Hoffmann's conjecture for totally singular forms of prime degree

Abstract

One of the most significant discrete invariants of a quadratic form $\varphi$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behavior of $\varphi$ under scalar extension to arbitrary overfields of $k$. A similarly important but more accessible variant of this notion is that of the Knebusch splitting pattern of $\varphi$, which captures the isotropy behavior of $\varphi$ as one passes over a certain prescribed tower of $k$-overfields. We determine all possible values of this latter invariant in the case where $\varphi$ is totally singular. This includes an extension of Karpenko’s theorem (formerly Hoffmann’s conjecture) on the possible values of the first Witt index to the totally singular case. Contrary to the existing approaches to this problem (in the nonsingular case), our results are achieved by means of a new structural result on the higher anisotropic kernels of totally singular quadratic forms. Moreover, the methods used here readily generalize to give analogous results for arbitrary Fermat-type forms of degree $p$ over fields of characteristic $p>0$.