In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 t ...
s 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 ''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 zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
and hence there are only finitely many equivalence classes in a genus.
The
Smith–Minkowski–Siegel mass formula 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 form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results ar ...
s there is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
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 ...
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 zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
. 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
References
*
External links
* {{SpringerEOM , title=Quadratic form
Quadratic forms