Investigation of Koecher-Maass series for various liftings of Siegel modular forms

针对 Siegel 模形式的各种提升的 Koecher-Maass 级数的研究

• 批准号: 13640003
• 负责人: KATSURADA Hidenori
• 金额: $ 2.62万
• 依托单位: Muroran Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640003/
• 关键词: Maass zeta function; Ikeda lifting; Eisenstein series; Squared Moebius function; Standard zeta function; Ikeda-Miyawaki life; Ikeda-Miyawaki lift

1. In 2000, we gave an explicit formula for the Koecher-Maass Dirichlet series for the Ikeda lifting of an elliptic cuspidal Hecke-eigenforms f jointly with T. Ibukiyama. We gave some result on the algebraicity of special values of a certain Dirichlet series attached to f appearing in such a formula. 2. We gave a good sufficient condition for two Siegel cuspidal Hecke-eigenforms to coincide with each other. This result is slightly stronger than the result which we gave in 1999. Furthermore, we proposed a certain conjecture on a refinement of the above result, and proved this conjecture under a certain condition on the non-vanishing of the Koecher-Maass series.3. By using the pullback formula for Siegel Eisenstein series, the differential operators due to Ibukiyama, and an explicit formula for Siegel series due to Katsurada, we gave an exact values of the standard zeta function of a modular form f twisted by a character χ in the following cases:(1) f is an elliptic cuspidal Hecke eigenform of neben type, and χ is a Dirichlet character of prime conductor.(2) f is a Siegel modular form of degree 2 of level 1, and χ is the trivial character.4. We proved that the p-adic Eisenstein series of degree 2 defined by Nagaoka is a true modular form (Joint work with S. Nagaoka.)5. We investigated the action of Hecke operator on the theta series (Joint work with R. Schulze-Pillot.)

1。在2000年，我们为Koecher-Maass Dirichlet系列提供了一个明确的公式，用于与T. Ibukiyama共同椭圆形的Hecke-eigenforms ikeda举起。我们对与F中出现的某个Dirichlet系列的特殊值的代数相结合。 2。我们给出了两个西格尔·苏皮（Siegel Cuspidal Hecke-eigenforms）彼此重合的良好条件。该结果比我们在1999年给出的结果稍强。此外，我们提出了一定的猜想，以对上述结果的完善进行了一定的猜想，并在一定条件下提供了有关Koecher-Maass系列的一定条件的猜想。3。 By using the pullback formula for Siegel Eisenstein series, the differential operators due to Ibukiyama, and an explicit formula for Siegel series due to Katsurada, we gave an exact value of the standard zeta function of a modular form f twisted by a character χ in the following cases:(1) f is an elliptic cuspidal hecke eigenform of neben type, and χ is a Dirichlet （2）F的特征是1级的2级的Siegel模块化形式，χ是微不足道的特征。4。我们证明了Nagaoka定义的2度2度的P-ADIC EISENSTEIN系列是真正的模块化形式（与S. Nagaoka的联合工作）5。我们调查了Hecke运营商对Theta系列的作用（与R. Schulze-Pillot的联合合作。）

项目成果

H. Katsurada: "Special values of the standard zeta functions. In Galois Theory and Modular Forms"Kluwer Academic Publishers. 337-356 (2003)

H. Katsurada：“标准 zeta 函数的特殊值。伽罗瓦理论和模形式”Kluwer 学术出版社。

H.Katsurada: "Exact values of the standard zeta functions"Proc.of Japanese-German Seminar "Explicit Structures of Modular Forms and Zeta Functions". 32-41 (2002)

H.Katsurada：“标准 zeta 函数的精确值”日德研讨会“模形式和 Zeta 函数的显式结构”的 Proc.。

竹ヶ原 裕元: "置換表現の数え上げと母関数"第47回代数学シンポジウム報告集. 5-16 (2002)

Hiromoto Takegahara：“排列表达式和生成函数的枚举”第 47 届代数研讨会报告 5-16 (2002)。

Y.Takegahara: "A generating function for the number of homomorphisms from a finitely generated abelian group to an alternating group"J.Algebra. 248. 554-574 (2002)

Y.Takegahara：“从有限生成的阿贝尔群到交替群的同态数的生成函数”J.Algebra。

H.Katsurada: "A remark on the coincidence of Hecke eigenforms"Comm.Math.Univ.St.Pauli. 52. 47-54 (2003)

H.Katsurada：“关于 Hecke 本征形的巧合性的评论”Comm.Math.Univ.St.Pauli。