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 spaceV 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 = gfpLp.
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.
spinor, genus, mathematics, spinor, genus, classification, quadratic, forms, lattices, over, ring, integers, introduced, martin, eichler, refines, genus, coarser, than, proper, equivalence, contents, definitions, results, also, referencesdefinitions, editwe, d. 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 Contents 1 Definitions 2 Results 3 See also 4 ReferencesDefinitions editWe 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 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 Results editAn important result is that for indefinite forms of dimension at least three each spinor genus contains exactly one proper equivalence class See also editGenus of a quadratic formReferences editCassels J W S 1978 Rational Quadratic Forms London Mathematical Society Monographs Vol 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 Vol 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 Retrieved from https en wikipedia org w index php title Spinor genus amp oldid 1034847461, wikipedia, wiki, book, books, library,
