# 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

## DefinitionsEdit

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 *f*_{p} of *V*_{p} of spinor norm 1 such that *M* = *g* *f*_{p}*L*_{p}.

A *spinor genus* is an equivalence class for this equivalence relation. Properly equivalent lattices are in the same spinor genus, and lattices in the same spinor genus are in the same genus. The number of spinor genera in a genus is a power of two, and can be determined effectively.

## ResultsEdit

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

## See alsoEdit

## ReferencesEdit

- Cassels, J. W. S. (1978).
*Rational Quadratic Forms*. London Mathematical Society Monographs.**13**. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029. - Conway, J. H.; Sloane, N. J. A.
*Sphere packings, lattices and groups*. Grundlehren der Mathematischen Wissenschaften.**290**. With contributions by Bannai, E.; Borcherds, R. E.; Leech, J.; Norton, S. P.; Odlyzko, A. M.; Parker, R. A.; Queen, L.; Venkov, B. B. (3rd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98585-9. Zbl 0915.52003.