Stabile Modulformen und Eisensteinreihen. (Stable modular forms and Eisenstein series)
Let \({\mathcal H}_ n:=\{Z=Z^{(n)}=Z'\); Im Z\(>0\}\) denote the Siegel half- space of degree n. Given \(0\leq j\leq n\), an even integer k and a cusp form g on \({\mathcal H}_ j\) of weight k, one defines the Klingen-Eisenstein series by \[ G(Z,s):=\sum_{M}g(\pi (M<Z>))\quad \det (CZ+D)^{-k}\quad (\frac{\det Im(\pi (M<Z>))}{\det Im(M<Z>)})^{-s}, \] where M runs through a set of representatives of the Siegel modular group over a certain subgroup and \(\pi\) denotes the projection of an \(n\times n\) matrix onto its upper left \(j\times j\) corner. G(Z,s) converges for complex s, whenever Re s is sufficiently large. It was proved by \textit{R. P. Langlands} [On the functional equations satisfied by Eisenstein series (Lect. Notes Math. 544) (1976; Zbl 0332.10018)] that G(Z,s) admits a meromorphic continuation to the whole complex s-plane. The author makes strong use of Langlands theory of Eisenstein series and spectral decomposition in order to derive properties of the functions, which are obtained by analytic continuation, and their behavior under the Siegel \(\Phi\)-operator. If \(k>(n+j+3)/2\) the Klingen-Eisenstein series G(Z,s) is regular at \(s=0\). The function \(G(Z):=G(Z,0)\) becomes a Siegel modular form satisfying \(\Phi^{n-j}G=g\). If \(k=(n+j+3)/2\) then G(Z,s) turns out to be regular at \(s=0\) again, but not always holomorphic. In the case \(k=(n+j+1)/2\) the function G(Z,0) proves to be holomorphic, but might identically vanish. If \(k<(n+j+1)/2\) the procedure of Hecke summation has to be modified. In this case \(G(Z):=_{s=s_ 0}G(Z,s)\), where \(s_ 0=(n+j+1)/2-k\), yields a holomorphic modular form, which might identically vanish. The results are used in order to characterize the subspace of stable modular forms. Especially the vector space of all holomorphic cusp forms on \({\mathcal H}_ n\) with respect to an irreducible representation \((V_{\rho},\rho)\) with weight k(\(\rho)\) consists of finite linear combinations of theta series \(\vartheta_{S,P}\) attached to positive definite even unimodular \(m\times m\) matrices S and pluriharmonic forms P belonging to \((V_{\rho},\rho)\), if \(m\equiv 0 mod 8\) and \(4n+8\leq m\leq 2k(\rho).\) On the other hand the results are specified for the Siegel-Eisenstein series, which correspond to \(j=0\) and \(g=1\). A version of the analytic part of the Siegel-Main-Theorem is derived, if the Eisenstein series are obtained by analytic continuation. \{An announcement of this inaugural lecture was published in Contemp. Math. 53, 535-542 (1986; Zbl 0595.10020).\}
scientific article; zbMATH DE number 3959538
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