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

\sum _{{{\binom {ab}{cd}}\in P_{r}\setminus \Gamma _{g}}}f\left({\frac {a\tau +b}{c\tau +d}}\right)\det(c\tau +d)^{{-k}}.

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.

References

Klingen, Helmut (1967), "Zum Darstellungssatz für Siegelsche Modulformen", Math. Z., 102: 30–43, doi:10.1007/bf01110283, MR 0219473

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