In mathematics, the spinor genus is a classification of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s and lattices over the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
, introduced by
Martin Eichler. It refines the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
but may be coarser than proper equivalence.
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 ''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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
. 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 form
Indefinite may refer to:
* the opposite of definite in grammar
** indefinite article
** indefinite pronoun
* Indefinite integral, another name for the antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, pr ...
s of dimension at least three, each spinor genus contains exactly one proper equivalence class.
See also
*
Genus of a quadratic form
References
*
* {{cite book , zbl=0915.52003 , last1=Conway , first1=J. H. , author1-link=John Horton Conway , last2=Sloane , first2=N. J. A. , author2-link=Neil Sloane , others=With contributions by Bannai, E.;