p-Signature

Diagonalize a form over the rationals to

$$\mathop{\rm diag}[p^a\cdot A, p^b\cdot B, \ldots],$$

where all the entries are Integers and $A$, $B$, ...are Relatively Prime to $p$. Then the $p$-signature of the form (for $p\not= -1, 2$) is

$$p^a+p^b+\ldots+4k {\rm\ (mod\ } 8),$$

where $k$ is the number of Antisquares. For $p=-1$, the $p$-signature is Sylvester's Signature.

See also Signature (Quadratic Form)