# 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 *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 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 formsEdit

For binary quadratic forms there is a group structure on the set *C* 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 alsoEdit

## ReferencesEdit

- Cassels, J.W.S. (1978).
*Rational Quadratic Forms*. London Mathematical Society Monographs.**13**. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.

## External linksEdit

- "Quadratic form",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]