On the proper spinor genus of a quadratic form

Abstract

The object of this paper is to exhibit a procedure and a setting in which lattices on a given quadratic space Vover a global field can be studied via lattices on another more tractable space. This will enable us to prescribe conditions under which the proper spinor genus of a lattice can be characterized without reference to the spinor norm. The method involves modification of Vby orthogonal adjunction and cancellation, and by suitable local changes. In effect this allows us to reap some of the benefits of isotropy in a situation that may be anisotropic.

Other descriptions of the proper spinor genus can be found in Cassels [2],Kneser[7],O’Meara [8], and Watson [12]; and a related concept has been defined by Jones [6]. See O'Meara's survey article [9] for historical details.

Our notation and terminology are those of [8].