Genus of a quadratic form

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In math, the genus groups quadratic forms and lattices over the integers. An integral quadratic form is a function on Z^n, or equivalently a free Z-module of rank n.

Two forms lie in the same genus if they look the same from every local viewpoint: they are equivalent when you allow p-adic integers Z_p for every prime p and also over the real numbers. If two forms are globally equivalent (by an integer change of variables), they are in the same genus, but not every pair in the same genus is globally equivalent. For example, x^2 + 82y^2 and 2x^2 + 41y^2 are in the same genus but not equivalent over Z.

Forms in the same genus have the same discriminant, so each genus contains only finitely many equivalence classes. The Smith–Minkowski–Siegel mass formula assigns a total weight to the classes in a genus; it adds up the reciprocals of the sizes of their automorphism groups (their symmetry groups).

In the special case of binary quadratic forms (n = 2), the set of equivalence classes with a fixed discriminant has a group-like structure. The genera are described by certain characters, and the principal genus is the one containing the principal form. This principal genus corresponds to a specific subgroup (the squares); the other genera are the cosets of that subgroup. Consequently, every genus in this case has the same number of classes.