Oleg Izhboldin and Alexander Vishik: Quadratic forms with absolutely Maximal Splitting

oleg@mathematik.uni-bielefeld.de, vishik@ias.edu

Submission: 2000, Feb. 14

published in: E. Bayer-Fluckiger e.a., Proceedings of the Conference on ``Quadratic Forms and their Applications'' in Dublin, July 4-10, 1999. Contemp. Math. 272 (2000), 103-125.

Let $F$ be a field and $\phi$ be a quadratic form over $F$. The higher Witt indices of $\phi$ are defined recursively by the rule $i_{k+1}(\phi)=i_{k}((\phi_{an})_{F(\phi_{an})})$, where $i_0(\phi)=i_W(\phi)$ is the usual Witt index of the form $\phi$. We say that $\phi$ has {\it absolutely maximal splitting} if $i_1(\phi)>i_k(\phi)$ for all $k>1$. One of the main results of this paper claims that for all anisotropic forms $\phi$ satisfying the condition $2^{n-1}+2^{n-3}<\dim\phi\le 2^n$, the following three conditions are equivalent: (i) the kernel of the natural homomorphism $H^n(F,\Z/2\Z)\to H^n(F(\phi),\Z/2\Z)$ is nontrivial, (ii) $\phi$ has absolutely maximal splitting, (iii) $\phi$ has maximal splitting (i.e., $i_1(\phi)=\dim\phi-2^{n-1}$). Moreover, we show that if we assume additionally that $\dim\phi\ge 2^n-7$, then these three conditions hold if and only if $\phi$ is an anisotropic $n$-fold Pfister neighbor. In our proof we use the technique developed by V. Voevodsky in his proof of Milnor's conjecture.

1991 Mathematics Subject Classification: 11E04, 11E81, 14C25, 19Exx

Keywords and Phrases: quadratic forms, quadrics, Milnor conjecture, Galois cohomology, Motives

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