|
Igusa Subgroup
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 Sp2''g''(Z) consists of the matrices :\beginA&B\\ C&D \end such that ''ABt'' and ''CDt'' are symmetric, and ''ADt − CBt'' = ''I'' (the identity matrix). The Igusa group Γg(''n'',2''n'') = Γ''n'',2''n'' consists of the matrices :\beginA&B\\ C&D \end in Sp2''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 ''ABt'' and ''CDt'' are congruent to 0 mod 2''n''. We have Γg(2''n'')⊆ Γg(''n'',2''n'') ⊆ Γg(''n'') where Γg(''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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Modular Group
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by . It is the symmetric space associated to the symplectic group . The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case ''g=1''. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper half-space in terms of its real and imaginary parts as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaplect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
