In
mathematics, the genus is a classification of
quadratic forms and
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
s 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 ...
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 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 g ...
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 are ...
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 ide ...
structure on the set ''C'' of
equivalence classes of forms with given
discriminant. 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