Twisted Pfister forms

  • Detlev W. Hoffmann

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Let FF be a field of characteristic 2\neq 2. In this paper we investigate quadratic forms \fii\fii over FF which are anisotropic and of dimension 2n,n22^n, n\geq 2, such that in the Witt ring WFWF they can be written in the form \fii=σπ\fii =\sigma -\pi where σ\sigma and π\pi are anisotropic nn- resp. mm-fold Pfister forms, 1m<n1\leq m <n. We call these forms twisted Pfister forms. Forms of this type with m=n1m=n-1 are of great importance in the study of so-called good forms of height 2, and such forms with m=1m=1 also appear in Izhboldin's recent proof of the existence of nn-fold Pfister forms τ\tau over suitable fields F,n3F, n\geq 3, for which the function field F(τ)F(\tau ) is not excellent over FF. We first derive some elementary properties and try to give alternative characterizations of twisted Pfister forms. We also compute the Witt kernel W(F(\fii)/F)W(F(\fii )/F) of a twisted Pfister form \fii\fii. Our main focus, however, will be the study of the following problems: For which forms ψ\psi does a twisted Pfister form \fii\fii become isotropic over F(ψ)F(\psi ) ? Which forms ψ\psi are equivalent to \fii\fii (i.e., the function fields F(\fii)F(\fii ) and F(ψ)F(\psi ) are place-equivalent over FF) ? 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.

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Detlev W. Hoffmann, Twisted Pfister forms. Doc. Math. 1 (1996), pp. 67–102

DOI 10.4171/DM/3