In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
Sp
2''g''(Z) consists of the matrices
:
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
:
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