Vol. 10, No. 5, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Hoffmann's conjecture for totally singular forms of prime degree

Stephen Scully

Vol. 10 (2016), No. 5, 1091–1132
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