this counterexample the quadratic form ϕ contains a 4-dimensional subform with trivial determinant (namely ⟨⟨a, b⟩⟩).

This makes natural to modify our conjecture, by means of introducing certain restrictions on ϕ. Counterexample considered above shows that one of the conditions on ϕ must be following: quadratic form ϕ contains no quadratic forms with trivial determinant. The main result of the present paper (Theorem 6.3) states that if dim(ϕ) ⩽ 8 then this restriction on the form ϕ is sufficient for our conjecture:

Theorem. Let ϕ and ψ be anisotropic quadratic forms over F. Suppose that ϕ contains no 4-dimensional subforms with trivial determinant and dim ϕ ⩽ 8. Then the isotropy of ϕ_{F (ψ)} implies that there is a homomorphism of algebras, C_{0}(ψ) → C_{0}(ϕ).

Among the corollaries of this theorem we point out the following (see Corollary 6.4): Let ϕ be a quadratic form over F with dim ϕ ⩽ 8. Then the following conditions are equivalent:

1) There exists a 3-fold Pfister form ψ such that ϕ_{F (ψ)} is isotropic,

2) ϕ contains a 5-dimensional Pfister neighbour (i.e., dim ϕ > 4 and there is a 4-dimensional subform in ϕ with trivial determinant).

In section §§5–6 we apply our main theorem to study of 7- and 8-dimensional forms ϕ over F ((t)) and F (t) such that ϕ = q ⊥ tμ, where q is an anisotropic Albert form over F and μ is an anisotropic form of dimension one or two (obviuosly that for these forms neither of the methods (A) and (B) can be applied). In particular, we prove the following assertion (Proposition 5.3, Corollary 5.5): Let q be an anisotropic Albert form over F. Consider the quadratic form ϕ = q ⊥ ⟨t⟩ over E = F ((t)). If ψ is a quadratic form over E such that ϕ_{E(ψ)} is isotropic, then dim(ψ) ⩽ 7. If, in addition, dim(ψ) = 7, then ψ is similar to ϕ.

The paper is organized as follows. In §1 the notation is introduced. In §2 we study the problem of existence of a homomorphism from one semisimple algebra to another one. In §3 we introduce a certain class of forms (G-forms) and is §4 prove our main technical result (Theorem 4.5). Applications to 7- and 8-dimensional forms are considered in §§5 and 6, respectively.

Acknowledgements. The work for this paper was done while the author was a visiting researcher at the Max-Planck-Institut für Mathematik, Bonn. I would like to thank this institute for its hospitality.

§1. Some basic facts and notation

1.1. Quadratic forms. We will use the following notation throughout the paper: by ϕ ⊥ ψ, ϕ ≅ ψ, and [ϕ] we denote respectively orthogonal sum of forms, isometry of forms, and the class of ϕ in the Witt ring W (F) of the field F. The maximal ideal of W (F) generated by the classes of even dimensional forms is denoted by I(F). We write ϕ ⊂ ψ if ϕ is isometric to a subform in ψ. We say that ψ contains ϕ if ϕ ⊂ ψ. With det ϕ, det_{±} ϕ ∈ F^{∗}/F^{∗2} we denote the determinant and signed determinant of ϕ respectively. We write ϕ ∼ ψ if ϕ is similar to ψ, i.e., kϕ = ψ for some k ∈ F^{∗}. The anisotropic part of ϕ is denoted by ϕ_{an} and i_{W} (ϕ) denotes the Witt index of ϕ. We denote by ⟨⟨a_{1}, …, a_{n}⟩⟩ the n-fold Pfister form

⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ .

Note that this definition is different from the one given in the Lam’s book [Lam]. The set of all n-fold Pfister forms (corr. all forms similar to n-fold Pfister forms) we denote by P_{n}(F) (corr. GP_{n}(F)). For any field extension L/F we put ϕ_{L} = ϕ ⊗ L, and D_{L}(ϕ) = {x ∈ L^{∗} | ⟨x⟩ ⊂ ϕ_{L}}.

Let ϕ be a quadratic form such that dim ϕ ⩾ 2, ϕ ≅ ℍ̸. The function field F (ϕ) of a form ϕ over F is the function field of the projective variety X_{ϕ} given by equation ϕ = 0. In the case where dim ϕ ⩽ 1 or ϕ ≅ ℍ, we set F (ϕ) =^{def} F.

Let ϕ be a form over F. In [Kn1], Knebusch have defined inductively fields F_{0}, F_{1}, …, and forms ϕ_{i} over F_{i} as follows. Let ϕ_{0} = ϕ_{an} and F_{0} = F. For i ⩾ 1 let F_{i} = F_{i−1}(ϕ_{i−1}) be the function field of F_{i−1} and ϕ_{i} = (ϕ_{F_{i}})_{an}. The smallest h for which dim ϕ_{h} ⩽ 1 is called the height of ϕ. The sequence of the fields F_{0}, F_{1}, …, F_{h} is called a generic splitting tower of ϕ. We need the following property of the field F_{i} (see [Kn1]): Let K/F be a field extension such that dim(ϕ_{K})_{an} ⩽ dim(ϕ_{F_{i}})_{an}. Then the free composite KF_{i} of K and F_{i} over F is purely transcendental over K.

1.2. Algebras. Let F be a field of characteristic different from 2. In our paper an F -algebra will always mean a finite-dimensional F -algebra. Following [Lam] we say that A is a SA (corr. CSA) over F if A is a simple algebra (corr. central simple algebra) over F. If A is an SA over F, then its center E = Z(A) is a subfield of A, and it is clear that A is a CSA over Z(A). The number \sqrt{dim_{Z(A)} A} is always an integer and is denoted by deg(A).

By the Wedderburn Main Theorem, every simple algebra over F has the form M_{m}(D), where D is a division SA. Moreover the algebra A determines integer m and algebra D uniquely (up to isomorphism). The integer m we denote by Coind A. The Schur index of A is ind(A) = deg(D) (or more detail ind_{E} A, where E = Z(A)). Clearly, deg(A) = Coind(A) ind(A).

In this paper we consider mostly the following three types of F -algebras: Type 1 : Central simple algebras of exponent 2 over F, Type 2 : Central simple algebra of exponent 2 over L, where L/F is a quadratic field

extension.

Type 3 : Algebras of the form A×A, where A is a central simple algebra of exponent

2 over F. If A is algebra of one of the above types, we write A ∈ Alg(F). For algebra

of “third type” we set ind(A × A) =^{def} ind A, Coind(A × A) =^{def} Coind A. If A belongs to Alg(F) and E/F is a field extension, then A_{E} = A ⊗_{F} E ∈ Alg(E) and ind A_{E} ⩽ ind A, Coind A_{E} ⩾ Coind A. Moreover, if E/F either unirational, or has the form E = F (ϕ), where ϕ is a quadratic form with dim ϕ ⩾ 3, then the “type” of A_{E} is the same as that of A. If E/F is an unirational field extension, then ind A_{E} = ind A and Coind A_{E} = Coind A.

If A is one of the three types, and B is a CSA over F, then the tensor product A ⊗_{F} B is an algebra of the same type as that of A.

We shall say that A is similar to A′, if there is n, n′ such that M_{n}(A) ≅ M_{n′}(A′). The similarity is an equivalence relation, and the equivalence class of A will be denoted by [A]. Obviously that the condition [A] = [A′] implies that Algebra A′ has the same type as that of A, and ind A = ind A′. If [A′] = [A] and k = deg A′/deg A is an integer, then A′ ≅ M_{k}(A).

If A ∈ Alg(F) and B is a CSA, then we denote by [A] + [B] the class of tensor product [A⊗_{F} B]. Obviously this operation is associative. The classes of equivalence of CSAs over F and the binary operation [A] + [B] constitute the Brauer group Br(F) of F.

1.3. Clifford algebras. Let ϕ be a quadratic form. We denote by C(ϕ) (corr. C_{0}(ϕ)) the Clifford algebra (corr. the even part of the Clifford algebra) of the quadratic form ϕ. We need the following properties of Clifford algebras (see [Lam, Ch. 5,Th. 2.5]).

1) If dim ϕ is odd, then C_{0}(ϕ) is a CSA over F.

2) If dim ϕ is even and det_{±} ϕ ≠ 1, then C_{0}(ϕ) is a CSA over the field L = F (\sqrt{det_{±} ϕ}).

3) If dim ϕ is even and det_{±} ϕ = 1, then C_{0}(ϕ) has the form A × A, where A is a CSA over F. Moreover, in this case we have C(ϕ) ≅ M_{2}(A).

It follows from these properties that C_{0}(ϕ) ∈ Alg(F). For any field extension E/F, we have C_{0}(ϕ_{E}) = C_{0}(ϕ)_{E}. If ϕ and ψ are quadratic forms over F such that ϕ ≡ ψ (mod I^{3}(F)), then [C_{0}(ϕ)] = [C_{0}(ψ)]. If γ = ϕ ⊥ k⟨⟨a, b⟩⟩, then [C_{0}(γ)] = [C_{0}(ϕ)] + [(a, b)], where (a, b) denotes the biquaternion algebra \binom{a,b}{F} (see [Lam, Ch.5, Corollary 2.9 and Corollary 3.3]).

1.4. Index reduction formula. Let A be a CSA over F and let E/F be a field extension. The quotient \frac{ind(A)}{ind(A_{E})} is called the index reduction factor of A relative to E/F (see [Dr]). The index reduction factor is always an integer. If the field extension E/F is unirational, then the index reduction factor is equal to one. If E/F is a finite extension, then the index reduction factor divides [E : F].

Let ϕ be a quadratic form over F and A be a CSA over F. By definition of F (ϕ), there is subfield E ⊂ F (ϕ) such that E/F is purely transcendental and [F (ϕ) : E] = 2. Hence the index reduction factor of A relative to F (ϕ)/F divides 2. Therefore either ind(A_{F (ϕ)}) = ind(A) or ind(A_{F (ϕ)}) = \frac{1}{2} ind(A).

Theorem 1.5 [Me]. Let A be a CSA over F, and ϕ a quadratic form over F of dimension d. Then ind(A_{F (ϕ)}) = \frac{1}{2} ind(A) if and only if one of the following conditions is fulfilled:

1) dim ϕ is odd and ind(A ⊗_{F} C_{0}(ϕ)) = \frac{ind A}{2^{(d−1)/2}} ;

2) dim ϕ is even, ϕ ∈ I^{2}(F), and ind(A ⊗_{F} C(ϕ)) = \frac{ind A}{2^{(d−2)/2}} ;

3) dim ϕ is even, ϕ ∉ I^{2}(F), and ind(A ⊗_{F} C_{0}(ϕ)) = \frac{ind A}{2^{d/2}}. □

In the case where the central simple algebra is a division algebra, we have the following

Theorem 1.6(see [Ti]). Let D be a division CSA over F, and ϕ a quadratic form over F. Then the following conditions are equivalent:

1) D_{F (ϕ)} is not a division algebra.

2) ind D_{F (ϕ)} = \frac{1}{2} ind D.

3) There is a homomorphism of F -algebras, C_{0}(ϕ) → D.

Corollary 1.7. Let ϕ be an anisotropic quadratic form over F. Suppose that C_{0}(ϕ) is either a division algebra or has the form D×D where D is a division algebra. Let ψ be a quadratic form such that ϕ_{F (ψ)} is isotropic. Then there is a homomorphism of F -algebras, C_{0}(ψ) → C_{0}(ϕ). □

If ϕ is a quadratic form of dimension less or equal to 6, then the condition that C_{0}(ϕ) is either a division algebra or has the form D × D, where D is a division algebra can be rewriting as follows: ϕ contains no 5-dimensional Pfister neighbours.

Thus we have the following

Corollary 1.8. Let ϕ and ψ be anisotropic quadratic forms over F. Suppose that ϕ contains no 5-dimensional Pfister neighbours and dim ϕ ⩽ 6. Then the isotropy of ϕ_{F (ψ)} implies that there is a homomorphism of F -algebras, C_{0}(ψ) → C_{0}(ϕ) □

§2. Homomorphisms of algebras

Let A and B be two semisimple algebras over F. Here we consider the question of whether there is a homomorphism of algebras, B → A.

Theorem 2.1. Let L/F be a field extension, A be a CSA over F, and B be a CSA over L. Then the following conditions are equivalent:

1) There is a homomorphism of F -algebras, B → A.

2) The number

k(A, B) =^{def} \frac{deg(A)}{deg(B) ind(B^{op} ⊗_{F} A) · [L : F]}

is an integer.

3) There is a homomorphism of L-algebras M_{n}(B) → A_{L}, where n = [L : F].

Proof. 1) ⇒ 2). Since B is a simple algebra, the homomorphism B → A is injective. So we can suppose that B ⊂ A. By the “Double Centralizer theorem”, we have dim_{F} C_{A}(B) · dim_{F} B = dim_{F} A = (deg A)^{2} (see [Pi, 12.7]). Since B and C_{A}(B) both are CSAs over L, we have dim_{F} (C_{A}(B)) = [L : F](deg C_{A}(B))^{2}, and dim_{F} (B) = [L : F](deg B)^{2}. Hence [L : F](deg C_{A}(B))^{2}[L : F](deg B)^{2} = (deg A)^{2}, i.e.,

deg C_{A}(B) deg B · [L : F] = deg A.

Using again the Centralizer theorem, we get [C_{A}(B)] = [B^{op} ⊗_{F} A] ∈ Br(L) (see [Dr, p. 42]). Hence

k(A, B) =\frac{deg(A)}{deg(B) ind(C_{A}(B)) · [L : F]} =
= Coind(C_{A}(B)) · \frac{deg(A)}{deg(B) deg(C_{A}(B)) · [L : F]} = Coind(C_{A}(B)).

Hence k(A, B) is an integer.

2) ⇒ 3). Let D be a division algebra over L such that [D] = [B^{op} ⊗_{F} A] ∈ Br(L). Let k = k(A, B). We have

deg(A_{L}) = deg(A) = k deg(B) ind(B^{op} ⊗_{F} A) · [L : F]
= kn deg(B) deg(D) = deg(M_{n}(B) ⊗_{L} M_{k}(D)).

Let A′ = M_{n}(B) ⊗_{L} M_{k}(D). We have

[A′] = [B] + [D] = [B] + [B^{op} ⊗_{F} A] = [B] + [B^{op} ⊗_{L} A_{L}] = [A_{L}] ∈ Br(L).

Since deg(A_{L}) = deg(A′), and [A_{L}] = [A′], we conclude that there is an isomorphism A′ ⟶^{∼} A_{L}. Hence one get a homomorphism M_{n}(B) → A_{L} as the composition

M_{n}(B) ↪ M_{n}(B) ⊗_{L} M_{k}(D) = A′ ⟶^{∼} A_{L}.

3) ⇒ 1). Since M_{n}(B) is a simple algebra, we can suppose that M_{n}(B) ⊂ A_{L}. Let D = C_{A_{L}}(M_{n}(B)) ⊗_{L} B. By the Centralizer theorem, we have M_{n}(D) ≅ A_{L}. Let E be an arbitrary subfield of D such that [D_{E}] = 0 (one can take E as a maximal subfield of D). Then [E : L] divides deg(D). We have [A_{E}] = [M_{n}(D_{E})] = 0 ∈ Br(E). Hence there exists a central simple algebra A′ over F such that [A] = [A′] and E is a strictly maximal subfield of A′ (see [Pi, 13.3]). Since

s = \frac{deg(A)}{deg(A′)} = \frac{deg(A_{L})}{[E : F]} = \frac{deg(M_{n}(D))}{[E : L] · [L : F]} = \frac{n deg(D)}{n[E : L]} = \frac{deg(D)}{[E : L]}

is an integer, we have A ≅ M_{s}(A′). Hence there is a homomorphism L → A, defined as the composition L ↪ E ↪ A′ ↪ M_{s}(A′) ⟶^{∼} A.

Thus we can suppose that L is a subfield of A. We have M_{n}(C_{A}(L)) ≅ A_{L} (see [Dr, p. 42]). Since M_{n}(D) ≅ A_{L} ≅ M_{n}(C_{A}(L)), we have D ≅ C_{A}(L). Therefore we get a homomorphism B → A as the composition B ↪ D ⟶^{∼} C_{A}(L) ↪ A. □

Lemma 2.2. Let A and B be semisimple algebras over F:

A = A_{1} × ⋯ × A_{s}, B = B_{1} × ⋯ × B_{r},

where A_{1}, …, A_{s}, B_{1}, …, B_{r} are simple algebras over F. Then the following conditions are equivalent:

1) There is a homomorphism B → A of F -algebras.

2) For any j (1 ⩽ j ⩽ s) there is a homomorphism B → A_{j} of F -algebras.

3) For any j (1 ⩽ j ⩽ s) there is i (1 ⩽ i ⩽ r) and a homomorphism B_{i} → A_{j} of F -algebras.

Proof. Obvious. □

Corollary 2.3. Let A be a CSA over F, and B = B_{1} × ⋯ × B_{r} be a semisimple algebra over F, where B_{1}, …, B_{r} are simple algebras over F. Then the following conditions are equivalent:

1) There is a homomorphism of F -algebras, B → A.

2) There is i (1 ⩽ i ⩽ r) such that k(A, B_{i}) is an integer.

Proof. Obvious. □

Lemma 2.4. Let A be a simple algebra and let L = Z(A). Let B be a semisimple algebra over F. Let B_{L} = B ⊗_{F} L. Then the following conditions are equivalent:

1) There is a homomorphism B → A of F -algebras,

2) There is a homomorphism B_{L} → A of L-algebras.

3) There is a homomorphism B_{L}/ Rad(B_{L}) → A of semisimple L-algebras.

Proof. 1) ⇒ 2). Let f: B → A be a homomorphism of F -algebras. We define f̃: B_{L} = B ⊗_{F} L → A by the rule f̃(b ⊗ l) = f(b)l.

2) ⇒ 3). Since A is a simple algebra, the homomorphism B_{L} → A factors through the algebra B_{L}/ Rad(B_{L}).

3) ⇒ 1). If B_{L}/ Rad(B_{L}) → A is a homomorphism of L-algebras, one can define B → A as the composition B → B_{L}/ Rad(B_{L}) → A. □

Lemma 2.5. Let A a central simple algebra and B be a simple algebra over F. Then:

1) For any positive integer n we have k(M_{n}(A), M_{n}(B)) = k(A, B);
2) If Q is a CSA over F, then k(A ⊗_{F} Q, B ⊗_{F} Q) = k(A, B);

3) If E/F is an unirational field extension, then k(A⊗_{F} E, B⊗_{F} E) = k(A, B);

Proof. Obvious. □

Proposition 2.6. Let A and B be semisimple algebras over F, and Q be a CSA over F.

Then the following conditions are equivalent:

1) There is a homomorphism of F -algebras B ⊗_{F} Q → A ⊗_{F} Q,

2) There is a homomorphism of F -algebras B → A.

Proof. 1) ⇒ 2). Let f: B ⊗_{F} Q → A ⊗_{F} Q be a homomorphism of F -algebras. Then there is a homomorphism

B ⊗_{F} Q ⊗_{F} Q^{op} → A ⊗_{F} Q ⊗_{F} Q^{op}.

Since Q is a CSA over F, we conclude that Q ⊗_{F} Q^{op} ≅ M_{n}(F), with n = dim_{F} Q. Hence we get a homomorphism M_{n}(B) → M_{n}(A). Using Lemma 2.2 one can reduce our problem to the case where A is a simple algebra. After this using Lemma 2.4 we reduce the problem to the case where A is a CSA. Now the required result follows immediately from Lemma 2.5 and Corollary 2.3.

2) ⇒ 1). Obvious. □

Proposition 2.7. Let A and B be semisimple algebras over F. Let E/F be a unirational field extension. Then the following conditions are equivalent:

1) There is a homomorphism of E-algebras, B ⊗_{F} E → A ⊗_{F} E;

2) There is a homomorphism of F -algebras, B → A.

Proof. 1) ⇒ 2). First of all, it is sufficient to consider only the case where E = F (t). As in a Proposition 2.6, we can reduce the proof to the case where A is a CSA over F, and the required result follows from Lemma 2.5 and Corollary 2.3.

2) ⇒ 1) Obvious. □

§3. G-forms

I this section we introduce a certain classes of qudararic form (see Definitions 3.1 and 3.3 bellow) and study properties of these classes.

Definition 3.1. Let F be a field, and n be a positive integer. Let Γ_{n}(F) be the set of all (2n + 1)-dimensional forms ϕ satisfying the following condition:

There are a, b ∈ F^{∗} and an n-dimensional quadratic form γ such that ⟨⟨a, b⟩⟩λ is anisotropic and ϕ ⊂ ⟨⟨a, b⟩⟩λ.

Example 3.2. Let F be a field and ϕ be an anisotropic quadratic form over F. Then:

1) ϕ ∈ Γ_{1}(F) if and only if dim ϕ = 3;

2) ϕ ∈ Γ_{2}(F) if and only if ϕ is a 5-dimensional Pfister neighbour.

Definition 3.3. Let ϕ be an anisotropic quadratic form over F. We say that ϕ is a G-form if it contains a subform belonging to one of the sets Γ_{2}(F), Γ_{4}(F), …, Γ_{2i}(F), . . ..

We denote by G(F) the set of all G-forms over F:

G(F) = {ϕ is anisotropic over F | ∃ψ ⊂ ϕ, ψ ∈ ∪_{i=1}^{∞}Γ_{2i}(F)}.

Example 3.4. Let F be a field and ϕ be an anisotropic quadratic form over F. Then:

1) If dim ϕ ⩽ 4, then ϕ ∉ G(F).

2) Let 5 ⩽ dim ϕ ⩽ 8. Then ϕ ∈ G(F) if and only if there is 4-dimensional

subform τ ⊂ ϕ such that τ ∈ GP_{2}(F). In other words, for any form ϕ with dim ϕ ⩽ 8 the condition ϕ ∈ G(F) holds if and only if ϕ contains a 5-dimensional Pfister neighbour.

Example 3.5. Let q be an anisotropic Albert form over F. Then:

1) q is not a G-form,

2) The quadratic form q ⊥ ⟨t⟩ over F ((t)) is not a G-form,

3) If a ∈ F^{∗} is such that q_{F (\sqrt{a})} is anisotropic, then the quadratic form q ⊥ t⟨⟨a⟩⟩ over F ((t)) is not a G-form.

Proof. Let μ = 0 in the first case, μ = ⟨1⟩ in the second case, and μ = ⟨⟨a⟩⟩ in the third case. Suppose that ϕ = q ⊥ tμ is a G-form. Then there is τ ⊂ ϕ such that τ ∈ GP_{2}(Ê) (see Example 3.4). Since Ê = F ((t)), we can write τ in the form τ = τ_{0} ⊥ tτ_{1}, where the quadratic forms τ_{0}, τ_{1} are defined over F. Since max(dim τ_{0}, dim τ_{1}) ⩾ 2, there is b ∈ F^{∗} such that τ_{Ê(\sqrt{b})} is isotropic. Since

τ ∈ GP_{2}(Ê), we see that τ_{Ê(\sqrt{b})} is hyperbolic. Hence i_{W} (ϕ_{Ê(\sqrt{b})}) ⩾ i_{W} (τ_{Ê(\sqrt{b})}) = 2. Obviously, i_{W} (ϕ_{Ê(\sqrt{b})}) = i_{W} (q_{F (\sqrt{b})})+i_{W} (μ_{F (\sqrt{b})}). Hence i_{W} (q_{F (\sqrt{b})})+i_{W} (μ_{F (\sqrt{b})}) ⩾ 2. Since q is an anisotropic Albert form, we have i_{W} (q_{F (\sqrt{b})}) ⩽ 1. Since dim μ ⩽ 2, we have i_{W} (μ_{F (\sqrt{b})}) ⩽ 1. Therefore i_{W} (q_{F (\sqrt{b})}) = i_{W} (μ_{F (\sqrt{b})}) = 1 and moreover dim μ = 2. Hence μ = ⟨⟨a⟩⟩ and a ≡ b ∈ F^{∗}/F^{∗2}. Therefore i_{W} (q_{F (\sqrt{a})}) = i_{W} (q_{F (\sqrt{b})}) = 1, i.e., the quadratic form q_{F (\sqrt{a})} is isotropic, a contradiction. □

Lemma 3.6. Let ϕ be an anisotropic quadratic form over field F. The following conditions are equivalent:

1) ϕ is a G-form.

2) There exist a quadratic form τ and a, b ∈ F^{∗} such that (a) dim τ < dim ϕ,

(b) ϕ ≡ τ (mod ⟨⟨a, b⟩⟩I(F)).

Proof. 1) ⇒ 2). Let ϕ ∈ G(F). By the definition of G(F), we can write ϕ in the form ϕ = ψ ⊥ ξ where ψ ∈ Γ_{2k}(F).

It follows from ψ ∈ Γ_{2k}(F) that dim ψ = 4k + 1 and there are a, b ∈ F^{∗} and a 2k-dimensional quadratic form λ over F such that ψ ⊂ ⟨⟨a, b⟩⟩λ. Let ψ′ be such that ψ ⊥ −ψ′ = ⟨⟨a, b⟩⟩λ. Obviously dim ψ′ = 4k−1 and ψ′ ≡ ψ (mod ⟨⟨a, b⟩⟩I(F)). Now it is sufficient to set τ = ψ′ ⊥ ξ.

2) ⇒ 1). We claim that [⟨⟨a, b⟩⟩] ≠ 0. Indeed, otherwise [ϕ] = [τ], and hence dim ϕ = dim ϕ_{an} = dim τ_{an} ⩽ dim τ, a contradiction.

Since ϕ ⊥ −τ ∈ ⟨⟨a, b⟩⟩I(F) it follows that (ϕ ⊥ −τ)_{F (⟨⟨a,b⟩⟩)} is hyperbolic. Hence there is a quadratic form λ such that (ϕ ⊥ −τ)_{an} = ⟨⟨a, b⟩⟩λ (see [Sch, Ch.4, Theorem 5.4(iv)]).

Suppose that λ ∉ I(F). Then [ϕ] − [τ] = [⟨⟨a, b⟩⟩λ] ≡ [⟨⟨a, b⟩⟩] ≢ 0 (mod I^{3}(F)). On the other hand, ϕ ≡ τ (mod ⟨⟨a, b⟩⟩I(F)) implies that [ϕ] − [τ] ∈ ⟨⟨a, b⟩⟩I(F) ⊂ I^{3}(F), a contradiction.

Therefore, λ ∈ I(F), i.e., dim λ is even. Let dim λ = 2k. It is clear, that there are quadratic forms ϕ_{0}, ρ_{0}, and ξ such that

ϕ = ϕ_{0} ⊥ ξ, τ = τ_{0} ⊥ ξ, (ϕ ⊥ −τ)_{an} = ϕ_{0} ⊥ −τ_{0}.

Then ϕ_{0} ⊥ −τ_{0} = ⟨⟨a, b⟩⟩λ. Hence dim ϕ_{0} + dim τ_{0} = 8k. Since dim ϕ > dim τ, we conclude that dim ϕ_{0} > dim τ_{0}. Hence dim ϕ_{0} ⩾ 4k + 1. Let ψ be any (4k + 1)- dimensional subform of ϕ_{0}. Since ψ ⊂ ϕ_{0} ⊂ ⟨⟨a, b⟩⟩λ and dim ψ = 4k + 1, we see that ψ ∈ Γ_{2k}. Finally since ψ ⊂ ϕ_{0} ⊂ ϕ, it follows that ϕ ∈ G(F). □

Lemma 3.7. Let ϕ be an anisotropic form over F. Then the following conditions are equivalent:

1) ϕ ∈ G(F),

2) ϕ_{F (t)} ∈ G(F (t)),
3) ϕ_{F ((t))} ∈ G(F ((t))).

Proof. 1)⇒2)⇒3). Obvious.

3)⇒ 1). It follows from Lemma 3.6, that there are quadratic forms τ̃ and λ̃ over F ((t)) and ã, b̃ ∈ F ((t))^{∗} such that dim τ̃ < dim ϕ, [ϕ̃ ⊥ −τ̃] = [⟨⟨ã, b̃⟩⟩λ̃], and λ̃ ∈ I(F ((t))). Since F ((t))^{∗}/F ((t))^{∗2} = F^{∗}/F^{∗2} × {1, t}, we can assume that

ã = at^{p}, b̃ = bt^{q}, τ̃ = ⟨k_{1}t^{i_{1}}, k_{2}t^{i_{2}}, …⟩ , λ̃ = ⟨l_{1}t^{j_{1}}, l_{2}t^{j_{2}}, …⟩ ,

where k_{1}, k_{2}, …, l_{1}, l_{2}, …, a, b ∈ F^{∗}, i_{1}, i_{2}, …, j_{1}, j_{2}, …, p, q ∈ {0, 1}.

Setting τ = ⟨k_{1}, k_{2}, …⟩, λ = ⟨l_{1}, l_{2}, …⟩, one can see that dim τ < dim ϕ, [ϕ ⊥ −τ] = [⟨⟨a, b⟩⟩λ], and λ ∈ I(F). By Lemma 3.6, we have ϕ ∈ G(F). □

Corollary 3.8. Let ϕ be an anisotropic form over F and E/F be an unirational field extension. Suppose that ϕ_{E} ∈ G(E). Then ϕ ∈ G(F). □

§4. Generalization of Tignol’s theorem

The purpose of this section is to generalizate a Crollory 1.7.

Proposition 4.1. Let ϕ be an anisotropic quadratic form over F and π = ⟨⟨a, b⟩⟩. Suppose that ϕ_{F (π)} is isotropic. Then at least one of the following conditions holds:

1) ϕ is a G-form.

2) Coind(C_{0}(ϕ) ⊗_{F} (a, b)) ⩾ 4.

Proof. Since the extension F (π)/F is excellent (see [ELW, Appendix II], or [Ro], [LWG]) there is a quadratic form τ over F such that (ϕ_{F (π)})_{an} = τ_{F (π)}. Hence (ϕ ⊥ −τ)_{an} = ⟨⟨a, b⟩⟩λ.

If λ ∈ I(F), then condition 2) in Lemma 3.6 holds. Then ϕ ∈ G(F).

Now let λ ∉ I(F). Then

[ϕ] − [τ] = [⟨⟨a, b⟩⟩] · [λ] ≡ [⟨⟨a, b⟩⟩] (mod I^{3}(F)).

Therefore, [C_{0}(ϕ) ⊗_{F} (a, b)] = [C_{0}(τ)].

Let dim ϕ = n. Then dim C_{0}(ϕ) = 2^{n−1}, and hence dim(C_{0}(ϕ) ⊗ (a, b)) = 2^{n+1}. Since dim τ ⩽ dim ϕ − 2 = n − 2, we have dim(C_{0}(τ)) ⩽ 2^{n−3}. Hence dim(C_{0}(ϕ) ⊗_{F} (a, b))/ dim C_{0}(τ) ⩾ 16. Since [C_{0}(ϕ) ⊗_{F} (a, b)] = [C_{0}(τ)], we have C_{0}(ϕ) ⊗ (a, b) = M_{k}(C_{0}(τ)), where k ⩾ \sqrt{16} = 4. Hence Coind(C_{0}(ϕ) ⊗_{F} (a, b)) ⩾ 4. □

Corollary 4.2. Let ϕ be an anisotropic quadratic form such that either C_{0}(ϕ) = M_{2}(A), or C_{0}(ϕ) = M_{2}(A) × M_{2}(A) where A is a simple algebra. Let π = ⟨⟨a, b⟩⟩. Suppose that ϕ_{F (π)} is isotropic. Then at least one of the following conditions holds:

1) ϕ is a G-form.

2) A ⊗_{F} (a, b) is not a division algebra.

Proof. We can suppose that ϕ ∉ G(F). By Proposition 4.1, we have Coind(C_{0}(ϕ)⊗_{F} (a, b)) ⩾ 4. Since Coind(C_{0}(ϕ)⊗_{F} (a, b)) = 2 Coind(A⊗(a, b)), we have Coind(A⊗_{F} (a, b)) ⩾ 2, i.e., A ⊗_{F} (a, b) is not a division algebra. □

Proposition 4.3. Let ϕ be an anisotropic quadratic form such that:

1) ϕ ∉ G(F),

2) Coind(C_{0}(ϕ)) = 2, i.e., either C_{0}(ϕ) = M_{2}(A) or C_{0}(ϕ) = M_{2}(A)×M_{2}(A),

where A is a division simple algebra. Let ψ be a quadratic form such that ϕ_{F (ψ)} is isotropic and dim ψ ⩾ 3.

Then there exist an extension E of the field F, a quadratic form γ over E and a quaternion algebra Q over E such that:

1) E(γ)/F is an unirational field extension,

2) A ⊗_{F} Q is a division E-algebra,

3) (A ⊗_{F} Q)_{E(γ)} is not a division E(γ)-algebra,

4) dim γ ⩾ dim ψ, γ ≡ ψ_{E} (mod I^{2}(E)), and [C_{0}(γ)] = [C_{0}(ψ) ⊗_{F} Q].

Proof. Let E_{0} = F (X, Y, Z) be the field of rational function in three variables, and γ_{0} = ψ ⊥ −Z⟨⟨X, Y ⟩⟩ be a quadratic form over E_{0}. Let E_{0}, E_{1}, …, E_{h} be the generic splitting tower of γ_{0} and let γ_{i} = ((γ_{0})_{L_{i}})_{an}.

Let n = dim ψ. Since dim γ_{0} = n + 4 > n − 2 ⩾ 1 ⩾ dim γ_{h}, there exists s (0 < s ⩽ r) such that dim γ_{s} ⩽ n − 2 < dim γ_{s−1}.

Lemma 4.4.

1) dim γ_{s} ⩽ n − 2 and dim γ_{0}, …, dim γ_{s−1} ⩾ n.

2) For any field extension K/E_{0} such that dim((γ_{0})_{K})_{an} ⩽ n − 2, the field extension (KE_{s})/K is purely transcendental.

3) The field extension E_{s}/F is unirational.

4) The quadratic form ψ_{E_{s}(⟨⟨X,Y ⟩⟩)} is isotropic.

Proof. 1) By the definition of γ_{0}, …, γ_{s}, we see that dim γ_{0} > ⋯ > dim γ_{s−1} > n−2 ⩾ dim γ_{s}. Since dim γ_{s−1} ≡ dim γ_{0} = n+4 ≡ n (mod 2) and dim γ_{s−1} > n−2, we conclude that dim γ_{s−1} ⩾ n.

2) Since dim γ_{s−1} ⩾ n ⩾ 3, the field extension E_{s} = E_{s−1}(γ_{s−1}) contains no nontrivial algebraic extensions of the field F. Therefore there exists a free composite KE_{s} of the fields K and E_{s} over F. Now it follows from properties of generic splitting tower that (KF_{s})/K is purely transcendental.

3) Since dim ψ ⩾ 3, we can write ψ in the form ψ = ⟨a_{1}, a_{2}, a_{3}⟩ ⊥ ρ. Let K = F (X, Y, Z)(\sqrt{\frac{Z}{a_{1}}} , \sqrt{\frac{−ZX}{a_{2}}} , \sqrt{\frac{−ZY}{a_{3}}} ). It is clear that K/F is purely transcendental. We have ⟨a_{1}, a_{2}, a_{3}⟩_{K} = ⟨Z, −ZX, −ZY ⟩_{K}. Hence

[γ_{K}] = [⟨a_{1}, a_{2}, a_{3}⟩_{K}] + [ρ_{K}] − [Z⟨⟨X, Y ⟩⟩_{K}] = [(ρ ⊥ ⟨−ZXY ⟩)_{K}].

Therefore dim(γ_{K})_{an} ⩽ dim(ρ ⊥ ⟨−ZXY ⟩) = n − 3 + 1 = n − 2. It follows from assertion 2), that (KE_{s})/K is purely transcendental. Since K/F is purely transcendental, we conclude that (KE_{s})/F is purely transcendental. Hence E_{s}/F is unirational.

4) Since [γ_{0}] = [ψ_{E_{0}}] − [Z⟨⟨X, Y ⟩⟩], we have [γ_{s}] = [ψ_{E_{s}}] − [Z⟨⟨X, Y ⟩⟩_{E_{s}}]. Hence [(γ_{s})_{E_{s}(⟨⟨X,Y ⟩⟩)}] = [ψ_{E_{s}(⟨⟨X,Y ⟩⟩)}]. Therefore

dim(ψ_{E_{s}(⟨⟨X,Y ⟩⟩)})_{an} = dim(γ_{E_{s}(⟨⟨X,Y ⟩⟩)})_{an} ⩽ dim(γ_{s})_{an} ⩽ n − 2 < dim ψ.

Hence ψ_{E_{s}(⟨⟨X,Y ⟩⟩)} is isotropic. The lemma is proved. □

Now we proceed to the proof of Proposition 4.3. Since ϕ_{F (ψ)} and ψ_{E_{s}(⟨⟨X,Y ⟩⟩)} are isotropic, ϕ_{E_{s}(⟨⟨X,Y ⟩⟩)} is isotropic too. Since E_{s}/F is unirational, we conclude that ϕ_{E_{s}} is anisotropic, and A_{E_{s}} is a division algebra. Thus anisotropic quadratic form ϕ_{E_{s}} becomes isotropic over the function field of quadratic form ⟨⟨X, Y ⟩⟩_{E_{s}}. Therefore all the conditions in Corollary 4.2 hold for the field E_{s}, quadratic forms ϕ_{E_{s}} and π = ⟨⟨X, Y ⟩⟩_{E_{s}}, and algebra A_{E_{s}}. Therefore at least one of the following conditions holds:

1) ϕ_{E_{s}} ∈ G(E_{s}),

2) A ⊗_{F} Q_{s} is not a division E_{s}-algebra, where Q_{s} denotes the quaternion algebra (X, Y) over the field E_{s}.

Consider first the case ϕ_{E_{s}} ∈ G(E_{s}). Since E_{s}/F is unirational, it follows from Corollary 3.8 that ϕ ∈ G(F), i.e., in this case Proposition 4.3 is proved.

Suppose now that ϕ_{E_{s}} ∉ G(E_{s}). Then A ⊗_{F} Q_{s} is not a division algebra. Since A ⊗_{F} Q_{0} is a division algebra over E_{0} = F (X, Y, Z), we conclude that there is k (0 ⩽ k ⩽ s − 1) such that A ⊗_{F} Q_{k} is a division algebra, and A ⊗_{F} Q_{k+1} is not a division algebra.

Let E = E_{k}, γ = γ_{k}, and Q = Q_{k}. We claim that the condition 1)–4) in Proposition 4.3 hold for E, γ and Q. Indeed:

1) E(γ)/F is unirational because E(γ) = E_{k+1} ⊂ E_{s}, and E_{s}/F is unirational. 2),3) A ⊗_{F} Q is a division algebra and (A ⊗_{F} Q)_{E(γ)} is not a division algebra by

definition of E = E_{k}.

4) dim γ = dim γ_{k} ⩾ dim γ_{s−1} ⩾ n = dim ψ. Since [γ] = [ψ_{E}] − [Z⟨⟨X, Y ⟩⟩_{E}], we conclude that γ ≡ ψ_{E} (mod I^{2}(E)) and [C_{0}(γ)] = [C_{0}(ψ) ⊗_{F} Q].

Thus Proposition 4.3 is proved. □

Theorem 4.5. Let ϕ and ψ be quadratic forms over F. Suppose that ϕ ∉ G(F) and Coind(C_{0}(ϕ)) ⩽ 2. Then the isotropy of ϕ_{F (ψ)} implies that there is a homomorphism of algebras, C_{0}(ψ) → C_{0}(ϕ).

Proof. If Coind(C_{0}(ϕ)) = 1, then C_{0}(ϕ) is either a division algebra or has the form A × A, where A a division algebra. In this case theorem is always proved (see Corollary 1.7).

Thus we can suppose that Coind(C_{0}(ϕ)) = 2.

Consider the case dim ψ = 2. Since ϕ_{F (ψ)} is isotropic, we conclude that there is k ∈ F^{∗} such that kψ ⊂ ϕ. Hence C_{0}(ψ) ≅ C_{0}(kψ) ⊂ C_{0}(ψ).

Thus we can suppose that dim ψ ⩾ 3. Then the conditions of Proposition 4.3 hold. Let A, E, γ, and Q be such as in Proposition 4.3. Let D = A ⊗_{F} Q. It follows from Proposition 4.3 that D is a division CSA over E. However D_{E(γ)} is not a division algebra.

By Theorem 1.6, there is a homomorphism of E-algebras, C_{0}(γ) → D. Since

[C_{0}(ψ_{E}) ⊗_{E} Q] = [C_{0}(γ)] = [C_{0}(γ) ⊗_{E} M_{2}(E)],

and

dim(C_{0}(ψ_{E}) ⊗_{E} Q) = 2^{dim ψ−1}4 ⩽ 2^{dim γ−1}4 = dim(C_{0}(γ) ⊗_{E} M_{2}(E)),

we conclude that there is a homomorphism of E algebras:

C_{0}(ψ_{E}) ⊗_{E} Q → C_{0}(γ) ⊗_{E} M_{2}(E)

Hence there is a homomorphism C_{0}(ψ_{E}) ⊗_{E} Q → C_{0}(ϕ_{E}) ⊗_{E} Q as the composition

C_{0}(ψ_{E}) ⊗_{E} Q → C_{0}(γ) ⊗_{E} M_{2}(E) → → D ⊗_{E} M_{2}(E) ≅ M_{2}(A_{E}) ⊗_{E} Q → C_{0}(ϕ_{E}) ⊗_{E} Q.

It follows from Corollary 2.6 that there is a homomorphism C_{0}(ψ_{E}) → C_{0}(ϕ_{E}) of E-algebras.

Since E/F is unirational, it follows from Corollary 2.7, that there is a homomorphism C_{0}(ψ) → C_{0}(ϕ) of F -algebras. □

§5. 7-dimensional forms

Theorem 5.1. Let ϕ and ψ be quadratic forms over F, and dim ϕ = 7. Suppose that the quadratic form ϕ_{F (ψ)} is isotropic. Then at least one of the following conditions holds.

1) There is a 4-dimensional subform in ϕ with trivial determinant.

2) There is a homomorphism of F -algebras, C_{0}(ψ) → C_{0}(ϕ).

Proof. Let ϕ̃ = ϕ ⊥ ⟨det ϕ⟩. If ind(C_{0}(ϕ)) = ind(C(ϕ̃)) ⩽ 2, then there is a ∈ F^{∗} such that ϕ̃_{F (\sqrt{a})} is hyperbolic (see [Kn2, Example 9.12]). Hence [ϕ_{F (\sqrt{a})}] = [⟨− det ϕ⟩]. Therefore i_{W} (ϕ_{F (\sqrt{a})}) = 3 > 2. Hence there is a 2-dimensional quadratic form μ such that ⟨⟨a⟩⟩μ ⊂ ϕ. Thus condition 1) holds.

Let now ind(C_{0}(ϕ)) > 2. Then Coind(C_{0}(ϕ)) ⩽ 8/4 = 2. If ϕ ∈ G(F), then condition 1) holds (see Example 3.4). If ϕ ∉ G(F), then the conditions of Theorem 4.5 hold. Hence there is a homomorphism, C_{0}(ψ) → C_{0}(ϕ). □

Remark. In the case where ϕ̃ = ϕ ⊥ ⟨det ϕ⟩ is anisotropic, dim ψ ⩾ 5 and ψ is not a Pfister neighbour of a 3-fold Pfister form, Theorem 5.1 can be easy proved by using results of Laghribi [La2,La3].

Corollary 5.2. Let ϕ be a quadratic form over F with dim ϕ ⩽ 7. Then the following conditions are equivalent:

1) There is a 3-fold Pfister form ψ such that ϕ_{F (ψ)} is isotropic.

2) ϕ contains a 5-dimensional Pfister neighbour (i.e., dim ϕ > 4 and there is a 4-dimensional subform in ϕ with trivial determinant).

Proof. In the cases where dim ϕ ⩽ 6 the required result follows immediately from results of Hoffmann [H1, H2]. Suppose now that dim ϕ = 7.

(1) ⇒ (2). Let ψ be a 3-fold Pfister form such that ϕ_{F (ψ)} is isotropic.

Suppose first that there is a homomorphism, C_{0}(ψ) → C_{0}(ϕ). Since C_{0}(ψ) = M_{8}(F) × M_{8}(F) and deg C_{0}(ϕ) = 8, we conclude that M_{8}(F) ≅ C_{0}(ϕ). Hence ϕ is a Pfister neighbor. Then ϕ contains a 5-dimensional Pfister neighbour.

If there is no a homomorphism from C_{0}(ψ) to C_{0}(ϕ), then by Theorem 5.1, we conclude that there is a 4-dimensional subform in ϕ with trivial determinant.

(2) ⇒ (1). Obvious. □

Proposition 5.3. Let q be an anisotropic Albert form over F and ϕ = q ⊥ ⟨t⟩ be a quadratic form over E = F ((t)). Let ψ be a 7-dimensional quadratic form over E such that ϕ_{E(ψ)} is isotropic. Then ψ is similar to ϕ.

To proof this proposition we need the following lemma.

Lemma 5.4. Let ϕ_{0}, ϕ_{1}, ψ_{0}, and ψ_{1} be anisotropic quadratic forms over F. Let E = F (t) and Ê = F ((t)). Let ϕ = ϕ_{0} ⊥ tϕ_{1}, ψ = ψ_{0} ⊥ tψ_{1} be quadratic forms over E = F (t).

1) The following conditions are equivalent:

(a) ϕ_{E(ψ)} is isotropic,

(b) ϕ_{Ê(ψ)} is isotropic.

2) If the both forms ψ_{0}, ψ_{1} are nonzero, then the following conditions are equivalent:

(a) ϕ_{E(ψ)} is isotropic,

(b) ϕ_{Ê(ψ)} is isotropic,

(c) For any field extension K/F, we have

D_{K}(ψ_{0})D_{K}(ψ_{1}) ⊂ D_{K}(ϕ_{0})D_{K}(ϕ_{1}).
3) If ϕ_{E(ψ)} (or ϕ_{Ê(ψ)}) is isotropic, then:

(a) if dim ψ_{0} ⩾ 2, then at least one of the forms ϕ_{0}, ϕ_{1} is isotropic over the field F (ψ_{0}),

(b) if dim ψ_{1} ⩾ 2, then at least one of the forms ϕ_{0}, ϕ_{1} is isotropic over

the field F (ψ_{1}),

4) If ϕ_{E(ψ)} (or ϕ_{Ê(ψ)}) is isotropic, and dim ϕ_{1} = dim ψ_{1} = 1, then ψ is similar to a subform in ϕ.

Proof. We start the proof of the lemma from assertions 2), and 3).

2), (a) ⇒ (b). Obvious.

2), (b) ⇒ (c). Obviously we can suppose that K = F. Let a ∈ D_{F} (ψ_{0})D_{F} (ψ_{1}). Then ψ_{0} ⊥ −aψ_{1} is isotropic. Obviously ψ_{Ê(\sqrt{−at})} = (ψ_{0} ⊥ −aψ_{1})_{Ê(\sqrt{−at})}. Hence ψ_{Ê(\sqrt{−at})} is isotropic. By the assumption, ϕ_{Ê(ψ)} is isotropic. Therefore ϕ_{Ê(\sqrt{−at})} is isotropic too. Hence (ϕ_{0} ⊥ −aϕ_{1})_{Ê(\sqrt{−at})} = ϕ_{Ê(\sqrt{−at})} is isotropic. Since the

quadratic form ϕ_{0} ⊥ −aϕ_{1} is defined over F, and Ê(\sqrt{−at}) is a complete discrete valuation field with the residue field F, we conclude that ϕ_{0} ⊥ −aϕ_{1} is isotropic. Hence a ∈ D_{F} (ϕ_{0})D_{F} (ϕ_{1}).

2), (c) ⇒ (a). Set K = E(ψ). Since (ψ_{0} ⊥ tψ_{1})_{K} is isotropic, we obviously have −t ∈ D_{K}(ψ_{0})D_{K}(ψ_{0}). Hence −t ∈ D_{K}(ϕ_{0})D_{K}(ϕ_{0}). Therefore (ϕ_{0} ⊥ tϕ_{1})_{K} is isotropic, i.e., ϕ_{E(ψ)} is isotropic.

3(a). By condition, ϕ_{E(ψ)} (or ϕ_{Ê(ψ)}) is isotropic. In any case ϕ_{Ê(ψ)} is isotropic.

Therefore ϕ_{Ê(ψ_{0})} is isotropic. Since Ê(ψ_{0}) ⊂ F (ψ_{0})((t)), we see that (ϕ_{0})_{F (ψ_{0})} ⊥ t(ϕ_{1})_{F (ψ_{0})} is isotropic over F (ψ_{0})((t)). Using Springer theorem, one can see that at least one of the forms (ϕ_{0})_{F (ψ_{0})} or (ϕ_{1})_{F (ψ_{0})} is isotropic.

3(b). The proof is the same as that of 3(a).

Assertion 1) of the lemma follows immediately from 2) and 3).

4). By our assumption and in view of assertion 1) of the lemma, we can suppose that ϕ_{E(ψ)} is isotropic. After scalaring, we can suppose that ϕ_{1} = ψ_{1} = ⟨1⟩. For any field extension K/F we have

D_{K}(ψ_{0}) = D_{K}(ψ_{0})D_{K}(ψ_{1}) ⊂ D_{K}(ϕ_{0})D_{K}(ϕ_{1}) = D_{K}(ϕ_{0}).

It follows from Third Representation Theorem [Lam, Th. 2.8], that ψ_{0} ⊂ ϕ_{0}. Therefore ψ = ψ_{0} ⊥ ⟨t⟩ ⊂ ϕ_{0} ⊥ ⟨t⟩ = ϕ. □

Proof of Proposition 5.3. It follows from Theorem 5.1 and Example 3.5 that there is a homomorphism C_{0}(ψ) → C_{0}(ϕ). Since dim(C_{0}(ψ)) = dim(C_{0}(ϕ)) and C_{0}(ψ) is a SA, this homomorphism is isomorphism. Therefore [C_{0}(ψ)] = [C_{0}(ϕ)].

Since E = F ((t)), we can write ψ in the form ρ ⊥ tτ, where ρ and τ are quadratic forms over F. We can suppose that dim τ is odd and dim ρ is even (otherwise we change ψ by tψ).

Let τ̃ = τ ⊥ ⟨det(τ ⊥ ρ)⟩, ψ̃ = ρ ⊥ tτ̃. By the definition we have ψ̃ ∈ I^{2}(E), [C(ψ̃)] = [C_{0}(ψ)]. Hence [C(ψ̃)] = [C_{0}(ϕ)] = [C(q)]. Therefore ψ̃ ≡ q (mod I^{3}(E)). Hence [ρ] = ∂_{t}^{0}(ψ) ∈ I^{2}(F), and [τ̃] = ∂_{t}^{1}(ψ) ∈ I^{2}(F).

By the assumption, ρ is an even-dimensional anisotropic quadratic form with dim ρ ⩽ dim ϕ = 7. Therefore dim ρ is equal to one of the numbers 0, 2, 4, or 6.

Suppose first that dim ρ = 0. Then ψ = tτ and E(ψ) = F (τ)(t). Since dim ψ = 7, the quadratic form q_{F (ψ)} is anisotropic (see [Le, Me]). Hence by Lemma 5.4, we conclude that ϕ_{E(ψ)} is anisotropic, a contradiction.

The case dim ρ = 2 is impossible, because of ρ ∈ I^{2}(F).

Suppose now that dim ρ = 4. It follows from ρ ∈ I^{2}(F), that ρ ∈ GP_{2}(F). Therefore ψ = ρ ⊥ tτ contains a 5-dimensional Pfister neighbour. Hence there is π ∈ P_{3}(E) such that ψ_{E(π)} is isotropic. Since ϕ_{E(ψ)} and ψ_{E(π)} are both isotropic, we conclude that ϕ_{E(π)} is isotropic. But this is impossible in view of Corollary 5.2 and Example 3.5.

Thus we see that dim(ρ) = 6. Therefore dim τ = 1. It follows from the assertion 4) of Lemma 5.4 that ϕ ∼ ψ. □

Corollary 5.5. Let q be an anisotropic Albert form over F and ϕ = q ⊥ ⟨t⟩ be a quadratic form over Ê = F ((t)). Let ψ be a 8-dimensional quadratic form over Ê. Then ϕ_{Ê(ψ)} is anisotropic.

Proof. Suppose at the moment that ϕ_{Ê(ψ)} is isotropic. Write ψ in the form ψ = ρ ⊥ tτ, where ρ and τ are defined over F. We can suppose that that dim ρ ⩾ 4 ⩾ dim τ (otherwise we change ψ by tψ). It follows from Lemma 5.4 that q_{F (ρ)} is isotropic. Hence dim ρ ⩽ 6 (see [Me, Le]).

Let ρ_{0} subform of ρ of dimension dim ρ_{0} = dim ρ − 1. Let ψ_{0} = ρ_{0} ⊥ tτ. Since ψ_{0} ⊂ ψ, we see that ϕ_{E(ψ_{0})} is isotropic. Hence by Proposition 5.3 we have ϕ ∼ ψ_{0}. Hence either dim ρ_{0} = 6 or dim ρ_{0} = 1. On the other hand, it follows from 4 ⩽ dim ρ ⩽ 6 that 3 ⩽ dim ρ_{0} ⩽ 5. This contradiction completes the proof. □

Corollary 5.6. Let q be an anisotropic Albert form over F and ϕ = q ⊥ ⟨t⟩ be a quadratic form over E = F (t). Let ψ be a 8-dimensional quadratic form over E. Then ϕ_{E(ψ)} is anisotropic. □

§6. 8-dimensional forms

Theorem 6.1. Let ϕ and ψ be quadratic forms over F, and dim ϕ = 8. Suppose that the quadratic form ϕ_{F (ψ)} is isotropic. Then at least one of the following conditions holds.

1) There is a 4-dimensional subform in ϕ with trivial determinant.

2) There is a homomorphism of F -algebras, C_{0}(ψ) → C_{0}(ϕ).

Proof. If Coind C_{0}(ϕ) ⩽ 2, the required result follows from Theorem 4.5 and Example 3.4. Thus we can suppose that Coind C_{0}(ϕ) ⩾ 4, i.e., ind C_{0}(ϕ) ⩽ 2. Now the required result follows immediately from the following lemma.

Lemma 6.2. Let ϕ be a 8-dimensional quadratic form such that ind(C_{0}(ϕ)) ⩽ 2. Then:

1) There is a ∈ F^{∗} such that i_{W} (ϕ_{F (\sqrt{a})}) ⩾ 2.

2) There is a 4-dimensional subform in ϕ with trivial determinant.

Proof. 1) If ϕ ∈ I^{2}(F), then the condition ind C_{0}(ϕ) ⩽ 2 implies that ϕ has the form ⟨⟨a⟩⟩q, with q a 4-dimensional quadratic form [Kn2, Example 9.12]. Hence i_{W} (ϕ_{F (\sqrt{a})}) = 4 ⩾ 2.

Suppose now that ϕ ∉ I^{2}(F). Let d = det_{±} ϕ and L = F (\sqrt{d}). Let γ = ϕ ⊥ ⟨⟨d⟩⟩. We have γ ∈ I^{2}(F). The condition ind(C(γ)_{L}) = ind(C_{0}(ϕ)) ⩽ 2 implies that ind(C(γ)) ⩽ 4. Hence there is a biquaternion algebra A such that [C(γ)] = [A]. Since ind(A_{L}) = ind(C(γ)_{L}) ⩽ 2, we can write A in the form A = (u, v)⊗(d, s). Let q = s ⟨−u, −v, uv, d, s, −sd⟩. We have [C(γ)] = A = [C(q)]. Hence γ − q ∈ I^{3}(F).

Let ρ = ϕ ⊥ −s ⟨−u, −v, uv, d⟩. We have

ρ = ϕ − s ⟨−u, −v, uv, d⟩ = ϕ + ⟨⟨d⟩⟩ − s ⟨−u, −v, uv, d, s, −sd⟩ = γ − q ∈ I^{3}(F)

Since dim(ρ) = 4 + 8 = 12 and ρ ∈ I^{3}(F), we can write ρ in the form ρ = ⟨⟨a⟩⟩τ, where τ a 6-dimension Albert form (see [Pf]). Then ρ_{F (\sqrt{a})} is hyperbolic and hence [ϕ_{F (\sqrt{a})}] = [s ⟨−u, −v, uv, d⟩_{F (\sqrt{a})}]. Therefore i_{W} (ϕ_{F (\sqrt{a})}) ⩾ 2.

2) Since i_{W} (ϕ_{F} (\sqrt{a})) ⩾ 2, there is 2-dimensional quadratic form λ such that ⟨⟨a⟩⟩λ ⊂ ϕ. Since dim(⟨⟨a⟩⟩λ) = 4 and det(⟨⟨a⟩⟩λ) = 1, we see that ϕ contains a 4-dimensional subform with trivial determinant. □

Remark. In the case where ϕ ∈ I^{2}(F), dim ψ ⩾ 5, Theorem 6.1 follows imediatelly from results of Laghribi [La2], [La3].

Now collected together the results of Corollary 1.8, Theorem 5.1 and Theorem 6.1 we get the following

Theorem 6.3. Let ϕ and ψ be anisotropic quadratic forms over F. Suppose that ϕ contains no 4-dimensional subforms with trivial determinant and dim ϕ ⩽ 8. Then the isotropy of ϕ_{F (ψ)} implies that there is a homomorphism of algebras, C_{0}(ψ) → C_{0}(ϕ). □

Corollary 6.4. Let ϕ be a quadratic form over F with dim ϕ ⩽ 8. Then the following conditions are equivalent:

1) There is a 3-fold Pfister form ψ such that ϕ_{F (ψ)} is isotropic.

2) ϕ contains a 5-dimensional Pfister neighbour (i.e., dim ϕ > 4 and there is a 4-dimensional subform in ϕ with trivial determinant).

Proof. 1)⇒2). In view of Corollary 5.2 it is sufficient to consider only the case dim ϕ = 8. By Theorem 6.3 we can suppose that there is a homomorphism, C_{0}(ψ) → C_{0}(ϕ). Since ψ is a 3-fold Pfister form, we have C_{0}(ψ) ≅ M_{8}(F)×M_{8}(F). Thus we get a homomorphism M_{8}(F) → C_{0}(ϕ). Therefore Coind(C_{0}(ϕ)) ⩾ 3 and hence ind(C_{0}(ϕ)) = 1. It follows from Lemma 6.2 that ϕ contains a 5-dimensional Pfister neighbour.

2)⇒1). Obvious. □

Remark. Corollary 6.4 gives rise to the following conjecture: if a quadratic form ϕ and a 3-fold Pfister form ψ are such that ϕ_{F (ψ)} is isotropic then ϕ contains a 5dimensional Pfister neighbour of ψ. In [H2] D. Hoffmann proved that this conjecture is true if dim ϕ ⩽ 6. However the conjecture is not true in the case where dim ϕ = 8 (see [Izh]). Therefore for any n ⩾ 8 there is a counterexample with dim ϕ = n. Thus the only unsolved problem is: if the conjecture true in the case dim ψ = 7?

Proof of the following result is similar to that of Proposition 5.3 and we omit it.

Example 6.5. Let q be anisotropic Albert form over F and a ∈ F^{∗} be such that q_{F (\sqrt{a})} is anisotropic. Let ϕ = q ⊥ t⟨⟨a⟩⟩ be a quadratic form over E = F ((t)). Let ψ be a 8-dimensional quadratic form over E such that ϕ_{E(ψ)} is isotropic. Then there is k ∈ F^{∗} such that ψ ∼ q ⊥ tk⟨⟨a⟩⟩. □

References

[ELW] Elman, R.; Lam, T.Y.; Wadsworth, A.R., Amenable fields and Pfister extensions., Proc. of Quadratic Forms Conference (ed. G. Orzech) 46 (1977), Queen’s Papers in Pure and Applied Mathematics, 445–491.

[H1] Hoffmann, D. W., Isotropy of 5-dimensional quadratic forms over the function field of a quadric, Proc. Symp. Pure Math. 58.2 (1995), 217–225.

[H2] Hoffmann, D. W., On 6-dimensional quadratic forms isotropic over the function field of a quadric, Comm. Alg. 22 (1994), 1999–2014.

[Izh] Izhboldin 0. T., On the Nonexcellence of Field Extensions F (π)/F, Doc. Math. 1 (1996), 127–136.

[Kn1] Knebusch, M., Generic splitting of quadratic forms, I, Proc. London Math. Soc. 33 (1976), 65–93.

[Kn2] Knebusch, M., Generic splitting of quadratic forms, II, Proc. London Math. Soc. 34 (1977), 1–31.

[Lam] Lam, T. Y., The algebraic Theory of Quadratic Forms, Massachusetts: Benjamin (revised printing 1980), 1973.

[La1] Formes quadratiques de dimension 6, Preprint (1996).

[La2] Isotropie de certaines formes quadratiques de dimensions 7 et 8 sur le corps des fonctions d’une quadrique, to appear in Duke Math. J. (1996).

[La3] Formes quadratiques en 8 variables dont l’algèbre de Clifford est d’indice 8, to appear in K-Theory J. (1996).

[Le] Leep, D., Function fields results, notes manuscriptes prises par T. Y. Lam (1989).

[Me] Merkurev A. S., Simple algebras and quadratic forms, Math. USSR Izvestiya 38 (1992), 215–221.

[LVG] Lewis, D.W.; Van Geel, J., Quadratic forms isotropic over the function field of a conic, Indag. Math. 5 (1994), 325–339.

[Pf] Pfister, A., Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116-132.

[Ro] Rost, M., Quadratic forms isotropic over the function field of a conic, Math Ann. 288 (1990), 511–513.

[Sh] Shapiro D. B., Similarities, quadratic forms, and Clifford algebra, These, Universite Californie, Berkley (1974).

[Sch] Scharlau, W., Quadratic and Hermitian Forms, Springer, Berlin, Heidelberg, New York, Tokyo, 1985.

[Ti] Tignol, J.-P., Réduction de l’indice d’une algèbre simple centrale sur le corps des fonctions d’une quadrique, série A 42 (1990), Bull. Soc. Math. Belgique, 735–745.

[Pi] Pirce, R. S., Associative Algebras, Springer-Verlag, New York, Heidelberg, Berlin, 1982.

[Dr] Draxl, P. K., Skew Fields, vol. 81, London Math Soc., Lecture Note Series, Campbidge University Press, 1983.

Department of Mathematics and Mechanics St.-Petersburg State University, Petrodvorets, 198904, Russia

Current address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, D-53225 Bonn, Germany

Email adresses: oleg@mpim-bonn.mpg.de, oleg@izh.usr.pu.ru