Dirichlet series corresponding to Siegel's modular forms
Dirichlet series corresponding to Siegel's modular forms
复制标题
对应于西格尔模形式的狄利克雷级数
DOI：
10.1007/bf01424773
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发表时间：
1978
影响因子：
1.4
通讯作者：
Tsuneo Arakawa
中科院分区：
文献类型：
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作者：
Tsuneo Arakawa
0.1. Koecher [KO]引入了与SiegeΓ模形式相对应的Dirichlet级数，并给出了Dirichlet级数的显式公式，这些公式以令人满意的方式表示极点和留数的位置，但没有成功地证明它们。马斯在他的讲义[马，§ 15]研究了这些狄利克雷级数在充分的一般性，并获得了解析延续和功能方程。他的方法是基于不变微分算子作用于真实的对称矩阵的理论，这为研究Dirichlet级数及其函数方程提供了有力的工具。然而，人们不能有精确的信息残留物的极点，他的方法。在[AR]中，我们利用Klingen的Eisenstein级数证明了Koecher的显式公式，并由Klingen [KL 1]证明了SiegeΓ模形式空间的结构定理。最近Weissauer [WE]研究了N阶SiegeΓ尖点型对应的Koecher-Maass Dirichlet级数，解决了一类关于具有Grüssen特征标的Dirichlet级数与SiegeΓ尖点型对应的匡威问题.本文的目的是在不使用Klingen的Eisenstein级数的情况下，证明N级SiegeΓ模形式（不一定是尖点形式）对应的Dirichlet级数的Koecher显式公式（Klingen在[KL 2，p.235]中提出了在不使用Klingen的Eisenstein级数的情况下获得Koecher公式的问题）.我们还得到了一个明确的公式的Epstein-Koecher zeta函数。另一个更算术方面的Koecher-Maass狄利克雷级数讨论在B Schlicherer [BO]。
0.1. Koecher [KO] introduced Dirichlet series corresponding to SiegeΓs modular forms and presented explicit formulas for the Dirichlet series which express the location of the poles and the residues in a satisfactory manner, but did not succeed in proving them. Maass in his lecture notes [MA, § 15] studied those Dirichlet series in full generality and obtained the analytic continuation and the functional equations for them. His method is based upon the theory of invariant differential operators acting on real symmetric matrices, which gives a powerful tool in investigating those Dirichlet series and their functional equations. However one cannot have precise information on the residues of the poles by his method. In [AR] we have proved Koecher's explicit formulas by using Klingen's Eisenstein series and the structure theorem for the space of SiegeΓs modular forms due to Klingen [KL1]. Recently Weissauer [WE] studied Koecher-Maass Dirichlet series corresponding to SiegeΓs cusp forms with level N and solved a certain converse problem concerning the correspondence between those Dirichlet series with grόssen characters and SiegeΓs cusp forms. Our aim of the present paper is to prove Koecher's explicit formulas for the Dirichlet series corresponding to SiegeΓs modular forms (not necessarily cusp forms) with level N without using Klingen's Eisenstein series (Klingen in [KL2, p. 235] suggested the problem of obtaining Koecher's formulas without the help of Klingen's Eisenstein series). We also obtain an explicit formula for the Epstein-Koecher zeta function. Another more arithmetic aspect of Koecher-Maass Dirichlet series is discussed in Bόcherer [BO].