The Pfister Factor Conjecture in Characteristic Two
Zusammenfassung der Projektergebnisse
The project investigated possible analogues of the Pfister Factor Conjecture for fields of characteristic 2. This is part of a larger research topic, aiming to generalise the algebraic theory of quadratic forms to the wider setting of algebras with involution and their associated linear algebraic groups. In particular, we wish to know how well totally decomposable algebras with involution generalise Pfister forms. The Pfister Factor Conjecture asks whether any totally decomposable orthogonal involution is adjoint to a Pfister form when the associated algebra is split. K.J. Becher proved in 2008 that this is indeed the case in characteristic different from 2. In characteristic 2, the notions of a quadratic form and a symmetric bilinear form diverge (otherwise they are equivalent). When working in characteristic 2, one must deal with two related, but again different theories when generalising the notion of Pfister form to this wider setting. Symmetric bilinear forms are associated with involutions, whereas quadratic forms are associated with quadratic pairs. Orthogonal involutions in characteristic 2 were investigated by A. Dolphin. It was shown that the behaviour of these involutions is highly unusual in characteristic 2, but that this unusual behaviour can be exploited to investigate these objects with relatively simple techniques. The isotropy behaviour of orthogonal involutions over generic splitting fields was completely determined using these techniques. In contrast, this isotropy behaviour is still an open question in characteristic different from 2. The results were used to show that totally decomposable orthogonal involutions in characteristic 2 share a characteristic property of bilinear Pfister forms, namely the extreme behaviour with respect to isotropy. They are either anisotropic, or as isotropic as possible. This implies an analogue of the Pfister Factor Conjecture, namely that totally decomposable orthogonal involutions on a split algebra are adjoint to bilinear Pfister forms. Quadratic pairs were investigated in collaboration with A. Dolphin. A notion of a totally decomposable quadratic pairs can also be defined, and it was shown that a totally decomposable quadratic pair on a split algebra is adjoint to a quadratic Pfister form. The proof obtained via the project is unique to characteristic 2, and it uses the results on orthogonal involutions in characteristic 2 described above. This method is able to capture more information on the resulting quadratic Pfister form than the proof currently available in characteristic different from 2. Results on the isotropy behaviour of quadratic pairs on an index 2 algebra after generic splitting and on the decomposition of the closely related generalised quadratic forms were also obtained. Results analogous to the Pfister Factor Conjecture for symplectic and unitary involutions are also known in characteristic different from 2. Specifically, that totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results were shown in characteristic 2 by A. Dolphin. In collaboration with A. Quéguiner-Mathieu, A. Dolphin investigated the effect on the isotropy of symplectic involutions and quadratic pairs after extending scalars to the function field of a conic in characteristic 2. Via a classification of these symplectic involutions in characteristic 2, a new proof for a result on minimal forms with respect to the function field of a conic was obtained. In collaboration with A. Chapman and A. Laghribi, A. Dolphin investigated linked quaternion algebras, that is quaternion algebras sharing a common subfield, in characteristic 2. Further A. Dolphin and A. Laghribi completed the classification of quadratic forms becoming isotropic over the function field of a 5-dimensional quadratic form.
Projektbezogene Publikationen (Auswahl)
- Decomposition of algebras with involution in characteristic 2, Journal of Pure and Applied Algebra, 217 (2013): 1620–1633
A. Dolphin
- Orthogonal Pfister involutions in characteristic two. Journal of Pure and Applied Algebra, 218 (2014): 1900–1915
A. Dolphin
(Siehe online unter https://doi.org/10.1016/j.jpaa.2014.02.013) - Non-hyperbolic splitting of quadratic pairs. Journal of Algebra and Its Applications, 14 (2015), 22p
K.J. Becher and A. Dolphin
(Siehe online unter https://doi.org/10.1142/S0219498815501388) - Totally decomposable quadratic pairs. Mathematische Zeitschrift
K.J. Becher and A. Dolphin
(Siehe online unter https://doi.org/10.1007/s00209-016-1648-3)
