Detlev W. Hoffmann

Quadratic forms and Pfister neighbors in characteristic 2

We study Pfister neighbors and their characterization over fields of characteristic 2, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form φ which is dominated by q. From this, we derive an analogue in characteristic 2 of a result by Knebusch saying that, in characteristic ¬= 2, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebuschs result generally fails in characteristic 2 for singular forms. As an application, we characterize certain forms of height 1 in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

Witt kernels of bilinear forms for algebraic extensions in characteristic 2

Let F be a field of characteristic 2 and let K/F be a purely inseparable extension of exponent 1. We determine the kernel W(K/F) of the natural restriction map W F → WK between the Witt rings of bilinear forms of F and K, respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension K/F. Based on this result, we will determine W(K/F) for a wide class of finite extensions which are not necessarily purely inseparable.

Quadratic Forms and Linear Algebraic Groups

Topics discussed at the workshop Quadratic forms and linear algebraic groups included besides the algebraic theory of quadratic and Her- mitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic as- pects pertaining to the theory of quadratic forms over function elds over local elds.