Klingen Eisenstein series
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 Klingen (1967).
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
- [math]\displaystyle{ \sum_{\binom{ab}{cd}\isin P_r\setminus\Gamma_g} f\left(\frac{a\tau+b}{c\tau+d}\right)\det(c\tau+d)^{-k}. }[/math]
It is a Siegel modular form of weight k and degree g. Here Pr is the integral points of a certain parabolic subgroup of the symplectic group, and Γr 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 Φg−r, where Φ is the Siegel operator.
References
- Klingen, Helmut (1967), "Zum Darstellungssatz für Siegelsche Modulformen", Math. Z. 102: 30–43, doi:10.1007/bf01110283
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