Constructive geometry of arithmetic quotients of symmetric spaces

对称空间算术商的构造几何

• 批准号: 14340004
• 负责人: ODA Takayuki
• 金额: $ 7.62万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340004/
• 关键词: Automorphic forms; Spherical functions; Whittaker functions; Green currents; modular cycle; Automorphic forms; Spherical functions; automorphic forms; spherical functions; modular cycles; modular forms; locally symmetric spaces; Green function

In the period of the project, we got the following results.1.Explicit formula for Whittaker functions of SL(3,R) belonging to non-spherical principal series. (Published in 2004).2.Given a pair of arithmetic quotients (X,Y), where X,Y are either type IV quotients, or quotients of the complex hyperball, and when Y is a divisor of X, we constructed the Green current of Y as an automorphic form. (Published in 2003).3.We proved the confluence from Siegel-Whittaker functions to Whittaker functions for P_J principal series representations of Sp(2,R). (In press).4.Together with Miki Hirano of Ehime University, we obtained the integral and power series expressions of the principal series Whittaker functions on GL(3,C) with minimal K-type. The project is still in progress.5.The project to get explicit formula for P_J principal series representations of Sp(3,R) : The contents are finished. We are preparing a joint paper together with Miki Hirano and Taku Ishii of Chiba Inst. of Technology.6.We maintained the monthly seminar on Automorphic Forms at Komaba.

1.非球面主级数SL(3，R)的Whittaker函数的显式公式。2.给出一对算术商(X，Y)，其中X，Y是IV型商，或者是复超球的商，当Y是X的除数时，我们将Y的绿流构造为自同构形式。3.证明了Sp(2，R)的P_J主级数表示的Siegel-Whittaker函数到Whittaker函数的合流性。与爱慕大学的平野美纪一起，我们得到了GL(3，C)上具有极小K型的主级数Whittaker函数的积分和级数表达式。5.Sp(3，R)的P_J主级数表示的显式表示：内容已完成。我们正在与千叶研究所的平野美树和石井拓共同准备一份联合论文。6.我们在Komaba每月举办一次关于自构形的研讨会。

项目成果

Siegel series and Spherical functions on O(2n)/(O(n)×O(n))

O(2n)/(O(n)×O(n)) 上的 Siegel 级数和球函数

• 发表时间: 2006
• 期刊: In "Automorphic forms and zeta functions", World Scientific
• 作者: 広中由美子;佐藤文広
• 通讯作者: 佐藤文広

Automorphic Green functions associated with the secondary spherical functions.

与二次球函数相关的自同构格林函数。

• 发表时间: 2003
• 期刊: Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, 451--533. 39
• 作者: 織田孝幸;都築正男
• 通讯作者: 都築正男

BirchとSwinnerton-Dyerの予想

伯奇和斯温纳顿-戴尔猜想

• 发表时间: 2003
• 期刊: 雑誌「数学」 55-1
• 作者: 織田 孝幸

Secondary Whittaker functions for $Psb J$-principal series representations of ${rm Sp}(3,R)$.

$Psb J$ 的辅助 Whittaker 函数 - ${rm Sp}(3,R)$ 的主级数表示。

• 发表时间: 2005
• 期刊: Proceedings of the Japan Academy 81-A-6
• 作者: Miki Hirano;Takayuki Oda
• 通讯作者: Takayuki Oda

浜畑芳紀: "The values of $J$-invariants for Drinfeld modules"Manuscripta Math.. 112. 93-108 (2003)

Yoshiki Hamahata：“Drinfeld 模块的 $J$-不变量的值”Manuscripta Math.. 112. 93-108 (2003)