Nikita A. Karpenko: Around 16-dimensional quadratic forms in Iq3

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Submission: 2015, Nov 29

We determine the indexes of all orthogonal Grassmannians of a generic 16-dimensional quadratic form in Iq3. This is applied to show that the 3-Pfister number of the form is at least 4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov-Merkurjev on proper subforms in Iq2 (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 Iq3 (originally available in characteristic not 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.

2010 Mathematics Subject Classification: 11E04; 20G15; 14C25

Keywords and Phrases: Quadratic forms over fields; algebraic groups; projective homogeneous varieties; Chow groups.

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