Izhboldin’s Results on Stably Birational Equivalence of Quadrics

Our main goal is to give proofs of all results announced by Oleg Izhboldin in [13]. In particular, we establish Izhboldin’s criterion for stable equivalence of 9-dimensional forms. Several other related results, some of them due to the author, are also included. All the fields we work with are those of characteristic different from 2. In these notes we consider the following problem: for a given quadratic form φ defined over some field F , describe all the quadratic forms ψ/F which are stably birational equivalent to φ. By saying “stably birational equivalent” we simply mean that the projective hypersurfaces φ = 0 and ψ = 0 are stably birational equivalent varieties. In this case we also say “φ is stably equivalent to ψ”(for short) and write φ st ∼ ψ. Let us denote by F (φ) the function field of the projective quadric φ = 0 (if the quadric has no function field, one set F (φ) = F ). Note that φ st ∼ ψ simply means that the quadratic forms φF (ψ) and ψF (φ) are isotropic (that is, the corresponding quadrics have rational points). For an isotropic quadratic form φ, the answer to the question raised is easily seen to be as follows: φ st ∼ ψ if and only if the quadratic form ψ is also isotropic. Therefore, we may assume that φ is anisotropic. One more class of quadratic forms for which the answer is easily obtained is given by the Pfister neighbors. Namely, for a Pfister neighbor φ one has: φ st ∼ ψ if and only if ψ is a neighbor of the same Pfister form as φ. Therefore, we may assume that φ is not a Pfister neighbor. Let φ be an anisotropic quadratic form which is not a Pfister neighbor (in particular, dim φ ≥ 4 since any quadratic form of dimension up to 3 is a Pfister neighbor) and assume that dimφ ≤ 6. Then φ st ∼ ψ (with an arbitrary quadratic form ψ) if and only if φ is similar to ψ (in dimension 4 this is due to Wadsworth, [42]; 5 is done by Hoffmann, [4, main theorem]; 6 in the case of the trivial discriminant is served by Merkurjev’s index reduction formula [33], see also [34, thm. 3]; the case of non-trivial discriminant is due to Laghribi, [32, th. 1.4(2)]). Date: 5 July 2002.