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 a ...
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 deno ...
, introduced by
Martin Eichler
Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German number theorist.
Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936.
Eichler and Goro Shimura developed a method to constr ...
. It refines the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
but may be coarser than proper equivalence.
Definitions
We define two Z-lattices ''L'' and ''M'' in a
quadratic space
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 ...
''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
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
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 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 In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Z''n'', or equivalently a free Z-module of finite rank. Two such forms are in the same ''ge ...
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.;
Borcherds, R. E.;
Leech, J.;
Norton, S. P.;
Odlyzko, A. M.; Parker, R. A.; Queen, L.; Venkov, B. B. , title=Sphere packings, lattices and groups , edition=3rd , series=Grundlehren der Mathematischen Wissenschaften , volume=290 , location=New York, NY , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, isbn=0-387-98585-9 , url-access=registration , url=https://archive.org/details/spherepackingsla0000conw_b8u0
Quadratic forms