Harder's conjecture and related topics

哈德猜想及相关话题

• 批准号: 21K03152
• 负责人: 桂田 英典
• 金额: $ 2.66万
• 依托单位: Hokkaido University (2022)Muroran Institute of Technology (2021)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2021
• 资助国家: 日本
• 起止时间: 2021-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21K03152/
• 关键词: Harder予想; Selmer群; ガロア表現; Klingen-Eisenstein級数; Gross-Keating invariant; Bloch-Kato予想

(1)Hader予想をある場合に証明した．Harder予想は次数が2で重さが(k+j,k)のベクトル値保型形式の合同に関する予想である．本研究ではkが偶数の時に再定式化し，特にjが4の倍数の時に証明した．その第１部は発刊予定．第２部もほぼ完成した．(跡部発，千田雅隆，伊吹山知義，山内卓也との共同研究.) また,それに関連して保形L関数の整数性についての論文を発刊した．jが4で割って余りが2の場合は現在検討中である．また，kが奇数のときにも予想を再定式化した．さらに，ある場合に，adjoint表現に付随するBloch-Kato予想の仮定の下でこれを証明した．これらを2023年1月京都で開催された研究集会で発表した．(2) Siegel保型形式にのあるガロア表現に付随するSelmer群のtorsion elementを構成した．(山内卓也との共同研究.)(3) Bergstrom-Farber-van-der-Geer予想の弱い形について考察しある場合に再定式化した．(4) Siegel級数の明示公式を用いてDuke-Imamoglu-Ikeda liftのフーリエ係数の良い評価を得た．これについて2022年6月に京都で開催された研究集会で発表した．論文も発刊予定である．また，2次形式のGross-Keating invariantの新しい特徴づけについて研究した．論文は投稿中である．さらにエルミーと形式においてもGross-Keating invariantを定義しその性質を研究した.(池田保との共同研究.) また，Gross-Keating invariantを具体的にもとめるためのアルゴリズムについて研究した.これは(2)で述べたHarder予想の具体例を与える際にも有用であった．(S. Cho, 池田保，C. H. Lee, 山内卓也との共同研究.)また，ここで得られた手法を応用してあるp進Siegel保型形式が真のSiegel保型形式になることを示した.(長岡昇勇との共同研究.)

(1) Hader wants to know how many times he wants to buy an insurance contract. (1) the number of times he or she wants to buy a contract. (1) the number of times you want to do this is the number of times you want to do it. (1) the number of times you want to do this is the number of times you want to do it. (1) the number of times you want to complete the contract is the number of times you want to do it. (Kombukata, Akira, Ikuyama, etc.) Yamaguchi also studied it together. The number of conformal figures is integer. J. 4. Cut. 2. Now there is an odd number of conformal figures. Adjoint shows that you want to make sure that you want to start a research conference in Kyoto in January 2023. (2) the form of Siegel protection shows that you want to pay with the Selmer group torsion element. (Yamato also co-studied it.) (3) Bergstrom-Farber-van-der-Geer wants to be weak. Please review the formula of Siegel data and then reformat it. (4) the express formula of the number of people using the express formula of the number of dollars. The number of people in Kyoto will be released in June 2022. The schedule of the conference will be published. In the form of "Gross-Keating invariant", "new", "special", "research", "Gross-Keating invariant", "definition", "research", "research". A joint study of Ikeda Ikeda. The specific information about Gross-Keating invariant is that of the research team. A specific example of how to imagine Harder is described in (2). (S. Cho, Ikeda Bao, C. H. Lee, co-research by Yamauchi Yamauchi.) You can use the Siegel protection form, the true Siegel protection form, the real Siegel protection form, and the display device. (long-term and courageous joint research.)

项目成果

Boundedness of denominators of special values of the L functions for modular forms

模形式的 L 函数特殊值分母的有界性

• 发表时间: 2022
• 期刊: Comment. Math. St. Pauli
• 作者: Hiraku Atobe;Masataka Chida;Tomoyoshi Ibukiyma;Hidenori Katsurada;and Takuya Yamauchi;小寺諒介;Tamotsu Ikeda and Hidenori Katsurada;R.Kodera and M. Ueda;Hidenori Katsurada
• 通讯作者: Hidenori Katsurada

Estimates for the Fourier coefficients of the Duke-Imamoglu-Ikeda lift

Duke-Imamoglu-Ikeda 升力的傅立叶系数估计

• DOI: 10.1515/forum-2022-0197
• 发表时间: 2023
• 期刊: Forum Math.
• 作者: Hiraku Atobe;Masataka Chida;Tomoyoshi Ibukiyma;Hidenori Katsurada;and Takuya Yamauchi;小寺諒介;Tamotsu Ikeda and Hidenori Katsurada
• 通讯作者: Tamotsu Ikeda and Hidenori Katsurada

Explicit formulas for the Rankin-Selberg series of the Ikeda type lifts and their application to the quantum unique ergodicity,

Ikeda 型升力的 Rankin-Selberg 级数的显式公式及其在量子唯一遍历性中的应用，

• 发表时间: 2022
• 作者: Hiraku Atobe;Masataka Chida;Tomoyoshi Ibukiyma;Hidenori Katsurada;and Takuya Yamauchi;小寺諒介;Tamotsu Ikeda and Hidenori Katsurada;R.Kodera and M. Ueda;Hidenori Katsurada;小寺諒介;三浦真人;Hidenori Katsurada;三浦真人;小寺諒介;桂田英典;三浦真人;Hidenori Katsurada
• 通讯作者: Hidenori Katsurada

An Inductive formula for the Gross-Keating invariant of a quadratic form

二次型 Gross-Keating 不变量的归纳公式

• 发表时间: 2025
• 期刊: Tohoku Math. J.
• 作者: Sungmun Cho;Tamotsu Ikeda;Hidenori Katsurada;Chul-hee Lee;and Takuya Yamauchi
• 通讯作者: and Takuya Yamauchi

Twisted Koecher-Maass series of the Ikeda type lift for the exceptional group of type E(7,3)

池田型电梯的扭曲 Koecher-Maass 系列，适用于 E(7,3) 型特殊组

• DOI: 10.1016/j.jnt.2023.01.010
• 发表时间: 2023
• 期刊: J. Number Theory
• 作者: Sungmun Cho;Tamotsu Ikeda;Hidenori Katsurada;Chul-hee Lee;and Takuya Yamauchi;Hidenori Katsurada and Henry H. Kim
• 通讯作者: Hidenori Katsurada and Henry H. Kim