Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms

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.