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# 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 more generally 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 forms

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 also

Spinor genus

References

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

External links

Hazewinkel, Michiel, ed. (2001), "Quadratic form", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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