Number Theory and Geometry related to Algebraic Groups

与代数群相关的数论和几何

• 批准号: 12440002
• 负责人: YUKIE Akihiko
• 金额: $ 5.5万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440002/
• 关键词: class number; density theorems; toric varieties; Q-curves; F-singularities; tight closure; elliptic curves; Mordell-Weil group; 代数群; 既均質ベクトル空間; 不定方程式; 概均質ベクトル空間; 量子群; キーワード6; キーワード7; キーワード8

(1) Yukie determined the density of the product of the class number and the regulator of biquadratic fields. Also he investigated other density theorems. He also obtained an upper bound for the number of quintic fields with bounded discriminant(2) Ishida worked on a generalization of the theory of complexes for rational fans to fans over real fields. In particular, he interpreted the ideal theory of commutative algebra theory in terms of fans, and worked on a construction of a process for real fans which is equivalent to blowups for algebraic varieties. He also formulated the notion of Zariski space for fans and established that it is possible to compactify real fans in the same manner as Nagata's completion of algebraic varieties(3) Among CM elliptic curves, Nakamura classified Q-curves which have nice properties with respect to the Galois group action and determined the structure of the Abelian varieties. obtained by such Q-curves. He also investigated the construction of singular Ab … More elian surfaces over the field of rational numbers(4) Ogata investigated defining equations of projective toric varieties and obtained an estimate of the number of tensors of an ample line bundle which give projectively normal imbeddings. He also obtained an estimate of the number of degrees of generators of the defining ideal and determined varieties which give rise to generators of the defining ideals of the highest degree(5) Hara introduced the notions in singularity theory in positive characteristic ring theoretically which should correspond to singularities in birational geometry and multiplier ideals in characteristic zero using the notion of Frobenius map and the tight closure. He also showed the relation between them and tried to find applications to algebraic geometry in positive characteristic(6) Sato investigated the distributions of the ranks of the Mordell-Well groups of quadratic twists of elliptic curves defined over number fields and related problems(7) Hasegawa investigated algebraic aspects of discrete integrable systems from the viewpoint of symmetry. In particular, he investigated quantization of discrete Painleve equations Less

(1)由纪子确定了密度的产品的类数和监管机构的双二次领域。他还调查了其他密度定理。他还获得了一个上限的数量五次领域有界判别式（2）石田工作的推广理论复杂的理性球迷在真实的领域。特别是，他解释了理想理论的交换代数理论的球迷，并致力于建设一个进程的真实的球迷这是相当于爆破代数品种。他还制定了概念的Zagliki空间的球迷和建立，这是可能的compactify真实的球迷以同样的方式永田的完成代数品种（3）在CM椭圆曲线，中村分类Q曲线有很好的性能方面的伽罗瓦群行动，并确定了结构的阿贝尔品种。通过这种Q曲线。他还研究了奇异Ab的构造 ...更多信息 （4）Ogata研究了射影复曲面簇的定义方程，得到了一个充要线丛中给出射影正规嵌入的张量个数的估计。他还获得了估计的数量度的发电机的定义理想和确定的品种，从而产生发电机的定义理想的最高程度（5）原介绍了概念，在奇异理论的积极特征环理论上应对应于奇异性的双有理几何和乘数理想的特征零使用的概念Frobenius地图和紧封闭。他还表明了它们之间的关系，并试图找到应用代数几何中的积极特征（6）佐藤调查的分布的行列的莫德尔，以及团体的二次扭曲的椭圆曲线定义的一些领域和相关问题（7）长谷川调查代数方面的离散可积系统的观点的对称性。特别是，他研究了离散Painleve方程的量子化

项目成果

A.C.Kable, A.Yukie: "The mean value of the product of class numbers of paired quadratic fields I"Tohoku Math. J.. J.54. 513-565 (2002)

A.C.Kable、A.Yukie：“成对二次域 I 的类数乘积的平均值”东北数学。

N.Hara, K-i. Watanabe: "F-regular and F-pure rigs vs. log terminal and log canonical singularities"J. Algebraic Geom. 11. 363-392 (2002)

N.Hara，K-i。

Anthony Kable, Akihiko Yukie: "The mean value of the product of class numbers of paired quadratic fields I"Tohoku Math J.. (To appear).

Anthony Kable、Akihiko Yukie：“成对二次域的类数乘积的平均值 I”Tohoku Math J..（待出）。

Atsushi Sato: "On the class numbers of certain number fields obtained from points on elliptic curves"Osaka J.Math. 38. 811-825 (2001)

Atsushi Sato：“关于从椭圆曲线上的点获得的某些数域的类数”Osaka J.Math。

N.Hara, K.Watanabe, K.Yoshida: "Rees algebras of F-regular type"J. Algebra. 247. 191-218 (2002)

N.Hara，K.Watanabe，K.Yoshida：“F-正则类型的里斯代数”J。