Twisted Pfister forms

Detlev W. Hoffmann

Documenta mathematica, Tome 1 (1996), pp. 67-102

Résumé

Let F be a field of characteristic =2. In this paper we investigate quadratic forms φ over F which are anisotropic and of dimension 2n,n≥2, such that in the Witt ring WF they can be written in the form φ=σ−π where σ and π are anisotropic n- resp. m-fold Pfister forms, 1≤mcall these forms twisted Pfister forms. Forms of this type with m=n−1 are of great importance in the study of so-called good forms of height 2, and such forms with m=1 also appear in Izhboldin's recent proof of the existence of n-fold Pfister forms τ over suitable fields F,n≥3, for which the function field F(τ) is not excellent over F. We first derive some elementary properties and try to give alternative characterizations of twisted Pfister forms. We also compute the Witt kernel W(F(φ)/F) of a twisted Pfister form φ. Our main focus, however, will be the study of the following problems: For which forms ψ does a twisted Pfister form φ become isotropic over F(ψ)? Which forms ψ are equivalent to φ (i.e., the function fields F(φ) and F(ψ) are place-equivalent over F)? We also investigate how such twisted Pfister forms behave over the function field of a Pfister form of the same dimension which then leads to a generalization of the result of Izhboldin mentioned above.

Détail

DOI : 10.4171/dm/3

Classification : 11E04, 11E81, 12F20