We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n, Z) which satisfy X2 + I = 0 (resp. X2 + X + I = 0) and the set of hermitian forms of rank n over Z[√-1] (resp. Z[(-1 + √-3)/2]) of determinant ±1. As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of m2 + 1 (resp. m + m + 1) can be represented by the quadratic form F(X, Y) = X2 + Y2 (resp. X2 + XY + Y2) with relatively prime integers X, Y: Suppose n ≤ 3 (resp. n ≤ 5). Then S can be represented over Z by F⊗n (n copies of F) if det S is represented by F as above. The proof is based on Siegel-Braun's Mass formula for hermitian forms.
ASJC Scopus subject areas
- Algebra and Number Theory