281 research outputs found
Symplectic Involutions, quadratic pairs and function fields of conics
In this paper we study symplectic involutions and quadratic pairs that become
hyperbolic over the function field of a conic. In particular, we classify them
in degree 4 and deduce results on 5 dimensional minimal quadratic forms, thus
extending to arbitrary fields some results of [24], which were only known in
characteristic different from 2.Comment: 17 page
Pfister's Theorem for orthogonal involutions of degree 12
We use the fact that a projective half-spin representation of has
an open orbit to generalize Pfister's result on quadratic forms of dimension 12
in to orthogonal involutions
The Arason invariant of orthogonal involutions of degree 12 and 8, and quaternionic subgroups of the Brauer group
Using the Rost invariant for torsors under Spin groups one may define an
analogue of the Arason invariant for certain hermitian forms and orthogonal
involutions. We calculate this invariant explicitly in various cases, and use
it to associate to every orthogonal involution with trivial discriminant and
trivial Clifford invariant over a central simple algebra of even co-index a
cohomology class of degree 3 with coefficients. This invariant
is the double of any representative of the Arason invariant; it vanishes
when the algebra has degree at most 10, and also when there is a quadratic
extension of the center that simultaneously splits the algebra and makes the
involution hyperbolic. The paper provides a detailed study of both invariants,
with particular attention to the degree 12 case, and to the relation with the
existence of a quadratic splitting field.Comment: A mistake pointed out by A. Sivatski in Section 5.3 has been
corrected in the new version of this preprin
Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties
An orthogonal involution on a central simple algebra , after
scalar extension to the function field of the Severi--Brauer
variety of , is adjoint to a quadratic form over
, which is uniquely defined up to a scalar factor. Some
properties of the involution, such as hyperbolicity, and isotropy up to an
odd-degree extension of the base field, are encoded in this quadratic form,
meaning that they hold for the involution if and only if they hold for
. As opposed to this, we prove that there exists non-totally
decomposable orthogonal involutions that become totally decomposable over
, so that the associated form is a Pfister form. We
also provide examples of nonisomorphic involutions on an index algebra that
yield similar quadratic forms, thus proving that the form does not
determine the isomorphism class of , even when the underlying algebra
has index . As a consequence, we show that the invariant for
orthogonal involutions is not classifying in degree , and does not detect
totally decomposable involutions in degree , as opposed to what happens for
quadratic forms
The J-invariant, Tits algebras and Triality
In the present paper we set up a connection between the indices of the Tits
algebras of a simple linear algebraic group and the degree one parameters
of its motivic -invariant. Our main technical tool are the second Chern
class map and Grothendieck's -filtration.
As an application we recover some known results on the -invariant of
quadratic forms of small dimension; we describe all possible values of the
-invariant of an algebra with orthogonal involution up to degree 8 and give
explicit examples; we establish several relations between the -invariant of
an algebra with orthogonal involution and the -invariant of the
corresponding quadratic form over the function field of the Severi-Brauer
variety of .Comment: 28p
Exemples de groupes semi-simples simplement connexes anisotropes contenant un sous-groupe unipotent
6 pages.Nous construisons des exemples des groupes semi-simples anisotropes de type G2 (resp. F4, E8) contenant un sous-groupe unipotent lisse non trivial définis sur un corps convenable d'exposant caractéristique 2 (resp. 3, 5). We show that there are anisotropic groups of type G2 (resp. F4, E8) containing a non-trivial smooth unipotent subgroup over a suitable field of characteristic 2 (resp. 3, 5)
Algebras with involution that become hyperbolic over the fonction field of a conic
We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra . We classify these algebras in degree~ and give an example of such a division algebra with orthogonal involution of degree~ that does not contain with its canonical involution, even though it contains and is totally decomposable into a tensor product of quaternion algebras with involution
Minimal quadratic forms for the function field of a conic in characteristic
In this note, we construct explicit examples of -minimal quadratic forms
of dimension and , where is the function field of a conic over a
field of characteristic . The construction uses the fact that any set of
cyclic algebras over a field of characteristic can be described
using only elements of the base field. It also uses a general result that
provides an upper bound on the Witt index of an orthogonal sum of two regular
anisotropic quadratic forms over a henselian valued field
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