Classification of quadratic pairs over fields with I^3=0 - Fatma Kader Bingöl

Talk by Fatma Kader Bingöl (Scuola Normale Superiore di Pisa), at the Antwerp Algebra, Geometry and Number Theory Seminar on December 13, 2024.

Quadratic forms over a field F are classified up to isometry by dimension, discriminant, and Clifford invariant when I^3F, the third power of the fundamental ideal in the Witt ring of F, vanishes, or in different terms, when the norm form of every quaternion algebra over F is surjective. In characteristic different from 2, orthogonal involutions on central simple algebras can be seen as twisted forms of quadratic forms, as in the split case, the involution is given explicitly by a quadratic form. The classification theorem has been extended to the setting of orthogonal involutions.When the base field has characteristic 2, the analog concept is that of a so-called quadratic pair, consisting of an involution and a linear form defined on the set of symmetric elements. It is not known whether the classification theorem holds for quadratic pairs. We show that this is the case when the underlying algebra has Schur index at most 4. This is partially joint work with Anne Quéguiner-Mathieu.