Igusa group

In mathematics, an Igusa group or Igusa subgroup is a subgroup of the Siegel modular group defined by some congruence conditions. They were introduced by Igusa (1964).

Definition

The symplectic group Sp_2g(Z) consists of the matrices

[math]\displaystyle{ \begin{pmatrix}A&B\\ C&D \end{pmatrix} }[/math]

such that AB^t and CD^t are symmetric, and AD^t − CB^t = I (the identity matrix).

The Igusa group Γ_g(n,2n) = Γ_n,2n consists of the matrices

[math]\displaystyle{ \begin{pmatrix}A&B\\ C&D \end{pmatrix} }[/math]

in Sp_2g(Z) such that B and C are congruent to 0 mod n, A and D are congruent to the identity matrix I mod n, and the diagonals of AB^t and CD^t are congruent to 0 mod 2n. We have Γ_g(2n)⊆ Γ_g(n,2n) ⊆ Γ_g(n) where Γ_g(n) is the subgroup of matrices congruent to the identity modulo n.

References

• Igusa, Jun-ichi (1964), "On the graded ring of theta-constants", Amer. J. Math. 86: 219–246, doi:10.2307/2373041