Around 16 - Dimensional Quadratic Forms in I

We determine the indexes of all orthogonal Grassmannians of a generic 16dimensional quadratic form in I q . This is applied to show that the 3-Pfister number of the form is ≥ 4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov–Merkurjev on proper subforms in I q (originally available in characteristic 0) as well as a new and characteristic-free proof of an old result by Hoffmann-Tignol and Izhboldin-Karpenko on 14-dimensional quadratic forms in I q (originally available in characteristic ̸= 2). We also suggest an extension of the method, based on investigation of the topological filtration on the Grothendieck ring of a maximal orthogonal Grassmanian, which applies to quadratic forms of dimension higher than 16. We work with non-degenerate quadratic forms over arbitrary fields. Recall that a quadratic form similar to a Pfister form is called a general Pfister form. We refer to [5] for general facts and terminology related to quadratic forms, especially for the definition of a (quadratic) Pfister form in arbitrary characteristic. We write Iq = Iq(F ) for the Witt group of classes of even-dimensional quadratic forms over a field F . Recall that Iq(F ) is a module over the Witt ring W (F ) of classes of non-degenerate symmetric bilinear forms. There is a filtration by submodules Iq = I 1 q ⊃ I q ⊃ . . . defined as follows: for any d ≥ 1, I q := I d−1(F ) · Iq(F ), where I(F ) ⊂ W (F ) is the fundamental ideal and Id−1(F ) is its power. Let φ be an even-dimensional non-degenerate quadratic form over a field F and let d ≥ 1 be an integer such that the Witt class [φ] ∈ Iq(F ) is in I q (F ). Then [φ] can be written as a sum of classes of general d-fold Pfister forms. The minimal possible number of the summands is denoted Pfd(φ) and called the d-Pfister number of φ, cf. [14, §9c]. Given a base field k and a positive even integer m, we are interested to determine Pfd(m) := sup φ Pfd(φ), where φ runs over m-dimensional quadratic forms defined over some field F ⊃ k and satisfying [φ] ∈ I q (F ). Trivially, Pf1(m) = m/2 for any m. Also, it is known (and relatively easy to show, cf. [5, Lemma 38.1]) that Pf2(m) = (m− 2)/2. In the present paper, we concentrate on the 3-Pfister number Pf3(m) which is known to be finite. Finiteness of Pfd(m) for d ≥ 4 is an open question. Date: 29 Novermber 2015. Revised: 4 June 2016.