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 Zⁿ, 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, x² + 82y² and 2x² + 41y² 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 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 C² and the genera are the cosets of C²: 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. Vol. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.

External links

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