Hoffmann's conjecture for totally singular forms of prime degree

TL;DR

This paper extends Hoffmann's conjecture to totally singular quadratic forms of prime degree, determining their Knebusch splitting patterns and generalizing results to Fermat-type forms in characteristic p.

Contribution

It provides a complete classification of Knebusch splitting patterns for totally singular forms and extends Karpenko's theorem to this case, introducing new structural insights.

Findings

Classified all possible Knebusch splitting patterns for totally singular forms.

Extended Karpenko's theorem on the first Witt index to totally singular forms.

Generalized results to Fermat-type forms over fields of characteristic p.

Abstract

One of the most significant discrete invariants of a quadratic form $\phi$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $\phi$ 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 $\phi$, which captures the isotropy behaviour of $\phi$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $\phi$ 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 generalise to give analogous results for arbitrary Fermat-type forms of degree $p$ over fields of characteristic $p>0$. Related problems concerning the Knebusch splitting of symmetric bilinear forms over fields of characteristic 2 and the full splitting of totally singular quadratic forms are also considered.