Twisted Pfister Forms

Detlev W. Hoffmann^{1}

Received: November 29, 1995 Revised: February 27, 1996

Communicated by Alexander S. Merkurjev

Abstract. Let F be a field of characteristic ≠ 2. In this paper we investigate quadratic forms φ over F which are anisotropic and of dimension 2^{n}, n ≥ 2, such that in the Witt ring W F they can be written in the form φ = σ − π where σ and π are anisotropic n- resp. m-fold Pfister forms, 1 ≤ m < n. We call these forms twisted Pfister forms. Forms of this type with m = n − 1 are of great importance in the study of so-called good forms of height 2, and such forms with m = 1 also appear in Izhboldin’s recent proof of the existence of n-fold Pfister forms τ over suitable fields F, n ≥ 3, for which the function field F (τ) is not excellent over F. We first derive some elementary properties and try to give alternative characterizations of twisted Pfister forms. We also compute the Witt kernel W (F (φ)/F) of a twisted Pfister form φ. Our main focus, however, will be the study of the following problems: For which forms ψ does a twisted Pfister form φ become isotropic over F (ψ) ? Which forms ψ are equivalent to φ (i.e., the function fields F (φ) and F (ψ) are place-equivalent over F) ? We also investigate how such twisted Pfister forms behave over the function field of a Pfister form of the same dimension which then leads to a generalization of the result of Izhboldin mentioned above.

1991 Mathematics Subject Classification: Primary 11E04; Secondary 11E81, 12F20.

1 Introduction

Let F be a field of characteristic ≠ 2. W F denotes the Witt ring of non-degenerate quadratic forms over F (which we will simply call forms over F). P_{n}F (resp. GP_{n}F) denotes the set of all forms isometric (resp. similar) to n-fold Pfister forms, i.e., forms of the type ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ = ⟨1, a_{1}⟩ ⊗ ⋯ ⟨1, a_{n}⟩. We say that φ is a Pfister neighbor if there exists π ∈ P_{n}F for some n such that φ is similar to a subform of π and dim φ > \frac{1}{2} dim π = 2^{n−1}. In this case, we say that φ is a Pfister neighbor of π. Any π ∈ P_{n}F can be written as π ≃ ⟨1⟩ ⊥ π′. The form π′ is called the pure part of π and it is uniquely determined up to isometry.

1This research has been carried out during stays at the Mathematics Department of the University of Kentucky, Lexington and the Sonderforschungsbereich 343: “Diskrete Strukturen in der Mathematik”, Universität Bielefeld. I would like to thank both institutions for their hospitality.

Documenta Mathematica 1 (1996) 67–102

An important part of the algebraic theory of quadratic forms deals with the behavior of forms over F under a field extension K/F. Of particular interest is the case where K = F (ψ) is the function field of a form ψ over F. If ψ is isotropic then F (ψ)/F is purely transcendental. This situation is of not much interest with regard to the questions we will consider since one of our main goals lies in determining whether an anisotropic form φ over F becomes isotropic over K, something which cannot happen if K/F is purely transcendental. The extension K/F is said to be excellent if for any form φ over F the anisotropic part (φ_{K})_{an} of φ over K is defined over F, i.e., there exists a form φ̃ over F such that (φ_{K})_{an} ≃ φ̃_{K}. Knebusch has shown in [K 2, Theorem 7.13] that if F (ψ)/F is excellent where ψ is an anisotropic form, then ψ is a Pfister neighbor. As for the converse of this statement, it suffices to consider Pfister forms. This is because if ψ is a Pfister neighbor of τ then F (ψ) and F (τ) are (place-)equivalent over F which implies that F (ψ) is excellent iff F (τ) is excellent. So let K = F (τ) for some anisotropic τ ∈ P_{n}F. It is easy to show that K/F is excellent for n = 1, and for n = 2 this was shown by Arason in [ELW 1, Appendix II]. It was an open problem whether K/F is always excellent for n ≥ 3 until recently, when Izhboldin [I] gave a negative answer. In fact, he proved the even stronger result that to any anisotropic τ ∈ P_{n}F, n ≥ 3, there always exists a field extension E/F, some σ ≃ ⟨1⟩ ⊥ σ′ ∈ P_{n}E and some d ∈ Ė = E \ {0} not a square such that σ′ ⊥ ⟨d⟩ is anisotropic, it becomes isotropic over E(τ), but its anisotropic part over E(τ) is not defined over E. In particular, E(τ)/E is not excellent.

Let us now turn to a seemingly unrelated problem. It is well-known that if φ is an anisotropic form over F then φ_{F (φ)} is hyperbolic iff φ ∈ GP_{n}F for some n. Going one step further, what can one say about an anisotropic form φ over F for which φ_{1} ≃ (φ_{F (φ)})_{an} does not vanish but where (φ_{1})_{F_{1}(φ_{1})} becomes hyperbolic where F_{1} = F (φ). Such a form is said to be of height 2. By the above, we know that φ_{1} ∈ GP_{m}F_{1} for some m ≥ 1 and we say that φ has degree m. We call φ good if there exists some ρ ∈ P_{m}F such that φ_{1} ≃ aρ_{F_{1}} for some a ∈ Ḟ_{1}. If one can choose a ∈ Ḟ already then φ is an excellent form in the sense of Knebusch [K 2, Section 7], and in this case one knows how φ has to look like (cf. [K 2, Lemma 10.1(i)]). An open problem is to classify anisotropic good non-excellent forms of height 2. It is believed that if φ is of that type and of degree n − 1 then there exists some α ∈ P_{n−2}F and some 4-dimensional form β over F such that φ ≃ α ⊗ β and α ⊗ ⟨⟨−d⟩⟩ is anisotropic where d = d_{±}β is the signed discriminant of β. This conjecture has been proved for n = 2 (cf. [K 2, Theorem 10.3]), n = 3 (cf. [F 2, Theorem 1.6]), and n = 4 (cf. [Ka, Théorème 2.12]). (It is easy to show that if φ is of this type α ⊗ β then φ is good non-excellent of height 2.)

What do these forms α ⊗ β of height 2 and Izhboldin’s examples σ′ ⊥ ⟨d⟩ have in common ? In both cases we are dealing with anisotropic forms φ of dimension 2^{n}. If φ ≃ α ⊗ β and if we write β ≃ ⟨d, u, v, uv⟩ (possibly after scaling), then in W F we have

φ = α ⊗ ⟨⟨u, v⟩⟩ − α ⊗ ⟨⟨−d⟩⟩ .

If φ ≃ σ′ ⊥ ⟨d⟩ then in W F we have

φ = σ − ⟨⟨−d⟩⟩ .

We observe that in these situations φ can be written as the difference of an n-fold Pfister form and an (n−1)-fold resp. 1-fold Pfister form. We aim at a unifying concept which includes both types of forms α ⊗ β and σ′ ⊥ ⟨d⟩. This leads us quite naturally to what we call twisted Pfister forms. φ is said to be a twisted Pfister form if φ is anisotropic of dimension 2^{n} for some n, such that in W F it can be written as φ = σ−π for some anisotropic forms σ ∈ P_{n}F and π ∈ P_{m}F, 1 ≤ m < n. The above examples represent twisted Pfister forms at the extreme ends of the spectrum: m = n − 1 and m = 1. These examples also serve as a motivation for our in-depth study of these forms. It should be emphasized that Izhboldin’s striking results in [I] and his clever constructions there gave the initial impulse to our present investigations.

As simple as the structure of twisted Pfister forms appears, this class of forms leads in our opinion to a wealth of interesting results and new problems as the above examples indicate. This is somewhat surprising considering their “proximity” to ordinary Pfister forms.

In the next section, we will recall some of the important facts about function fields and generic splitting of quadratic forms which we will need rather extensively in what will follow. Starting with the basic notion of linkage of Pfister forms in Section 3, we will then give the precise definition of twisted Pfister forms and derive some of their fundamental properties as well as some alternative characterizations. For completeness’ sake, we included a short Section 4 in which we compute the Witt kernel W (F (φ)/F) of a twisted Pfister form φ. These results have been previously obtained by Fitzgerald [F 1]. In Section 5 we attack the problem of determining those forms ψ for which a twisted Pfister form φ becomes isotropic over F (ψ). In Section 6 we will determine in some cases the equivalence class of a twisted Pfister form φ (here, we mean that φ is equivalent to ψ, φ ∼ ψ, if φ_{F (ψ)} and ψ_{F (φ)} are isotropic). Some of our results in Sections 5 and 6 apply to an even bigger class of forms than twisted Pfister forms. The results in these two sections can be regarded as an extension and generalization of our earlier work in [H 4]. In Section 7, we consider the case of a twisted Pfister form φ of dimension 2^{n} and an anisotropic τ ∈ P_{n}F. We generalize Izhboldin’s results in [I] on when (φ_{F (τ)})_{an} is defined over F and add some remarks about so-called F (τ)-minimal forms. Finally, in Section 8, we explicitly construct τ ∈ P_{n}F, n ≥ 3, such that F (τ)/F is not excellent where F is purely transcendental of degree n−1 over ℚ. We also generalize Izhboldin’s construction of a field extension E/F such that E(τ)/E is not excellent where one starts with an arbitrary field F permitting an anisotropic Pfister form τ ∈ P_{n}F, n ≥ 3. Our construction is still based on Izhboldin’s original ideas used in [I].

2 Some basic facts

In our notations and terminology we follow Lam’s book [L 1] and Scharlau’s book [S]. φ ≃ ψ denotes isometry of the forms φ and ψ over F, whereas φ = ψ stands for equality in the Witt ring W F. We write φ_{an} for the anisotropic part of φ and i_{W} (φ) for its Witt index. Thus, if we denote the hyperbolic plane ⟨1, −1⟩ by ℍ and put i = i_{W} (φ), we have φ ≃ φ_{an} ⊥ (i × ℍ). If φ is a subform of ψ, i.e., if there exists a form η such that ψ ≃ φ ⊥ η, then we write φ ⊂ ψ for short.

If K/F is a field extension and if φ is a form over F, then we denote the form which one obtains from φ by scalar extension by φ_{K}. The Witt kernel W (K/F) is the kernel of the natural map W F → W K induced by scalar extension. We put D_{K}(φ) = {a ∈ K̇ | ⟨a⟩ ⊂ φ_{K}} and G_{K}(φ) = {a ∈ K̇ | aφ_{K} ≃ φ_{K}} (we omit the subscript if K = F). K/F is said to be excellent if for any form φ over F there exists a form φ̃ over F such that (φ_{K})_{an} ≃ φ̃, i.e., the anisotropic part of φ over K is defined over F. A form φ over F is called K-minimal if φ is anisotropic, φ_{K} is isotropic, and η_{K} is anisotropic for any η ⊂ φ with dim η < dim φ. Two field extensions K and L of F are called equivalent if there exist F -places λ : K → L ∪ ∞ and μ : L → K ∪ ∞, we write K ∼ L for short. In this situation, K/F is excellent iff L/F is excellent (this follows from [K 1, Proposition 3.1], see also [ELW 1, Corollary 2.8]), and K-minimal forms are exactly the L-minimal forms. φ is said to be round (or multiplicative) if D(φ) = G(φ). If φ is a Pfister form then φ is multiplicative and either anisotropic or hyperbolic (cf. [L 1, Ch. 10, Corollaries 1.6, 1.7] or [S, Ch. 4, Corollary 1.5]).

Let now φ be a form over F such that dim φ ≥ 2 and φ ≄ ℍ. The function field F (φ) of φ is the function field of the projective quadric defined by φ = 0. To avoid case distinctions, we put F (φ) = F if dim φ ≤ 1 or φ ≃ ℍ. If dim φ = n ≥ 2 then F (φ)/F is a purely transcendental extension of degree n − 2 over F followed by a quadratic extension, and F (φ)/F is purely transcendental iff φ is isotropic ([S, Ch. 4, Remark 5.2(vi)]). F (φ) is a generic zero (or isotropy) field of φ over F, i.e., if K is any field extension of F with φ_{K} isotropic then there exists a place λ : F (φ) → K ∪∞ over F. We say that two forms φ and ψ are equivalent if F (φ) ∼ F (ψ), and we write φ ∼ ψ. In the following proposition, we collect some more results about function fields of quadratic forms which we will need later on.

Proposition 2.1 Let φ and ψ be anisotropic forms over F.

(i) ([K 1, Theorem 3.3].) φ_{F (ψ)} and ψ_{F (φ)} are both isotropic iff F (φ) ∼ F (ψ), i.e., iff φ ∼ ψ.

(ii) ([S, Ch. 4, Theorem 5.4(i)].) φ_{F (φ)} is hyperbolic iff φ ∈ GP_{n}F for some

n ≥ 1. (iii) ([L 1, Ch. 7, Lemma 3.1], [S, Ch. 2, Lemma 5.1].) If dim ψ = 2 then φ_{F (ψ)} is

isotropic iff aψ ⊂ φ for some a ∈ Ḟ. (iv) (Cassels-Pfister subform theorem, [S, Ch. 4, Theorem 5.4(ii)].) If φ_{F (ψ)} is

hyperbolic then aψ ⊂ φ for any a ∈ D(φ)·D(ψ).

(v) ([S, Ch. 4, Theorem 5.4(iv)].) If ψ is a Pfister neighbor of the Pfister form π, then φ_{F (ψ)} is hyperbolic iff there exists a form γ over F such that φ ≃ π ⊗γ. In particular, W (F (ψ)/F) = πW F.

(vi) ([H 3, Theorem 1].) If dim φ ≤ 2^{n} < dim ψ for some n then φ_{F (ψ)} stays

anisotropic. (vii) ([H 3, Proposition 2].) If ψ is a Pfister neighbor of the Pfister form π then

φ ∼ ψ iff φ is a Pfister neighbor of π. (viii) ([L 2, Theorem 10.1].) Let ρ be another form over F. If φ_{F (ψ)} is isotropic

and if ψ_{F (ρ)} is isotropic then φ_{F (ρ)} is isotropic.

(ix) ([K 2, Theorem 7.13] or [ELW 1, Examples 2.2(i)].) If F (ψ)/F is excellent

then ψ is a Pfister neighbor.

(x) (Cf. part (iii) of this proposition and [ELW 1, Appendix II by Arason].) If ψ is a Pfister neighbor of an n-fold Pfister form, n = 1 or 2, then F (ψ)/F is excellent.

Let now φ be a form over F which is not hyperbolic. We define inductively fields F_{i}, i ≥ 0, and forms φ_{i} over F_{i} as follows. Let φ_{0} ≃ φ_{an} and F_{0} = F. For i ≥ 1 we put F_{i} = F_{i−1}(φ_{i−1}) and φ_{i} ≃ ((φ_{i−1})_{F_{i−1}})_{an}. The smallest h for which dim φ_{h} ≤ 1 is called the height of φ. The tower F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} is called a generic splitting tower of φ over F, F_{h} is a generic splitting field of φ over F, and F_{h−1} is called the leading field of φ over F. φ_{j} is called the j-th kernel form of φ and i_{W} (φ_{F_{j}}) = i_{j}(φ) the j-th Witt index of φ. By the splitting pattern of φ we mean the sequence {dim φ_{0}, dim φ_{1}, ⋯, dim φ_{h}} (this definition is different from the one given in [HuR]). The degree of φ is defined as follows. If dim φ is odd we put deg φ = 0. Otherwise, we know by Proposition 2.1(ii) that φ_{h−1} ∈ GP_{n}F_{h−1} for some n ≥ 1. In this case we put deg φ = n. Let τ ∈ P_{n}F_{h−1} such that φ_{h−1} is similar to τ. Then τ is called the leading form of φ. If the leading form is defined over F we say that φ is a good form (in this case there actually exists σ ∈ P_{n}F such that τ ≃ σ_{F_{h−1}}, cf. [K 2, Proposition 9.2]).

There are two natural filtrations of the Witt ring. One is given by the n-th powers I^{n}F of the ideal IF of even-dimensional forms in W F. I^{n}F is additively generated by the n-fold Pfister forms. One has I^{2}F = {φ ∈ IF | d_{±}φ = 1 ∈ Ḟ /Ḟ^{2}}, where d_{±}φ denotes the signed discriminant of a form φ, and by Merkurjev’s theorem [M] one has I^{3}F = {φ ∈ I^{2}F | c(φ) = 1} where c(φ) denotes the Clifford invariant of φ which is an element in the Brauer group BrF of F. The other filtration is given by the ideals J_{n}F = {φ ∈ W F | deg φ ≥ n} (cf. [K 1, Theorem 6.4] for the fact that these sets are ideals, see also [S, Ch. 4, Theorem 7.3]). One has I^{n}F ⊂ J_{n}F for all n ≥ 0 (cf. [K 1, Corollary 6.6], [S, p. 164]). This is essentially the Arason-Pfister Hauptsatz which in its original form states that if 0 ≠ φ ∈ I^{n}F is anisotropic then dim φ ≥ 2^{n}, and furthermore, if φ ∈ I^{n}F is anisotropic and dim φ = 2^{n} then φ ∈ GP_{n}F (see [AP, Hauptsatz and Korollar 3]). If we define deg′ φ = n if φ ∈ I^{n}F \ I^{n+1}F, we thus have deg′ φ ≤ deg φ. It is still an open problem whether I^{n}F = J_{n}F for all n and all F. This is known to be true for n ≤ 4 (cf. [Ka, Théorème 2.8] and the references there). We will mainly work with the ideals J_{n}F.

Proposition 2.2 Let φ and ψ be forms over F with φ not hyperbolic, and let F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} be a generic splitting tower of φ as defined above.

(i) ([K 1, Proposition 6.9 and Corollary 6.10], see also [S, Ch. 4, Theorem 7.5].) I^{m}F J_{n}F ⊂ J_{m+n}F for all m, n ≥ 0. Furthermore, deg(φ ⊗ ψ) = deg φ iff dim ψ is odd.

(ii) ([AK, Satz 18].) If deg φ_{F (ψ)} > deg φ then dim ψ ≤ 2^{n}, and if furthermore dim ψ = 2^{n} then ψ ∈ GP_{n}F and φ ≡ ψ (mod J_{n+1}F). In particular, ψ_{F_{h−1}} is similar to the leading form of φ.

(iii) ([K 1, Corollary 3.9 and Proposition 5.13].) Let K/F be a field extension. Let K ·F_{j} be the free composite of K and F_{j} over F. If i_{W} (φ_{K}) ≥ i_{j}(φ) then K·F_{j} is purely transcendental over K.

3 Twisted Pfister forms

The following result is well-known (cf. [EL, Theorem 1.4]). Since we will use it quite often without always referring to it explicitly, we will include a proof at this point for the reader’s convenience.

Lemma 3.1 Let α ∈ W F be a round form, i.e., G(α) = D(α). Let φ ∈ W F. If α ⊗ φ represents a ∈ Ḟ, then there exists ψ ∈ W F with a ∈ D(ψ) such that α ⊗ φ ≃ α ⊗ ψ. Furthermore, if dim φ ≥ 2 and α ⊗ φ is isotropic, then then there exists an isotropic ψ ∈ W F such that α ⊗ φ ≃ α ⊗ ψ.

Proof. Let φ ≃ ⟨a_{1}, ⋯, a_{n}⟩ so that α ⊗ φ ≃ a_{1}α ⊥ ⋯ ⊥ a_{n}α. Since a ∈ D(α ⊗ φ) there are x_{i} ∈ D(α), not all 0, such that a = a_{1}x_{1} + ⋯ + a_{n}x_{n}. Say, x_{1}, ⋯, x_{m} ≠ 0, m ≤ n. As α is round, x_{i}α ≃ α for 1 ≤ i ≤ m. Thus, α ⊗ ⟨a_{1}, ⋯, a_{m}⟩ ≃ α ⊗ ⟨a_{1}x_{1}, ⋯, a_{m}x_{m}⟩. By the above, a is represented by ⟨a_{1}x_{1}, ⋯, a_{m}x_{m}⟩. Hence, ⟨a_{1}x_{1}, ⋯, a_{m}x_{m}⟩ ≃ ⟨a, a′_{2}, ⋯, a′_{m}⟩ and thus α⊗φ ≃ α⊗⟨a, a′_{2}, ⋯, a′_{m}, a_{m+1}, ⋯, a_{n}⟩.

Now suppose that dim φ ≥ 2 and that α ⊗ φ is isotropic. Write φ ≃ φ′ ⊥ ⟨−x⟩. By assumption, there exists y ∈ Ḟ such that y is represented both by α ⊗ φ′ and by xα. This is clear if both forms are anisotropic because their difference is isotropic. If either one of them is isotropic, it is universal and therefore represents any non-zero element represented by the other form. By the above, α ⊗ φ′ ≃ α ⊗ ψ′ with y ∈ D(ψ′) and xα ≃ yα. In particular, the form ψ ≃ ψ′ ⊥ ⟨−y⟩ is isotropic and α ⊗ φ ≃ α ⊗ ψ. □

To gain a better understanding of the definition of twisted Pfister forms which we will give later it seems useful to recall another well-known result due to Elman and Lam [EL, Theorem 4.5].

Lemma 3.2 Let σ ∈ P_{n}F and π ∈ P_{m}F be anisotropic with m ≤ n. Let a, b ∈ Ḟ. Then i := i_{W} (aσ ⊥ bπ) = 0 or 2^{r} for some integer r with 0 ≤ r ≤ m. Furthermore, i ≥ 1 iff there exists x ∈ Ḟ such that (aσ ⊥ bπ)_{an} ≃ x(σ ⊥ −π)_{an}. If i = 2^{r} ≥ 1 then there exist α ∈ P_{r}F, σ_{1} ∈ P_{n−r}F, and π_{1} ∈ P_{m−r}F such that σ ≃ α ⊗ σ_{1} and π ≃ α ⊗ π_{1}.

Proof. We may assume that i ≥ 1. Then there exist u ∈ D(σ) and v ∈ D(π) such that au + bv = 0. The roundness of σ and π implies that uσ ≃ σ and vπ ≃ π. Thus, with x = au = −bv, we have

aσ ⊥ bπ ≃ auσ ⊥ bvπ ≃ xσ ⊥ −xπ.

Thus, (aσ ⊥ bπ)_{an} ≃ x(σ ⊥ −π)_{an}.

Now if i = 1 there is nothing else to show. So let us assume that i ≥ 2 and let α′ be a common Pfister neighbor of σ and π of maximal dimension. Since i ≥ 2 we have dim α′ ≥ 2 as both forms have at least a common 2-dimensional form, and every such 2-dimensional form is trivially a Pfister neighbor. Say, α′ is a Pfister neighbor of α ∈ P_{n}F. Since α′ becomes isotropic over F (α), it follows that σ_{F (α)} and π_{F (α)} are also isotropic and hence hyperbolic. By the Cassels-Pfister subform theorem and because 1 is represented by σ, π, and α, there exist forms σ_{0} and π_{0} such that σ ≃ α ⊥ σ_{0} and π ≃ α ⊥ π_{0}, cf. Proposition 2.1(iv). The maximality of dim α′ implies that dim α = dim α′. Suppose i > dim α. Then σ_{0} ⊥ −π_{0} is isotropic and there exists a w ∈ Ḟ which is represented both by σ_{0} and π_{0}. In particular, the Pfister neighbor α ⊥ ⟨w⟩ of α⊗⟨⟨w⟩⟩ is a common subform of σ and π, a contradiction to the maximality of dim α′ = dim α. Thus, i = dim α = 2^{r} for some r ≥ 1. As for the remaining statement, there is nothing else to show if dim α = dim σ. So suppose dim σ_{0} > 0 and let v ∈ D(σ_{0}). Thus, the Pfister neighbor α ⊥ ⟨v⟩ of α ⊗ ⟨⟨v⟩⟩ is a subform of σ, and by an argument similar to above, we get that σ ≃ α⊗⟨⟨v⟩⟩ ⊥ σ̃. The existence of σ_{1} ∈ P_{n−r}F now follows by an easy induction on dim σ_{0}. The existence of π_{1} can be shown in the same way. □

Definition 3.3 Let σ, π be anisotropic Pfister forms. If i_{W} (σ ⊥ −π) = 2^{r}, r ≥ 0, then r is called the linkage number of σ and π. We write ln(σ, π) = r. A form α ∈ P_{r}F such that σ ≃ α ⊗ σ_{1} and π ≃ α ⊗ π_{1} for suitable Pfister forms σ_{1}, π_{1} is called a link of σ and π.

It should be remarked that a link α is generally not uniquely determined up to isometry.

We now consider the case of an anisotropic form φ of dimension 2^{n} such that in W F we have φ = aσ + bπ, where a, b ∈ Ḟ and σ ,π are Pfister forms with dim σ ≥ dim π. In view of Lemma 3.2, we then have that φ ≃ x(σ ⊥ −π)_{an} for some x ∈ Ḟ. We want to exclude the case where φ ∈ GP_{n}F. An easy check then shows that we may assume σ ∈ P_{n}F and π ∈ P_{m}F are both anisotropic with 1 ≤ m < n and we have ln(σ, π) = m − 1. We now come to the definition of twisted Pfister forms and of what we will call weakly twisted Pfister forms, a type of form which will also appear frequently throughout the paper.

Definition 3.4 (i) Let 1 ≤ m < n. A form φ over F is called a twisted (n, m)- Pfister form (or simply (n, m)-Pfister form) if there exist anisotropic forms σ ∈ P_{n}F and π ∈ P_{m}F such that ln(σ, π) = m − 1 and such that φ ≃ (σ ⊥ −π)_{an}. In this case we say that the (n, m)-Pfister form φ is defined by (σ, π). The set of all forms isometric (resp. similar) to (n, m)-Pfister forms is denoted by P_{n,m}F (resp. GP_{n,m}F). φ is called a twisted Pfister form if φ ∈ GP_{n,m}F for some (n, m) with 1 ≤ m < n.

(ii) Let 1 ≤ m < n. A form φ over F is called a weakly twisted (n, m)-Pfister form if φ is anisotropic, dim φ = 2^{n}, and φ ≡ π ⊗ η (mod J_{n}F) for some anisotropic π ∈ P_{m}F and some odd-dimensional η ∈ W F. We call π the twist of φ. The set of all weakly twisted (n, m)-Pfister forms will be denoted by P_{n,m}^{w}F.

Remark 3.5 (i) Let 1 ≤ m < n. In view of Lemma 3.2 and by the remarks preceeding the definition, φ ∈ GP_{n,m}F iff φ is anisotropic, dim φ = 2^{n}, and there exist anisotropic σ ∈ GP_{n}F and π ∈ GP_{m}F such that φ = σ + π in W F. If this is the case, then φ ∉ GP_{n}F. In fact, φ ≡ σ + π ≡ π ≢ 0 (mod J_{n}F) because σ ∈ GP_{n}F ⊂ J_{n}F and π ∈ GP_{m}F is anisotropic and thus, since dim π = 2^{m} < 2^{n} and by the Arason-Pfister Hauptsatz, π ∉ J_{n}F. Similarly, φ ≡ π ≢ 0 (mod I^{n}F).

(ii) Let now φ ∈ P_{n,m}F be defined by (σ, π). Let α be a link of σ and π, i.e., α ∈ P_{m−1}F and there exist σ_{1} ∈ P_{n−m+1}F and d ∈ Ḟ such that σ ≃ α ⊗ σ_{1} and π ≃ α ⊗ ⟨⟨−d⟩⟩. Let σ′_{1} denote the pure part of σ_{1}, i.e., σ_{1} ≃ ⟨1⟩ ⊥ σ′_{1}. Then φ ≃ α ⊗ (⟨d⟩ ⊥ σ′_{1}).

(iii) If φ ∈ P_{n,m}F is defined by (σ, π) then φ ≡ σ −π ≡ −π (mod J_{n}F). Hence, GP_{n,m}F ⊂ P_{n,m}^{w}F. This is generally a proper inclusion if m ≤ n − 3 (see, e.g., Example 5.13), but it is an equality if 1 ≤ n − 2 ≤ m ≤ n − 1 ≤ 3 (cf. Proposition 3.17 below).

Before we continue, let us mention some of the properties of twisted Pfister forms which will be useful later. In fact, we state these results for a possibly wider class of forms (see also Conjecture 3.9 below).

Proposition 3.6 Let 1 ≤ m < n. Let φ ∈ W F be anisotropic and dim φ = 2^{n}. Suppose that φ ≡ xπ (mod J_{n}F) for some anisotropic π ∈ P_{m}F and some x ∈ Ḟ. Then the following holds.

(i) φ_{F (π)} is anisotropic and in GP_{n}F (π). In particular, if φ ∈ P_{n,m}F is defined by (σ, π), then φ_{F (π)} ≃ σ_{F (π)} is anisotropic. (ii) φ is good with leading form defined by π. We have

ht(φ) = \{ \begin{matrix}2 & if & m = n − 1, \\ 3 & if & m < n − 1. \end{matrix}

In particular, i_{1}(φ) = 2^{m−1} and the splitting pattern of φ is {2^{n}, 2^{n−1}, 0} if

m = n − 1 and {2^{n}, 2^{n} − 2^{m}, 2^{m}, 0} if m < n − 1.

These statements hold in particular if φ ∈ GP_{n,m}F.

Proof. (i) First note that π_{F (π)} = 0 and thus φ_{F (π)} ∈ J_{n}F (π). If φ_{F (π)} is anisotropic then, since dim φ = 2^{n}, this implies φ_{F (π)} ∈ GP_{n}F (π). So suppose φ_{F (π)} is isotropic. Then dim(φ_{F (π)})_{an} < 2^{n} and by the Arason-Pfister Hauptsatz, φ_{F (π)} is hyperbolic. Thus, there exists γ ∈ W F, dim γ = 2^{n−m}, such that φ ≃ π ⊗ γ. Since n > m we have that dim γ is even, i.e., γ ∈ IF. But π ∈ P_{m}F ⊂ I^{m}F. Hence, φ ≃ π ⊗ γ ∈ I^{m+1}F ⊂ J_{m+1}F. But clearly, φ ≡ xπ ≢ 0 (mod J_{m+1}F), a contradiction.

If φ ∈ P_{n,m}F is defined by (σ, π), then in W F we have φ = σ − π and thus, in W F (π), φ_{F (π)} = σ_{F (π)}. Now φ ≡ −π (mod J_{n}F) and by Remark 3.5(i) and by the above, it is clear that φ_{F (π)} ≃ σ_{F (π)} is anisotropic.

(ii) Since dim φ = 2^{n} but φ ∉ GP_{n}F, we have ht(φ) ≥ 2. Let F_{0} = F, F_{1}, and F_{2} be the first three fields in a splitting tower of φ, and let φ_{1} and φ_{2} be the first two kernel forms of φ. Clearly, 0 < dim φ_{1} < 2^{n} and (φ_{1})_{F_{1}(π)} ≡ 0 (mod J_{n}F_{1}(π)). Hence, by the Arason-Pfister Hauptsatz, (φ_{1})_{F_{1}(π)} is hyperbolic. Thus, there exists γ ∈ W F_{1} such that φ_{1} ≃ π_{F_{1}} ⊗ γ. Comparing dimensions shows that 1 ≤ dim γ ≤ 2^{n−m} −1. This shows in particular that i_{1}(φ) ≥ 2^{m−1}. Define ψ ∈ W F_{1} by ψ ≃ φ_{1} ⊥ −xπ_{F_{1}} ≃ π_{F_{1}} ⊗ (γ ⊥ ⟨−x⟩). Note that dim ψ ≤ 2^{n} and

ψ ≡ φ_{1} − xπ_{F_{1}} ≡ φ_{F_{1}} − xπ_{F_{1}} ≡ xπ_{F_{1}} − xπ_{F_{1}} ≡ 0 (mod J_{n}F).

Thus, by the Arason-Pfister Hauptsatz, either ψ is hyperbolic or ψ is anisotropic and in GP_{n}F_{1}.

Suppose that ψ is hyperbolic. Let μ ≃ (φ ⊥ −xπ)_{an} over F. By definition, μ ≡ 0 (mod J_{n}F). Note that φ and π are anisotropic and dim φ = 2^{n} > dim π = 2^{m}.

Hence, 0 < 2^{n} − 2^{m} ≤ dim μ ≤ 2^{n} + 2^{m} < 2^{n+1}. Therefore, by the Arason-Pfister Hauptsatz, 2^{n} ≤ dim μ < 2^{n+1} and we must have deg μ = n. Over F_{1} = F (φ) we have μ_{F_{1}} = φ_{F_{1}} − xπ_{F_{1}} = ψ_{F_{1}} = 0 and thus deg μ_{F_{1}} = ∞ > deg μ = n. Now dim φ = 2^{n} and Proposition 2.2(ii) yields φ ∈ GP_{n}F, obviously a contradiction.

It follows that ψ ≃ φ_{1} ⊥ −xπ_{F_{1}} ∈ GP_{n}F_{1} is anisotropic. Furthermore, dim φ_{1} = 2^{n} − 2^{m}. In particular, ψ_{F_{1}(π)} is isotropic and hence hyperbolic and thus (φ_{1})_{F_{1}(π)} is hyperbolic as well. If m = n − 1 then dim φ_{1} = dim π = 2^{n−1} which immediately yields that φ_{1} is similar to π_{F_{1}}, which in turn implies that φ is good of height 2 with leading form defined by π. Now if m < n − 1 then dim φ_{1} = 2^{n} − 2^{m} > 2^{n−1}. Thus, φ_{1} is a Pfister neighbor with complementary form −xπ_{F_{1}}. It follows readily that φ_{2} ≃ xπ_{F_{2}} and that φ is a good form of height 3 with leading form defined by π. In fact, the second kernel form is defined by xπ already over F. □

It should be remarked that the fact that the leading form of φ is defined by π also follows directly from φ ≡ xπ ≡ π (mod J_{m+1}F) by [K 2, Theorem 9.6]. In our proof, we also wanted to determine the height of φ explicitly.

Corollary 3.7 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Then φ_{F (π)} is anisotropic and in GP_{n}F (π), i_{1}(φ) = 2^{m−1}, and φ is good with leading form defined by π. Furthermore, if F (π)/F is excellent, then there exists σ ∈ GP_{n}F such that φ_{F (π)} ≃ σ_{F (π)}.

Proof. Write φ ≡ π ⊗ η (mod J_{n}F) with dim η odd. That φ_{F (π)} is anisotropic and in GP_{n}F (π) can be shown as in Proposition 3.6, using the fact that π ⊗ η ≢ 0

(mod J_{n}F) as π is anisotropic and dim η is odd and hence deg(π⊗η) = deg π = m < n (see Proposition 2.2(i)). Similarly as before, we get that i_{1}(φ) ≥ 2^{m−1}. We want to show that we have equality and also that φ is good with leading form defined by π. We may assume that after scaling d_{±}η = 1. It is clear that φ ≡ π ⊗η ≡ π (mod J_{m+1}F) and it follows from [K 2, Theorem 9.6] that φ is good with leading form defined by π. Let L be the leading field of π ⊗ η and K = F (π). Since (π ⊗ η)_{K} = 0 we have that the free composite KL is purely transcendental over K (see Proposition 2.2(iii)). Since φ_{K} is anisotropic, we therefore have that φ_{KL} is anisotropic and hence φ_{L} is anisotropic as well. Now (π ⊗ η)_{L} = π_{L} in W L by [K 1, Proposition 6.12]. Hence, φ_{L} is anisotropic, dim φ_{L} = 2^{n}, and φ_{L} ≡ π_{L} (mod J_{n}L). By Proposition 3.8, we have i_{1}(φ_{L}) = 2^{m−1}. But i_{1}(φ_{L}) ≥ i_{1}(φ) ≥ 2^{m−1}. Hence, i_{1}(φ) = 2^{m−1}.

Finally, if F (π)/F is excellent, then the existence of some σ ∈ GP_{n}F such that φ_{F (π)} ≃ σ_{F (π)} follows from [ELW 1, Proposition 2.11]. □

Corollary 3.8 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F. Let ψ ⊂ aφ for some a ∈ Ḟ and dim ψ > 2^{n} − 2^{m−1}. Then ψ_{F (φ)} is isotropic and φ ∼ ψ.

Proof. We have i_{1}(φ) = i_{W} (φ_{F (φ)}) = 2^{m−1} by the previous proposition. Since ψ is similar to a subform of φ and dim ψ > dim φ − i_{1}(φ), it follows readily that ψ_{F (φ)} is isotropic. Clearly, ψ_{F (ψ)} and hence φ_{F (ψ)} are isotropic as well. Thus, φ ∼ ψ. □

We finish this section with some conjectures and a characterization of forms in GP_{n,m}F. As already remarked, if φ ∈ P_{n,m}F is defined by (σ, π), then φ ≡ −π

(mod J_{n}F). It would be interesting to know whether a converse of this also holds, i.e., is the following conjecture always true?

Conjecture 3.9 Let 1 ≤ m < n. Let φ be an anisotropic form over F with dim φ = 2^{n}. If there exists an anisotropic π ∈ GP_{m}F such that φ ≡ π (mod J_{n}F) then φ ∈ GP_{n,m}F.

This conjecture is related to the following well-known conjecture (see, for example, [Ka, Conj. 9]).

Conjecture 3.10 Let ψ ∈ J_{n}F be anisotropic and dim ψ < 2^{n} + 2^{n−1}. Then dim ψ = 2^{n} and ψ ∈ GP_{n}F.

By the definition of degree it is clear that dim ψ ≥ 2^{n} and that ψ ∈ GP_{n}F if dim ψ = 2^{n}. Note also that if ψ ≃ (π_{1} ⊥ −π_{2})_{an} where π_{i} ∈ P_{n}F and ln(π_{1}, π_{2}) = n − 2, then dim ψ = 2^{n} + 2^{n−1}. So the conjecture essentially states that there is a gap in the dimensions of anisotropic forms in J_{n}F between 2^{n} and 2^{n} + 2^{n−1}. We have the following results concerning these two conjectures.

Proposition 3.11 (i) Conjecture 3.10 implies Conjecture 3.9.

(ii) Conjecture 3.10 holds for n ≤ 4. (iii) Conjecture 3.9 holds for n ≤ 4.

Proof. (i) Let φ be anisotropic, dim φ = 2^{n}, and φ ≡ π (mod J_{n}F) for some anisotropic π ∈ GP_{m}F where 1 ≤ m < n. If m = n − 1 then, for all x ∈ Ḟ, π ≡ xπ

(mod J_{n}F) so that in this case we may assume (after possibly scaling) that there exists u ∈ D(φ) ∩ D(π). Consider σ ≃ (φ ⊥ −π)_{an}. We clearly have σ ∈ J_{n}F. Furthermore, 0 < 2^{n} − 2^{m} ≤ dim σ ≤ 2^{n} + 2^{n−1} − 2. The last inequality is obvious if m < n − 1, and it follows for m = n − 1 since we assumed that φ and π represent a common element u ∈ Ḟ. If Conjecture 3.10 holds, we have that σ ∈ GP_{n}F. Hence, in W F, φ = σ − π with σ ∈ GP_{n}F and π ∈ GP_{m}F. By Remark 3.5(i) we have φ ∈ GP_{n,m}F.

(ii) The case n = 2 is trivial and the case n = 3 is essentially due to Pfister (cf. [P, Satz 14] or [S, Ch. 2, Theorem 14.4], the result is usually given in terms of I^{3}F). The case n = 4 can be found in [H 7], again in terms of I^{4}F. Here, we use that I^{n}F = J_{n}F for n ≤ 4.

(iii) follows from (ii) and (i). □

The next little result shows that Conjecture 3.9 is at least “stably” true.

Proposition 3.12 Let 1 ≤ m < n. Let φ be an anisotropic form over F with dim φ = 2^{n}. If there exists an anisotropic π ∈ GP_{m}F such that φ ≡ π (mod J_{n}F) then φ ∈ GP_{n,m}K for some field extension K/F.

Proof. Let K be the leading field of φ ⊥ −π. Since 0 ≠ φ ⊥ −π ∈ J_{n}F and dim(φ ⊥ −π) < 2^{n+1}, we have that deg(φ ⊥ −π) = n, i.e., ((φ ⊥ −π)_{K})_{an} ≃ σ ∈ GP_{n}K. Since K is obtained by taking function fields of dimension > 2^{n} (in case K ≠ F), φ_{K} and π_{K} are anisotropic by [H 3, Theorem 1]. Also, φ_{K} = σ + π_{K} in W K. It is now obvious by Remark 3.5(i) that φ ∈ GP_{n,m}K. □

Corollary 3.13 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Then φ ∈ GP_{n,m}K for some field extension K/F.

Proof. Write φ ≡ π ⊗ η (mod J_{n}F) with dim η odd. Without loss of generality, we may assume that d_{±}η = 1. Let L be the leading field of π ⊗ η. As in the proof of Corollary 3.7, we have that φ_{L} is anisotropic and φ_{L} ≡ π_{L} (mod J_{n}L). The claim now follows immediately from Proposition 3.12. □

We know by Proposition 3.11(i) that Conjecture 3.10 implies Conjecture 3.9. It would be interesting to know whether the converse holds as well, i.e., whether these two conjectures are equivalent. At least a partial answer is given by the following.

Proposition 3.14 (i) If Conjecture 3.9 holds for (n, m) = (n, 1), n ≥ 3, then there are no anisotropic forms of dimension 2^{n} + 2 in J_{n}F.

(ii) If Conjecture 3.9 holds for (n, m) = (n, 1) and for (n, m) = (n, 2), n ≥ 4, then there are no anisotropic forms of dimension 2^{n} + 2 and 2^{n} + 4 in J_{n}F.

Proof. (i) Let ψ ∈ J_{n}F and suppose that dim ψ = 2^{n} + 2 and that ψ is anisotropic. Write ψ ≃ φ ⊥ −π with dim π = 2. Obviously, both π ∈ GP_{1}F and φ are anisotropic, dim φ = 2^{n} and φ ≡ π (mod J_{n}F). By assumption, this implies that φ ∈ GP_{n,1}F. Thus, there exist σ ∈ GP_{n}F and τ ∈ GP_{1}F such that in W F we have φ = σ + τ. Hence, ψ = σ + τ − π ∈ W F. Now ψ and σ ∈ J_{n}F. Therefore, τ − π ∈ J_{n}F. But dim τ +dim π = 4 < 2^{n} which, by the Arason-Pfister Hauptsatz, yields that τ −π = 0 in W F. Hence, in W F we have ψ = σ. But dim σ = 2^{n} < dim ψ. Thus, ψ is isotropic, a contradiction.

(ii) By part (i), forms of dimension 2^{n} + 2 in J_{n}F are isotropic. So let ψ ∈ J_{n}F and suppose that dim ψ = 2^{n} + 4 and that ψ is anisotropic. Write ψ ≃ ψ′ ⊥ −δ with dim δ = 2. Let d = d_{±}δ so that δ is similar to ⟨1, −d⟩. Let L = F (\sqrt{d}). We have that ψ_{L} = ψ′_{L} ∈ J_{n}L. Now dim ψ′ = 2^{n} + 2 and by assumption and part (i), we have that ψ′_{L} is isotropic. Hence, ψ′ contains a subform similar to δ, say, ψ′ ≃ φ ⊥ xδ. Let −π ≃ δ ⊥ xδ ∈ GP_{2}F. Then we have ψ ≃ φ ⊥ −π and thus, φ ≡ π (mod J_{n}F) with anisotropic π ∈ GP_{2}F, anisotropic φ, dim φ = 2^{n}. By assumption, this implies that φ ∈ GP_{n,2}F. With a reasoning analogous to the one in the proof of part (i), we conclude again that ψ is isotropic, a contradiction. □

The following result shows that the existence of a large enough Pfister neighbor as a subform of the form φ in Conjecture 3.9 is equivalent to φ being in GP_{n,m}F.

Proposition 3.15 Let 1 ≤ m < n. Let φ be an anisotropic form over F with dim φ = 2^{n}. Suppose there exists an anisotropic π ∈ GP_{m}F such that φ ≡ π

(mod J_{n}F). Then φ ∈ GP_{n,m}F iff φ contains a Pfister neighbor of dimension 2^{n−1} + 1.

Proof. Say, φ ∈ P_{n,m}F. Then it follows readily from Remark 3.5(ii) (and with the notations there) that φ contains the Pfister neighbor α⊗σ′_{1} of dimension 2^{n} −2^{m−1} ≥ 2^{n−1} + 1.

Conversely, let μ ⊂ φ be a Pfister neighbor of dimension 2^{n−1} + 1 of, say, σ ∈ P_{n}F, and let x ∈ Ḟ such that μ ⊂ xσ. Define ψ ≃ (φ ⊥ −xσ)_{an}. Then dim ψ ≤ dim φ + dim σ − 2 dim μ = 2^{n} − 2. Note that ψ ≡ φ − xσ ≡ φ ≡ π (mod J_{n}F) and hence ψ_{F (π)} ≡ 0 (mod J_{n}F (π)). By the Arason-Pfister Hauptsatz, this implies that ψ_{F (π)} = 0 in W F (π) and there exists η ∈ W F such that ψ ≃ π ⊗ η. Since dim ψ ≤ 2^{n} − 2 we must therefore have dim ψ ≤ 2^{n} − 2^{m}. As ψ ⊥ −π ∈ J_{n}F and dim(ψ ⊥ −π) ≤ 2^{n}, the Arason-Pfister Hauptsatz yields two cases. Either ψ ⊥ −π = 0 in W F. Then φ = ψ +xσ = π +xσ in W F and thus φ ∈ GP_{n,m}F by Remark 3.5(i). Or ψ ⊥ −π ≃ τ ∈ GP_{n}F is anisotropic. In this case, φ = ψ−xσ = τ −xσ+π = γ +π, where γ ≃ (τ ⊥ −xσ)_{an}. Now dim φ = 2^{n} and dim π = 2^{m}, τ, xσ ∈ GP_{n}F, and φ, π, and γ are all anisotropic. By Lemma 3.2, dim γ = 0, 2^{n}, or ≥ 2^{n} + 2^{n−1}. We consider two cases. If m ≤ n − 2 then in order to have φ = γ + π, i.e., γ = φ − π, we must have dim γ = 2^{n} as

0 < 2^{n} − 2^{m} ≤ dim(φ ⊥ −π)_{an} ≤ 2^{n} + 2^{m} < 2^{n} + 2^{n−1} .

But γ ∈ I^{n}F and thus necessarily γ ∈ GP_{n}F. Since φ = γ + π we therefore have φ ∈ GP_{n,m}F. If m = n−1 then ψ ⊥ −π ∈ GP_{n}F implies that ψ ≃ yπ for some y ∈ Ḟ. Hence, φ = ψ+xσ = yπ+xσ in W F which readily implies φ ∈ GP_{n,m}F = GP_{n,n−1}F. □

Let us finish this section by showing that in certain cases “weakly twisted” implies “twisted” as mentioned already in Remark 3.5(iii). First, we show a very easy lemma.

Lemma 3.16 Let n ≥ 2, π ∈ P_{n−2}F and η̃ ∈ W F. Then there exists a form η ∈ W F with dim η ≤ 2 and dim η ≡ dim η̃ (mod 2) such that π ⊗ η̃ ≡ π ⊗ η (mod J_{n}F).

Proof. We may assume that dim η̃ ≥ 3. Let us first consider the case where dim η̃ is even. Let a = d_{±}η̃. Then η̃ ⊥ −⟨⟨−a⟩⟩ ∈ I^{2}F and thus, since π ∈ J_{n−2}F, π ⊗ (η̃ ⊥ −⟨⟨−a⟩⟩) ∈ J_{n}F or π ⊗ η̃ ≡ π ⊗ ⟨⟨−a⟩⟩ (mod J_{n}F) and we put η ≃ ⟨⟨−a⟩⟩.

Let us now consider the case where dim η̃ is odd. After scaling, we may assume that η̃ ≃ ⟨1⟩ ⊥ η′. Let now a = d_{±}η′. By a similar argument as above, we get

π ⊗ η̃ ≡ π + π ⊗ η′ ≡ π + π ⊗ ⟨⟨−a⟩⟩ ≡ π ⊗ ⟨⟨1, −a⟩⟩ + aπ ≡ aπ (mod J_{n}F)

because π ⊗ ⟨⟨1, −a⟩⟩ ∈ GP_{n}F ⊂ J_{n}F. Here, we put η ≃ ⟨a⟩. □

The final result in this section now follows readily from this lemma together with Proposition 3.11.

Proposition 3.17 Let 1 ≤ n − 2 ≤ m ≤ n − 1. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Then there exists x ∈ Ḟ such that φ ≡ xπ (mod J_{n}F). In particular, if Conjecture 3.9 holds for (n, m) (m as above) then P_{n,m}^{w}F = GP_{n,m}F. Thus, this equality holds whenever 1 ≤ n − 2 ≤ m ≤ n − 1 ≤ 3.

4 The Witt kernel of the function field of a twisted Pfister form

We already used several times that if π ∈ P_{n}F is anisotropic and if φ ∈ W (F (π)/F) is anisotropic, then there exists a form γ over F such that φ ≃ π ⊗ γ. In particular, W (F (π)/F) is a strong n-Pfister ideal, i.e., every anisotropic form in W (F (π)/F) is isometric to an orthogonal sum of forms similar to n-fold Pfister forms in W (F (π)/F), in this case forms similar to π itself. We will show that if φ ∈ P_{n,m}F then W (F (φ)/F) is a strong (n + 1)-Pfister ideal, and we will determine the (n + 1)-fold Pfister forms in W (F (φ)/F). These results are implicitly contained in the work of Fitzgerald [F 1]. We will nevertheless provide a proof for the reader’s convenience.

Theorem 4.1 Let φ ∈ P_{n,m}F be defined by (σ, π). Let α ∈ P_{m−1}F be a link of σ and π and let d ∈ Ḟ such that π ≃ α ⊗ ⟨⟨−d⟩⟩. Let η be an anisotropic form over F. Then η ∈ W (F (φ)/F) if and only if there exist an integer k ≥ 1, r_{i}, s_{i} ∈ Ḟ, 1 ≤ i ≤ k, such that s_{i} ∈ D(⟨d⟩ ⊥ −α) and

η ≃ ⊥_{i=1}^{k} r_{i}σ ⊗ ⟨⟨s_{i}⟩⟩ .

This theorem follows from the following more general result.

Theorem 4.2 Let σ ≃ ⟨1⟩ ⊥ σ′ ∈ P_{n}F, n ≥ 2, be anisotropic and let γ_{1}, γ_{2} ∈ W F such that γ_{1} ⊂ σ′, dim γ_{1} > 2^{n−1}, and γ_{1} ⊥ γ_{2} ≃ σ. Let d ∈ Ḟ such that ψ ≃ γ_{1} ⊥ ⟨d⟩ is anisotropic and not a Pfister neighbor. Let η be an anisotropic form over F. Then η ∈ W (F (ψ)/F) if and only if there exist an integer k ≥ 1, r_{i}, s_{i} ∈ Ḟ, 1 ≤ i ≤ k, such that s_{i} ∈ D(⟨d⟩ ⊥ −γ_{2}) and

η ≃ ⊥_{i=1}^{k} r_{i}σ ⊗ ⟨⟨s_{i}⟩⟩ .

Proof. To show the “if”-part, it suffices to show that if s ∈ D(⟨d⟩ ⊥ −γ_{2}) then σ ⊗ ⟨⟨s⟩⟩ ∈ W (F (ψ)/F). Now s being represented by ⟨d⟩ ⊥ −γ_{2} is equivalent to d being represented by ⟨s⟩ ⊥ γ_{2} (Witt cancellation!). Now clearly sσ represents s. Hence,

ψ ≃ γ_{1} ⊥ ⟨d⟩ ⊂ γ_{1} ⊥ γ_{2} ⊥ ⟨s⟩ ⊂ σ ⊥ sσ ≃ σ ⊗ ⟨⟨s⟩⟩ .

It is now obvious that σ ⊗ ⟨⟨s⟩⟩ is isotropic and hence hyperbolic over F (ψ).

As for the converse, let η ∈ W (F (ψ)/F) be anisotropic. Since γ_{1} ⊂ ψ we have that η also becomes hyperbolic over F (γ_{1}). But γ_{1} is a Pfister neighbor of σ, i.e., γ_{1} ∼ σ and thus η ∈ W (F (σ)/F). Hence, there exists a form τ over F with η ≃ σ ⊗τ. After scaling, we may assume that τ represents 1, i.e., τ ≃ ⟨1⟩ ⊥ τ′ and η ≃ σ ⊥ σ⊗τ′. Now dim ψ > 2^{n−1} and ψ is not a Pfister neighbor. In particular, ψ is not similar to a subform of σ ∈ P_{n}F and therefore σ_{F (ψ)} stays anisotropic. Hence, we must have dim τ′ ≥ 1. As η_{F (ψ)} = 0, the Cassels-Pfister subform theorem yields that for every a ∈ D(η)·D(ψ) we have aψ ⊂ η. Since ψ and σ and therefore also η have the subform γ_{1} in common, they represent common elements. Hence, we may choose a = 1 and we get that ψ ⊂ η, i.e.,

ψ ≃ γ_{1} ⊥ ⟨d⟩ ⊂ η ≃ σ ⊥ σ ⊗ τ′ ≃ γ_{1} ⊥ γ_{2} ⊥ σ ⊗ τ′ .

Hence, there exists u ∈ D(γ_{2}) ∪ {0} and s ∈ D(σ ⊗ τ′) ∪ {0} such that d = u + s. Note that d ∉ D(γ_{2}) because otherwise ψ ≃ γ_{1} ⊥ ⟨d⟩ ⊂ γ_{1} ⊥ γ_{2} ≃ σ, i.e., ψ is a Pfister neighbor of σ, in contradiction to the definition of ψ. Hence, we must have s ≠ 0, i.e., s ∈ D(σ ⊗ τ′). By Lemma 3.1, we may in fact assume that s ∈ D(τ′) so that τ′ ≃ ⟨s⟩ ⊥ τ′′. Hence, we get

η ≃ σ ⊥ sσ ⊥ σ ⊗ τ′′ ≃ σ ⊗ ⟨⟨s⟩⟩ ⊥ σ ⊗ τ′′ .

Now s = d − u ∈ D(⟨d⟩ ⊥ −γ_{2}). We have already shown that in this case, σ ⊗ ⟨⟨s⟩⟩ becomes hyperbolic over F (ψ). Therefore, σ ⊗ τ′′ also has to become hyperbolic over F (ψ) because η does. The proof can now easily be finished by induction on dim τ. □

Proof of Theorem 4.1. As in Remark 3.5(ii), we write σ ≃ α ⊗ σ_{1} for some σ_{1} ≃ ⟨1⟩ ⊥ σ′_{1} ∈ P_{n−m+1}F, so that we get φ ≃ α ⊗ (⟨d⟩ ⊥ σ′_{1}). Let ψ ≃ α ⊗ σ′_{1} ⊥ ⟨d⟩ ⊂ φ. We have dim ψ = 2^{n} − 2^{m−1} + 1. Hence, by Corollary 3.8, φ ∼ ψ and therefore W (F (φ)/F) = W (F (ψ)/F). Note that ψ is not a Pfister neighbor because φ is not a Pfister neighbor and the only forms equivalent to Pfister neighbors are Pfister neighbors themselves (cf. Proposition 2.1(vii)). Note also that σ ≃ α ⊗ σ′_{1} ⊥ α. Now α ≃ ⟨1⟩ ⊥ α′ and hence σ′ ≃ α ⊗ σ′_{1} ⊥ α′ contains α ⊗ σ′_{1} as a subform. The assumptions in Theorem 4.2 on ψ are then fulfilled by putting γ_{1} ≃ α⊗σ′_{1} and γ_{2} ≃ α. The claim of the theorem now follows immediately from Theorem 4.2 with σ, d, ψ, γ_{1} and γ_{2} given as above. □

5 Isotropy of twisted Pfister forms over function fields of quadratic forms

Let φ be an anisotropic Pfister form over F and ψ ∈ W F be anisotropic with dim ψ ≥ 2. The fact that Pfister forms are either anisotropic or hyperbolic plus the Cassels-Pfister subform theorem imply that φ becomes isotropic over F (ψ) iff ψ is similar to a subform of φ. Now suppose φ is a twisted Pfister form and ψ is as above. When is φ isotropic over F (ψ) ? The problem turns out to be considerably more complicated and we are only able to obtain partial results. Let us start with a useful observation.

Proposition 5.1 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Let ψ ∈ W F be anisotropic with dim ψ ≥ 2 and assume that D(φ) ∩ D(ψ) ≠ ∅. Then φ_{F (ψ)} is isotropic iff ψ_{F (π)} ⊂ φ_{F (π)}. In particular, if φ ∈ P_{n,m}F is defined by (σ, π) and if D(φ) ∩ D(ψ) ≠ ∅ or D(σ) ∩ D(ψ) ≠ ∅, then φ_{F (ψ)} is isotropic iff ψ_{F (π)} ⊂ σ_{F (π)}.

Proof. The second statement clearly follows from the first one since if φ ∈ P_{n,m}F is defined by (σ, π) then φ ≡ σ − π ≡ −π (mod J_{n}F) and φ_{F (π)} ≃ σ_{F (π)}, see Proposition 3.6(i).

To prove the first statement, we note that by Corollary 3.7 we have that φ_{F (π)} ∈ GP_{n}F (π) is anisotropic. If φ_{F (ψ)} is isotropic then φ_{F (π)(ψ)} is also isotropic and hence hyperbolic, and the Cassels-Pfister subform theorem together with D(φ) ∩ D(ψ) ≠ ∅ implies that ψ_{F (π)} ⊂ φ_{F (π)}.

Conversely, suppose that ψ_{F (π)} ⊂ φ_{F (π)}. Clearly, φ_{F (π)(ψ)} is isotropic and hence hyperbolic because φ_{F (π)} ∈ GP_{n}F (π). Note that F (π)(ψ) ≃ F (ψ)(π). Suppose φ_{F (ψ)} is anisotropic. Then, by Proposition 2.1(v) and since φ_{F (ψ)(π)} = 0, there exists γ ∈ W F (ψ) such that φ_{F (ψ)} ≃ γ ⊗π_{F (ψ)}. Now dim γ = 2^{n−m} is even and π ∈ GP_{m}F. In particular, γ ⊗π_{F (ψ)} ∈ J_{m+1}F (ψ) by Proposition 2.2(1). Now if we write φ ≡ π⊗η

(mod J_{n}F) with dim η odd, we readily get φ ≡ π ⊗ η ≡ π (mod J_{m+1}F). Hence we have

φ_{F (ψ)} ≡ π_{F (ψ)} ≡ γ ⊗ π_{F (ψ)} ≡ 0 (mod J_{m+1}F (ψ))

which yields π_{F (ψ)} = 0 in W F (ψ). But then F (ψ)(π)/F (ψ) is purely transcendental. Thus, the anisotropic form φ_{F (ψ)} stays anisotropic over F (ψ)(π), a contradiction to φ_{F (ψ)(π)} = 0. □

This result gives us a criterion to decide whether φ becomes isotropic over F (ψ), however, it only works over F (π). Although function fields of Pfister forms have a somewhat nicer behavior than function fields of arbitrary forms, it seems desirable to find criteria which, at least in principle, work over F itself. What we would like to have is some sort of descent from F (π) to F where π is an anisotropic Pfister form. This can easily be achieved if F (π)/F is an excellent field extension which is always the case for m = 1 and 2, but generally not for m ≥ 3, see also Proposition 2.1(x) and Corollary 8.4.

Proposition 5.2 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Let ψ ∈ W F be anisotropic with dim ψ ≥ 2 and assume that D(φ) ∩ D(ψ) ≠ ∅. Suppose furthermore that F (π)/F is excellent. Then φ_{F (ψ)} is isotropic iff there exists a form ψ̃ ∈ W F, dim ψ̃ = 2^{n}, such that ψ ⊂ ψ̃ and ψ̃_{F (π)} ≃ φ_{F (π)}. In particular, if φ ∈ P_{n,m}F is defined by (σ, π) and if D(φ) ∩ D(ψ) ≠ ∅ or D(σ) ∩ D(ψ) ≠ ∅, then φ_{F (ψ)} is isotropic iff there exists a form ψ̃ ∈ W F, dim ψ̃ = 2^{n}, such that ψ ⊂ ψ̃ and ψ̃_{F (π)} ≃ σ_{F (π)} ≃ φ_{F (π)}.

Proof. The “if”-part follows directly from Proposition 5.1 even without the excellence assumption. As for the converse, consider φ ⊥ −ψ. Since φ_{F (ψ)} is isotropic, we know by Proposition 5.1 that ψ_{F (π)} ⊂ φ_{F (π)}. This plus the excellence of F (π)/F imply that there exists χ ∈ W F, dim χ = 2^{n} − dim ψ, such that φ_{F (π)} ≃ ψ_{F (π)} ⊥ χ_{F (π)}, and with ψ̃ ≃ ψ ⊥ χ over F, we get ψ̃_{F (π)} ≃ φ_{F (π)}. □

Remark 5.3 The previous proposition provides indeed a criterion which, at least in principle, can be checked over F. This is because ψ̃_{F (π)} ≃ φ_{F (π)} means that ψ̃ ⊥ −φ ∈ W (F (π)/F). In other words, with φ and ψ as above, φ_{F (ψ)} is isotropic iff there exist forms ψ̃ and τ in W F with dim ψ̃ = 2^{n} such that ψ ⊂ ψ̃ and (ψ̃ ⊥ −φ)_{an} ≃ π⊗τ.

We do not know whether Proposition 5.2 holds in general without the assumption on F (π)/F being excellent. However, this result at least indicates that in order to decide if φ becomes isotropic over F (ψ), it seems to be important to characterize those forms ψ over F of dimension 2^{n} for which ψ_{F (π)} ≃ φ_{F (π)}. This will be the focus of most of the remainder of this section.

We will eventually be interested in characterizing those forms ψ ∈ W F of dimension 2^{n} which become isometric to some φ ∈ P_{n,m}F over F (π), where φ is defined by (σ, π). Our aim is to make this description as precise as possible, something which, in general, doesn’t seem to be easy and which we will only do in the cases m = n − 1 and m = n − 2. In fact, the case m = n − 1 has been dealt with in [H 4, Theorem 3.3]. We have the following result.

Theorem 5.4 Let φ ∈ P_{n,n−1}F be defined by (σ, π). Let ψ ∈ W F with dim ψ = 2^{n}. Then the following are equivalent.

(i) φ_{F (ψ)} is isotropic.

(ii) ψ_{F (π)} is similar to φ_{F (π)} ≃ σ_{F (π)}.

(iii) Either ψ is similar to φ or ψ is similar to some τ ∈ P_{n}F and φ contains a Pfister neighbor of τ.

Proof. The equivalence of (i) and (ii) is clear from Proposition 5.1. Clearly, (iii) implies (i). That (iii) follows from any of the other statements was shown in [H 4, Theorem 3.3] under the additional assumption that ψ contains a Pfister neighbor of dimension 2^{n−1} + 2^{n−2}. By Proposition 5.8, (ii) implies that there exist α ∈ P_{n−2}F and ψ_{1} ∈ W F, dim ψ_{1} = 4, such that ψ ≃ α ⊗ ψ_{1}. Let ψ′ ⊂ ψ_{1} with dim ψ′ = 3. Then ψ′ is a Pfister neighbor of some β ∈ P_{2}F and α ⊗ ψ′ ⊂ ψ is a Pfister neighbor of dimension 2^{n−1} + 2^{n−2} of α ⊗ β ∈ P_{n}F and we can apply [H 4, Theorem 3.3] as desired. □

Corollary 5.5 Let φ ∈ P_{n,n−1}F be defined by (σ, π) and suppose that F (π)/F is excellent (which always holds if n − 1 = 1 or 2). Let ψ ∈ W F with dim ψ ≥ 2. Then φ_{F (ψ)} is isotropic iff ψ is similar to a subform of φ or ψ is similar to a subform of some τ ∈ P_{n}F and φ contains a Pfister neighbor of τ.

Proof. This is a direct consequence of Proposition 5.2 and the previous theorem. □

Part (iii) of the previous theorem tells us that in order to decide whether φ_{F (ψ)} is isotropic (where dim ψ = 2^{n}), it suffices to look only at φ and ψ and how they relate to each other over F. The form π isn’t really needed explicitly. It turns out that if φ ∈ P_{n,m}F with m < n − 1 then the situation is not quite so nice anymore as the form π will play a more prominent role. Let us start with a simple observation.

Proposition 5.6 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Let ψ ∈ W F be anisotropic with dim ψ = 2^{n}. Suppose that φ_{F (ψ)} is isotropic. Then there exists μ ∈ W F such that ψ ≡ π ⊗ μ (mod J_{n}F). In particular, deg ψ ≥ m, and deg ψ = m iff dim μ is odd (i.e., iff ψ ∈ P_{n,m}^{w}F with twist π). Furthermore, If m = n − 1 (resp. m = n − 2) then there exists such μ with dim μ ≤ 1 (resp. dim μ ≤ 2).

Proof. Write φ ≡ π ⊗ η (mod J_{n}F) with dim η odd. After scaling, we may assume that D(ψ) ∩ D(φ) ≠ ∅. Since φ_{F (ψ)} is isotropic we have by Proposition 5.1 that ψ_{F (π)} ≃ φ_{F (π)}. Let χ ≃ (ψ ⊥ −φ)_{an}. Then χ ∈ W (F (π)/F) and there exists μ̃ ∈ W F with χ ≃ π ⊗ μ̃. Let us put μ ≃ μ̃ ⊥ η. Then we have

ψ ≡ χ + φ ≡ π ⊗ μ̃ + π ⊗ η ≡ π ⊗ μ (mod J_{n}F) .

We have deg(π ⊗ μ) = deg π = m if dim μ is odd (cf. Proposition 2.2(i)), in which case deg ψ = deg(π ⊗ μ) = m as m < n. If dim μ is even, we have deg(π ⊗ μ) ≥ m + 1. Since n ≥ m + 1 we thus also have deg ψ ≥ m + 1.

The remaining statements for n − 2 ≤ m ≤ n − 1 follow readily from Lemma 3.16. □

Remark 5.7 In the above proof, we have dim χ ≤ 2^{n+1} − 2 and one obtains dim μ̃ ≤ 2^{n+1−m} − 1 or dim μ ≤ 2^{n+1−m} − 1 + dim η. If φ ∈ GP_{n,m}F or if F (π)/F is excellent, then we know that there exists σ ∈ GP_{n}F such that φ_{F (π)} ≃ σ_{F (π)} ∈ GP_{n}F (π) (cf. Corollary 3.7 in the case where F (π)/F is excellent). We can slightly improve the estimate of dim μ for general m in this case. After scaling, we may assume that σ ∈ P_{n}F and D(σ) ∩ D(ψ) ≠ ∅. Since φ_{F (ψ)} is isotropic we then have by Proposition 5.1 that ψ_{F (π)} ≃ σ_{F (π)}. Let χ ≃ (ψ ⊥ −σ)_{an}. Then dim χ ≤ 2^{n+1} − 2 as D(ψ) ∩ D(σ) ≠ ∅, and also χ ∈ W (F (π)/F). Hence, there exists μ ∈ W F with χ ≃ π ⊗ μ. Since 2^{m} dim μ = dim χ ≤ 2^{n+1} − 2 we have dim μ ≤ 2^{n+1−m} − 1. Furthermore, σ ∈ P_{n}F and we get

χ ≡ ψ − σ ≡ ψ ≡ π ⊗ μ (mod J_{n}F) .

In the case where φ is a twisted Pfister form, we can be more precise about how ψ has to look like.

Proposition 5.8 Let φ ∈ P_{n,m}F be defined by (σ, π). Let α ∈ P_{m−1}F, σ_{1} ∈ P_{n−m+1}F, and d ∈ Ḟ such that σ ≃ α ⊗ σ_{1} and π ≃ α ⊗ ⟨⟨−d⟩⟩ (see Remark 3.5(ii)). Let ψ ∈ W F with dim ψ = 2^{n}. Then the following holds.

(i) If ψ_{F (π)} ≃ φ_{F (π)} then there exists ψ_{1} ∈ W F, dim ψ_{1} = 2^{n−m+1}, such that ψ ≃ α ⊗ ψ_{1}. In particular, ψ ∈ I^{m}F, i.e., deg′ ψ ≥ m.

(ii) If ψ ∈ P_{n}F and ψ_{F (π)} ≃ φ_{F (π)} then there exist s ∈ Ḟ, σ_{2}, ψ_{2} ∈ P_{n−m}F, such that σ ≃ α ⊗ ⟨⟨s⟩⟩ ⊗ σ_{2} and ψ ≃ α ⊗ ⟨⟨s⟩⟩ ⊗ ψ_{2}. In particular, ln(ψ, σ) ≥ m and ln(ψ, π) = m − 1.

Proof. First, let us recall that φ_{F (π)} ≃ σ_{F (π)}, so that in both parts we actually assume that ψ_{F (π)} ≃ σ_{F (π)} and therefore, we may assume that ψ represents 1 already over F after possibly scaling (this is of course always true if ψ ∈ P_{n}F).

(i) If m = 1, i.e., dim α = 1, there is nothing to show. So let us assume that m ≥ 2 so that we have α_{F (α)} = 0. Since ψ_{F (π)} ≃ σ_{F (π)} we clearly have ψ_{F (α)(π)} ≃ σ_{F (α)(π)} ≃ (α⊗σ_{1})_{F (α)(π)} = 0. Similarly, π_{F (α)} = 0 which implies that F (α)(π)/F (α) is purely transcendental. Hence, we must already have ψ_{F (α)} = 0. Note that ψ is anisotropic as ψ_{F (π)} ≃ φ_{F (π)} is anisotropic. Hence, by Proposition 2.1(v), there exists ψ_{1} ∈ W F, dim ψ_{1} = 2^{n−m+1}, such that ψ ≃ α ⊗ ψ_{1}. Since α ∈ I^{m−1}F and dim ψ_{1} even, i.e., ψ_{1} ∈ IF, we have ψ ∈ I^{m}F, i.e., deg′ ψ ≥ m.

(ii) Let γ ≃ (ψ ⊥ −σ)_{an} ∈ I^{n}F. By assumption, γ ∈ W (F (π)/F). Also, dim γ ≤ 2^{n+1} − 2 as 1 ∈ D(ψ) ∩ D(σ). Thus, by Proposition 2.1(v), there exists β ∈ W F, dim β < 2^{n+1−m}, such that γ ≃ π ⊗ β. Since γ ∈ I^{n}F and π ∈ P_{m}F with 1 ≤ m < n, we must have that dim β is even. Therefore, dim β ≤ 2^{n+1−m} − 2, i.e., dim γ ≤ 2^{n+1} − 2^{m+1}. Hence, i_{W} (ψ ⊥ −σ) ≥ 2^{m} and the existence of s ∈ Ḟ, σ_{2}, ψ_{2} ∈ P_{n−m}F such that σ ≃ α ⊗ ⟨⟨s⟩⟩ ⊗ σ_{2} and ψ ≃ α ⊗ ⟨⟨s⟩⟩ ⊗ ψ_{2} now follows from Lemma 3.2 (cf., in particular, the proof of Lemma 3.2 and use the fact that we already have σ ≃ α ⊗ σ_{1} and ψ ≃ α ⊗ ψ_{1} by part (i)).

Clearly, ln(ψ, σ) ≥ m since α ⊗ ⟨⟨s⟩⟩ ∈ P_{m}F divides both ψ and σ. It is also obvious that m ≥ ln(ψ, π) ≥ m − 1 as α ∈ P_{m−1}F divides both ψ and π. Now ln(ψ, π) = m would imply that π ⊂ ψ and thus ψ_{F (π)} ∈ P_{n}F (π) would be hyperbolic, a contradiction to ψ_{F (π)} ≃ φ_{F (π)} being anisotropic. □

Before we state our theorem about forms in P_{n,n−2}F which parallels in a certain sense Theorem 5.4, we will provide a lemma which we will need in the proof of this theorem.

Lemma 5.9 Let 1 ≤ m ≤ n − 2 and let φ ∈ W F be anisotropic with dim φ = 2^{n} such that φ ≡ π (mod J_{n}F) for some anisotropic π ∈ GP_{m}F. Assume furthermore that φ ≃ α ⊗ β for some α ∈ P_{m−1}F and some β ∈ W F, dim β = 2^{n−m+1}. If Conjecture 3.9 holds for (n, m + 1) then φ ∈ GP_{n,m}F.

Proof. After scaling, we may assume that π ∈ P_{m}F. First, we note that there exists d ∈ Ḟ such that π ≃ α⊗⟨⟨−d⟩⟩. This is obvious if m = 1, i.e., α ≃ ⟨1⟩ ∈ P_{0}F. If m > 1 we have that α_{F (α)} = 0, thus φ_{F (α)} = 0 as well, which in turn yields π_{F (α)} ∈ J_{n}F (α). Since π ∈ P_{m}F and m < n we must have π_{F (α)} = 0. The existence of d ∈ Ḟ such that π ≃ α ⊗ ⟨⟨−d⟩⟩ follows immediately from Proposition 2.1(v). Write β ≃ ⟨x⟩ ⊥ β′ and define β̃ ≃ ⟨xd⟩ ⊥ β′ and φ̃ ≃ α ⊗ β̃. Then dim φ̃ = dim φ = 2^{n} and φ̃ = φ − xπ in W F. In particular, one gets φ̃_{F (π)} ≃ φ_{F (π)} which is anisotropic by Proposition 3.6. Hence, φ̃ is anisotropic. Furthermore,

φ̃ ≡ φ − xπ ≡ π − xπ ≡ π ⊗ ⟨⟨−x⟩⟩ (mod J_{n}F) .

We have two cases. If π ⊗ ⟨⟨−x⟩⟩ is isotropic and hence hyperbolic, then φ̃ ∈ J_{n}F and thus φ̃ ∈ GP_{n}F as dim φ̃ = 2^{n}. In W F, we get φ = φ̃ + xπ which readily implies that φ ∈ GP_{n,m}F as φ̃ ∈ GP_{n}F and xπ ∈ GP_{m}F (cf. Remark 3.5(i)). If π ⊗ ⟨⟨−x⟩⟩ is anisotropic then by our assumption φ̃ ∈ GP_{n,m+1}F and there exists σ ∈ GP_{n}F and ρ ∈ GP_{n,m+1}F such that φ̃ = σ + ρ in W F. In particular, φ̃ ≡ ρ ≡ π ⊗ ⟨⟨−x⟩⟩

(mod J_{n}F), and it readily follows that there exists y ∈ Ḟ such that ρ ≃ yπ ⊗ ⟨⟨−x⟩⟩ (recall that m + 1 < n). Hence, in W F,

\begin{aligned}φ − π & = & φ̃ + xπ − π = σ + yπ ⊗ ⟨⟨−x⟩⟩ + xπ − π \\ & = & σ + π ⊗ ⟨y, −xy, x, −1⟩ = σ − π ⊗ ⟨⟨−x, −y⟩⟩ . \end{aligned}

Suppose first that m + 2 < n. Then φ − π ≡ 0 ≡ −π ⊗ ⟨⟨−x, −y⟩⟩ (mod J_{n}F), i.e., π ⊗ ⟨⟨−x, −y⟩⟩ ∈ J_{n}F, which implies π ⊗ ⟨⟨−x, −y⟩⟩ = 0 as π ⊗ ⟨⟨−x, −y⟩⟩ ∈ P_{m+2}F and m + 2 < n. We then have φ = σ + π with σ ∈ GP_{n}F and π ∈ P_{m}F which yields that φ ∈ GP_{n,m}F as desired.

Finally, if m + 2 = n, we have that σ, π ⊗ ⟨⟨−x, −y⟩⟩ ∈ GP_{n}F. By Lemma 3.6 we then get dim(σ ⊥ −π ⊗ ⟨⟨−x, −y⟩⟩)_{an} = 0, 2^{n}, or ≥ 2^{n} + 2^{n−1}. On the other hand,

0 < 2^{n} − 2^{m} = dim φ − dim π ≤ dim(φ ⊥ −π)_{an} ≤
≤ dim φ + dim π = 2^{n} + 2^{m} < 2^{n} + 2^{n−1} .

Now (σ ⊥ −π ⊗ ⟨⟨−x, −y⟩⟩)_{an} ≃ (φ ⊥ −π)_{an} and we therefore must have dim(σ ⊥ −π ⊗ ⟨⟨−x, −y⟩⟩)_{an} = 2^{n}. As σ ⊥ −π ⊗ ⟨⟨−x, −y⟩⟩ ∈ J_{n}F it follows that (σ ⊥ −π ⊗ ⟨⟨−x, −y⟩⟩)_{an} ≃ τ ∈ GP_{n}F. Hence, φ = τ + π with τ ∈ GP_{n}F and π ∈ P_{m}F and again φ ∈ GP_{n,m}F. □

Theorem 5.10 Suppose that Conjecture 3.9 holds for (n, n − 1). Let φ ∈ P_{n,n−2}F be defined by (σ, π). Let ψ ∈ W F with dim ψ = 2^{n}. Then the following are equivalent.

(i) φ_{F (ψ)} is isotropic.

(ii) ψ_{F (π)} is similar to φ_{F (π)} ≃ σ_{F (π)}.

(iii) There exists τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and either

• ψ is similar to τ, or

• there exist x ∈ Ḟ and ρ ∈ P_{n,n−1}F such that ρ is defined by (τ, π⊗⟨⟨x⟩⟩) and ψ is similar to ρ, or

• there exists χ ∈ P_{n,n−2}F such that χ is defined by (τ, π) and ψ is similar to χ.

Proof. The equivalence of (i) and (ii) is clear from Proposition 5.1. One readily checks that (iii) implies that ψ_{F (π)} is similar τ_{F (π)} ≃ σ_{F (π)} and we are in (ii). Finally,

(i) implies by Proposition 5.6 that ψ ≡ π ⊗ μ (mod J_{n}F) for some μ ∈ W F, 0 ≤ dim μ ≤ 2. If dim μ ∈ {0, 2} then ψ ∈ GP_{n}F or ψ ∈ GP_{n,n−1}F (the latter only if π ⊗ μ ≠ 0 and because we assumed that Conjecture 3.9 holds for (n, n − 1)). If dim μ = 1 we have ψ ∈ GP_{n,n−2}F by Lemma 5.9 together with Proposition 5.8(i). All this together with the fact that ψ_{F (π)} is similar to σ_{F (π)} readily imply (iii) and we leave the details to the reader. □

Corollary 5.11 Let φ ∈ P_{n,n−2}F be defined by (σ, π). Let ψ ∈ W F with dim ψ = 2^{n}. Suppose that n ≤ 4 or that ψ contains a Pfister neighbor of dimension 2^{n−1} + 1. Then the following are equivalent.

(i) φ_{F (ψ)} is isotropic.
(ii) ψ_{F (π)} is similar to φ_{F (π)} ≃ σ_{F (π)}.

(iii) There exists τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and either • ψ is similar to τ, or

• there exist x ∈ Ḟ and ρ ∈ P_{n,n−1}F such that ρ is defined by (τ, π⊗⟨⟨x⟩⟩) and ψ is similar to ρ, or

• there exists χ ∈ P_{n,n−2}F such that χ is defined by (τ, π) and ψ is similar to χ.

Proof. This is an immediate consequence of the previous theorem and Propositions 3.11 and 3.15. □

Corollary 5.12 Suppose that Conjecture 3.9 holds for (n, n − 1). Let φ ∈ P_{n,n−2}F be defined by (σ, π) and suppose that F (π)/F is excellent. Let ψ ∈ W F with dim ψ ≥ 2. Then the following are equivalent.

(i) φ_{F (ψ)} is isotropic.

(ii) ψ_{F (π)} is similar to a subform of φ_{F (π)} ≃ σ_{F (π)}.

(iii) There exists τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and • ψ is similar to a subform of τ, or

• there exist x ∈ Ḟ and ρ ∈ P_{n,n−1}F such that ρ is defined by (τ, π⊗⟨⟨x⟩⟩) and ψ is similar to a subform of ρ, or

• there exists χ ∈ P_{n,n−2}F such that χ is defined by (τ, π) and ψ is similar to a subform of χ.

In particular, the equivalence of (i), (ii) and (iii) always holds for n ≤ 4.

Proof. This is an immediate consequence of Theorem 5.10 and Proposition 5.2. Furthermore, if n ≤ 4 then Conjecture 3.9 holds by Proposition 3.11, and F (π)/F is excellent since π is of fold ≤ 2. □

Corollaries 5.5 and 5.12 give us a fairly complete picture for which forms ψ ∈ W F a given form φ ∈ P_{n,m}F, which is defined by (σ, π), becomes isotropic over F (ψ) in the cases (n, m) ∈ {(2, 1), (3, 1), (3, 2), (4, 2)}. In a certain sense, we know this in general in the case (n, m) = (n, 1) or (n, 2) by Propositions 5.1 and 5.2. It comes down to characterizing those forms ψ̃ of dimension 2^{n} for which ψ̃_{F (π)} ≃ σ_{F (π)}. In the cases (2, 1) and (3, 2) we have a very precise description by Corollary 5.5. In the cases (3, 1) and (4, 2) we can essentially reduce this problem to the determination of those τ ∈ P_{n}F with τ_{F (π)} ≃ σ_{F (π)} or τ ⊥ −σ ∈ W (F (π)/F). This narrows down the set of forms we have to look at quite considerably.

The following example shows that if φ ∈ P_{n,m}F with n − m ≥ 3 and if ψ ∈ W F, dim ψ = 2^{n}, then φ_{F (ψ)} being isotropic does generally not imply that ψ is similar to a Pfister form or a twisted Pfister form, something which cannot happen in the cases considered above.

Example 5.13 Let F = ℝ(t) be the rational function field in one variable t over the reals. Let m ≥ 1 and n − m ≥ 3. Let σ ≃ ⟨⟨1, ⋯, 1⟩⟩ ∈ P_{n}F and π ≃ ⟨⟨1, ⋯, 1, −t⟩⟩ ∈ P_{m}F. We then have

φ ≃ (σ ⊥ −π)_{an} ≃ ⟨\underbrace{1, ⋯, 1}_{2^{n}−2^{m−1}}, \underbrace{t, ⋯, t}_{2^{m−1}}⟩ ∈ P_{n,m}F .

Let

ψ ≃ (σ ⊥ −⟨1, 1, 1⟩ ⊗ π)_{an} ≃ ⟨ \underbrace{1, ⋯, 1}_{2^{n}−3·2^{m−1}}, \underbrace{t, ⋯, t}_{3·2^{m−1}}⟩ .

One easily sees that φ and ψ are anisotropic (for example by passing to the power series field ℝ((t)) ⊃ F and applying Springer’s theorem [L 1, Ch. 6, Proposition 1.9], [S, Ch. 6, Corollary 2.6(i)]). Clearly, ψ_{F (π)} ≃ σ_{F (π)} ≃ φ_{F (π)}. Thus, by Proposition 5.1, φ_{F (ψ)} is isotropic. We claim that ψ is neither similar to a Pfister form nor to a twisted Pfister form. First, using that σ, ⟨⟨1, 1⟩⟩ ⊗ π ∈ J_{m+2}F, we note that

ψ ≡ σ − ⟨1, 1, 1⟩ ⊗ π ≡ −⟨1, 1, 1⟩ ⊗ π + ⟨⟨1, 1⟩⟩ ⊗ π ≡ π ≢ 0 (mod J_{m+2}F) .

Hence, deg ψ = deg π = m. Clearly, ψ is not similar to a Pfister form. Furthermore, ψ is also not similar to twisted Pfister form. For otherwise, since deg ψ = m, we have ψ ∈ GP_{n,m}F and by definition, there exist anisotropic forms τ ∈ GP_{n}F and ρ ∈ GP_{m}F such that ψ = τ +ρ in W F. Thus, τ +ρ = σ −⟨1, 1, 1⟩⊗π or ρ+⟨1, 1, 1⟩⊗π = σ −τ in W F and we get dim(ρ ⊥ ⟨1, 1, 1⟩ ⊗ π)_{an} = dim(σ ⊥ −τ)_{an}. Now dim(σ ⊥ −τ)_{an} = 0 or ≥ 2^{n} by Lemma 3.2. We also have dim ρ = 2^{m} and dim ⟨1, 1, 1⟩ ⊗ π = 3·2^{m}. Thus,

0 < 2^{m+1} = dim ⟨1, 1, 1⟩ ⊗ π − dim ρ ≤ dim(ρ ⊥ ⟨1, 1, 1⟩ ⊗ π)_{an} ≤
≤ dim ⟨1, 1, 1⟩ ⊗ π + dim ρ = 2^{m+2} < 2^{n} .

This obviously yields a contradiction. Note, however, that ψ ∈ P_{n,m}^{w}F. □

6 The equivalence class of a twisted Pfister form

Recall that two forms φ and ψ over F are called equivalent, we write φ ∼ ψ, if φ_{F (ψ)} and ψ_{F (φ)} are isotropic. Since the function field of an isotropic form is purely transcendental over the ground field and since anisotropic forms stay anisotropic over purely transcendental extensions, the question whether φ ∼ ψ holds is of interest only in the case of anisotropic forms. Let us denote the equivalence class of a form φ over F with respect to “∼” by Equiv(φ).

We know by Proposition 2.1(vii) that if φ is an anisotropic Pfister form then Equiv(φ) = {ψ ∈ W F | ψ is a Pfister neighbor of φ}. The equivalence classes of forms of dimension ≤ 5 and certain forms of dimension 6, 7, and 8 have been determined in [W], [H 1], [H 2], [H 4], [Lag]. Furthermore, for forms in P_{n,n−1}F we have the following result (cf. [H 4, Corollary 3.4, Theorem 4.4]).

Theorem 6.1 Let n ≥ 2 and let φ ∈ P_{n,n−1}F.

(i) Let ψ ∈ W F with dim ψ = 2^{n}. Then φ ∼ ψ iff ψ is similar to φ.

(ii) Let n ≤ 3. Then

Equiv(φ) = {ψ ∈ W F | xψ ⊂ φ for some x ∈ Ḟ and dim ψ > 2^{n} − 2^{n−2}} .

In view of part (ii) of this theorem, the following conjecture seems natural (see also [H 4, Conjecture 4.3]).

Conjecture 6.2 Let n ≥ 2 and φ ∈ P_{n,n−1}F. Then

Equiv(φ) = {ψ ∈ W F | xψ ⊂ φ for some x ∈ Ḟ and dim ψ > 2^{n} − 2^{n−2}} .

Let us right away state what we propose as the corresponding conjecture for forms in P_{n,n−2}F and which we will prove to be correct in the cases n ≤ 4 (see Corollary 6.11).

Conjecture 6.3 Let n ≥ 3 and let φ ∈ P_{n,n−2}F be defined by (σ, π). Let ψ ∈ W F. Then the following statements are equivalent.

(i) ψ ∈ Equiv(φ).

(ii) There exists χ ∈ P_{n,n−2}F such that

• χ is defined by (τ, π) for some τ ∈ P_{n}F with τ_{F (π)} ≃ σ_{F (π)},
• xψ ⊂ χ for some x ∈ Ḟ, and
• dim ψ > 2^{n} − 2^{n−3}.

It will be crucial to determine first those ψ ∈ W F of dimension 2^{n} such that ψ ∼ φ, and we will start with some more general results.

Theorem 6.4 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Let ψ ∈ W F be anisotropic with dim ψ = 2^{n}. Then the following are equivalent.

(i) ψ ∈ Equiv(φ) (i.e., ψ ∼ φ).

(ii) ψ_{F (π)} is similar to φ_{F (π)} and ψ ∈ P_{n,m}^{w}F with twist π.

(iii) ψ_{F (π)} is similar to φ_{F (π)} and deg ψ = m.

Proof. We clearly may assume that ψ is anisotropic.

(ii)⇒(i). Since ψ_{F (π)} is similar to φ_{F (π)}, and since φ ≡ π ⊗ η (mod J_{n}F) and ψ ≡ π ⊗ μ (mod J_{n}F) with dim η ≡ dim μ ≡ 1 (mod 2), it follows directly from Proposition 5.1 and the symmetry of the situation that φ_{F (ψ)} and ψ_{F (φ)} are both isotropic. Hence, φ ∼ ψ.

(i)⇒(ii). Let now φ ∼ ψ. Then, because φ_{F (ψ)} is isotropic, ψ_{F (π)} is similar to φ_{F (π)} by Proposition 5.1 and ψ ≡ π⊗μ (mod J_{n}F) for some μ ∈ W F by Proposition 5.6. Suppose dim μ is even so that we have deg(π ⊗ μ) ≥ m + 1. Let K = F (π) and let L be the generic splitting field of π ⊗ μ as defined in Section 2. Then (π ⊗ μ)_{K} = 0 in W K and it follows that the free composite M = KL is purely transcendental over K (cf. Proposition 2.2(iii)). Since φ_{K} is anisotropic we have that φ_{KL} is also anisotropic and thus, φ_{L} is anisotropic as well. Since φ ∼ ψ, it follows that ψ_{L} stays also anisotropic. But ψ_{L} ≡ (π ⊗ μ)_{L} ≡ 0 (mod J_{n}L) and dim ψ = 2^{n}. This yields that ψ ∈ GP_{n}L. Now φ ∼ ψ also implies that φ_{L} ∼ ψ_{L} and we conclude that φ_{L} is similar to ψ_{L}, in particular, φ_{L} ∈ GP_{n}L and deg φ_{L} = n > m = deg φ. But [AK, Satz 20] implies that deg φ_{L} = deg φ = m because L is a generic splitting field of π ⊗ μ and deg(π ⊗ μ) ≥ m + 1 > m since dim μ is even. This is clearly a contradiction and we therefore have that dim μ is odd.

(ii)⇔(iii). The condition that ψ_{F (π)} is similar to φ_{F (π)}, which appears in both statements, implies that φ_{F (ψ)} is isotropic by Proposition 5.1, and thus we get ψ ≡ π ⊗ μ (mod J_{n}F) for some μ ∈ W F in both (ii) and (iii). The equivalence of (ii) and (iii) now follows from the easy observation that deg ψ = m iff deg(π ⊗ μ) = m iff dim μ is odd. □

Corollary 6.5 Let n ≥ 3. Let φ ∈ P_{n,n−2}F be defined by (σ, π). Let ψ ∈ W F be anisotropic with dim ψ = 2^{n}. Then φ ∼ ψ iff ψ_{F (π)} is similar to φ_{F (π)} and ψ ≡ xπ

(mod J_{n}F) for some x ∈ Ḟ.

In particular, if Conjecture 3.9 holds for (n, n − 2) or (n, n − 1) (which is fulfilled if n ≤ 4), or if ψ contains a Pfister neighbor of dimension 2^{n−1} + 1, then φ ∼ ψ iff ψ is similar to some χ ∈ P_{n,n−2}F which is defined by (τ, π) such that τ_{F (π)} ≃ σ_{F (π)}.

Proof. This follows from Theorem 6.4 together with Propositions 3.6, 3.11, 3.15, 5.6, 5.8 and Lemma 5.9. We leave the details to the reader. □

Definition 6.6 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. We define 𝔈(φ) to be the set of all ψ ∈ W F with dim ψ > 2^{n} − 2^{m−1} for which there exist ψ̃ ∈ P_{n,m}^{w}F with twist π such that

• ψ ⊂ ψ̃, and

• ψ̃_{F (π)} is similar to φ_{F (π)}.

In view of Theorem 6.4 and Corollary 3.8, we conjecture the following.

Conjecture 6.7 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Then Equiv(φ) = 𝔈(φ).

Proposition 6.8 Let 1 ≤ m < n. Let φ ∈ P_{n,m}^{w}F with twist π ∈ P_{m}F. Then 𝔈(φ) ⊂ Equiv(φ).

Proof. Let ψ ∈ 𝔈(φ). Then there exist ψ̃, μ ∈ W F with dim ψ̃ = 2^{n}, dim μ odd, such that ψ̃_{F (π)} is similar to φ_{F (π)} and ψ̃ ≡ π ⊗ μ (mod J_{n}F). By Theorem 6.4 this implies ψ̃ ∼ φ. Furthermore, ψ̃ has the property that ψ ⊂ ψ̃, and we also have dim ψ > 2^{n} − 2^{m−1}. Hence, ψ ∼ ψ̃ by Corollary 3.8 and therefore ψ ∼ ψ̃ ∼ φ, i.e., ψ ∈ Equiv(φ). □

The main result of this section is the following.

Theorem 6.9 Conjecture 6.7 holds for m ≤ 2.

Before we prove the theorem, we consider a special case.

Lemma 6.10 Let φ ∈ P_{n,m}F be defined by (σ, π) and let ψ ⊂ φ with dim ψ ≤ 2^{n} − 2^{m−1}. Then there exists a field extension K/F such that φ_{K} ∈ P_{n,m}K is defined by (σ_{K}, π_{K}) and ψ_{K} ⊂ σ_{K}. In particular, ψ_{F (φ)} is anisotropic. If m = 1 such a K can be chosen to be of the form K = F (β) for some β ∈ P_{2}F.

Proof. Let K/F be a field extension. If φ_{K}, σ_{K} and π_{K} all stay anisotropic then one easily concludes that one still has φ_{K} ∈ P_{n,m}K and that it is defined by (σ_{K}, π_{K}). To prove this lemma, we may assume that dim ψ = 2^{n} − 2^{m−1}. Let ψ′ ∈ W F, dim ψ′ = 2^{m−1} such that φ ≃ ψ ⊥ ψ′. Then, in W F, we have φ = σ − π = ψ + ψ′ or σ ⊥ −ψ = ψ′ ⊥ π. Note that dim ψ′ = 2^{m−1} = \frac{1}{2} dim π = 2^{m}. By [H 3, Remark 1 and Theorem 4], there exists a field K in the generic splitting tower of ψ′ ⊥ π such that i_{W} ((ψ′ ⊥ π)_{K}) = 2^{m−1}, i.e., −ψ′_{K} ⊂ π_{K}, and π_{K} is anisotropic (see also [HuR, Corollaries 1.9 and 1.12]). In particular, dim((ψ′ ⊥ π)_{K})_{an} = 2^{m−1}. By comparing dimensions, we get σ_{K} ≃ ψ_{K} ⊥ ((ψ′ ⊥ π)_{K})_{an} and hence ψ_{K} ⊂ σ_{K}.

Note that if m = 1 then dim(ψ′ ⊥ π) = 3, so ψ′ ⊥ π is a Pfister neighbor of some β ∈ P_{2}F. In our construction, the field K in the splitting tower of ψ′ ⊥ π is either F itself if ψ′ ⊥ π is already isotropic in which case F (β)/F is purely transcendental, or it is F (ψ′ ⊥ π) if ψ′ ⊥ π is anisotropic. In this case, the field F (ψ′ ⊥ π) is equivalent to F (β) since ψ′ ⊥ π is a Pfister neighbor of β. In any case, the field in the splitting tower which we consider is equivalent to F (β) and thus we may as well choose K = F (β).

It remains to show that φ_{K} and σ_{K} are anisotropic. Since dim ψ′ = 2^{m−1} < dim π, it follows from [H 3, Theorem 1] that ψ′_{F (π)} is anisotropic. Now π_{F (π)} = 0 and thus ((ψ′ ⊥ π)_{F (π)})_{an} ≃ ψ′_{F (π)} and we have i_{W} ((ψ′ ⊥ π)_{F (π)}) = 2^{m−1} = i_{W} ((ψ′ ⊥ π)_{K}). By Proposition 2.2(iii), we have that L = K·F (π) is purely transcendental over F (π). Now φ_{F (π)} ≃ σ_{F (π)} is anisotropic. Hence, φ_{L} ≃ σ_{L} stays anisotropic which clearly implies that φ_{K} and σ_{K} are anisotropic. By our remark at the beginning, we have that φ_{K} ∈ P_{n,m}K is defined by (σ_{K}, π_{K}). Since φ_{K} ∈ P_{n,m}K we have that φ_{K} cannot be similar to a subform of σ_{K}. Therefore, σ_{K(φ)} stays anisotropic and thus, ψ_{K(φ)} stays also anisotropic. This obviously yields that ψ_{F (φ)} is anisotropic. □

We added the additional statement in the case m = 1 because we will need this particular fact later on in the proof of Proposition 7.8

Proof of Theorem 6.9. Let 1 ≤ m ≤ 2 and m < n. Let φ ∈ W F be anisotropic and dim φ = 2^{n}. Suppose that φ ≡ π ⊗ η (mod J_{n}F) for some anisotropic π ∈ P_{m}F and some η ∈ W F with dim η odd. By Proposition 6.8, it remains to show that Equiv(φ) ⊂ 𝔈(φ).

So let ψ ∈ W F with ψ ∼ φ. Clearly, dim ψ ≥ 2. Now ψ ∼ φ implies that φ_{F (ψ)} is isotropic. Since m ≤ 2 we have that F (π)/F is excellent. Proposition 5.2 implies that then there exists ψ̃ ∈ W F, dim ψ̃ = 2^{n}, such that ψ ⊂ ψ̃ and, possibly after scaling, ψ̃_{F (π)} ≃ φ_{F (π)}. By Proposition 5.1, we have that φ_{F (ψ̃)} is isotropic. Now ψ̃_{F (ψ)} is isotropic as ψ ⊂ ψ̃. We also have that ψ_{F (φ)} is isotropic because ψ ∼ φ. Hence, ψ̃_{F (φ)} is isotropic as well (cf. Proposition 2.1(viii)), and therefore φ ∼ ψ̃. By Theorem 6.4, there exists μ ∈ W F, dim μ odd, such that ψ̃ ≡ π ⊗ μ (mod J_{n}F). By Corollary 3.13, there exists a field extension K/F such that ψ̃_{K} ∈ GP_{n,m}K. We have already seen that ψ ∼ φ ∼ ψ̃. In particular, ψ_{F (ψ̃)} is isotropic which clearly yields that ψ_{K(ψ̃)} is also isotropic. Now ψ_{K} ⊂ ψ̃_{K} ∈ GP_{n,m}K. Lemma 6.10 implies that dim ψ > 2^{n} − 2^{m−1}. This completes the proof. □

Corollary 6.11 Conjecture 6.3 holds for n ≤ 4.

Proof. This is an immediate consequence of Corollary 6.5 and Theorem 6.9. □

Example 6.12 We return to the forms φ and ψ over F = ℝ(t) which we defined in Example 5.13. We had φ ∈ P_{n,m}F being defined by (σ, π) where n − m ≥ 3. We showed that ψ_{F (π)} ≃ φ_{F (π)} and by our construction we had that ψ ≡ −⟨1, 1, 1⟩ ⊗ π

(mod J_{n}F). Hence, by Theorem 6.4, φ ∼ ψ. However, we also showed that ψ ∉ GP_{n,m}F. This shows that if φ ∈ P_{n,m}F and ψ ∼ φ with dim ψ = dim φ = 2^{n} then generally this does not imply ψ ∈ GP_{n,m}F if n − m ≥ 3. □

We know that two anisotropic Pfister forms are equivalent iff they are isometric. To finish this section, we would like to say something about equivalence of twisted Pfister forms. Since their dimensions are always 2-powers, equivalent twisted Pfister forms must be of the same dimension. We have the following.

Proposition 6.13 Let 1 ≤ m, ℓ < n. Let φ ∈ P_{n,m}F be defined by (σ, π), and let ψ ∈ P_{n,ℓ}F be defined by (τ, ρ). Then φ ∼ ψ iff π ≃ ρ and σ_{F (π)} ≃ τ_{F (π)}. Furthermore, if this is the case then we have the following.

(i) If m = n − 1 then φ is similar to ψ.

(ii) If m ≤ n − 2 then φ is similar to ψ iff σ ≃ τ.

Proof. We have deg φ = deg π = m, deg ψ = deg ρ = ℓ and φ ≡ −π (mod J_{n}F), ψ ≡ −ρ (mod J_{n}F). If φ ∼ ψ then, by Theorem 6.4, m = ℓ and π ≡ φ ≡ ψ ≡ ρ

(mod J_{m+1}F) which readily yields π ≃ ρ. Also by Theorem 6.4, we have that φ_{F (π)} ≃ σ_{F (π)} is similar to ψ_{F (π)} ≃ τ_{F (π)} (here, we already use π ≃ ρ), which immediately implies σ_{F (π)} ≃ τ_{F (π)}. This shows the “only if” part. The converse follows also easily from Theorem 6.4.

Let us now assume that φ ∼ ψ. If m = n − 1 we know from Theorem 6.1 that φ is similar to ψ. So let us finally assume that m ≤ n − 2. If σ ≃ τ then obviously φ ≃ ψ by definition of a twisted Pfister form. Conversely, suppose that φ ≃ aψ for some a ∈ Ḟ. Then, in W F,

\begin{aligned}0 & = & φ − aψ = σ − π − a(τ − π) \\ & = & σ − aτ − π ⊗ ⟨⟨−a⟩⟩ . \end{aligned}

Now σ, τ ∈ P_{n}F and π ⊗ ⟨⟨−a⟩⟩ ∈ P_{m+1}F with m + 1 < n. We therefore get 0 ≡ −π⊗⟨⟨−a⟩⟩ (mod I^{n}F) and the Arason-Pfister Hauptsatz implies that π⊗⟨⟨−a⟩⟩ = 0. Hence, 0 = σ − aτ or σ ≃ aτ which implies that σ ≃ τ as σ and τ are both n-fold Pfister forms. □

This little result has a nice application. It is of interest to determine Equiv(φ) for a given anisotropic form φ ∈ W F, dim φ ≥ 2. Clearly, if ψ is similar to φ then ψ ∼ φ. More generally, if aψ ⊂ φ for some a ∈ Ḟ and dim ψ > dim φ − i_{1}(φ), then ψ_{F (φ)} is easily seen to be isotropic. Obviously, so is φ_{F (ψ)}. Hence, ψ ∼ φ. Even more generally, if there exists an anisotropic γ ∈ W F such that aφ ⊂ γ and bψ ⊂ γ for some a, b ∈ Ḟ such that dim φ, dim ψ > dim γ − i_{1}(γ), then by the same reasoning as above, φ ∼ γ ∼ ψ.

Another situation where we have φ ∼ ψ (both forms anisotropic) is when dim φ = dim ψ ≥ 2 and there exists a ∈ Ḟ such that φ ⊥ aψ is similar to some π ∈ P_{n}F. Clearly, we have dim φ = dim ψ = 2^{n−1}. Then π is isotropic and hence hyperbolic over F (φ) and F (ψ). In particular, φ_{F (φ)} ≃ −aψ_{F (φ)} and φ_{F (ψ)} ≃ −aψ_{F (ψ)}. Comparing dimensions and Witt indices, we conclude that φ_{F (ψ)} and ψ_{F (φ)} are both isotropic, i.e., φ ∼ ψ.

This leads to the following definitions.

Definition 6.14 Let φ, ψ ∈ W F be anisotropic. Then φ and ψ are neighbors if there exists an anisotropic γ ∈ W F, dim γ ≥ 2, such that φ and ψ are similar to subforms of γ and dim φ, dim ψ > dim γ − i_{1}(γ).

φ and ψ are called conjugate if dim φ = dim ψ and there exists a ∈ Ḟ such that φ ⊥ aψ ∈ GP_{n}F for some n.

If in the definition of neighbor the form γ is a similar to a Pfister form, then we have that φ and ψ are both Pfister neighbors of the same Pfister form. So this definition of neighbor is a natural generalization of a Pfister neighbor. Our definition of conjugate forms is slightly more general than the definition of conjugate forms in [K 2, Definition 8.7].

Remark 6.15 Let φ and ψ be anisotropic forms over F.

(i) If φ and ψ are similar, say, φ ≃ aψ, then φ and ψ are neighbors. If in addition dim φ = dim ψ = 2^{n} then φ and ψ are also conjugate. This is because φ ⊥ −aψ is isometric to the hyperbolic (n + 1)-fold Pfister form.

(ii) Suppose that φ and ψ are neighbors and that dim φ = dim ψ = 2^{n}. If dim γ > 2^{n} then φ_{F (γ)} and ψ_{F (γ)} are anisotropic, see Proposition 2.1(vi). A form γ as in the definition above with dim γ > 2^{n} can therefore not exists. So if γ is such a form as in the definition, we must have dim γ = 2^{n} which immediately implies that φ is similar to ψ. Hence, two anisotropic forms of dimension 2^{n} are similar iff they are neighbors.

(iii) Suppose that dim φ = dim ψ = 2^{n}. Then φ and ψ are similar or conjugate iff there exists an a ∈ Ḟ such that φ ⊥ aψ ∈ W (F (φ)/F) ∩ W (F (ψ)/F) (cf. [K 2, Theorem 8.8]).

Generally, conjugate forms are not similar. In a forthcoming paper, we will investigate such examples and the relationship between conjugacy and similarity.

The first examples known to us of forms φ and ψ with φ ∼ ψ but where φ and ψ are neither neighbors nor conjugate were given by twisted Pfister forms.

Proposition 6.16 Let n ≥ 3 and 1 ≤ m ≤ n − 2. Let φ, ψ ∈ P_{n,m}F be defined by (σ, π) and (τ, π), respectively, and assume that σ_{F (π)} ≃ τ_{F (π)} but σ ≄ τ. Then φ ∼ ψ but φ and ψ are neither neighbors nor conjugate.

Proof. By Proposition 6.13, we know that φ ∼ ψ and that φ is not similar to ψ. By Remark 6.15(ii), φ and ψ are not neighbors.

Suppose that φ and ψ are conjugate, i.e., φ ⊥ aψ ∈ GP_{n+1}F for some a ∈ Ḟ. Then

0 ≡ φ + aψ ≡ σ − π + a(τ − π) ≡ −π ⊗ ⟨⟨a⟩⟩ (mod I^{n}F)

because φ ⊥ aψ ∈ GP_{n+1}F ⊂ I^{n}F and σ, τ ∈ P_{n}F ⊂ I^{n}F. Since dim(π ⊗ ⟨⟨a⟩⟩) = 2^{m+1} < 2^{n}, the Arason-Pfister Hauptsatz implies π⊗⟨⟨a⟩⟩ = 0 and thus φ ⊥ aψ = σ ⊥ aτ in W F. Comparing dimensions and because φ ⊥ aψ ≃ ρ ∈ GP_{n+1}F we get that σ ⊥ aτ ≃ ρ ∈ GP_{n+1}F. Hence, ρ_{F (σ)} becomes isotropic and therefore hyperbolic. Thus, in W F (σ),

0 = ρ_{F (σ)} = σ_{F (σ)} + aτ_{F (σ)} = aτ_{F (σ)}

which yields that σ is similar to a subform of τ. This in turn implies that σ ≃ τ, a contradiction. □

In the last section, we will construct examples of forms σ, τ, and π which satisfy the conditions in Proposition 6.16. Let us conclude this section with another example of equivalent forms which are neither neighbors nor conjugate.

Example 6.17 Let F = ℝ(t) and let φ and ψ be the anisotropic forms in Example 5.13. Then φ ∼ ψ but φ and ψ are neither neighbors nor conjugate. We leave the details to the reader. □

7 Twisted Pfister forms over the function field of a Pfister form

Let φ ∈ P_{n,m}F be defined by (σ, π), i.e., σ ∈ P_{n}F and π ∈ P_{m}F are anisotropic, ln(σ, π) = m − 1, and φ ≃ (σ ⊥ −π)_{an}. Let τ ∈ P_{n}F. We know by Proposition 5.1 that σ_{F (τ)} is isotropic iff σ_{F (π)} ≃ τ_{F (π)}. So let us from now on assume that σ_{F (τ)} is isotropic, i.e., σ_{F (π)} ≃ τ_{F (π)}. This also implies by Proposition 5.8 that there exists α ∈ P_{m−1}F which divides π, σ, and τ.

Izhboldin [I] used twisted Pfister forms in his construction of Pfister forms which yield non-excellent function field extensions. More precisely, he essentially showed that for φ and τ as above and in the particular case where m = 1 and ln(σ, τ) = 1, then (φ_{F (τ)})_{an} is not defined over F. It is our aim to generalize this result. First, let us note the following.

Lemma 7.1 Let φ ∈ P_{n,m}F be defined by (σ, π) and let τ ∈ P_{n}F with σ_{F (π)} ≃ τ_{F (π)}, i.e., φ_{F (τ)} is isotropic. Then

dim(φ_{F (τ)})_{an} = \{ \begin{matrix}2^{m} & if & σ ≃ τ \\ 2^{n} − 2^{m} & if & σ ≄ τ \end{matrix}

Proof. Suppose first that σ ≃ τ. Then σ_{F (τ)} = 0 and π_{F (τ)} is anisotropic because m < n, and in W F (τ) we have φ_{F (τ)} = σ_{F (τ)}−π_{F (τ)} = −π_{F (τ)}. It follows immediately that (φ_{F (τ)})_{an} ≃ −π_{F (τ)} and dim(φ_{F (τ)})_{an} = 2^{m}.

Now suppose that σ ≄ τ. Then σ_{F (τ)} is anisotropic and thus we have, using φ = σ −π in W F and φ_{F (τ)} isotropic, that 2^{n} > dim(φ_{F (τ)})_{an} ≥ dim σ −dim π = 2^{n} −2^{m}. By Proposition 3.6 we therefore have dim(φ_{F (τ)})_{an} = 2^{n} − 2^{m}. □

We now come to the main result of this section

Theorem 7.2 Let φ ∈ P_{n,m}F be defined by (σ, π) and let τ ∈ P_{n}F with σ_{F (π)} ≃ τ_{F (π)}, i.e., φ_{F (τ)} is isotropic. Then the following are equivalent.

(i) There exists a Pfister neighbor χ of τ such that χ ⊂ φ.

(ii) There exists a Pfister neighbor χ of τ with dim χ = 2^{n−1} + 2^{m−1} such that

χ ⊂ φ. (iii) (φ_{F (τ)})_{an} is defined over F. (iv) ln(σ, τ) ≥ n − 1.

Proof. (i)⇒(ii). Let a ∈ Ḟ and χ ⊂ φ with dim χ ≥ 2^{n−1} + 1 such that χ ⊂ aτ. Let η ≃ (φ ⊥ −aτ)_{an}. Then dim η ≤ dim φ + dim τ − 2 dim χ ≤ 2^{n} − 2. Note that we have η ≡ φ − aτ ≡ σ − π − aτ ≡ −π (mod I^{n}F) as σ, τ ∈ P_{n}F ⊂ I^{n}F. Thus, we get η_{F (π)} ≡ 0 (mod I^{n}F (π)) and the Arason-Pfister Hauptsatz implies η_{F (π)} = 0 or η_{F (π)} ∈ W (F (π)/F). Hence, by Proposition 2.1(v), there exists μ ∈ W F such that η ≃ μ ⊗ π. Thus, dim π = 2^{m} divides dim η and therefore dim η = dim(φ ⊥ −aτ)_{an} ≤ 2^{n} − 2^{m} or i_{W} (φ ⊥ −aτ) ≥ 2^{n−1} + 2^{m−1}. In particular, φ and aτ have a common subform of dimension 2^{n−1} + 2^{m−1}.

(ii)⇒(iii). If σ ≃ τ we have already seen in the proof of Lemma 7.1 that (φ_{F (τ)})_{an} ≃ −π_{F (τ)} and we are done. Hence, we may assume that σ ≄ τ and thus dim(φ_{F (τ)})_{an} = 2^{n} − 2^{m} by Lemma 7.1. Let χ ⊂ φ such that dim χ = 2^{n−1} + 2^{m−1} and χ ⊂ aτ for some a ∈ Ḟ. Write φ ≃ χ ⊥ φ̃ and aτ ≃ χ ⊥ τ̃ for suitable φ̃, τ̃ ∈ W F. In W F (τ), we have

φ_{F (τ)} = (φ ⊥ −aτ)_{F (τ)} = (φ̃ ⊥ −τ̃)_{F (τ)} .

Now an easy check shows that dim(φ̃ ⊥ −τ̃) = 2^{n} − 2^{m} = dim(φ_{F (τ)})_{an}. Therefore, we must have (φ_{F (τ)})_{an} ≃ (φ̃ ⊥ −τ̃)_{F (τ)} and we see that (φ_{F (τ)})_{an} is defined over F by φ̃ ⊥ −τ̃.

(iii)⇒(iv). Let η ∈ W F such that (φ_{F (τ)})_{an} ≃ η_{F (τ)}. By Lemma 7.1 we have 2^{m} ≤ dim η ≤ 2^{n} − 2^{m} and thus 0 < dim(φ ⊥ −η)_{an} < 2^{n+1}. Also, by our choice of η, (φ ⊥ −η)_{an} ∈ W (F (τ)/F) and by Proposition 2.1(v) there exists a ∈ Ḟ such that (φ ⊥ −η)_{an} ≃ aτ. Hence, in W F we get aτ = φ − η = σ − π − η or σ ⊥ −aτ = π ⊥ η. Now dim(π ⊥ η) ≤ 2^{m} + 2^{n} − 2^{m} = 2^{n} and we get that i_{W} (σ ⊥ −aτ) ≥ 2^{n−1}. By Lemma 3.2, ln(σ, τ) ≥ n − 1.

(iv)⇒(i) This is rather obvious in the case where σ ≃ τ, i.e., ln(σ, τ) = n.

So let us assume that ln(σ, τ) = n − 1 and let ρ ≃ (σ ⊥ −τ)_{an}. Then dim ρ = 2^{n} and in fact ρ ∈ GP_{n}F as ρ ∈ I^{n}F. Let ψ ≃ (φ ⊥ −τ)_{an}. Then, in W F,

ψ = φ − τ = σ − π − τ = ρ − π .

This yields

2^{n} − 2^{m} = dim ρ − dim π ≤ dim ψ ≤ dim ρ + dim π = 2^{n} + 2^{m} .

On the other hand, in W F (π),

ψ_{F (π)} = φ_{F (π)} − τ_{F (π)} = σ_{F (π)} − τ_{F (π)} − π_{F (π)} = 0

as σ_{F (π)} ≃ τ_{F (π)} and π_{F (π)} = 0. Thus, there exists γ ∈ W F with ψ ≃ γ ⊗ π by Proposition 2.1(v). Now dim γ must be odd for otherwise ψ ∈ I^{m+1}F, but ψ ≡ σ − π − τ ≡ −π ≢ 0 (mod I^{m+1}F). Hence, there are two cases. Either dim ψ = 2^{n} − 2^{m} or dim ψ = 2^{n} + 2^{m}. If dim ψ = 2^{n} − 2^{m} then by the definition of ψ we get i_{W} (φ ⊥ −τ) = \frac{1}{2}(dim φ + dim τ − dim ψ) = 2^{n−1} + 2^{m−1}. Thus, φ and τ have a common subform of dimension 2^{n−1} + 2^{m−1} and we are done.

So let us finally assume that dim ψ = 2^{n} + 2^{m} so that in fact ψ ≃ (φ ⊥ −τ)_{an} ≃ ρ ⊥ −π. Now π divides ψ and we have that π also divides ρ. But ρ ∈ GP_{n}F. Hence, there exist δ ∈ P_{n−m}F and x ∈ Ḟ such that ρ ≃ xπ ⊗ δ. Thus,

xψ ≃ π ⊗ (δ ⊥ ⟨−x⟩) ⊂ π ⊗ δ ⊗ ⟨⟨−x⟩⟩ ∈ P_{n+1}F .

This shows that ψ is a Pfister neighbor of β ≃ π ⊗ δ ⊗ ⟨⟨−x⟩⟩ ∈ P_{n+1}F. Since ψ is anisotropic, β is anisotropic as well. Also, ψ_{F (τ)} = φ_{F (τ)} − τ_{F (τ)} = φ_{F (τ)} in W F (τ). Comparing dimensions, we conclude that ψ_{F (τ)} is isotropic and that therefore also β_{F (τ)} is isotropic and hence hyperbolic, which in turn implies that β ≃ τ ⊗ ⟨⟨t⟩⟩ for some t ∈ Ḟ. Since xψ ⊂ β we get ψ ⊂ xβ ≃ τ ⊗ ⟨x, xt⟩. Now ψ ⊥ τ = φ in W F and by comparing dimensions we see that ψ ⊥ τ is isotropic. In particular, there exists y ∈ D(ψ) ⊥ D(−τ). Since y ∈ D(ψ) ⊂ D(τ ⊗ ⟨x, xt⟩) and since τ ∈ P_{n}F, we may assume by Lemma 3.1 that for suitable z ∈ Ḟ we have τ ⊗ ⟨x, xt⟩ ≃ τ ⊗ ⟨y, z⟩.

But −y ∈ D(τ) = G(τ). If we write τ ⊗ ⟨x, xt⟩ ≃ ψ ⊥ μ for suitable μ ∈ W F with dim μ = 2^{n+1} − dim ψ = 2^{n} − 2^{m}, we obtain

ψ ⊥ μ ≃ τ ⊗ ⟨x, xt⟩ ≃ yτ ⊥ zτ ≃ −τ ⊥ zτ

and thus, in W F,

μ = zτ − τ − ψ = zτ − φ

(here we use ψ = φ − τ.) In particular, (zτ ⊥ −φ)_{an} ≃ μ (recall that ψ ⊥ μ is similar to the anisotropic Pfister form β and that therefore μ is also anisotropic), which implies that i_{W} (zτ ⊥ −φ) = \frac{1}{2}(dim τ + dim φ − dim μ) = 2^{n−1} + 2^{m−1}, i.e., φ and zτ have a common subform of dimension 2^{n−1} + 2^{m−1} which is obviously a Pfister neighbor of τ. □

Remark 7.3 If φ contains a Pfister neighbor of τ of dimension > 2^{n−1} + 2^{m−1} then σ ≃ τ. For in this case, there exists a ∈ Ḟ such that i_{W} (φ ⊥ −aτ) > 2^{n−1} + 2^{m−1} or dim(φ ⊥ −aτ)_{an} < 2^{n} − 2^{m}. But in W F we have φ ⊥ −aτ = σ ⊥ −aτ ⊥ −π and we must necessarily have dim(σ ⊥ −aτ)_{an} < 2^{n} which, by Lemma 3.1, implies ln(σ, τ) = n, in other words σ ≃ τ.

Conversely, if σ ≃ τ then the largest Pfister neighbor of τ contained in φ has dimension 2^{n} − 2^{m−1}. That φ contains such a Pfister neighbor of this dimension follows readily from Remark 3.5(ii) (using the notation there, one may take α ⊗ σ′_{1}). On the other hand φ does not contain any Pfister neighbor of dimension > 2^{n} −2^{m−1}. For suppose otherwise. Let χ ⊂ φ be such a Pfister neighbor of some χ̃ ∈ P_{n}F with dim χ > 2^{n} − 2^{m−1}. Then χ̃ ∼ χ ∼ φ, the first equivalence because χ is a Pfister neighbor of χ̃, the second one by Corollary 3.8. But this is absurd since φ is clearly not a Pfister neighbor of χ̃.

Corollary 7.4 Let φ ∈ P_{n,n−1}F and τ ∈ P_{n}F. Then (φ_{F (τ)})_{an} is defined over F.

Proof. If φ_{F (τ)} stays anisotropic then there is nothing to show. So let us assume that φ_{F (τ)} is isotropic. By Proposition 5.1, we have that φ_{F (π)} ≃ σ_{F (π)} ≃ τ_{F (π)}. It follows immediately from Proposition 5.8(ii) that ln(σ, τ) ≥ n − 1. The desired result follows now from Theorem 7.2. □

Statements (i) resp. (ii) of Theorem 7.2 essentially say that the obstruction to (φ_{F (τ)})_{an} being defined over F is the non-existence of a Pfister neighbor of τ as a subform of φ, and by Corollary 7.4 this can only happen if n ≥ 3 and (n, m) ≠ (n, n − 1). This is not at all obvious as the case of the function field of a 2-fold Pfister form η shows. There are many examples of fields F with anisotropic forms ψ ∈ W F and η ∈ P_{2}F such that ψ_{F (η)} is isotropic but ψ does not contain a Pfister neighbor of η (for such examples we refer to [LVG], [HLVG], [HVG]). However, since F (τ)/F is excellent we have, by definition of excellence, that (ψ_{F (η)})_{an} is defined over F. Conversely, if τ ∈ P_{n}F ≥ 3 such that F (τ)/F is not excellent then there might still be many forms ψ which contain Pfister neighbors of τ but where (ψ_{F (τ)})_{an} is not defined over F. For example, let φ be such that (φ_{F (τ)})_{an} is not defined over F and put ψ ≃ τ ⊥ φ. Then (ψ_{F (τ)})_{an} ≃ (φ_{F (τ)})_{an} is not defined over F, but ψ contains τ itself as a subform.

Twisted Pfister forms also yield new non-trivial examples of F (τ)-minimal forms where τ ∈ P_{n}F, n ≥ 3. Recall that for a field extension K/F we say that φ is K-minimal if φ is anisotropic, φ_{K} is isotropic, and if η ⊂ φ with dim η < dim φ then η_{K} is anisotropic. We are interested in the case where K = F (τ) for some anisotropic τ ∈ P_{n}F. If n = 1 the K-minimal forms are exactly the scalar multiples of τ, cf. Proposition 2.1(iii). For n = 2, one can show that K-minimal forms are always of odd dimension ≥ 3. One can even construct a field F with some τ ∈ P_{2}F such that to each odd integer m ≥ 3 there exists a K-minimal form of dimension m, cf. [HVG].

Not much is known about K-minimal forms in the case K = F (τ) with τ ∈ P_{n}F, n ≥ 3. The following is known.

Theorem 7.5 ([H 3, Theorem 3], [H 2, Corollary 4.2].) Let τ ∈ P_{n}F be anisotropic and K = F (τ).

(i) The K-minimal forms of dimension ≤ 2^{n−1} +1 are exactly the Pfister neighbors of τ of dimension 2^{n−1} + 1.

(ii) If n ≤ 3 then the K-minimal forms of dimension ≤ 2^{n−1} + 2 are exactly the Pfister neighbors of τ of dimension 2^{n−1} + 1. In particular, if φ ∈ W F is anisotropic with dim φ ≤ 2^{n−1} + 2, then φ_{K} is isotropic iff φ contains a Pfister neighbor of τ.

In view of this result, Lemma 7.1 and Theorem 7.2, we get the following.

Proposition 7.6 Let n ≥ 3 and 1 ≤ m ≤ n − 2. Let φ ∈ P_{n,m}F be defined by (σ, π) and let τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and ln(σ, τ) ≤ n − 2. Then each F (τ)- minimal form χ contained in φ has dimension 2^{n−1} + 2 ≤ dim χ ≤ 2^{n} − 2^{m−1} + 1. Moreover, if (n, m) = (3, 1) then 7 ≤ dim χ ≤ 8.

Proof. By Lemma 7.1, i_{W} (φ_{F (τ)}) = 2^{m−1}. Thus, any subform of φ of dimension 2^{n} − 2^{m−1} + 1 becomes isotropic over F (τ). Hence, if χ ⊂ φ is F (τ)-minimal we must necessarily have dim χ ≤ 2^{n} − 2^{m−1} + 1. We know by Theorem 7.2 that φ does not contain any Pfister neighbor of τ. Therefore, by Theorem 7.5(i), we must have dim χ ≥ 2^{n−1} + 2, and if n = 3 then Theorem 7.5(ii) even implies that dim χ ≥ 7. □

In fact, Izhboldin [I] showed that with φ, σ, τ as in Proposition 7.6, if m = 1 and if ln(σ, τ) = 1 then φ itself is F (τ)-minimal. This leads us to conjecture the following.

Conjecture 7.7 Let n ≥ 3 and 1 ≤ m ≤ n − 2. Let φ ∈ P_{n,m}F be defined by (σ, π) and let τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and ln(σ, τ) = m. Let χ ⊂ φ. Then χ is F (τ)-minimal iff dim χ = 2^{n} − 2^{m−1} + 1.

Note that τ_{F (π)} ≃ σ_{F (π)} implies that ln(σ, τ) ≥ m, cf. Proposition 5.8. In our conjecture, we require that the linkage of σ and τ is at the lower end, i.e., ln(σ, τ) = m. This will be needed in the proof of the conjecture in the case m = 1 and it is for this reason that we imposed this condition in the conjecture.

Proposition 7.8 (Izhboldin [I].) Let n ≥ 3. Let φ ∈ P_{n,1}F be defined by (σ, π) and let τ ∈ P_{n}F such that τ_{F (π)} ≃ σ_{F (π)} and ln(σ, τ) = 1. Then φ is F (τ)-minimal.

Proof. φ_{F (τ)} is isotropic by Proposition 5.1. To prove that φ is F (τ)-minimal it suffices to show that if η ⊂ φ and dim η = 2^{n} − 1, then η_{F (τ)} stays anisotropic.

By Lemma 6.10, there exists β ∈ P_{2}F such that for K = F (β) we have that σ_{K} is anisotropic and η_{K} ⊂ σ_{K}. Suppose η_{F (τ)} is isotropic. Then η_{K(τ)} is isotropic and hence, σ_{K(τ)} is isotropic and therefore hyperbolic. This implies that τ_{K} is similar and thus isometric to σ_{K}. Let ψ ≃ (σ ⊥ −τ)_{an}. Clearly, ψ ∈ W (K/F) and hence there exists γ ∈ W F such that ψ ≃ β ⊗ γ. Now ln(σ, τ) = 1 and thus dim ψ = 2^{n+1} − 4. Hence, dim γ = 2^{n−1} − 1 is odd and one readily concludes that ψ ≡ β ⊗ γ ≡ β ≢ 0

(mod I^{3}F) as β ∈ P_{2}F is anisotropic (if β were isotropic, the anisotropic form ψ would stay anisotropic over K = F (β)). But ψ = σ − τ ∈ I^{n}F with n ≥ 3, a contradiction. □

Remark 7.9 The reason why this proof works so smoothly in the case m = 1 is that the field K from Lemma 6.10 is of a very nice form which just fits the situation. One would hope that with the field K from Lemma 6.10 one could give a similar proof also for m > 1. Consider the situation in Conjecture 7.7. To show that the conjecture is true it suffices to show that if η ⊂ φ with dim η = 2^{n} − 2^{m−1} then η_{F (τ)} is anisotropic. One might want to proceed as in the proof above. Let K be as in Lemma 6.10 such that η_{K} ⊂ σ_{K} and φ_{K}, σ_{K}, π_{K} stay anisotropic. The problem is to show that τ_{K} ≄ σ_{K}. This worked for m = 1 because one can choose K = F (β) for some β ∈ P_{2}F. If m > 1 then our construction in the proof of Lemma 6.10 generally leads to a field K for which it is not so clear why τ_{K} ≄ σ_{K} should hold.

8 Constructions of twisted Pfister forms

In this section we explicitly construct examples of φ ∈ P_{n,m}F defined by (σ, π), and τ ∈ P_{n}F such that ln(σ, τ) = k for some m ≤ k ≤ n − 2 such that σ_{F (π)} ≃ τ_{F (π)}. By Proposition 5.8 we then have ln(τ, π) = m − 1 and thus, we get a form ψ ∈ P_{n,m}F defined by (τ, π) simply by putting ψ ≃ (τ ⊥ −π)_{an}. By Proposition 6.16 this shows the existence of φ, ψ ∈ P_{n,m}F such that φ ∼ ψ but φ and ψ are neither neighbors nor conjugate, and by Theorem 7.2 it also shows the existence of φ ∈ P_{n,m}F and τ ∈ P_{n}F such that (φ_{F (τ)})_{an} is not defined over F. Note that by the symmetry of the situation we also have that (ψ_{F (σ)})_{an} is not defined over F. Hence, F (τ)/F and F (σ)/F are both non-excellent field extensions.

In the first example, we will achieve this over a purely transcendental extension of the rationals ℚ, and in the second example we will actually generalize Izhboldin’s approach in [I].

Example 8.1 Let 1 ≤ m ≤ k ≤ n−2. To simplify notations, let ℓ = n−2−k so that k + ℓ + 2 = n. Let F = ℚ(x_{1}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}) be the rational function field in the k + ℓ + 1 = n − 1 variables x_{i} and y_{j} over the rationals ℚ. Let p_{0}, ⋯, p_{ℓ} be distinct prime numbers with p_{i} ≡ 7 (mod 8). We now define Pfister forms σ, τ ∈ P_{n}F and π ∈ P_{m}F as follows:

\begin{aligned}σ & ≃ & ⟨⟨1, x_{1}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩ ; \\ τ & ≃ & ⟨⟨2, x_{1}, ⋯, x_{k}, p_{0}y_{0}, ⋯, p_{ℓ}y_{ℓ}⟩⟩ ; \\ π & ≃ & ⟨⟨x_{1}, ⋯, x_{m−1}, −x_{m}⟩⟩ . \end{aligned}

One easily sees that σ, τ, and π are anisotropic (for instance by passing to the iterated power series field in the variables x_{i}, y_{j}, and then repeatedly applying Springer’s theorem [L 1, Ch. 6, Proposition 1.9], [S, Ch. 6, Corollary 2.6(i)]).

Claim 1: ln(σ, τ) = k. Proof.

σ ⊥ −τ ≃ ⟨⟨x_{1}, ⋯, x_{k}⟩⟩ ⊗ \underbrace{(⟨⟨1, y_{0}, ⋯, y_{ℓ}⟩⟩ ⊥ −⟨⟨2, p_{0}y_{0}, ⋯, p_{ℓ}y_{ℓ}⟩⟩)}_{γ} .

For ∅ ≠ I ⊂ {0, ⋯, ℓ} we define Y_{I} = ∏_{i∈I} y_{i} and P_{I} = ∏_{i∈I} p_{i}. We find that

γ ≃ ⟨1, 1, −1, −2⟩ ⊥ ⊥_{∅≠I⊂{0,···,ℓ}}Y_{I}⟨1, 1, −P_{I}, −2P_{I}⟩ .

By Springer’s theorem, we get

i_{W} (γ) = i_{W} (⟨1, 1, −1, −2⟩) + ∑_{∅≠I⊂{0,···,ℓ}} i_{W} (⟨1, 1, −P_{I}, −2P_{I}⟩) ,

where the Witt indices of the forms on the right hand side is computed over ℚ. Now i_{W} (⟨1, 1, −1, −2⟩) = 1 as ⟨1, 1, −1, −2⟩ ≃ ℍ ⊥ ⟨1, −2⟩ and ⟨1, −2⟩ is anisotropic over ℚ. By passing to the local field ℚ_{p_{i}} for some i ∈ I, we get for the Legendre symbols \binom{−2}{p_{i}} ≠ 1 and \binom{−1}{p_{i}} ≠ 1 as p_{i} ≡ 7 (mod 8). Hence, ⟨1, 1⟩ and ⟨1, 2⟩ are anisotropic over ℚ_{p_{i}} and thus also ⟨1, 1, −P_{I}, −2P_{I}⟩ because p_{i} divides P_{I} exactly to the first power (note that all the p_{j}’s in the product P_{I} are distinct !). Hence, i_{W} (⟨1, 1, −P_{I}, −2P_{I}⟩) = 0 and we have i_{W} (γ) = 1. Again by Springer’s theorem, we readily conclude that

i_{W} (σ ⊥ −τ) = i_{W} (⟨⟨x_{1}, ⋯, x_{k}⟩⟩ ⊗ γ) = 2^{k}i_{W} (γ) = 2^{k} .

Hence, ln(σ, τ) = k.

Claim 2: ln(σ, π) = ln(τ, π) = m − 1. In particular, φ ≃ (σ ⊥ −π)_{an}, ψ ≃ (τ ⊥ −π)_{an} ∈ P_{n,m}F. Proof. This can be shown in a similar way as before.

σ ⊥ −π ≃ ⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩ ⊗ (⟨⟨1, x_{m}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩ ⊥ −⟨⟨−x_{m}⟩⟩) ,

and by Springer’s theorem we obtain that

⟨⟨1, x_{m}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩ ⊥ −⟨⟨−x_{m}⟩⟩ ≃ ℍ ⊥ ⟨⟨1, x_{m}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩′ ⊥ ⟨x_{m}⟩

has Witt index 1 as ⟨⟨1, x_{m}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩′ ⊥ ⟨x_{m}⟩ is anisotropic (here, ρ′ denotes the pure part of a Pfister form ρ), and that therefore

i_{W} (σ ⊥ −π) = dim ⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩ = 2^{m−1}

which in turn implies that ln(σ, π) = m−1. A similar argument shows that ln(τ, π) = m − 1 and we omit the details. It is now obvious that φ ≃ (σ ⊥ −π)_{an} and ψ ≃ (τ ⊥ −π)_{an} have dimension 2^{n} and are in P_{n,m}F.

Claim 3: σ_{F (π)} ≃ τ_{F (π)}. Proof. Let K = F (π). We have

0 = π_{K} = ⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩_{K} ⊥ −x_{m}⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩_{K}

and therefore ⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩_{K} ≃ x_{m}⟨⟨x_{1}, ⋯, x_{m−1}⟩⟩_{K}. Hence,

⟨⟨x_{1}, ⋯, x_{m−1}, x_{m}⟩⟩_{K} ≃ ⟨⟨x_{1}, ⋯, x_{m−1}, 1⟩⟩_{K} .

Note also that ⟨⟨1, 2⟩⟩ ≃ ⟨⟨1, 1⟩⟩ and that ⟨⟨1, 1⟩⟩ ≃ a⟨⟨1, 1⟩⟩ for any positive a ∈ ℚ̇. In particular, ⟨⟨1, 1, y_{i}⟩⟩ ≃ ⟨⟨1, 1, p_{i}y_{i}⟩⟩. All this together yields

σ_{K} ≃ ⟨⟨1, x_{1}, ⋯, x_{m−1}, x_{m}, x_{m+1}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩_{K}
≃ ⟨⟨1, x_{1}, ⋯, x_{m−1}, 1, x_{m+1}, ⋯, x_{k}, y_{0}, ⋯, y_{ℓ}⟩⟩_{K}
≃ ⟨⟨1, x_{1}, ⋯, x_{m−1}, 1, x_{m+1}, ⋯, x_{k}, p_{0}y_{0}, ⋯, p_{ℓ}y_{ℓ}⟩⟩_{K}
≃ ⟨⟨2, x_{1}, ⋯, x_{m−1}, 1, x_{m+1}, ⋯, x_{k}, p_{0}y_{0}, ⋯, p_{ℓ}y_{ℓ}⟩⟩_{K}
≃ ⟨⟨2, x_{1}, ⋯, x_{m−1}, x_{m}, x_{m+1}, ⋯, x_{k}, p_{0}y_{0}, ⋯, p_{ℓ}y_{ℓ}⟩⟩_{K}
≃ τ_{K} . □

This completes our first example. Similar examples can be constructed also if one replaces ℚ by any global field 𝕂 of characteristic ≠ 2 and with the iterated power series field F = 𝕂((x_{1})) ⋯ ((x_{k}))((y_{0})) ⋯ ((y_{ℓ})). For the u-invariant of such a field we get u(F) = 2^{k+ℓ}u(𝕂) = 4·2^{n−1} = 2^{n+1}. Thus, we can construct examples of (non-formally real) fields F with u(F) = 2^{n+1} and τ ∈ P_{n}F such that F (τ)/F is not excellent. If F is non-formally real and u(F) < 2^{n} + 2^{n−1} then examples of the type we constructed above cannot exist. For in order for examples of this type to exist one needs two n-fold Pfister forms σ, τ such that ln(σ, τ) ≤ n − 2 or dim(σ ⊥ −τ)_{an} ≥ 2^{n} + 2^{n−1}. So the question is: Are there (non-formally real) fields F with u(F) < 2^{n} + 2^{n−1} such that there exists τ ∈ P_{n}F with F (τ)/F not excellent ? In [H 5] it was shown that if F is linked (which implies that u(F) ∈ {0, 1, 2, 4, 8}) then F (τ)/F is excellent for all Pfister forms τ over F (here, F may be formally real or non-formally real). Among the many other results in [H 5] let us only mention that if τ is a Pfister form over a field F and if the Hasse number of F, ũ(F), is ≤ 6 or if dim τ ≥ 2ũ(F), then F (τ)/F is excellent. This is of course mainly of interest in the case where F is formally real. For if F is non-formally real then there are no anisotropic forms of dimension > ũ(F).

Corollary 8.2 To each n ≥ 3 there exists a field F such that there are anisotropic n-fold Pfister forms ρ, σ over F with F (ρ)/F excellent and F (σ)/F not excellent.

Proof. We only show this for n = 3 to keep the notations simple. Let F = ℚ((x))((y)). The previous example shows that for σ ≃ ⟨⟨1, x, y⟩⟩ we have that F (σ)/F is not excellent. Let ρ ≃ ⟨⟨1, 1, 1⟩⟩. Since ρ is defined over ℚ, it is not hard to see that the field E = F (ρ) = ℚ((x))((y))(ρ) is contained in L = ℚ(ρ)((x))((y)). Let ψ be an anisotropic form over F. By Springer’s theorem, we can write ψ ≃ ψ_{0} ⊥ xψ_{1} ⊥ yψ_{2} ⊥ xyψ_{3} where the ψ_{i} are forms over ℚ which are uniquely determined up to isometry over ℚ. Let K = ℚ(ρ) ⊂ E. It is known that function fields of Pfister forms over global fields are always excellent (cf. [ELW 2], [H 5], see also the remarks preceding this corollary). Hence, there are forms μ_{i} defined over ℚ such that (μ_{i})_{K} ≃ ((ψ_{i})_{K})_{an}. In W E we obviously have

(ψ_{0})_{E} ⊥ x(ψ_{1})_{E} ⊥ y(ψ_{2})_{E} ⊥ xy(ψ_{3})_{E} = (μ_{0})_{E} ⊥ x(μ_{1})_{E} ⊥ y(μ_{2})_{E} ⊥ xy(μ_{3})_{E}.

The right hand side is defined over F by μ_{0} ⊥ xμ_{1} ⊥ yμ_{2} ⊥ xyμ_{3}. To show excellence, it remains to show that (μ_{0})_{E} ⊥ x(μ_{1})_{E} ⊥ y(μ_{2})_{E} ⊥ xy(μ_{3})_{E} is anisotropic. Indeed, this form is anisotropic over the bigger field L. This is because L = K((x))((y)), the (μ_{i})_{K} are anisotropic and by Springer’s theorem we have that (μ_{0})_{L} ⊥ x(μ_{1})_{L} ⊥ y(μ_{2})_{L} ⊥ xy(μ_{3})_{L} is anisotropic. Thus, we have shown that (ψ_{F (ρ)})_{an} is defined over F by μ_{0} ⊥ xμ_{1} ⊥ yμ_{2} ⊥ xyμ_{3}, which in turn proves the excellence of F (ρ)/F. □

The next example generalizes Izhboldin’s construction in [I] which was carried out only in the case m = ln(σ, τ) = 1. However, we will show that his arguments can be applied, after some minor modifications, to the more general situation where m can be any positive integer ≤ n − 2 and ln(σ, τ) = k can be any integer with m ≤ k ≤ n − 2.

Example 8.3 Let n ≥ 3 and let 1 ≤ m ≤ k ≤ n−2. Let τ ∈ P_{n}F be anisotropic. We will construct a unirational field extension E/F such that there exist anisotropic forms π ∈ P_{m}E and σ ∈ P_{n}E with ln(σ, π) = ln(τ_{E}, π) = m − 1 and ln(σ, τ_{E}) = k (note that τ will stay anisotropic over any unirational field extension), and furthermore σ_{E(π)} ≃ τ_{E(π)}. We then get a form φ ≃ (σ ⊥ −π)_{an} ∈ P_{n,m}E, and by Theorem 7.2 we have that (φ_{E(τ)})_{an} is not defined over E. In particular, E(τ)/E is not excellent. Our construction will involve various field extensions of F. Their relations among each other are shown in a diagram below.

As for the construction of E, let this time ℓ = n − k ≥ 2 and write

τ ≃ ⟨⟨a_{1}, ⋯, a_{k}, b_{1}, ⋯, b_{ℓ}⟩⟩

for suitable a_{i}, b_{j} ∈ Ḟ. Let F_{0} = F (y_{1}, ⋯, y_{ℓ}) and F_{1} = F_{0}(x) = F (x, y_{1}, ⋯, y_{ℓ}) be rational function fields in the variables x, y_{1}, ⋯, y_{ℓ} over F. Let

\begin{aligned}σ & ≃ & ⟨⟨a_{1}, ⋯, a_{k}, y_{1}, ⋯, y_{ℓ}⟩⟩ ∈ P_{n}F_{0} \\ π & ≃ & ⟨⟨x, a_{2}, ⋯, a_{m}⟩⟩ ∈ P_{m}F_{1} \end{aligned}

After passing to the iterated power series field in the variables x, y_{1}, ⋯, y_{ℓ} and by repeatedly applying Springer’s theorem, one readily checks that τ_{F_{1}}, σ_{F_{1}}, π_{F_{1}} are anisotropic and that ln(σ_{F_{1}}, τ_{F_{1}}) = k and ln(σ_{F_{1}}, π_{F_{1}}) = ln(τ_{F_{1}}, π_{F_{1}}) = m − 1. We leave the details to the reader. The aim is to construct E/F_{1} such that σ_{E(π)} ≃ τ_{E(π)}, such that this form and π_{E} stay anisotropic and such that ln(σ_{E}, τ_{E}) = k. Note that we will have m ≥ ln(σ_{E}, π_{E}) ≥ ln(σ_{F_{1}}, π_{F_{1}}) = m − 1. Now ln(σ_{E}, π_{E}) = m implies that π_{E} divides σ_{E} and thus σ_{E(π)} = 0, a contradiction to its anisotropy. Hence, we will still have ln(σ_{E}, π_{E}) = m − 1 and similarly ln(τ_{E}, π_{E}) = m − 1.

To get this field E, we first define the following forms over F_{1} which again are easily seen to be anisotropic:

\begin{aligned}τ̃ & ≃ & ⟨⟨x, a_{2}, ⋯, a_{k}, b_{1}, ⋯, b_{ℓ}⟩⟩ \\ σ̃ & ≃ & ⟨⟨x, a_{2}, ⋯, a_{k}, y_{1}, ⋯, y_{ℓ}⟩⟩ . \end{aligned}

Let E be the generic splitting field of the anisotropic form defined by (σ −τ)−(σ̃ −τ̃) in W F_{1}. Then, in W E, (σ − τ)_{E} − (σ̃ − τ̃)_{E} = 0 or (σ − τ)_{E} = (σ̃ − τ̃)_{E}. As π divides both σ̃ and τ̃, we get that σ̃_{E(π)} = τ̃_{E(π)} = 0, hence, (σ − τ)_{E(π)} = 0, i.e., σ_{E(π)} ≃ τ_{E(π)}.

We first show that σ_{E(π)} ≃ τ_{E(π)} is anisotropic. Let F_{2} = F_{1}(\sqrt{−x}). Then F_{2}/F_{0} is purely transcendental and thus σ_{F_{2}} and τ_{F_{2}} stay anisotropic and we still have ln(σ_{F_{2}}, τ_{F_{2}}) = ln(σ_{F_{0}}, τ_{F_{0}}) = k. Furthermore, ⟨⟨x⟩⟩_{F_{2}} = ⟨⟨−1⟩⟩_{F_{2}} = 0 and hence π_{F_{2}} = σ̃_{F_{2}} = τ̃_{F_{2}} = 0 in W F_{2}. Let K be the generic splitting field over F_{2} of (σ ⊥ −τ)_{F_{2}}. Clearly, (σ − τ)_{K} = 0 in W K, i.e., σ_{K} ≃ τ_{K}. We claim that σ_{K} ≃ τ_{K} is anisotropic. Now ln(σ_{F_{2}}, τ_{F_{2}}) = k < n and thus σ_{F_{2}} ≄ τ_{F_{2}}, i.e., (σ ⊥ −τ)_{F_{2}} ≠ 0. Clearly, deg(σ ⊥ −τ)_{F_{2}} ≥ n. Suppose that σ_{K} ≃ τ_{K} = 0. Then by [AK, Satz 20] it follows that deg(σ ⊥ −τ)_{F_{2}} = n and (σ ⊥ −τ)_{F_{2}} ≡ σ_{F_{2}} (mod J_{n+1}F_{2}). Hence, −τ_{F_{2}} ≡ 0 (mod J_{n+1}F_{2}) and the Arason-Pfister Hauptsatz implies that τ_{F_{2}} = 0, a contradiction.

Obviously, σ̃_{K(π)} = τ̃_{K(π)} = 0. Thus, (σ − τ)_{K(π)} − (σ̃ − τ̃)_{K(π)} = 0. Since E/F_{1} is the generic splitting field of (σ −τ)−(σ̃ −τ̃) ∈ W F_{1}, we have by Proposition 2.2(iii) that E ·K(π)/K(π) is purely transcendental. But K(π)/K is purely transcendental as well because π_{K} = 0. Hence, E ·K(π)/K is purely transcendental and therefore σ_{E·K(π)} ≃ τ_{E·K(π)} is anisotropic because σ_{K} ≃ τ_{K} is anisotropic. Since E(π) ⊂ E·K(π) we have that σ_{E(π)} ≃ τ_{E(π)} is anisotropic.

Let now F_{3} = F_{1}(\sqrt{a_{1}X}). Again, we clearly have that F_{3}/F_{0} is purely transcendental. Furthermore, σ_{F_{3}} ≃ σ̃_{F_{3}} and τ_{F_{3}} ≃ τ̃_{F_{3}} as a_{1} = X in Ḟ_{3}/Ḟ_{3}^{2}. Hence, (σ − τ)_{F_{3}} − (σ̃ − τ̃)_{F_{3}} = 0 in W F_{3} and we have that E·F_{3}/F_{3} is purely transcendental by the same reason as before. Hence, E ·F_{3}/F_{0} is purely transcendental as well and thus, since F_{0} ⊂ E ⊂ E ·F_{3}, we conclude that E/F_{0} is unirational. Therefore, ln(σ_{E}, τ_{E}) = ln(σ_{F_{0}}, τ_{F_{0}}) = k as desired. Obviously, E/F is also unirational as F_{0}/F is purely transcendental. This completes Izhboldin’s construction. □

[figure: F; F_{0} = F (y_{1}, ⋯, y_{ℓ}); F_{1} = F_{0}(x); E; F_{3} = F_{1}(\sqrt{a_{1}x}); E·F_{3}; E(π); F_{2} = F_{1}(\sqrt{−x}); K; K(π); E·K(π)]

Corollary 8.4 (Izhboldin [I].) Let τ ∈ P_{n}F be anisotropic, n ≥ 3. Then there exists a unirational field extension E/F such that E(τ)/E is not excellent.

References

[AK] Arason, J.Kr.; Knebusch, M.: Über die Grade quadratischer Formen. Math. Ann. 234 (1978), 167–192.

[AP] Arason, J.Kr.; Pfister, A.: Beweis des Krullschen Durschnittsatzes für den Wittring. Invent. Math. 12 (1971), 173–176.

[EL] Elman, R.; Lam, T.Y.: Pfister forms and K-theory of fields. J. Algebra 23 (1972), 181–213.

[ELW 1] Elman, R.; Lam, T.Y.; Wadsworth, A.R.: Amenable fields and Pfister extensions. Proc. of Quadratic Forms Conference (ed. G. Orzech). Queen’s Papers in Pure and Applied Mathematics No. 46 (1977), 445–491.

[ELW 2] Elman, R.; Lam, T.Y.; Wadsworth, A.R.: Function fields of Pfister forms. Invent. Math. 51 (1979), 61–75.

[F 1] Fitzgerald, R.W.: Witt kernels of function field extensions. Pacific J. Math. 109 (1983), 89–106.

[F 2] Fitzgerald, R.W.: Quadratic forms of height two. Trans. Amer. Math. Soc. 283 (1984), 339–351.

[H 1] Hoffmann, D.W.: Isotropy of 5-dimensional quadratic forms isotropic over the function field of a quadric. K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Proceedings of the 1992 Santa Barbara Summer Research Institute (eds. B. Jacob, A. Rosenberg). Proc. Symp. Pure Math. 58.2 (1995), 217–225.

[H 2] Hoffmann, D.W.: On 6-dimensional quadratic forms isotropic over the function field of a quadric. Comm. Alg. 22 (1994), 1999–2014.

[H 3] Hoffmann, D.W.: Isotropy of quadratic forms over the function field of a quadric. Math. Z. 220 (1995), 461–476.

[H 4] Hoffmann, D.W.: On quadratic forms of height 2 and a theorem of Wadsworth. To appear in: Trans. Amer. Math. Soc.

[H 5] Hoffmann, D.W.: Amenability and excellence of function fields of Pfister forms. Preprint 1995.

[H 6] Hoffmann, D.W.: Splitting of quadratic forms, I: Forms of small dimension. Preprint 1995.

[H 7] Hoffmann, D.W.: On the dimensions of anisotropic quadratic forms with trivial invariants. SFB 343, Universität Bielefeld, Preprint no. 88 (1995).

[HLVG] Hoffmann, D.W.; Lewis, D.W.; Van Geel, J.: Minimal forms for function fields of conics. K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Proceedings of the 1992 Santa Barbara Summer Research Institute (eds. B. Jacob, A. Rosenberg). Proc. Symp. Pure Math. 58.2 (1995), 227–237.

[HVG] Hoffmann, D.W.; Van Geel, J.: Minimal forms with respect to function fields of conics. Manuscripta Math. 86 (1995), 23–48.

[HuR] Hurrelbrink, J.; Rehmann, U.: Splitting patterns of quadratic forms. Math. Nachr. 176 (1995), 111–127.

[I] Izhboldin, O.T.: On the nonexcellence of field extensions F (π)/F. Doc. Math. J. DMV 1 (1996), 127–136 (cf. also Preprint 1995).

[Ka] Kahn, B.: Formes quadratiques de hauteur et de degré 2. To appear in: Indag. Math.

[K 1] Knebusch, M.: Generic splitting of quadratic forms I. Proc. London Math. Soc. 33 (1976), 65–93.

[K 2] Knebusch, M.: Generic splitting of quadratic forms II. Proc. London Math. Soc. 34 (1977), 1–31.

[Lag] Laghribi, A.: Isotropie de certaines formes quadratiques de dimension 7 et 8 sur le corps des fonctions d’une quadrique. To appear in: Duke Math. J.

[L 1] Lam, T.Y.: The Algebraic Theory of Quadratic Forms. Reading, Massachusetts: Benjamin 1973 (revised printing 1980).

[L 2] Lam, T.Y.: Ten lectures on quadratic forms over fields. Proc. of Quadratic Forms Conference (ed. G. Orzech). Queen’s Papers in Pure and Applied Mathematics No. 46 (1977), 1–102.

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

[M] Merkurjev, A.S.: On the norm residue symbol of degree 2. Dokladi Akad. Nauk. SSSR 261 (1981), 542–547. English translation: Soviet Math. Doklady 24 (1981), 546–551.

[P] Pfister, A.: Quadratische Formen in beliebigen Körpern. Invent. Math. 1 (1966), 116–132.

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

[W] Wadsworth, A.R.: Similarity of quadratic forms and isomorphisms of their function fields. Trans. Amer. Math. Soc. 208 (1975), 352–358.

Detlev W. Hoffmann

Laboratoire de Mathématiques Faculté des Sciences Université de Franche-Comté 25030 Besançon Cedex France detlev@math.univ-fcomte.fr