Quadratic forms with absolutely maximal splitting (Q2704200)

Let \(F\) be a field of characteristic \(\neq 2\) and let \(\phi\) be an anisotropic quadratic form over \(F\). The higher Witt indices \(i_k(\phi)\) are defined recursively as follows: \(i_1(\phi)\) is the Witt index of \(\phi\) over its own function field \(F_1=F(\phi)\), and for \(k\geq 1\) let \(\phi_k\) be the anisotropic part of \(\phi\) over \(F_k\), \(F_{k+1}=F_k(\phi_k)\), and \(i_{k+1}(\phi)\) the Witt index of \(\phi\) over \(F_{k+1}\). By a theorem of the reviewer [Math.\ Z. 220, 461-476 (1995; Zbl 0840.11017)], if \(n\) and \(m\) are integers such that \(\dim\phi =2^{n-1}+m\) with \(1\leq m\leq 2^{n-1}\), then \(i_1(\phi)\leq m\), and if equality holds, \(\phi\) is said to have maximal splitting. Pfister neighbors are known to have maximal splitting, but there are counterexamples for all \(m\leq 2^{n-3}\) (\(n\geq 3\)). The authors are interested in the characterization of forms with maximal splitting, in particular in the conjecture that if \(m> 2^{n-3}\), then maximal splitting implies that the form is a Pfister neighbor. This conjecture was proved by the first author for \(n\leq 4\) [St. Petersbg. Math. J. 9, 219-224 (1998; Zbl 0892.11013)]. NEWLINENEWLINENEWLINEIn the present paper, this conjecture is related to conjectures on the Galois cohomology group \(H^n(F)\) of \(F\) with coefficients in \({\mathbb Z}/2{\mathbb Z}\) and the kernel \(H^n(F(\phi)/F)\) of the map \(H^n(F)\to H^n(F(\phi))\), namely that if \(\dim\phi >2^{n-1}\), then this kernel is nonzero iff \(\phi\) is a Pfister neighbor of some \(n\)-fold Pfister form \(\pi\), in which case it is isomorphic to \({\mathbb Z}/2{\mathbb Z}\), generated by \(e^n(\pi)\). With the same notations as above, it is shown that \(m\geq 2^{n-1}-7\) plus maximal splitting implies that \(\phi\) is a Pfister neighbor (\(n\geq 5\)), and that for \(m > 2^{n-3}\), there is equivalence between \(\phi\) having maximal splitting and \(H^n(F(\phi)/F)\neq 0\) (here, it is required that \(F\) be of characteristic \(0\)). NEWLINENEWLINENEWLINEA major tool is the concept of forms with absolutely maximal splitting (AMS-forms) introduced by the authors, where \(\phi\) is said to be an AMS-form if \(i_1(\phi)>i_k(\phi)\) for all \(k>1\). It is shown that (at least in characteristic \(0\)), absolutely maximal splitting implies maximal splitting, and that \(H^n(F(\phi)/F)\neq 0\) if \(\phi\) as above is an AMS-form. NEWLINENEWLINENEWLINEA crucial ingredient in the proofs is the study of the motivic decomposition of AMS-quadrics. This uses methods developed by the second author in his thesis [Preprint Max-Planck-Institut für Mathematik, Bonn, MPI-1998-13] which in turn are based on techniques developed by Voevodsky in his proof of the Milnor conjecture. To keep the paper self-contained, some of Voevodsky's results which are used in the proofs are added in an appendix.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].