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In mathematics, a Klingen Eisenstein series is a Siegel modular form of weight ''k'' and degree ''g'' depending on another Siegel cusp form ''f'' of weight ''k'' and degree ''r''<''g'', given by a series similar to an Eisenstein series. It is a generalization of the Siegel Eisenstein series, which is the special case when the Siegel cusp form is 1. Klingen Eisenstein series is introduced by . ==Definition== Suppose that ''f'' is a Siegel cusp form of degree ''r'' and weight ''k'' with ''k'' > ''g'' + ''r'' + 1 an even integer. The Klingen Eisenstein series is : It is a Siegel modular form of weight ''k'' and degree ''g''. Here ''P''''r'' is the integral points of a certain parabolic subgroup of the symplectic group, and Γ''r'' is the group of integral points of the degree ''g'' symplectic group. The variable τ is in the Siegel upper half plane of degree ''g''. The function ''f'' is originally defined only for elements of the Siegel upper half plane of degree ''r'', but extended to the Siegel upper half plane of degree ''g'' by projecting this to the smaller Siegel upper half plane. The cusp form ''f'' is the image of the Klingen Eisenstein series under the operator Φ''g''−''r'', where Φ is the Siegel operator. 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「Klingen Eisenstein series」の詳細全文を読む スポンサード リンク
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