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Classification of Pseudo-reductive Groups (AM-191)$
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Brian Conrad and Gopal Prasad

Print publication date: 2015

Print ISBN-13: 9780691167923

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691167923.001.0001

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Constructions with regular degenerate quadratic forms

Constructions with regular degenerate quadratic forms

Chapter:
7 Constructions with regular degenerate quadratic forms
Source:
Classification of Pseudo-reductive Groups (AM-191)
Author(s):

Brian Conrad

Gopal Prasad

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691167923.003.0007

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG = 1 and the Cartan k-subgroups of G are tori. It begins with an overview of the lemma and propositions for regular degenerate quadratic forms, coupled with two examples. It then considers the conformal isometry between quadratic spaces over a field, which is a linear isomorphism that respects the quadratic forms up to a nonzero scaling factor. It also introduces a proposition that provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO(q) for a regular quadratic form q. Finally, it describes all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k.

Keywords:   degenerate quadratic form, Severi–Brauer variety, pseudo-simple k-group, root system, Cartan k-subgroup, conformal isometry, quadratic space, linear isomorphism, automorphism, quadrics

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