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In mathematics, a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties. It is the group of automorphisms of Z2''n'' preserving a non-degenerate skew symmetric form. The name "paramodular group" is often used to mean one of several standard matrix representations of this group. The corresponding group over the reals is called the parasymplectic group and is conjugate to a (real) symplectic group. A paramodular form is a Siegel modular form for a paramodular group. Paramodular groups were introduced by and named by . ==Explicit matrices for the paramodular group== There are two conventions for writing the paramodular group as matrices. In the first (older) convention the matrix entries are integers but the group is not a subgroup of the symplectic group, while in the second convention the paramodular group is a subgroup of the usual symplectic group (over the rationals) but its coordinates are not always integers. These two forms of the symplectic group are conjugate in the general linear group. Any nonsingular skew symmetric form on Z2''n'' is equivalent to one given by a matrix : where ''F'' is an ''n'' by ''n'' diagonal matrix whose diagonal elements ''F''''ii'' are positive integers with each dividing the next. So any paramodular group is conjugate to one preserving the form above, in other words it consists of the matrices : of GL2''n''(Z) such that : The conjugate of the paramodular group by the matrix : (where ''I'' is the identity matrix) lies in the symplectic group Sp2''n''(Q), since : though its entries are not in general integers. This conjugate is also often called the paramodular group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Paramodular group」の詳細全文を読む スポンサード リンク
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