Birational equivalence of n-Pfister neighbors

ABSTRACT

Let φ₁ and φ₂ be quadratic forms over a field K of characteristic different from 2. In this paper, we complete the proof of the following result : if $dim\left({\phi }_1\right)=dim\left({\phi }_2\right)=7$, then φ₁ is isotropic over K(φ₂) and φ₂ is isotropic over K(φ₁) if and only if K(φ₁) and K(φ₂) are isomorphic over K. To this end, we introduce a generalization of neighbors of Pfister forms to multiples of Pfister forms and study some of their function field and isotropy properties. We also apply this results to the seven- and nine-dimensional cases of the quadratic Zariski cancellation problem.

KEYWORDS:

• Birational equivalence
• function fields
• Quadratic forms
• Zariski cancellation problem

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11E04
• 11E81
• 14E05
• 12F20

Acknowledgments

I would like to thank my advisor, Detlev Hoffmann, for having suggested this problem. I also wish to express my gratitude to Burt Totaro for his interest in my work and his encouragement. I am grateful to the referee for numerous suggestions on how to improve this paper.