Motivic Equivalence

and Similarity of Quadratic Forms

To Sasha Merkurjev on the occasion of his 60th birthday

Detlev W. Hoffmann

Received: September 25, 2014 Revised: November 20, 2014

Abstract. A result by Vishik states that given two anisotropic quadratic forms of the same dimension over a field of characteristic not 2, the Chow motives of the two associated projective quadrics are isomorphic iff both forms have the same Witt indices over all field extensions, in which case the two forms are called motivically equivalent. Izhboldin has shown that if the dimension is odd, then motivic equivalence implies similarity of the forms. This also holds for even dimension ≤ 6, but Izhboldin also showed that this generally fails in all even dimensions ≥ 8 except possibly in dimension 12. The aim of this paper is to show that motivic equivalence does imply similarity for fields over which quadratic forms can be classified by their classical invariants provided that in the case of formally real such fields the space of orderings has some nice properties. Examples show that some of the required properties for the field cannot be weakened.

2010 Mathematics Subject Classification: Primary: 11E04; Secondary: 11E81, 12D15, 14C15 Keywords and Phrases: quadratic form, quadric, function field of a quadric, generic splitting, similarity, motivic equivalence, formally real field, effective diagonalization

1. Introduction

Throughout this note, we will consider only fields of characteristic not 2. By a form over F we will mean a finite dimensional nondegenerate quadratic form over F, and by a quadric over F a smooth projective quadric X_{φ} = {φ = 0} for some form φ over F. An important theme in the theory of quadratic forms is the study of forms in terms of geometric properties of their associated quadrics. Suppose, for example, that for two given forms φ and ψ over F one has that the motives M(X_{φ}) and M(X_{ψ}) are isomorphic in the category of Chow motives, in which case we call φ and ψ motivically equivalent and we write φ ∼^{mot} ψ. Does this already imply that the quadrics are isomorphic as projective varieties ? The converse is of course trivially true. It is well known that the quadrics X_{φ} and X_{ψ} are isomorphic iff φ and ψ are similar (see, e.g. [18, Th. 2.2]), i.e. there exists c ∈ F^{×} = F \{0} with φ ≅ cψ in which case we write φ ∼^{sim} ψ. The above question then reads as follows: Let φ and ψ be forms of the same dimension over F. Does φ ∼^{mot} ψ imply φ ∼^{sim} ψ? In fact, Izhboldin has shown that the answer is yes if dim φ is odd ([14, Cor. 2.9]) or even and at most 6 ([14, Prop. 3.1]), and that there are counterexamples in every even dimension ≥ 8 except possibly 12 over suitably chosen fields ([15, Th. 0.1]). To our knowledge, it seems to be still open if such counterexamples exist in dimension 12. The purpose of the present note is to give criteria for fields that guarantee that motivic equivalence implies similarity in all dimensions. We show that it holds for fields over which forms of a given dimension can be classified by their classical invariants determinant, Clifford invariant and signatures provided that in the case of formally real fields the space of orderings satisfies a certain property called effective diagonalization ED (which will be defined below). We show furthermore that there are counterexamples once the condition ED is only slightly weakened. Rather than working with motives of quadrics, we will use an alternative criterion for motivic equivalence due to Vishik [24, Th. 1.4.1] (see also Vishik [25, Th. 4.18] or Karpenko [16, § 5]). If we denote the Witt index of a form φ by i_{W} (φ), this important criterion reads as follows.

Vishik’s Criterion 1.1. Let φ and ψ be forms over F with dim φ = dim ψ. Then φ ∼^{mot} ψ if and only if i_{W} (φ_{E}) = i_{W} (ψ_{E}) for every field extension E/F.

Let us remark that while Vishik formulated his criterion in terms of integral Chow motives, it still holds for Chow motives with ℤ/2ℤ coefficients, see [8]. The proofs of our results will concern mainly formally real fields (in the sequel we will call such fields real for short). For nonreal fields, the results are still valid but can often be shown in a much quicker and simpler fashion. The real case will involve various arguments concerning the space of orderings X_{F} of a real field and the signatures sgn_{P} (φ) of a form φ over F with respect to an ordering P ∈ X_{F}.

Motivic Equivalence and Similarity of Quadratic Forms 267

Consider the Witt ring W F and the torsion ideal W_{t}F (we have W F = W_{t}F iff F in nonreal). By Pfister’s local-global principle (see, e.g., [20, Ch. VIII, Th. 3.2]), a form φ is torsion iff sgn_{P} (φ) = 0 for all P ∈ X_{F}. We call a form totally indefinite if | sgn_{P} (φ)| < dim φ for all P ∈ X_{F}. Also, we will use the fact that the Witt ring only contains 2-primary torsion. Let IF be the fundamental ideal in W F generated by even-dimensional forms in F and let I^{n}F = (IF)^{n}. We define I_{t}^{n}F = I^{n}F ∩ W_{t}F. A real field F is said to satisfy effective diagonalization (ED) if any form φ over F has a diagonalization ⟨a_{1}, …, a_{n}⟩ such that for all 1 ≤ i < n and for all P ∈ X_{F} one has a_{i} <_{P} 0 ⟹ a_{i+1} <_{P} 0 (see [26] or [23]). Recall that the u-invariant and the Hasse number ũ are defined as follows:

u(F) = sup{dim φ | φ is anisotropic and φ ∈ W_{t}F }
ũ(F) = sup{dim φ | φ is anisotropic and totally indefinite}

For nonreal F, we thus have u(F) = ũ(F). It is also well known that these invariants cannot take the values 3, 5, 7 (see [5, Ths. F–G] for the more involved case ũ for real fields). Our main result reads as follows.

Main Theorem 1.2. Let F be an ED-field and let φ, ψ be anisotropic forms over F of the same dimension. If φ ∼^{mot} ψ then there exists x ∈ F^{×} such that φ ⊥ −xψ ∈ I_{t}^{3}F.

Corollary 1.3. Let F be an ED-field with I_{t}^{3}F = 0 and let φ, ψ be anisotropic forms over F of the same dimension. Then φ ∼^{mot} ψ if and only if φ ∼^{sim} ψ.

Recall that fields with I_{t}^{3}F = 0 are exactly those fields over which quadratic forms can be classified by their classical invariants dimension, (signed) determinant, Clifford invariant and signatures, see [4]. Now fields with finite ũ are always ED (see, e.g., [7, Th. 2.5]). By the Arason-Pfister Hauptsatz (see, e.g., [20, Ch. X, 5.1]) we thus get

Corollary 1.4. Let F be a field with ũ(F) ≤ 6 and let φ, ψ be anisotropic forms over F of the same dimension. Then φ ∼^{mot} ψ if and only if φ ∼^{sim} ψ.

This corollary applies to global fields for which ũ = 4 (this follows from the well known Hasse-Minkowski theorem) and fields of transcendence degree one over a real closed field for which ũ = 2 (see, e.g., [5, Th. I]). However, for each k ∈ {2n | n ∈ ℕ} ∪ {∞} there exist ED-fields F (in fact, fields F with a unique ordering) with ũ(F) = k and I_{t}^{3}F = 0 (see [13, Th. 2.7] or [11, Th. 3.1]) to which Corollary 1.3 can still be applied. In § 2, we investigate how determinants and Clifford invariants behave under motivic equivalence. The third section does the same for signatures and there we also prove the main theorem by putting all this together. In § 4, we give a few examples that show that under weakening some of the imposed conditions, one cannot expect any longer that motivic equivalence implies similarity.

Acknowledgment. I am grateful to the anonymous referees for their speedy work and their valuable comments and suggestions that helped to improve the exposition of this article.

2. Comparing determinants and Clifford invariants

We will freely use without reference various basic facts from the algebraic theory of quadratic forms in characteristic ≠ 2. All such facts and any unexplained terminology can be found in the books [20] or [3]. If φ is a form defined on an F -vector space V, we put D_{F} (φ) = {φ(x) | x ∈ V }∩F^{×}. We use the convention ⟨⟨a_{1}, …, a_{n}⟩⟩ to denote the n-fold Pfister form ⟨1, −a_{1}⟩⊗…⊗⟨1, −a_{n}⟩. A form φ over a field F is called a Pfister neighbor if there exists a Pfister form π over F and some a ∈ F^{×} such that aφ is a subform of π (i.e. there exists another form ψ over F with aφ ⊥ ψ ≅ π) and 2 dim φ > dim π. Since such a Pfister form π is known to be either anisotropic or hyperbolic, it follows that a Pfister neighbor φ of π is anisotropic iff π is anisotropic. We call two forms φ and ψ over F half-neighbors if there exist an integer n ≥ 0, a, b ∈ F^{×} and an (n + 1)- fold Pfister form π such that dim φ = dim ψ = 2^{n} and aφ ⊥ −bψ ≅ π. Now in this situation, if E is any field extension of F over which φ or ψ is isotropic then π_{E} is hyperbolic and thus aφ_{E} ≅ bψ_{E} and it readily follows that φ ∼^{mot} ψ. Thus, a good way to construct examples of nonsimilar motivically equivalent forms is to find nonsimilar half-neighbors, see § 4. The function field F (φ) of a form φ is defined to be the function field of the associated quadric F (X_{φ}) (we put F (φ) = F if dim φ = 1 or φ a hyperbolic plane). In the sequel, we state some definitions and facts concerning generic splitting of quadratic forms. We refer to Knebusch’s original paper [17] on that topic for details. Let φ be a form over F. The generic splitting tower of φ is constructed inductively as follows. Let F = F_{0} and φ_{0} = φ_{an} be its anisotropic part over F. Suppose that for i ≥ 0 we have constructed the field extension F_{i}/F. Consider the anisotropic form φ_{i} ≅ (φ_{F_{i}})_{an}. If dim φ_{i} ≥ 2 we put F_{i+1} = F_{i}(φ_{i}) and φ_{i+1} ≅ (φ_{F_{i+1}})_{an}. Note that if dim φ_{i} ≥ 2, we have 2i_{W} (φ_{F_{i}}) = dim φ − dim φ_{i} < 2i_{W} (φ_{F_{i+1}}) or, equivalently, dim φ_{i} > dim φ_{i+1}. The smallest h such that dim φ_{h} ≤ 1 is called the height of φ. The generic splitting tower of φ is then given by

F = F_{0} ⊂ F_{1} ⊂ … ⊂ F_{h−1} ⊂ F_{h} .

F_{h−1} is called the leading field of φ. It is known that

𝔖_{a}(φ) := {i_{W} (φ_{E}) | E/F field extension} = {i_{W} (φ_{F_{i}}) | 0 ≤ i ≤ h} .

We call 𝔖_{a}(φ) the absolute splitting pattern of φ. In the literature, it has often proved to be of advantage to consider instead the relative splitting pattern 𝔖_{r}(φ) defined as follows. If 𝔖_{a}(φ) = {i_{ℓ} = i_{W} (φ_{F_{ℓ}}) | 0 ≤ ℓ ≤ h}, then put j_{m} = i_{m} − i_{m−1}, 1 ≤ m ≤ h, the increase of the Witt index at the m-th step in the splitting tower. Then 𝔖_{r}(φ) = (j_{1}, …, j_{h}) as an ordered sequence, but we won’t need this here.

The degree deg(φ) is defined as follows. If the dimension of φ is odd, then deg(φ) = 0. If φ is hyperbolic one defines deg(φ) = ∞. So suppose φ is not hyperbolic and dim φ is even. Then the anisotropic form φ_{h−1} over F_{h−1} becomes hyperbolic over its own function field F_{h} = F_{h−1}(φ_{h−1}) and is thus similar to an n-fold Pfister form for some n ≥ 1. We then define deg(φ) = n. Now the above implies that if φ is not hyperbolic then

2^{deg(φ)} = min{dim(φ_{E})_{an} | E/F is a field extension with φ_{E} not hyperbolic} ,

and it follows that if dim(φ_{E})_{an} = 2^{deg(φ)}, then (φ_{E})_{an} is similar to an n-fold Pfister form over E. An important and deep theorem which we will also use states that I^{n}F = {φ ∈ W F | deg φ ≥ n}, see [22, Th. 4.3]. While part (i) of the following lemma is rather trivial, part (ii) is a bit less so and seems to be due to Izhboldin (see [16, Remark 2.7]) but to our knowledge a proof was not yet in the literature, so we included one for the reader’s convenience.

Lemma 2.1. Let φ and ψ be anisotropic forms over F with φ ∼^{mot} ψ. Then

(i) deg(φ) = deg(ψ);
(ii) For every a ∈ F^{×} we have deg(φ ⊥ −aψ) > deg(φ).

Proof. Part (i) follows immediately from the definition of degree and Vishik’s criterion for motivic equivalence. Let now deg(φ) = deg(ψ) = n. Part (ii) is trivial for n = 0, so assume n ≥ 1. If φ ⊥ −aψ is hyperbolic there is nothing to show. So assume τ ≅ (φ ⊥ −aψ)_{an} ≠ 0. By the degree characterization of I^{n}F, we have τ ∈ I^{n}F and hence deg(τ) ≥ n. Suppose deg(τ) = n. Let E/F be the leading field of φ. By what was said preceding the lemma, (φ_{E})_{an} and (ψ_{E})_{an} are anisotropic n-fold Pfister forms which are clearly motivically equivalent and thus similar (this follows readily from, e.g., [20, Ch. X, Cor. 4.9]). Hence, there exist an n-fold Pfister form π over E and x, y ∈ E^{×} such that in W E, φ_{E} = xπ, ψ_{E} = yπ. Thus, τ_{E} = ⟨x, −ay⟩⊗π ∈ I^{n+1}E and therefore deg(τ) = n < n+1 ≤ deg(τ_{E}). But this implies deg(φ) ≤ n − 2 by [1, Satz 19], a contradiction. □

The signed determinant of a form φ over F will be denoted by d(φ). For a diagonalization φ ≅ ⟨a_{1}, …, a_{n}⟩ we have d(φ) = (−1)^{n(n−1)/2} ∏_{i=1}^{n} a_{i} ∈ F^{×}/F^{×2} and the map φ ↦ d(φ) induces an isomorphism IF/I^{2}F → F^{×}/F^{×2}. The Clifford invariant c(φ) of φ is defined as follows. The Clifford algebra C(φ) is a central simple algebra over F if dim φ is even, and its even part C_{0}(φ) is central simple if dim φ is odd. In both cases, these algebras are Brauer-equivalent to a tensor product of quaternion algebras and thus their classes lie in the 2-torsion part Br_{2}(F) of the Brauer group of F. One defines

c(φ) = \{ \begin{matrix}[C(φ)] ∈ Br_{2}(F) & if dim φ even \\ [C_{0}(φ)] ∈ Br_{2}(F) & if dim φ odd \end{matrix}

By Merkurjev’s theorem [21], c induces an isomorphism I^{2}F/I^{3}F → Br_{2}(F).

Corollary 2.2. Let φ and ψ be forms over F of even dimension dim φ = dim ψ. Let d = d(φ) ∈ F^{×}/F^{×2} and K = F if d = 1 and K = F (\sqrt{d}) if d ≠ 1. If φ ∼^{mot} ψ then d = d(φ) = d(ψ) and c(φ_{K}) = c(ψ_{K}).

Proof. We have φ, ψ ∈ IF and also φ ⊥ −ψ ∈ I^{2}F and thus φ ≡ ψ mod I^{2}F since φ ∼^{mot} ψ and by Lemma 2.1. The above isomorphism IF/I^{2}F ≅ F^{×}/F^{×2} immediately implies d(φ) = d(ψ). Now over K we then have φ_{K}, ψ_{K} ∈ I^{2}K since d(φ_{K}) = d(ψ_{K}) = 1. This time, Lemma 2.1 yields φ_{K} ≡ ψ_{K} mod I^{3}K and by invoking Merkurjev’s theorem we readily get c(φ_{K}) = c(ψ_{K}). □

Corollary 2.3. Let φ and ψ be forms over F of even dimension dim φ = dim ψ. Let d = d(φ) ∈ F^{×}/F^{×2} and suppose that φ ∼^{mot} ψ.

(i) There exists a ∈ F^{×} such that φ ⊥ −ψ ≡ ⟨⟨a, d⟩⟩ mod I^{3}F.

(ii) With a as in (i), if b ∈ F^{×}, then φ ⊥ −bψ ≡ ⟨⟨ab, d⟩⟩ mod I^{3}F.

In particular, with a as before, we have φ ⊥ −aψ ∈ I^{3}F.

Proof. (i) If d = 1 then Corollary 2.2 together with Merkurjev’s theorem implies φ, ψ ∈ I^{2}F and φ ⊥ −ψ ≡ 0 mod I^{3}F. The result follows since ⟨⟨a, d⟩⟩ = ⟨⟨a, 1⟩⟩ = 0 in W F for any a ∈ F^{×}. If d ≠ 1, we still have φ ⊥ −ψ ∈ I^{2}F since d(ψ) = d and this time for K = F (\sqrt{d}) that (φ ⊥ −ψ)_{K} ∈ I^{3}K. Hence, the central simple F -algebra C(φ ⊥ −ψ) splits over the quadratic extension K, so its index is at most 2 and it is well known that then there exists a quaternion algebra (a, d)_{F} for some a ∈ F^{×} such that C(φ ⊥ −ψ) ∼ (a, d)_{F} in Br_{2}(F). Hence, it follows again readily from Merkurjev’s theorem and the fact that c(⟨⟨a, d⟩⟩) = [(a, d)_{F}] that we have φ ⊥ −ψ ≡ ⟨⟨a, d⟩⟩ mod I^{3}F. (ii) We have φ ⊥ −ψ, ψ ⊥ −bψ ∈ I^{2}F and −ψ ⊥ ψ = 0 ∈ W F. Furthermore, by denoting the class of a quaternion algebra by its own symbol and using well known rules for manipulating Clifford invariants (see, e.g., [20, p. 118]), we get

c(φ ⊥ −bψ) = c(φ ⊥ −ψ ⊥ ψ ⊥ −bψ)
= c(φ ⊥ −ψ)c(ψ ⊥ −bψ)
= (a, d)_{F} c(ψ)c(−dbψ)
= (a, d)_{F} c(ψ)c(ψ)(−db, d)_{F}
= (ab, d)_{F} .

We conclude as in (i) that now φ ⊥ −bψ ≡ ⟨⟨ab, d⟩⟩ mod I^{3}F. □

3. Comparing signatures and proof of the Main Theorem

The following lemma compares signatures of motivically equivalent forms.

Lemma 3.1. Let φ and ψ be forms of the same dimension over a real field F. If φ ∼^{mot} ψ then | sgn_{P} (φ)| = | sgn_{P} (ψ)| for all P ∈ X_{F}.

Proof. We first note that if γ is any form of dimension ≥ 2 over any real field K and if Q ∈ X_{K}, then for L = K(γ) we have that Q extends to an ordering Q′ ∈ X_{L} iff γ is indefinite at Q, i.e. dim γ > | sgn_{Q}(γ)| (see, e.g. [6, Th. 3.5]). In this case, we clearly have sgn_{Q}(γ) = sgn_{Q′}(γ_{L}) which implies dim(γ_{L})_{an} ≥ | sgn_{Q′}(γ_{L})| = | sgn_{Q}(γ)|. Applied to φ, ψ and P ∈ X_{F}, it now follows readily that there exists an extension E/F with E in the generic splitting tower of φ such that P extends to P′ ∈ X_{E} and

dim(φ_{E})_{an} = | sgn_{P′}(φ_{E})| = | sgn_{P} (φ)| .

By motivic equivalence, we have dim(φ_{E})_{an} = dim(ψ_{E})_{an} and hence

| sgn_{P} (φ)| = dim(ψ_{E})_{an} ≥ | sgn_{P′} ψ_{E}| = | sgn_{P} ψ| .

By symmetry, we also have | sgn_{P} ψ| ≥ | sgn_{P} (φ)|. □

Remark 3.2. The above proof also shows that \frac{1}{2}(dim φ − | sgn_{P} (φ)|) ∈ 𝔖_{a}(φ), a fact that was already noticed in [9, Prop. 2.2].

We need a few properties regarding spaces of orderings of real fields. For more details regarding the following, we refer to [19], [7], [23]. Recall that the space of orderings X_{F} is a topological space whose topology has as sub-basis the socalled Harrison sets H(a) = {P ∈ X_{F} | a >_{P} 0} for a ∈ F^{×}. These are clopen sets, and F has the strong approximation property SAP if each clopen set is a Harrison set. F has the property S_{1} if every binary torsion form represents a totally positive element. SAP and S_{1} together are equivalent to ED, see [23, Th. 2].

Lemma 3.3. Let F be a real SAP field and let φ and ψ be forms over F of the same dimension with φ ∼^{mot} ψ. Then there exist a, b ∈ F^{×} such that sgn_{P} (aφ) = sgn_{P} (bψ) ≥ 0 for all P ∈ X_{F}.

Proof. Let U = {P ∈ X_{F} | sgn_{P} (φ) < 0}. Then U ⊂ X_{F} is clopen and SAP implies that there exists a ∈ F^{×} with U = H(−a). Then sgn_{P} (aφ) ≥ 0 for all P ∈ X_{F}. Similarly, there exists b ∈ F^{×} with sgn_{P} (bψ) ≥ 0 for all P ∈ X_{F}. Since aφ ∼^{mot} φ ∼^{mot} ψ ∼^{mot} bψ, we have sgn_{P} (aφ) = sgn_{P} (bψ) for all P ∈ X_{F} by Lemma 3.1. □

Let ∑^{×} F^{2} denote the set of nonzero sums of squares in F. If F is nonreal, then it is well known that F^{×} = ∑^{×} F^{2}.

Lemma 3.4. Let F be a real S_{1} field and let φ and ψ be forms over F of the same dimension with φ ∼^{mot} ψ and sgn_{P} (φ) = sgn_{P} (ψ) for all P ∈ X_{F}. Then there exists s ∈ ∑^{×} F^{2} with φ ⊥ −sψ ∈ I_{t}^{3}F.

Proof. Note first that the signatures don’t change by scaling with an s ∈ ∑^{×} F^{2}. Hence φ ⊥ −sψ has total signature zero for any such s and thus φ ⊥ −sψ ∈ W_{t}F. On the other hand, by Corollary 2.3, there exists a ∈ F^{×} with φ ⊥ −ψ ≡ ⟨⟨a, d⟩⟩ mod I^{3}F where d = d(φ) = d(ψ) ∈ F^{×}/F^{×2}. Now if P ∈ X_{F} and if π is an n-fold Pfister form over F, then sgn_{P} (π) ∈ {0, 2^{n}}, hence, for τ ∈ I^{n}F we have sgn_{P} (τ) ≡ 0 mod 2^{n}. Now comparing signatures mod 8 immediately yields that ⟨⟨a, d⟩⟩ ≅ ⟨1, −a, −d, ad⟩ has total signature zero and is therefore torsion. Consider the n-fold Pfister form σ_{n} ≅ 2^{n} ×⟨1⟩. For n large enough, the (n+2)- fold Pfister form σ_{n} ⊗ ⟨1, −a, −d, ad⟩ will now be hyperbolic, so its Pfister neighbor σ_{n} ⊗ ⟨1, −d⟩ ⊥ ⟨−a⟩ will be isotropic. It follows readily that there exist u, v ∈ D_{F} (σ_{n}) ⊆ ∑^{×} F^{2} with ⟨u, −a, −dv⟩ isotropic, so in particular, au ∈ D_{F} (⟨1, −duv⟩). Since uv ∈ ∑^{×} F^{2}, we can apply the characterization of S_{1} in [12, Lemma 2.2(iii)] to find t ∈ ∑^{×} F^{2} such that aut ∈ D_{F} (⟨1, −d⟩). But then s := ut ∈ ∑^{×} F^{2} and ⟨1, −as, −d⟩ is isotropic. Therefore the Pfister form ⟨⟨as, d⟩⟩ is hyperbolic, i.e. ⟨⟨as, d⟩⟩ = 0 in W F. By the above and Corollary 2.3, we now have φ ⊥ −sψ ∈ W_{t}F ∩ I^{3}F = I_{t}^{3}F as desired. □

Proof of Main Theorem 1.2. Let F be an ED-field and let φ, ψ be anisotropic forms over F of the same dimension n with φ ∼^{mot} ψ. We have to show that there exists x ∈ F^{×} such that φ ⊥ −xψ ∈ I_{t}^{3}F. The theorem is trivial for odd n by Izhboldin’s result because it implies φ ∼^{sim} ψ. So we may assume that n is even. If F is nonreal (in which case I_{t}^{3}F = I^{3}F and ED is an empty condition), the result follows already from Corollary 2.3 with x = b = a. So suppose that F is real. Now ED is equivalent to SAP plus S_{1}. Because of SAP, we may assume by Lemma 3.3 that, possibly after scaling, sgn_{P} (φ) = sgn_{P} (ψ) for all P ∈ X_{F}. Since we also have S_{1}, we can apply Lemma 3.4 to conclude. □

4. Examples

The following two examples show that in Corollary 1.3 the condition I_{t}^{3}F = 0 does not suffice for motivic equivalence to imply similarity once the condition ED is only slightly weakened.

Example 4.1. Let F = ℝ((x))((y)) be the iterated power series field in two variables x, y over the reals. It is well known that S = {±1, ±x, ±y, ±xy} is a set of representatives of F^{×}/F^{×2}. Let τ_{n} ≅ n × ⟨1⟩ (where we allow the 0-dimensional form τ_{0}). Then Springer’s theorem implies that up to isometry the anisotropic forms over F are exactly the forms of type

ϵ_{1}τ_{n_{1}} ⊥ ϵ_{2}xτ_{n_{2}} ⊥ ϵ_{3}yτ_{n_{3}} ⊥ ϵ_{4}xyτ_{n_{4}}

with ϵ_{i} ∈ {±1} and n_{i} ≥ 0, and that the isometry type is uniquely determined by the four pairs (ϵ_{i}, n_{i}) (see, e.g., [20, Ch. VI, Cor. 1.6, Prop 1.9]). Since u(ℝ) = 0, it also follows from the above that u(F) = 0, in particular W_{t}F = I_{t}^{3}F = 0. Now consider the anisotropic forms

φ ≅ ⟨1, 1, 1, x, x, x, y, y⟩ and ψ ≅ ⟨1, x, y, y, xy, xy, xy, xy⟩ .

We have φ ⊥ ψ ≅ ⟨⟨−1, −1, −x, −y⟩⟩, so φ and ψ are half-neighbors and thus φ ∼^{mot} ψ. However, one also readily sees that there is no s ∈ S with sφ ≅ ψ, hence φ ≁^{sim} ψ. Of course, it is also well known that F lacks the property SAP and thus ED as, for example, the totally indefinite form ⟨1, x, y, −xy⟩ is not weakly isotropic. We can be more precise. Recall that the reduced stability index st(F) of a field F can be characterized as the least n such that I^{n+1}F = 2I^{n}F mod W_{t}F, and that SAP is equivalent to st(F) ≤ 1 (see [2]). For F = ℝ((x))((y)), we trivially have property S_{1} since W_{t}F = 0, and one also readily sees that st(F) = 2. Now Corollary 1.3 applies to fields with I_{t}^{3}F = 0, S_{1} and st(F) ≤ 1, but the above shows that generally, it cannot be extended to fields satisfying I_{t}^{3}F = 0, S_{1} and st(F) = 2. □

In [7], the property S_{1} has been generalized as follows. A field F is said to have property S_{n} for n ≥ 1 if for every n-fold Pfister form π ≅ ⟨1⟩ ⊥ π′ over F and every a ∈ ∑^{×} F^{2} there exists an m ≥ 1 with

D_{F} (⟨1, −a⟩) ∩ D_{F} (⟨\underbrace{1, …, 1}_{m}⟩ ⊗ π′) ≠ ∅ .

Example 4.2. It is not difficult to construct real fields K with |K^{×}/K^{×2}| = 4 and where the square classes are represented by {±1, ±2} (see, e.g., [20, Remark II.5.3]). Clearly, K is uniquely ordered and u(K) = ũ(K) = 2. Consider F = K((t)). Then u(F) = 4, so in particular I_{t}^{3}F = 0, F has two orderings (see, e.g., [20, Prop. VIII.4.11]) and thus is SAP. Furthermore, one readily checks that F has property S_{2}. Now consider the anisotropic forms

φ ≅ ⟨1, 1, 1, 1, 1, 1⟩ ⊥ t⟨1, 2⟩ and ψ ≅ ⟨1, 1⟩ ⊥ t⟨1, 1, 1, 1, 1, 2⟩ .

Since ⟨1, 1⟩ ≅ ⟨2, 2⟩ we have φ ⊥ ψ ≅ ⟨⟨−1, −1, −1, −t⟩⟩. So φ and ψ are halfneighbors and hence φ ∼^{mot} ψ. On the other hand, since 2 ∉ F^{×2}, it follows readily that φ ≁^{sim} ψ. Hence, in general, Corollary 1.3 cannot be extended to fields satisfying I_{t}^{3}F = 0, S_{2} and SAP (i.e. st(F) ≤ 1). □

Note that the two forms in the previous example also provide motivically equivalent nonsimilar forms over ℚ((t)), a field that also satisfies S_{2} and SAP. However, this would give a weaker counterexample in the sense that I_{t}^{4}ℚ((t)) = 0 but I_{t}^{3}ℚ((t)) ≠ 0 as can be readily seen.

Example 4.3. If F is nonreal and u(F) < 2^{n+1}, then (n + 1)-fold Pfister forms will always be hyperbolic over F and thus half-neighbors of dimension 2^{n} will always be similar. However, in [10, Cor. 3.6], it was shown that for any n ≥ 3 there exist nonreal fields F with u(F) = 2^{n+1} over which one can find nonsimilar half-neighbors of dimension 2^{n}. In fact, one can take any nonreal field E with u(E) = 4 and take F = E((x_{1})) … ((x_{n−1})). As a consequence, there exist nonreal fields F with u(F) = 16 and motivically equivalent nonsimilar forms of dimension 8. □

It should be noted that to our knowledge, all constructions of nonsimilar motivically equivalent forms over nonreal fields (e.g. in [15]) require the existence of anisotropic 4-fold Pfister forms, so for these fields one would have I^{4}F ≠ 0 and in particular u(F) ≥ 16. Thus, also in view of the above examples, we ask the following.

Question 4.4. Are there fields F with u(F) < 16 which in the real case also satisfy ED, such that there exist nonsimilar motivically equivalent forms over F ?

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Detlev W. Hoffmann Fakultät für Mathematik Technische Universität Dortmund D-44221 Dortmund Germany detlev.hoffmann@math.tudortmund.de 276

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