Motivic Equivalence of Quadratic Forms

Oleg T. Izhboldin^{1}

Received: December 21, 1998

Communicated by Ulf Rehmann

Abstract. Let X_{ϕ} and X_{ψ} be projective quadrics corresponding to quadratic forms ϕ and ψ over a field F. If X_{ϕ} is isomorphic to X_{ψ} in the category of Chow motives, we say that ϕ and ψ are motivic isomorphic and write ϕ ∼^{m} ψ. We show that in the case of odd-dimensional forms the condition ϕ ∼^{m} ψ is equivalent to the similarity of ϕ and ψ. After this, we discuss the case of even-dimensional forms. In particular, we construct examples of generalized Albert forms q_{1} and q_{2} such that q_{1} ∼^{m} q_{2} and q_{1} ≁ q_{2}.

Keywords and Phrases: Quadratic form, quadric, Pfister form, Chow motives

1991 Mathematics Subject Classification: Primary 11E81; Secondary 19E15

Let F be a field of characteristic ≠ 2 and ϕ be a quadratic form of dimension ≥ 3 over F. By X_{ϕ} we denote the projective variety given by the equation ϕ = 0. It is well known that the variety X_{ϕ} determines the form ϕ uniquely up to similarity. More precisely, the condition X_{ϕ} ≃ X_{ψ} holds if and only if ϕ ≃ kψ for a suitable element k ∈ F^{∗}. Now, let M : V_{F} → C be an arbitrary functor from the category V_{F} of smooth projective F -varieties to a category C. Is it possible to say anything specific about ϕ and ψ if we know that M(X_{ϕ}) ≃ M(X_{ψ})? Clearly, the answer depends on the category C and the functor M. In the present paper, we mainly consider the example of the category C = MV_{F} of Chow motives. In this particular case, we set M(X) = M(X), where M(X) denotes the motive of X in the category of Chow motives. If M(X_{ϕ}) ≃ M(X_{ψ}), we say that ϕ is motivic equivalent to ψ (and we write

ϕ ∼^{m} ψ).

Recently, Alexander Vishik has proved that ϕ ∼^{m} ψ iff dim ϕ = dim ψ and i_{W} (ϕ_{L}) = i_{W} (ψ_{L}) for all extensions L/F (see [27]). His proof uses deep results concerning the Voevodsky motivic category. In [10], Nikita Karpenko found a new, more elementary, proof that, in contrast to Vishik’s proof, deals only with Chow motives. In §2, we give an elementary proof of Vishik’s theorem in the case of odddimensional forms. In fact, we prove a more precise result. Namely, we show that, in the case of odd-dimensional forms, the condition ϕ ∼^{m} ψ is equivalent to the similarity of the forms ϕ and ψ (here we do not use any results of the paper of Vishik). In other words, we prove that the condition M(X_{ϕ}) ≃ M(X_{ψ}) is equivalent to the condition X_{ϕ} ≃ X_{ψ} for the odd-dimensional quadrics X_{ϕ} and X_{ψ}. In the proof we use some results of §1 concerning low dimensional forms belonging to W (F (ϕ)/F).

1Supported by TMR-Network Project ERB FMRX CT-97-0107

In §3, we show that the condition ϕ ∼^{m} ψ is equivalent to the condition ϕ ∼ ψ for all forms of dimension ≤ 7. Besides, we discuss the case of even-dimensional forms of dimension ≥ 8. This case is much more complicated. For instance, for all n ≥ 3, there exists an example of anisotropic 2^{n}-dimensional forms ϕ and ψ such that ϕ ∼^{m} ψ but ϕ ≁ ψ. In §4, for any n and m such that 0 ≤ m ≤ n − 3, we construct generalized Albert forms q_{1} and q_{2} such that dim(q_{1})_{an} = dim(q_{2})_{an} = 2(2^{n} − 2^{m}), q_{1} ∼^{m} q_{2} but q_{1} ≁ q_{2}. This example gives a negative answer to a question stated by T. Y. Lam [18].

Some words about terminology and notation. Mainly we use the same terminology and notation as in the book of T. Y. Lam [17], W. Scharlau [23], and the fundamental papers of M. Knebusch [11, 12]. However, there exist several differences. We use the notation ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ for the Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ (in [17] and [23], ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩). We write ϕ ∼ ψ if there exists an element k ∈ F such that kϕ ≃ ψ (i.e., if ϕ is similar to ψ). We say that ϕ and ψ are half-neighbors if dim ϕ = dim ψ and there exist s, r ∈ F such that π = sϕ ⊥ rψ is a Pfister form (see, e.g., [6]). In this case, we will write ϕ ∼^{hn} ψ and we say that ϕ and ψ are half-neighbors of π. Our definition differs from the original definition of Knebusch [12]. However, we prefer to use the new definition since we want to regard any pair ϕ, ψ of 2^{n}-dimensional similar forms as half-neighbors. We denote by P_{n}(F) the set of all n-fold Pfister forms. The set of all forms similar to n-fold Pfister forms is denoted by GP_{n}(F). We also use the notation P_{∗}(F) = ∪_{n}P_{n}(F) and GP_{∗}(F) = ∪_{n}GP_{n}(F).

Acknowledgments. This work was supported by TMR-Network Project ERB FMRX CT-97-0107. Also, the author would like to thank the Universität Bielefeld, and the Université de Franche-Comté for their hospitality and support. The author wishes to thank Nikita Karpenko for useful discussions.

1. Low dimensional forms in W (F (ϕ)/F)

In this section, we give slight generalizations of some results of M. Knebusch. In fact, we modify some proofs of [12] by using Hoffmann’s theorem [5] ^{2}. We recall that Hoffmann’s theorem asserts that for a pair of anisotropic quadratic forms ϕ and ψ satisfying the condition dim ϕ ≤ 2^{n} < dim ψ, the form ϕ remains anisotropic over F (ψ).

Proposition 1.1. Let ϕ and ψ be anisotropic quadratic forms over F such that dim ϕ ≥ dim ψ. Suppose that the form π =^{Def} ϕ ⊥ ψ belongs to the group W (F (ϕ)/F). Then

(1) if π is isotropic, then π is hyperbolic,

(2) if π is anisotropic, then π is similar to a Pfister form.

Proof. (1) Assume that π is isotropic but not hyperbolic. This means that 0 < dim π_{an} < dim π. In the Witt ring W (F), we have π − ϕ = ψ. Therefore,

dim(π_{an} ⊥ −ϕ)_{an} = dim ψ ≤ dim ϕ < dim π_{an} + dim ϕ = dim(π_{an} ⊥ −ϕ).

2see also [6, Prop. 2.4] and [3, Th. 1.6]

Consequently, the form π_{an} ⊥ −ϕ is isotropic. Hence the set D_{F} (π_{an}) ∩ D_{F} (ϕ) is nonempty.

Since π_{F (ϕ)} is hyperbolic, it follows that ((π)_{an})_{F (ϕ)} is also hyperbolic. Since the set D_{F} (π_{an}) ∩ D_{F} (ϕ) is nonempty, the Cassels–Pfister subform theorem implies that ϕ ⊂ π_{an}. Therefore,

dim(π_{an} ⊥ −ϕ)_{an} = dim π_{an} − dim ϕ < dim π − dim ϕ = dim ψ.

This contradicts to the relation dim(π_{an} ⊥ −ϕ)_{an} = dim ψ proved above.

(2) Assume that π is not isotropic. To prove that π is similar to a Pfister form, it suffices to prove that π_{F (π)} is hyperbolic (see [12]).

Let F̃ = F (π), π̃ = πF̃, ϕ̃ = ϕF̃, and ψ̃ = ψF̃. Since dim ψ ≤ \frac{1}{2} dim π, Hoffmann’s theorem implies that the form ψ̃ = ψ_{F (π)} is anisotropic. If we assume that ϕ̃ is anisotropic, then we can apply item (1) of Proposition 1.1 to the F̃-forms ϕ̃, ψ̃, and π̃. Then we conclude that π̃ is hyperbolic. Now, we assume that ϕ̃ = ϕ_{F (π)} is isotropic. Since π_{F (ϕ)} is hyperbolic and ϕ_{F (π)} is isotropic, it follows that π_{F (π)} is hyperbolic. Thus, the form π_{F (π)} is hyperbolic in any case and the proposition is proved.

Corollary 1.2. (Fitzgerald, [3, Th. 1.6]). Let ϕ be an F -form, and let π ∈ W (F (ϕ)/F) be an anisotropic nonzero form of dimension ≤ 2 dim ϕ. Then π ∈ GP_{∗}(F) and one of the following conditions holds:

• ϕ is a Pfister neighbor of π,

• ϕ is a half-neighbor of π,

Proof. Since π is anisotropic and π_{F (ϕ)} is hyperbolic, the form ϕ is similar to a subform of π. Multiplying ϕ by a scalar, we may assume that ϕ ⊂ π. Let ψ be the complement of ϕ in π. Then all hypotheses of Proposition 1.1 hold. Since π is anisotropic, Proposition 1.1 implies π ∈ GP_{∗}(F). The rest of the proof is an immediate consequence of the definitions of Pfister neighbors and half-neighbors, and the Cassels-Pfister subform theorem.

Corollary 1.3. (cf. [12, Th. 8.9]). Let ϕ and η be anisotropic forms such that dim ϕ ≥ dim η and (ϕ_{F (ϕ)})_{an} ≃ (η_{F (ϕ)})_{an}. Then either ϕ ≃ η or ϕ ⊥ −η ∈ GP_{∗}(F).

Proof. Let ψ = −η and π = ϕ ⊥ −η = ϕ ⊥ ψ. All the hypotheses of Proposition 1.1 hold. In the case where π is isotropic, Proposition 1.1 implies that π is hyperbolic. Then ϕ = η in the Witt ring. Since ϕ and η are anisotropic, we have ϕ ≃ η. If π is anisotropic, Proposition 1.1 implies that ϕ ⊥ −η = π ∈ GP_{∗}(F).

2. Motivic equivalence of odd-dimensional forms

Definition 2.1. To any field F, let be assigned an equivalence relation ∼^{∗}_{F} on the set of all quadratic forms over F such that the following conditions hold:

(i) If ϕ and ψ are forms over F such that ϕ ∼ ψ, then ϕ ∼^{∗}_{F} ψ. (ii) If ϕ and ψ are forms over F such that ϕ ∼^{∗}_{F} ψ, then, for any extension E/F,

we have ϕ_{E} ∼^{∗}_{E} ψ_{E}.

(iii) If ϕ and ψ are forms over a field F such that ϕ ∼^{∗}_{F} ψ, then dim ϕ = dim ψ and i_{W} (ϕ) = i_{W} (ψ). A collection of equivalence relations ∼^{∗}_{F} satisfying properties (i)–(iii) will be

called a good equivalence relation on quadratic forms (over all fields).

Below we will drop the index F at ∼^{∗}_{F} and write simply ∼^{∗}.

Definition 2.2. Let ϕ and ψ be F -forms. We say that the quadratic form ϕ is equivalent to the quadratic form ψ in the sense of Vishik if dim ϕ = dim ψ and for any field extension E/F we have i_{W} (ϕ_{E}) = i_{W} (ψ_{E}). In this case, we write ϕ ∼^{v} ψ.

The following lemma is obvious.

Lemma 2.3. The equivalence relation ∼^{v} is a minimal good equivalence relation. More precisely,

• The equivalence relation ∼^{v} is a good relation.

• For any good relation ∼^{∗}, the condition ϕ ∼^{∗} ψ implies ϕ ∼^{v} ψ.

Example 2.4. Let X be a smooth variety over F. By M(X) we denote the motive of X in the category of Chow motives. Let us define the equivalence ∼^{m} of quadratic forms ϕ and ψ as follows:

ϕ ∼^{m} ψ if M(X_{ϕ}) ≃ M(X_{ψ}).

Then ∼^{m} is a good equivalence relation.

Proof. Clearly, conditions (i) and (ii) in Definition 2.1 are fulfilled. We need to verify only condition (iii). Let X = X_{ϕ}, and let F̄ denote the algebraic closure of F. By [9, Item (2.2) and Prop. 2.6] ^{3}

• dim ϕ coincides with the largest integer m such that CH_{m−2}(X) ≠ 0,

• the integer i_{W} (ϕ) coincides with the largest integer m satisfying the conditions m ≤ \frac{1}{2} dim ϕ and coker(CH_{m−1}(X) → CH_{m−1}(XF̄)) = 0.

Thus, it suffices to show that the groups coker(CH^{j}(X) → CH^{j}(XF̄)) and CH^{j}(X) depend only on the motive of X. This can easily be proved if we observe that the functor CH^{j} is representable in the category of Chow motives. Namely, CH^{j}(X) = Hom_{MV_{F}} (M(pt_{F})(j), M(X)), where M(pt_{F}) is the motive of pt_{F} = Spec(F) and the object M(pt_{F})(j) is defined, e.g., in [24]. Thus, CH^{j}(X) depends only on the motive of X. Now, we consider the base change functor Φ : MV_{F} → MVF̄. Since the homomorphism CH^{j}(X) → CH^{j}(XF̄) coincides with the homomorphism

Φ : Hom_{MV_{F}}(M(pt_{F})(j), M(X)) → Hom_{MVF̄}(Φ(M(pt_{F})(j)), Φ(M(X))),

it follows that the group coker(CH^{j}(X) → CH^{j}(XF̄)) also depends only on M(X).

Theorem 2.5. Let ∼^{∗} be a good equivalence relation. Let ϕ and ψ be odd-dimensional quadratic forms over a field. Then the condition ϕ ∼^{∗} ψ is equivalent to the condition ϕ ∼ ψ.

Proof. We start the proof with three lemmas

Lemma 2.6. Let ϕ and ψ be odd-dimensional anisotropic forms of dimension ≥ 3 such that dim ϕ = dim ψ and (ϕ_{F (ϕ)})_{an} ≃ (ψ_{F (ϕ)})_{an}. Then ϕ ≃ ψ.

Proof. If ϕ ≄ ψ, Corollary 1.3 shows that ϕ ⊥ −ψ ∈ GP_{∗}(F). Since dim ϕ = dim ψ, we conclude that dim ψ is a power of 2. Since dim ψ ≥ 3, we see that dim ψ is even. We get a contradiction to the assumption of the lemma.

3see also [22, Prop. 2] and [25].

The following lemma is obvious.

Lemma 2.7. Let ϕ and ψ be odd-dimensional forms such that dim ϕ = dim ψ and det ϕ = det ψ. Then the condition ψ ∼ ϕ is equivalent to the condition ϕ ≃ ϕ.

Lemma 2.8. Let ϕ and ψ be odd-dimensional forms such that dim ϕ_{an} = dim ψ_{an} ≥ 3. Suppose that ϕ_{F (ϕ_{an})} ∼ ψ_{F (ϕ_{an})}. Then ϕ ∼ ψ.

Proof. Replacing first ϕ and ψ by ϕ_{an} and ψ_{an}, respectively, we may assume that ϕ and ψ are anisotropic. Replacing then ϕ by \frac{1}{det ϕ}ϕ and ψ by \frac{1}{det ψ} ψ, we may assume that det ϕ = 1 = det ψ. Since ϕ_{F (ϕ)} ∼ ψ_{F (ϕ)}, Lemma 2.7 implies that ϕ_{F (ϕ)} ≃ ψ_{F (ϕ)}. By Lemma 2.6, we have ϕ ≃ ψ.

Now, we return to the proof of Theorem 2.5. We use induction on n = dim ϕ_{an} = dim ψ_{an}. The case where n = 1 is obvious. So we may assume that n ≥ 3. Since ϕ ∼^{∗} ψ, we have ϕ_{F (ϕ_{an})} ∼^{∗} ψ_{F (ϕ_{an})}. By the induction assumption, we have ϕ_{F (ϕ_{an})} ∼ ψ_{F (ϕ_{an})}. Now, Lemma 2.8 implies that ϕ ∼ ψ.

Corollary 2.9. Let ϕ and ψ be odd-dimensional quadratic forms over a field. Then

ϕ ∼^{v} ψ iff ϕ ∼^{m} ψ iff ϕ ∼ ψ.

3. Even-dimensional forms

In this section, we study the relation ∼^{m} in the case of even-dimensional forms. If quadratic forms ϕ and ψ of dimension ≥ 2 satisfy the condition ϕ ∼^{v} ψ, then ϕ_{F (ψ)} and ψ_{F (ϕ)} are isotropic (because ϕ_{F (ϕ)} and ψ_{F (ψ)} are isotropic).

Proposition 3.1. Let ϕ and ψ be quadratic forms of dimension < 8. Then

ϕ ∼^{v} ψ iff ϕ ∼^{m} ψ iff ϕ ∼ ψ.

Proof. In view of Corollary 2.9, we may assume that d = dim ϕ = dim ψ is even. Thus, it suffices to consider the cases d = 2, 4, and 6. The implications ϕ ∼ ψ ⇒ ϕ ∼^{m} ψ ⇒ ϕ ∼^{v} ψ are obvious. Therefore, we must verify only that ϕ ∼^{v} ψ implies ϕ ∼ ψ. Since ϕ ∼^{v} ψ, the forms ϕ_{F (ψ)} and ψ_{F (ϕ)} are isotropic. In the case d = 2, this obviously means that ϕ ∼ ψ. If d = 4, then ϕ ∼ ψ by Wadsworth’s theorem [28]. Thus, we may assume that d = 6. We need the following assertion concerning the isotropy of 6-dimensional forms.

Lemma 3.2. (see [4, 13, 16, 21]). Let ϕ and ψ be anisotropic 6-dimensional forms such that ϕ_{F (ψ)} is isotropic. Then either ϕ ∼ ψ or ψ is a 3-fold Pfister neighbor.

In view of this lemma, we may assume that ψ is a Pfister neighbor of a 3-fold Pfister form π. Since ψ_{F (ϕ)} is isotropic, it follows that π_{F (ϕ)} is isotropic. Hence ϕ is a Pfister neighbor of π. Therefore, ϕ ∼ (π − ⟨ ; ⟨d_{±}ϕ⟩ ; ⟩)_{an} and ψ ∼ (π ⊥ − ⟨ ; ⟨d_{±}ψ⟩ ; ⟩)_{an}. Thus, it suffices to verify that d_{±}ϕ = d_{±}ψ. This is a consequence of the following chain of equivalent conditions

a = d_{±}ϕ ⇔ i_{W} (ϕ_{F (\sqrt{a})}) = 3 ⇔ i_{W} (ψ_{F (\sqrt{a})}) = 3 ⇔ a = d_{±}ψ

The proof is complete.

Now, we begin to study even-dimensional forms of dimension ≥ 8.

Lemma 3.3. (see, e.g., [27]). Let ϕ and ψ be half-neighbors. Then ϕ ∼^{v} ψ.

For the reader’s convenience, we cite the proof (which, in fact, is trivial).

Proof. The condition ϕ ∼^{hn} ψ means that dim ϕ = dim ψ, and there exist s, r ∈ F^{∗} such that sϕ ⊥ rψ = π ∈ P_{∗}(F). Let L/F be a field extension. If both ϕ_{L} and ψ_{L} are anisotropic, then i_{W} (ϕ_{L}) = 0 = i_{W} (ψ_{L}). If at least one of the forms ϕ_{L} or ψ_{L} is isotropic, then π_{L} is also isotropic. Taking into account the condition π ∈ P_{∗}(F), we conclude that π_{L} is hyperbolic. Therefore, sϕ_{L} = −rψ_{L} in the Witt ring. Since dim ϕ = dim ψ, we have sϕ_{L} ≃ −rψ_{L}. Hence i_{W} (ϕ_{L}) = i_{W} (ψ_{L}).

The following lemma shows that there exist examples of nonsimilar halfneighbors.

Lemma 3.4. (see [6], [8]). For any n ≥ 3, there exists a field F and 2^{n}-dimensional half-neighbors ϕ and ψ such that ϕ ≁ ψ.

As a consequence of this result, we see that, for any n ≥ 3, there exists a pair of 2^{n} dimensional forms ϕ and ψ such that ϕ ∼^{v} ψ and ϕ ≁ ψ. In particular, Proposition 3.1 cannot always be generalized for 8-dimensional forms.

Nevertheless, for 8-dimensional forms with trivial determinant, we have the following

Proposition 3.5. Let ϕ and ψ be 8-dimensional forms with trivial determinant. Then the following conditions are equivalent:

(1) ϕ ∼^{v} ψ ;

(2) ϕ_{F (ψ)} and ψ_{F (ϕ)} are isotropic;

(3) ϕ and ψ are half-neighbors.

Proof. The implications (3)⇒(1)⇒(2) are obvious. The implication (2)⇒(3) follows immediately from the results of A. Laghribi [16], [15], [14].

4. Generalized Albert forms

In this section, we construct examples of nonsimilar ∼^{v}-equivalent forms based on the so-called generalized Albert forms.

Definition 4.1. A generalized Albert form (or n-Albert form) is a form of type q = π′ ⊥ −τ′, where π′ and τ′ are pure parts of n-fold Pfister forms π and τ.

Remark 4.2. • Any n-Albert form has dimension 2(2^{n} − 1).

• Suppose that q is an n-Albert form. By [2, Proof of Prop. 4.4], the anisotropic part q_{an} looks like q_{an} = ⟨ ; ⟨a_{1}, …, a_{m}⟩ ; ⟩ q′, where q′ is an anisotropic (n − m)- Albert form. In particular, dim q_{an} has dimension 2^{m}·2(2^{n−m}−1) = 2(2^{n}−2^{m}), where 0 ≤ m ≤ n. We say that m is the linkage number of the n-Albert from q.

• Every 1-Albert form has the form q = ⟨ ; ⟨a⟩ ; ⟩′ ⊥ − ⟨ ; ⟨b⟩ ; ⟩ = ⟨−a, b⟩. Hence any 2-dimensional form is a 1-Albert form.

• Every 2-Albert form has the form

q = ⟨ ; ⟨a_{1}, a_{2}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨b_{1}, b_{2}⟩ ; ⟩′ = ⟨−a_{1}, −a_{2}, a_{1}a_{2}, b_{1}, b_{2}, −b_{1}b_{2}⟩ .

Thus, a 2-Albert form is the “classical” 6-dimensional Albert form.

Our interest in n-Albert forms is motivated by the following observation of A. Vishik (see [27]): if q_{1} and q_{2} are n-Albert forms such that q_{1} ≡ q_{2} (mod I^{n+1}(F)),

then q_{1} ∼^{v} q_{2}.

The following question is due to Lam [18, Item (6.6), Page 28].

Question 4.3. Let q_{1} and q_{2} be n-Albert forms such that q_{1} ≡ q_{2} (mod I^{n+1}(F)). Is it always true that q_{1} ∼ q_{2} ?

The answer to this question is obviously positive in the case n = 1. In the case n = 2, the answer is also positive. This is a version of a Jacobson’s theorem (see, e.g., [19, Prop. 2.4]). In this section, we construct a counterexample to this question for any n ≥ 3.

Theorem 4.4. There exists a field F and anisotropic 3-Albert forms q_{1} and q_{2} over F such that q_{1} ≡ q_{2} (mod I^{4}(F)) and q_{1} ≁ q_{2}. In particular, the answer to Question 4.3 is negative in the case n = 3.

Proof. We need the following theorem of Hoffmann.

Theorem 4.5. (see [6, Th. 4.3]). There exists a field k and anisotropic 8-dimensional quadratic forms over k,

ϕ_{1} = s_{1} ⟨ ; ⟨a_{1}, b_{1}⟩ ; ⟩ ⊥ −k_{1} ⟨ ; ⟨c_{1}, d_{1}⟩ ; ⟩ ,
ϕ_{2} = s_{2} ⟨ ; ⟨a_{2}, b_{2}⟩ ; ⟩ ⊥ −k_{2} ⟨ ; ⟨c_{2}, d_{2}⟩ ; ⟩

such that ϕ_{1} ≡ ϕ_{2} (mod I^{4}(k)), ind C(ϕ_{1}) = ind C(ϕ_{2}) = 4 and ϕ_{1} ≁ ϕ_{2}.

Remark 4.6. In fact, the formulation of Theorem 4.3 in [6] differs from the one presented above. In his theorem, Hoffmann has constructed a pair ϕ, ψ ∈ I^{2}(k) of 8-dimension quadratic forms such that ϕ ≁ ψ and ϕ ∼^{hn} ψ. Clearly, changing ψ by a scalar, we may always assume that ϕ ≡ ψ (mod I^{4}(k)). To obtain Theorem 4.5, it suffices to show that we may always take ϕ and ψ in the form of direct sums of forms belonging to GP_{2}(k). In the proof of [6, Theorem 4.3] it is so for the form ϕ (the explicit formula for ϕ in [6] shows that ϕ contains a subform a ⟨1, x, y, xy⟩). The required statement concerning ψ is obvious since i_{W} (ψ_{k(\sqrt{−x})}) = i_{W} (ϕ_{k(\sqrt{−x})}) ≥ 2.

Now we return to the proof of Theorem 4.4. Under the conditions of this theorem, we obviously have (a_{1}, b_{1}) + (c_{1}, d_{1}) = c(ϕ_{1}) = c(ϕ_{2}) = (a_{2}, b_{2}) + (c_{2}, d_{2}). Hence there exists an Albert form ρ (of dimension 6) such that c(ϕ_{1}) = c(ϕ_{2}) = c(ρ). Hence ind C(ρ) = ind C(ϕ_{1}) = 4. By an Albert’s theorem, ρ is anisotropic (see [1, Th. 3] or [26, Th. 3]). Since (a_{i}, b_{i}) + (c_{i}, d_{i}) = c(ρ) for i = 1, 2, there exist r_{1} and r_{2} such that

⟨ ; ⟨a_{1}, b_{1}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨c_{1}, d_{1}⟩ ; ⟩′ ≃ r_{1}ρ,
⟨ ; ⟨a_{2}, b_{2}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨c_{2}, d_{2}⟩ ; ⟩′ ≃ r_{2}ρ.

In the Witt ring W (k(t)), we have

tρ − ϕ_{i} = tr_{i}(⟨ ; ⟨a_{i}, b_{i}⟩ ; ⟩ − ⟨ ; ⟨c_{i}, d_{i}⟩ ; ⟩) − (s_{i} ⟨ ; ⟨a_{i}, b_{i}⟩ ; ⟩ − k_{i} ⟨ ; ⟨c_{i}, d_{i}⟩ ; ⟩)

= tr_{i}(⟨ ; ⟨a_{i}, b_{i}⟩ ; ⟩ − tr_{i}s_{i} ⟨ ; ⟨a_{i}, b_{i}⟩ ; ⟩) − tr_{i}(⟨ ; ⟨c_{i}, d_{i}⟩ ; ⟩ − tr_{i}k_{i} ⟨ ; ⟨c_{i}, d_{i}⟩ ; ⟩)

= tr_{i}(⟨ ; ⟨a_{i}, b_{i}, tr_{i}s_{i}⟩ ; ⟩ − ⟨ ; ⟨c_{i}, d_{i}, tr_{i}k_{i}⟩ ; ⟩).

We set q_{i} = ⟨ ; ⟨a_{i}, b_{i}, tr_{i}s_{i}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨c_{i}, d_{i}, tr_{i}k_{i}⟩ ; ⟩′ and F = k(t). Since tρ−ϕ_{i} = tr_{i}q_{i} in the

Witt ring W (F) and dim(tρ ⊥ −ϕ_{i}) = 6 + 8 = 14 = dim q_{i}, we have tρ ⊥ −ϕ_{i} ≃ tr_{i}q_{i}.

Since ρ and ϕ_{i} are anisotropic, q_{i} is also anisotropic by Springer’s theorem (see [17, Ch. 6, Th. 1.4] or [23, Ch. 6, Cor. 2.6]).

Now, we need the following obvious assertion.

Lemma 4.7. (see, e.g., [6, Lemma 3.1]). Let μ_{1}, μ_{2}, ν_{1}, ν_{2} be anisotropic quadratic forms over k. Suppose that the form μ_{1} ⊥ tν_{1} is similar to μ_{2} ⊥ tν_{2} over the field of rational functions k(t). Then

• either μ_{1} ∼ μ_{2} and ν_{1} ∼ ν_{2},

• or μ_{1} ∼ ν_{2} and ν_{1} ∼ μ_{2}.

Since ϕ_{1} ≁ ϕ_{2} and dim ρ < dim ϕ_{1} = dim ϕ_{2}, Lemma 4.7 shows that (tρ ⊥ −ϕ_{1}) ≁ (tρ ⊥ −ϕ_{2}). Hence q_{1} ≁ q_{2}. On the other hand, the conditions q_{1}, q_{2} ∈ I^{3}(F) and ϕ_{1} ≡ ϕ_{2} (mod I^{4}(F)) imply that

q_{1} ≡ tr_{1}q_{1} ≡ (tρ ⊥ −ϕ_{1}) ≡ (tρ ⊥ −ϕ_{2}) ≡ tr_{2}q_{2} ≡ q_{2} (mod I^{4}(F)).

Thus, we have proved that q_{1} and q_{2} are anisotropic 3-Albert forms such that q_{1} ≡ q_{2} (mod I^{4}(F)) and q_{1} ≁ q_{2}. The theorem is proved.

Corollary 4.8. For any n ≥ 3, there exists a field E and n-Albert forms γ_{1} and γ_{2} over E such that γ_{1} ≡ γ_{2} (mod I^{n+1}(E)) and γ_{1} ≁ γ_{2}. In other words, the answer to Question 4.3 is negative for any n ≥ 3.

Proof. Let q_{1}, q_{2} and F be as in Theorem 4.4. We write q_{1} and q_{2} in the form q_{1} = π′_{1} ⊥ −τ′_{1}, q_{2} = π′_{2} ⊥ −τ′_{2} with π_{1}, π_{2}, τ_{1}, τ_{2} ∈ P_{3}(F) and put E = F (x_{1}, …, x_{n−3}) and

γ_{1} = (π_{1} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩)′ ⊥ −(τ_{1} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩)′,
γ_{2} = (π_{2} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩)′ ⊥ −(τ_{2} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩)′.

Obviously, γ_{i} = q_{i} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩ in the Witt ring W (E). Since q_{1} ≡ q_{2} (mod I^{4}(F)),

we have γ_{1} ≡ γ_{2} (mod I^{n+1}(E)). Since q_{1} ≁ q_{2}, we have q_{1} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩ ≁

q_{2} ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩ (see, e.g., Lemma 4.7). Hence γ_{1} ≁ γ_{2}.

We have constructed a pair of n-Albert forms γ_{1} and γ_{2} such that γ_{1} ∼^{m} γ_{2} and γ_{1} ≁ γ_{2}. Obviously, in our example, we have dim(γ_{i})_{an} = 2^{n−3} · 14 = 2^{n−3}(2^{3} − 2) = 2(2^{n} − 2^{n−3}). In other words, both n-Albert forms γ_{1} and γ_{2} are (n − 3)-linked. We can generalize this example as follows.

Theorem 4.9. For any n ≥ 3 and m such that 0 ≤ m ≤ n − 3, there exists a field F and n-Albert forms q_{1} and q_{2} over F such that q_{1} ≡ q_{2} (mod I^{n+1}(F)), q_{1} ≁ q_{2}, and dim(q_{1})_{an} = dim(q_{2})_{an} = 2(2^{n} − 2^{m}).

Here we only outline the proof of the theorem.

Step 1. It suffices to prove this theorem only in the case m = 0 (this means that q_{1} and q_{2} are anisotropic). After this, the general case can be obtained in the same way as Corollary 4.8.

Step 2. Consider a field E and n-Albert forms γ_{1} and γ_{2} as in Corollary 4.8. Since γ_{1} ≡ γ_{2} (mod I^{n+1}(E)), there exist π_{1}, …, π_{N} ∈ P_{n+1}(E) for some integer N such that γ_{1} − γ_{2} = ∑_{i=1}^{N} π_{i}. We consider the quadratic forms

q̃_{1} = ⟨ ; ⟨x_{1}, …, x_{n}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨y_{1}, …, y_{n}⟩ ; ⟩′ ,
q̃_{2} = ⟨ ; ⟨z_{1}, …, z_{n}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨t_{1}, …, t_{n}⟩ ; ⟩′ ,
τ =⊥_{i=1}^{N} ⟨ ; ⟨u_{i,1}, …, u_{i,n+1}⟩ ; ⟩ .

over the field of rational functions

Ẽ = E(x_{1}, …, x_{n}, y_{1}, …, y_{n}, z_{1}, …, z_{n}, t_{1}, …, t_{n}, u_{1,1}, …, u_{N,n+1}).

Obviously there exists a place s̃ : Ẽ → E such that q̃_{1} ↦ γ_{1}, q̃_{2} ↦ γ_{2}, and

⟨ ; ⟨u_{i,1}, …, u_{i,n+1}⟩ ; ⟩ ↦ π_{i} for all i = 1, …, N. Since γ_{1} − γ_{2} = ∑_{i=1}^{N} π_{i}, the form

s̃_{∗}(q̃_{1} ⊥ −q̃_{2} ⊥ −τ) is hyperbolic.

Step 3. We define the field F as a “generic” extension F/Ẽ such that (q̃_{1})_{F} − (q̃_{2})_{F} = τ_{F}. More precisely, we set F = Ẽ_{h}, where Ẽ_{0}, Ẽ_{1}, …, Ẽ_{h} is the generic splitting tower for the Ẽ-form q̃_{1} ⊥ −q̃_{2} ⊥ −τ. We claim that the F -forms q_{1} =^{Def} (q̃_{1})_{F} and q_{2} =^{Def} (q̃_{1})_{F} satisfy the hypotheses of Theorem 4.9. Since q_{1} − q_{2} = τ_{F}, we have q_{1} ≡ q_{2} (mod I^{n+1}(F)). Thus, it suffices to verify that q_{1} and q_{2} are anisotropic and q_{1} ≁ q_{1}.

Step 4. Using properties of generic splitting fields (see [23, Ch. 4, Cor. 6.10] or [11, Th. 5.1]), we can extend s̃: Ẽ → E to a place s : F → E. Obviously, s_{∗}(q_{1}) = γ_{1} and s_{∗}(q_{2}) = γ_{2}. Therefore, the condition γ_{1} ≁ γ_{2} implies q_{1} ≁ q_{2}.

Step 5. To prove that q_{1} and q_{2} are anisotropic, it suffices to construct a field extension K/Ẽ with the same key property as F (i.e., (q̃_{1})_{K} − (q̃_{2})_{K} = τ_{K}) and such that (q̃_{1})_{K} and (q̃_{2})_{K} are anisotropic. Since F/Ẽ is a “generic” extension, we necessarily get that q_{1} = (q̃_{1})_{F} and q_{2} = (q̃_{2})_{F} are anisotropic. The following extension K/Ẽ has the required properties:

K = Ẽ(\sqrt{\frac{x_{1}}{z_{1}}} , …, \sqrt{\frac{x_{n}}{z_{n}}} , \sqrt{\frac{y_{1}}{t_{1}}} , …, \sqrt{\frac{y_{n}}{t_{n}}} , \sqrt{u_{1,1}}, …, \sqrt{u_{N,1}}).

The “sketch” of the proof is complete. In fact, Steps 4 and 5 are the most difficult points. We refer the reader to the paper [7, Proof of Lemma 2.2], where similar arguments (as in Step 5) are presented with complete proofs.

Corollary 4.10. For any m and n such that 0 ≤ m ≤ n − 3, there exists a field F and anisotropic 2(2^{n} − 2^{m})-dimensional forms q_{1} and q_{2} over F such that q_{1} ∼^{v} q_{2} and q_{1} ≁ q_{2}.

5. Open questions

Obviously, Theorem 4.9 cannot be generalized to the cases m = n − 1 and m = n because in these cases the anisotropic parts of n-Albert forms either belong to GP_{n}(F) or are zero. There is only one case, where we cannot say anything definite. Namely, m = n − 2. For this reason, we propose the following modification of Lam’s Question 4.3.

Conjecture 5.1. Let q_{1} and q_{2} be Albert forms (i.e., 6-dimensional forms with trivial discriminants). Let ϕ_{1} = ⟨ ; ⟨a_{1}, …, a_{k}⟩ ; ⟩ q_{1} and ϕ_{2} = ⟨ ; ⟨b_{1}, …, b_{k}⟩ ; ⟩ q_{2}. Suppose that ϕ_{1} ≡ ϕ_{2} (mod I^{k+3}(F)). Then ϕ_{1} ∼ ϕ_{2}.

We note that, in this conjecture, we always may assume that a_{i} = b_{i} for i =

1, …, k. Indeed, putting π = ⟨ ; ⟨a_{1}, …, a_{k}⟩ ; ⟩, we obtain (ϕ_{2})_{F (π)} ≡ (ϕ_{1})_{F (π)} = 0

(mod I^{k+1}(F (π))). By the Arason–Pfister theorem, we conclude that ϕ_{2} is hyperbolic over the field F (π). Hence ϕ_{2} has the form ϕ_{2} = πq′_{2} = ⟨ ; ⟨a_{1}, …, a_{k}⟩ ; ⟩ q′_{2}. Comparing dimensions, we get dim q′_{2} = 6. Let us write q′_{2} in the form q′_{2} = ⟨c_{1}, …, c_{6}⟩ and set q′′_{2} = ⟨c_{1}, …, c_{5}, c′_{6}⟩, where c′_{6} = −c_{1} … c_{5}. We have π ⟨c_{6}, −c′_{6}⟩ = πq′_{2} − πq′′_{2} = ϕ_{2} − πq′′_{2} ∈ I^{k+2}(F) + I^{k}(F) · I^{2}(F) = I^{k+2}(F). Since dim π ⟨c_{6}, −c′_{6}⟩ = 2^{k} · 2 < 2^{k+2}, the Arason–Pfister theorem shows that π ⟨c_{6}, −c′_{6}⟩ is hyperbolic. Hence πq′_{2} = πq′′_{2}. Therefore, ϕ_{2} = πq′′_{2} = ⟨ ; ⟨a_{1}, …, a_{k}⟩ ; ⟩ q′′_{2}. Since q′′_{2} is an Albert form, we have proved, that the conjecture reduces to the case where b_{i} = a_{i}.

Another question concerning the ∼^{v}-equivalence is motivated by the results of §3 and §4. First of all, in view of Lemma 3.4 and Corollary 4.10, we have the following assertion.

Proposition 5.2. Let d be an integer belonging to the set

{2^{n}|n ≥ 3} ∪ {2^{i}(2^{j} − 1)|i ≥ 1, j ≥ 3}

Then there exist anisotropic d-dimensional quadratic forms ϕ and ψ over a suitable field such that ϕ ∼^{v} ψ and ϕ ≁ ψ.

Here we state the following

Problem 5.3. Describe the set VE of all integers d for which there exist anisotropic d-dimensional quadratic forms ϕ and ψ over a suitable field such that ϕ ∼^{v} ψ and ϕ ≁ ψ.

We know almost the full answer to this problem. The results of the previous sections imply that VE ⊂ {8, 10, 12, …, 2i, … }. Besides, we can prove that any even integer ≥ 8 (except possibly 12) belongs to VE.

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Oleg Izhboldin

Department of Mathematics and Mechanics St.-Petersburg State University Petrodvorets, 198904 Russia oleg@mathematik.uni-bielefeld.de 352

Documenta Mathematica 3 (1998)