Nikita A. Karpenko: A relation between higher Witt indices

Submission: 2004, Jun 27

Let i_1, i_2, ..., i_h be the higher Witt indices of an arbitrary non-degenerate quadratic form over a field of characteristic not 2 (where h is the height of the form). We show that for any integer q in [1, h-1] one has v_2(i_q)+1 is greater or equal to the minimum of v_2(i_{q+1}), ..., v_2(i_h), where v_2 is the 2-adic order. Besides we show that v_2(i_q) is less or equal to the maximum of v_2 (i_{q+1}), ..., v_2(i_h) provided that i_q+2(i_{q+1}+...+i_h) is not a power of 2.

These inequalities are applied by Hurrelbrink and Rehmann to the problem of determination of the smallest possible height of an anisotropic quadratic form of any given dimension. The first inequality formally implies Vishik's conjecture on dim I^n.

The method of the proof involves the Steenrod operations on the modulo 2 Chow groups of some direct powers of the projective quadric. It produces not only the above inequalities, but also some other relations between the higher Witt indices.

2000 Mathematics Subject Classification: 11E04; 14C25

Keywords and Phrases: Quadratic forms, Witt indices, Chow groups, Steenrod operations, correspondences.

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