Spinor genus

In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler.

It refines the genus but may be coarser than proper equivalence.

We define two Z-lattices L and M in a quadratic space V over Q to be spinor equivalent if there exists a transformation g in the proper orthogonal group O+(V) and for every prime p there exists a local transformation fp of Vp of spinor norm 1 such that M = g fpLp.

The number of spinor genera in a genus is a power of two, and can be determined effectively.

An important result is that for indefinite forms of dimension at least three, each spinor genus contains exactly one proper equivalence class.