Hoffmann’s conjecture for totally singular forms of prime degree

Scully, Stephen

doi:10.2140/ant.2016.10.1091

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Hoffmann's conjecture for totally singular forms of prime degree

Stephen Scully

Vol. 10 (2016), No. 5, 1091–1132

DOI: 10.2140/ant.2016.10.1091

Abstract

One of the most significant discrete invariants of a quadratic form $ϕ$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behavior of $ϕ$ 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 $ϕ$ , which captures the isotropy behavior of $ϕ$ as one passes over a certain prescribed tower of $k$ -overfields. We determine all possible values of this latter invariant in the case where $ϕ$ 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 $p > 0$ .

Keywords

quadratic forms, quasilinear $p$-forms, splitting patterns, canonical dimension

Mathematical Subject Classification 2010

Primary: 11E04

Secondary: 14E05, 15A03

Milestones

Received: 7 August 2015

Revised: 24 February 2016

Accepted: 24 March 2016

Published: 28 July 2016

Authors

Stephen Scully
Department of Mathematical and Statistical Sciences University of Alberta Edmonton AB T6G 2G1 Canada

Vol. 10, No. 5, 2016