Spinor genus Contents Definitions Results See also References Navigation menu0395.100290915.52003

Quadratic forms


quadratic formsring of integersMartin Eichlergenusquadratic spacespinor normequivalence relationindefinite forms




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 References




Definitions


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.


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


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



See also


  • Genus of a quadratic form


References



  • Cassels, J. W. S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em


  • 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.


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