In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the 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. An
integral 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 ...
is a quadratic form on Z
''n'', or equivalently a free Z-module of finite rank. Two such forms are in the same ''genus'' if they are equivalent over the local rings Z
''p'' for each prime ''p'' and also equivalent over R.
Equivalent forms are in the same genus, but the converse does not hold. For example, ''x''
2 + 82''y''
2 and 2''x''
2 + 41''y''
2 are in the same genus but not equivalent over Z. Forms in the same genus have equal
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
and hence there are only finitely many equivalence classes in a genus.
The
Smith–Minkowski–Siegel mass formula In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism gr ...
gives the ''weight'' or ''mass'' of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.
Binary quadratic forms
For
binary quadratic forms there is a
group structure on the set ''C'' of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of forms with given
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
. The genera are defined by the ''generic characters''. The principal genus, the genus containing the principal form, is precisely the subgroup ''C''
2 and the genera are the cosets of ''C''
2: so in this case all genera contain the same number of classes of forms.
See also
*
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 of a quadratic form, genus but may be coarser than proper equivalence.
Definitions
We ...
References
*
External links
* {{SpringerEOM , title=Quadratic form
Quadratic forms