Igusa group explained
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 .
Definition
The symplectic group Sp2g(Z) consists of the matrices
\begin{pmatrix}A&B\ C&D\end{pmatrix}
such that
ABt and
CDt are symmetric, and
ADt − CBt =
I (the identity matrix).
The Igusa group Γg(n,2n) = Γn,2n consists of the matrices
\begin{pmatrix}A&B\ C&D\end{pmatrix}
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
ABt and
CDt are congruent to 0 mod 2
n.We have Γ
g(2
n)⊆ Γ
g(
n,2
n) ⊆ Γ
g(
n) where Γ
g(
n) is the subgroup of matrices congruent to the identity modulo
n
