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).

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 Sp2g(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 2n. We have Γg(2n)⊆ Γg(n,2n) ⊆ Γg(n) where Γg(n) is the subgroup of matrices congruent to the identity modulo n.
Igusa, Jun-ichi (1964), "On the graded ring of theta-constants", Amer. J. Math., 86: 219–246, doi:10.2307/2373041, MR 0164967

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