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2016 Hoffmann's conjecture for totally singular forms of prime degree
Stephen Scully
Algebra Number Theory 10(5): 1091-1132 (2016). DOI: 10.2140/ant.2016.10.1091


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.


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Stephen Scully. "Hoffmann's conjecture for totally singular forms of prime degree." Algebra Number Theory 10 (5) 1091 - 1132, 2016.


Received: 7 August 2015; Revised: 24 February 2016; Accepted: 24 March 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1350.11049
MathSciNet: MR3531363
Digital Object Identifier: 10.2140/ant.2016.10.1091

Primary: 11E04
Secondary: 14E05, 15A03

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.10 • No. 5 • 2016
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