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.