Research on special values of the standard L-functions of Siegel modular forms and related fields

西格尔模形式标准L函数的特殊值及相关领域研究

• 批准号: 15540003
• 负责人: KATSURADA Hidenori
• 金额: $ 2.37万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540003/
• 关键词: Koecher-Maass series; Saito-Kurokawa lift; Special values; Congruence of modular forms; Standard zeta function; Basis Problem; Doi-Hida-Ishi conjecture; Block-Kato conjecture; スタンダードL関数; アイゼンシュタイン級数の引き戻し公式; ジーゲル保型形式; 局所密度; テータ級数

(1)I considered the relation between special values of the standard zeta functions and the congruence a Siegel cusp form of arbitrary degree. In particular, I proposed a conjecture concerning the congruence of the Saito-Kurokawa lift of a cuspidal Hecke eigenform with respect to SL(2,Z) and the special value of its L-function, and proved it under certain condition. Furthermore we gave some numerical evidence for our conjecture. I gave talks on these topics at the workshop held at Hakuba on October 2004, and at the conference held at Karatsu on March 2005.(2)Let M and N be square free positive integers, and φ and χ be Dirichlet characters mod M and N, respectively. For a cuspidal Hecke eigenform f of neben type φ I gave an explicit formula for the special values of the standard zeta function of f twisted by χ. I gave numerical examples for these values and checked some results derived from the Block-Kato conjecture.(3)I proved that the space of Eisenstein series of square free level of quadratic nebentype is spanned by the genus theta series jointly with R. Schulze-Pillot. By using this result, I proved that the space of cusp forms of square free level of quadratic nebentype is spanned by the theta series jointly with S.Boecherer and R.Schulze-Pillot. They gave talks on these topics at the symposium held at Rikkyo University on September 2004.(4)I gave an explicit form of the Koecher-Maass series for the real analytic Eisenstein series jointly with T.Ibukiyama. I gave a talk on this topic at the symposium held at Rikkyo University on September 2004.

(1)考虑了标准zeta函数的特殊值与任意次Siegel尖形式的同余的关系。特别地，我提出了关于锯齿Hecke特征型的Saito-Kurokawa升对SL（2，Z）的同余性及其l函数的特殊值的一个猜想，并在一定条件下证明了它。此外，我们还给出了一些数值证据来证明我们的猜想。我在2004年10月在白波举行的研讨会和2005年3月在卡拉津举行的会议上就这些主题进行了演讲。(2)设M和N为自由正整数，φ和χ分别为对M和N模取的狄利克雷字符。对于neben型φ I的锯齿Hecke特征型f，给出了f被χ扭曲的标准ζ函数的特殊值的显式公式。我给出了这些值的数值例子，并检查了由Block-Kato猜想得出的一些结果。(3)与R. Schulze-Pillot共同证明了二次nebenttype的平方自由水平的Eisenstein级数的空间是由格级数张成的。利用这一结果，我证明了二次型nebentype的平方自由水平的尖形空间是由S.Boecherer和R.Schulze-Pillot共同张成的。他们于2004年9月在立教大学举行的研讨会上发表了上述内容。(4)与T.Ibukiyama共同给出了实解析型爱森斯坦级数的koecher - mass级数的显式形式。2004年9月，我在立教大学举行的研讨会上发表了这一主题的演讲。

项目成果

Hall's relations in finite groups.

有限群中的霍尔关系。

• 发表时间: 2004
• 期刊: Journal of Algebra 371
• 作者: TAKAGAHARA;Yugen
• 通讯作者: Yugen

An explicit formula for the Koecher-Maass Dirichlet series for the Ikeda lifting

池田提升的 Koecher-Maass Dirichlet 级数的显式公式

• 发表时间: 2004
• 期刊: Abhand.Math.Sem.der Univ.Hamburg 74
• 作者: T.Ibukiyama;H.Katsurada
• 通讯作者: H.Katsurada

T.Ibukiyama: "An explicit form of the Koecher-Maass Dirichlet series for klingen's Eisenstein series"J.Number Theory. 71. 223-256 (2003)

T.Ibukiyama：“克林根爱森斯坦级数的 Koecher-Maass Dirichlet 级数的显式形式”J.数论。

An explicit form of the Koecher-Maass Dirichlet series for Klingen's Eisenstein series

克林根爱森斯坦级数的 Koecher-Maass Dirichlet 级数的显式形式

• 发表时间: 2003
• 期刊: J.Number Theory 71
• 作者: Masaaki Amou;Masanori Katsurada;T.Ibukiyama
• 通讯作者: T.Ibukiyama

H.Katsurada: "Pullback formula of Eisenstein series and its applications"第2回保型形式周辺分野スプリングコンファレンス報告集. 167-204 (2004)

H. Katsurada：“爱森斯坦级数的回拉公式及其应用”第二届自守形式春季会议报告 167-204 (2004)。