STUDIES ON MODULAR FORMS, NUMBER THEORY AND ALGEBRAIC STRUCTURES

模形式、数论和代数结构研究

• 批准号: 10640015
• 负责人: HATADA Kazuyuki
• 金额: $ 2.11万
• 依托单位: GIFU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640015/
• 关键词: p-adic limits; modular forms of nebentypus; l-adic modular forms; Siegel modular forms; common eigen-functions of Hecke operators; The Ramanujan Conjecture; representations of Galois groups; Abelian varieties; 原始形式; mod lガロワ表現; ヘッケ作用素の同時固有函数; 楕

1. Kazuyuki Hatada's studies: (1) Let g be any integer【greater than or equal】2. Hatada has obtained that there exist infinitely many Siegel cusp eigenforms of degree g which satisfy the Ramanujan Conjecture and that any of them has a Galois representation Gal(QィイD4-ィエD4/Q) →Gspinn(2g+1,QィイD4-ィエD4ィイD2lィエD2) with the expected property. Hatada has obtained also such an infinite family of Siegel cusp eigenforms of degree g which don't satisfy the Ramanujan Conjecture that any of the family has a good Galois representation of the above type. (2) Hatada has applied his method of (1) to differential forms of the first kind on Abelian varieties. (3) Let m be any integer【greater than or equal】0, l be any prime number, and N be any integer with (N,l)=1. Hatada has obtained that any holomorphic modular form of level NlィイD1mィエD1 m of nebentypus is an l-adic modular form of level N. Hatada has obtained that the mod l Galois representation of any primitive form on ΓィイD21ィエD2(NlィイD1mィエD1) is that of a primitive form on ΓィイD21ィエD2(N) for any 【greater than or equal】l2. (4) Hatada has extended Leopoldt's p-adic limit formula for the generalized Bernoulli numbers to the case of algebraic number fields. (5) Hatada has obtained a new method to construct Q without artificiality.2. Yoshio Fujimoto's studies: Let X be a non-singular projective threefold of Kodaira dimension κ(X)【greater than or equal】0 with a non-isomorphic surjective endomorphism. Fujimoto obtained, the minimal, models of X are non-singular; Some finite etale covering Y of X is isomorphic to the product of an Abelian variety and a non-singular projective variety if κ(X)=0,2.3. Toyohiko Aiki showed well-posedness for Caginalp and Penrose-Fife types of phase-field equations, and uniqueness of weak solutions of the shape memory alloy problems under a weaker condition than before.4. Shigeru Takeuchi studied Category of DR/CR spaces/algebras. Goro Chuman and Keiji Iwata studied Education of Mathematics.

1. Kazuyuki Hatada的研究： (1) 设g为任意整数【大于或等于】2。半乳糖(Q-D4-D4/Q) →Gspinn(2g+1,QD4-D4-D2D2) 具有预期的性质，也获得了 g 级西格尔尖点本征形的无限族，该族不满足拉马努金猜想，即该族中的任何一个都具有良好的伽罗瓦。 (2) Hatada 将他的方法 (1) 应用于阿贝尔簇的第一类微分形式。 (3) 让 m 。为任意整数【大于或等于】0，l为任意素数，N为(N,l)=1的任意整数。 N 级的模形式。 Hatada 已经获得了任何原始形式的 mod l Galois 表示ГiiD21ieD2(NliiiD1mieD1) 是 ГiiD21ieD2(N) 上任意【大于或等于】l2 的原始形式。 (4) Hatada 将广义伯努利数的利奥波德 p 进极限公式扩展到代数数域的情况。 (5)Hatada获得了一种无需人工构造Q的新方法。2． Yoshio Fujimoto的研究：设X为小平维κ(X)【大于或等于】0的非奇异射影三倍，并获得非同构满射自同态，则X的最小模型是非奇异的；如果 κ(X)=0,2.3，覆盖 X 的 Y 的有限 etale 与阿贝尔簇和非奇异射影簇的乘积同构。 Toyohiko Aiki表现出了Caginalp和Penrose-Fife型相场方程的适定性，以及在比以前更弱的条件下形状记忆合金问题的弱解的唯一性。 4．竹内茂研究了DR/CR空间/代数的范畴。 Goro Chuman 和 Keiji Iwata 研究数学教育。

项目成果

Kazuyuki HATADA: "A formula for the traces of the Hecke operators on certain spaces of new forms"by C. Hamer, Mathematical Reviews. Vol. 99b. 797-798 (1999)

Kazuyuki HATADA：“新形式某些空间上赫克算子踪迹的公式”，C. Hamer，《数学评论》。

Shigeru TAKEUCHI: "Category of DR spaces"Science Reports of the Faculty of Education Gifu University. 24巻1号. 7-12 (1999)

武内茂：《DR空间的分类》岐阜大学教育学部科学报告第24卷第1.7-12期（1999年）

Yoshio Fujimoto(Eiichi Satoと共著): "On smooth projective threefolds with non-trivial surjective endomorphisms"Proceedings of the Japan Academy. 74 ser.A. 143-145 (1998)

Yoshio Fujimoto（与 Eiichi Sato 合着）：“具有非平凡满射自同态的平滑射影三重”，日本学院学报 74 ser.A（1998）。

Toyohiko AIKI: "Weak solutions for Folk's model of shape memory alloys"Mathematical Methods in the Applied Sciences. (発表予定).

Toyohiko AIKI：“形状记忆合金 Folk 模型的弱解”应用科学中的数学方法（待提交）。

Toyohiko AIKI: "Weak solutions for Falk's model of shape memory alloys."Mathematical Methods in the Applied Sciences.. (to be published).

Toyohiko AIKI：“福克形状记忆合金模型的弱解。”应用科学中的数学方法..（待出版）。