Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. (A rationality theorem for formal Hecke series related to the Siegel modular group). (German) Zbl 0613.10026
The author proves the rationality of a formal Hecke series related to the Siegel modular group \(\Gamma_ n=Sp(n; {\mathbb{Z}})\). Given a prime number p, one deals with the p-component of the Hecke algebra related to the pair \((\Gamma_ n, Sp(n;{\mathbb{Q}}))\) and the attached isomorphism Q in the sense of E. Freitag [Siegelsche Modulfunktionen (Grundl. Math. Wiss. 254) (1983; Zbl 0498.10016)]. Using purely algebraic methods, it is demonstrated that the identity
\[
\sum_{D}Q(\Gamma_ n\left( \begin{matrix} D\\ 0\end{matrix} \begin{matrix} 0\\ D^{-1}\end{matrix} \right)\Gamma_ n)\quad T^{\alpha}=\frac{1-T}{1-p^ nT}\prod^{n}_{i=1}\frac{(1- p^{2i}T^ 2)}{(1-X_ ip^ nT)(1-X_ i^{-1}p^ nT)}
\]
holds formally in T, where the summation is taken over all elementary divisor matrices D of the form having \(p^{\alpha_ 1},...,p^{\alpha_ n}\), respectively, as diagonal entries with \(0\leq \alpha_ 1\leq...\leq \alpha_ n\); \(\alpha =\sum^{n}_{i=1}\alpha_ i\). This rationality theorem yields several applications: a simple approach to the Andrianov identities [A. N. Andrianov, Math. USSR, Sb. 34, 259-300 (1978); translation from Mat. Sb., Nov. Ser. 105(147), 291-341 (1978; Zbl 0389.10022)], explicit formulas for the action of certain Hecke operators on theta series and a representation of the standard zeta-function related to a Siegel modular form, which is a simultaneous eigenform under all Hecke operators. Moreover the solution of the basis problem for Siegel modular forms given by the author [Math. Z. 183, 21-46 (1983; Zbl 0497.10020), ibid. 189, 81-110 (1985; Zbl 0558.10022)] can be simplified.
Reviewer: A.Krieg
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
Keywords:
rationality; Siegel modular group; Hecke algebra; Hecke operators; theta series; standard zeta-function; Siegel modular form; basis problemReferences:
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