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, 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

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Sp_2''g''(Z) consists of the matrices :

\beginA&B\\ C&D \end

such that ''AB^t'' and ''CD^t'' are symmetric, and ''AD^t − CB^t'' = ''I'' (the identity matrix). The Igusa group Γ_g(''n'',2''n'') = Γ_''n'',2''n'' consists of the matrices :

\beginA&B\\ C&D \end

in Sp_2''g''(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 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''.

References

*{{citation, mr=0164967 , last=Igusa, first= Jun-ichi , title=On the graded ring of theta-constants , journal=Amer. J. Math., volume= 86 , year=1964, pages= 219–246, doi=10.2307/2373041 Automorphic forms