On a Conjecture of Izhboldin

on Similarity of Quadratic Forms

Detlev W. Hoffmann^{1}

Received: January 4, 1999

Communicated by Ulf Rehmann

Abstract. In his paper Motivic equivalence of quadratic forms, Izhboldin modifies a conjecture of Lam and asks whether two quadratic forms, each of which isomorphic to the product of an Albert form and a k-fold Pfister form, are similar provided they are equivalent modulo I^{k+3}. We relate this conjecture to another conjecture on the dimensions of anisotropic forms in I^{k+3}. As a consequence, we obtain that Izhboldin’s conjecture is true for k ≤ 1.

1991 Mathematics Subject Classification: Primary 11E81; Secondary 11E04. Keywords and Phrases: Quadratic form, Pfister form, Albert form, similarity of quadratic forms.

In what follows, we will adhere to the same terminology and notations used in Izhboldin’s article [I] mentioned in the abstract. In particular, if two quadratic forms ϕ and ψ are similar, we will write ϕ ∼ ψ. Let F be a field of characteristic ≠ 2. Recall that an Albert form α over F is a 6-dimensional quadratic form over F with signed discriminant 1 ∈ F^{∗}/F^{∗2} (i.e. α ∈ I^{2}F), and an n-fold Pfister form over F is a form of type ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ := ⟨1, −a_{1}⟩⊗⋯⊗⟨1, −a_{n}⟩, a_{i} ∈ F^{∗}. In his paper [I], Izhboldin states the following conjecture :

Conjecture 1 (Cf. Conjecture 5.1 in [I].) Let q_{1} and q_{2} be Albert forms over F and let π_{1} and π_{2} be two k-fold Pfister forms over F (k ≥ 0) such that q_{i}⊗π_{i}, i = 1, 2 is anisotropic and q_{1}⊗π_{1} ≡ q_{2}⊗π_{2} mod I^{k+3}F. Then q_{1}⊗π_{1} ∼ q_{2}⊗π_{2}.

1This research has been carried out in the framework of TMR Network ERB FMRX CT-97-0107 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures.”

Documenta Mathematica 4 (1999) 61–64

In fact, this conjecture is a special case of a question asked by Lam [L, (6.6)]. Lam’s original question was as follows. Suppose σ_{i}, ρ_{i} ∈ P_{n}F, i = 1, 2, and let ϕ_{i} = (σ_{i} ⊥ −ρ_{i})_{an} be the anisotropic part of σ_{i} ⊥ −ρ_{i}. If ϕ_{1} ≡ ϕ_{2} mod I^{n+1}F, does it then follow that ϕ_{1} ∼ ϕ_{2} ? By a result of Elman and Lam [EL, Theorem 4.5], it is known that dim ϕ_{i} ∈ {2^{n+1} − 2^{m}, 1 ≤ m ≤ n + 1}, and that if dim ϕ_{i} = 2^{n+1} − 2^{m}, then ρ_{i} and σ_{i} are (m − 1)-linked, i.e. there exists an (m − 1)-fold Pfister form which divides both ρ_{i} and σ_{i}. It is an easy exercise to show that Lam’s question has a positive answer if dim ϕ_{1} (or dim ϕ_{2}) equals 0 of 2^{n} (i.e. m = n + 1 or m = n). In [I, Section 4], Izhboldin constructs counterexamples with dim ϕ_{1} (or dim ϕ_{2}) equal to 2^{n+1}−2^{m} with 1 ≤ m ≤ n−2. The only remaining case m = n − 1 boils down to Conjecture 1 above. It turns out that this conjecture would have a positive answer if another wellknown conjecture on quadratic forms were true, this other conjecture being

Conjecture 2 Let n ≥ 2 and let q be an anisotropic form in I^{n}F. If dim q > 2^{n} then dim q ≥ 2^{n} + 2^{n−1}.

Proposition 1 Conjecture 2 for n = k + 3 implies Conjecture 1 for k.

It was shown in [H 2] that Conjecture 2 holds for n ≤ 4. As a consequence, we have

Corollary Conjecture 1 holds for k ≤ 1.

Note that for k = 0 this is essentially Jacobson’s theorem saying that two Albert forms are similar if and only if their associated biquaternion algebras are isomorphic (see [MS] for a quadratic form-theoretic proof of Jacobson’s theorem).

Proof of Proposition 1. Suppose that Conjecture 2 holds for k + 3. Let q_{1} and q_{2} be Albert forms over F and let π_{1} and π_{2} be two k-fold Pfister forms over F (k ≥ 0) such that q_{1} ⊗ π_{1} ≡ q_{2} ⊗ π_{2} mod I^{k+3}F and such that q_{i} ⊗ π_{i} is anisotropic for i = 1, 2. First, we note that we may assume π_{1} = π_{2} (cf. the remarks following Conjecture 5.1 in [I]). We denote this k-fold Pfister form by π. Since q_{i} ⊗ π ∈ I^{k+2}F, we can scale q_{i} (and thus q_{i} ⊗π) without changing the equivalence mod I^{k+3}F, and we may thus assume that q_{i} ≅ ⟨1⟩ ⊥ q′_{i}, dim q′_{i} = 5 for i = 1, 2. This yields q′_{1} ⊗ π ≡ q′_{2} ⊗ π mod I^{k+3}F. In particular, π ⊗ (q′_{1} ⊥ −q′_{2}) is a form of dimension 2^{k}(2^{3} + 2) = 2^{k+3} + 2^{k+1} in I^{k+3}F. By Conjecture 2, π ⊗ (q′_{1} ⊥ −q′_{2}) is isotropic. In particular, there exists x ∈ F^{∗} such that x is represented by both π ⊗ q′_{1} and π ⊗ q′_{2}. Using the multiplicativity of Pfister forms (cf. [EL, Theorem 1.4]), there exist 4dimensional forms q′′_{i}, i = 1, 2, such that π ⊗ q′_{i} ≅ π ⊗ (⟨x⟩ ⊥ q′′_{i}). From this, it follows readily that π ⊗ q′′_{1} ≡ π ⊗ q′′_{2} mod I^{k+3}F. Note that dim(π ⊗ q′′_{i}) = 2^{k+2}, so that π ⊗ q′′_{1} and π ⊗ q′′_{2} are (anisotropic) half-neighbors. As a consequence, π ⊗ q′′_{1} becomes isotropic over the function field of π ⊗ q′′_{2} (see, e.g., [H 3, Corollary 2.6] or [I, Lemma 3.3]). By [H 1, Theorem 1.4], this implies that π ⊗ q′′_{1} and π ⊗ q′′_{2} are similar, so that there exists some y ∈ F^{∗} such that π ⊗ q′′_{1} ≅ yπ ⊗ q′′_{2}. Thus, we obtain

π ⊗ q_{1} ≡ π ⊗ ⟨1, x⟩ ⊥ π ⊗ q′′_{1}  modI^{k+3}F
≡ π ⊗ q_{2}  modI^{k+3}F
≡ yπ ⊗ q_{2}  modI^{k+3}F
≡ yπ ⊗ ⟨1, x⟩ ⊥ yπ ⊗ q′′_{2} modI^{k+3}F
≡ yπ ⊗ ⟨1, x⟩ ⊥ π ⊗ q′′_{1} modI^{k+3}F

and hence π⊗⟨1, x⟩ ≡ yπ⊗⟨1, x⟩ mod I^{k+3}F. Now dim(π⊗⟨1, x⟩) = 2^{k+1}, and the Arason-Pfister Hauptsatz therefore implies that π ⊗ ⟨1, x⟩ ≅ yπ ⊗ ⟨1, x⟩. We conclude that

π ⊗ q_{1} ≅ π ⊗ ⟨1, x⟩ ⊥ π ⊗ q′′_{1}
≅ yπ ⊗ ⟨1, x⟩ ⊥ yπ ⊗ q′′_{2}
≅ yπ ⊗ q_{2} .

Note that we didn’t really make use of the fact that q_{1} and q_{2} are Albert forms. However, it is not difficult to show that if π is a k-fold Pfister form and q = q′ ⊥ ⟨a⟩ ∈ IF such that π ⊗ q ∈ I^{k+2}F, then if one chooses b ∈ F^{∗} such that q̃ = q′ ⊥ ⟨b⟩ ∈ I^{2}F, one has π ⊗ q ≅ π ⊗ q̃. So what is essential is the fact that π ⊗ q_{i} is in I^{k+2}F, in which case we may as well assume by what we just mentioned that q_{i} is an Albert form. In the proof of Conjecture 2 for n = 4 in [H 2], one makes use of a certain property P D_{2}. It turns out that this property can be used to establish Conjecture 1 for k = 1 without invoking Conjecture 2 for n = 4. Let us recall the general definition of property P D_{n}.

Definition Let n be an integer ≥ 1. The field F is said to have the Pfister decomposition property for Pfister forms of fold ≤ n, P D_{n} for short, if for each m (1 ≤ m ≤ n), for each anisotropic π ∈ P_{m−1}F, for each r ∈ Ḟ, and each anisotropic φ ∈ πW F, there exist forms σ and τ over F such that for ρ := π ⊗ ⟨⟨r⟩⟩ one has φ ≅ π ⊗ σ ⊥ ρ ⊗ τ and (φ_{F (ρ)})_{an} ≅ (π ⊗ σ)_{F (ρ)}.

Proposition 2 Suppose that F has P D_{n} for some n ≥ 1. Then Conjecture 1 holds for k = n − 1.

Proof. Suppose that F has P D_{n} for n = k + 1. As in the previous proof, we may assume that we are in the situation where π ⊗ q_{1} ≡ π ⊗ q_{2} mod I^{k+3}F with Albert forms q_{i}, i = 1, 2, a k-fold Pfister form π and with π ⊗ q_{i} being anisotropic for i = 1, 2. After scaling, we may assume that q_{1} ≅ ⟨1, −r⟩ ⊥ q′_{1} for some r ∈ F^{∗}. It follows that π ⊗ q_{1} contains the subform ρ = π ⊗ ⟨⟨r⟩⟩. In particular, π ⊗ q_{1} becomes isotropic over the function field F (ρ), and thus π ⊗q_{2} also becomes isotropic over F (ρ) (cf. [I, Theorem 4.3]). Property P D_{k+1} then implies that π ⊗q_{2} contains a subform similar to ρ, and since we may scale π ⊗ q_{2} ∈ I^{k+2}F without changing the equivalence modI^{k+3}F, we may assume that π ⊗ q_{2} ≅ π ⊗ (⟨1, −r⟩ ⊥ q′_{2}) for some 4-dimensional form q′_{2}. It follows that π ⊗ q′_{1} ≡ π ⊗ q′_{2} mod I^{k+3}F. As in the proof of Proposition 1, this implies that π ⊗ q′_{1} and π ⊗ q′_{2} are similar, and thus that π ⊗ q_{1} and π ⊗ q_{2} are also similar.

It was proved by Rost that each field has property P D_{2} (see [H 2, Lemma 2.6]). Again, we can conclude that Conjecture 1 holds for k ≤ 1, this time by invoking P D_{2}. In the case n ≥ 3, we do not know whether P D_{n} holds for all fields nor whether P D_{n} for a field F implies that Conjecture 2 holds for F for n+2 (or vice versa).

References

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

[H 1] Hoffmann, D.W.: On quadratic forms of height 2 and a theorem of Wadsworth. Trans. Amer. Math. Soc. 348 (1996), 3267–3281.

[H 2] Hoffmann, D.W.: On the dimensions of anisotropic quadratic forms in I^{4}. Invent. Math. 131 (1998), 185–198.

[H 3] Hoffmann, D.W.: Similarity of quadratic forms and half-neighbors. J. Algebra 204 (1998), 255–280.

[I] Izhboldin, O.T.: Motivic equivalence of quadratic forms. Doc. Math. J. DMV 3 (1998), 341–351.

[L] Lam, T.Y.: Fields of u-invariant 6 after A. Merkurjev. Israel Math. Conf. Proc. Vol. 1: Ring Theory 1989 (in honor of S.A. Amitsur) (ed. L. Rowen). pp.12–31. Jerusalem: Weizmann Science Press 1989.

[MS] Mammone, P.; Shapiro, D.B.: The Albert quadratic form for an algebra of degree four. Proc. Amer. Math. Soc. 105 (1989), 525–530.

Detlev W. Hoffmann Équipe de Mathématiques UMR 6623 du CNRS Université de Franche-Comté 16, route de Gray 25030 Besançon Cedex, France detlev@math.univ-fcomte.fr