Resolution of singularities of defining equations of number field and its relation to zeta function

数场定义方程奇点的解析及其与zeta函数的关系

• 批准号: 22654003
• 负责人: MAEDA Hironobu
• 金额: $ 0.82万
• 依托单位: Tokyo University of Agriculture and Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Challenging Exploratory Research
• 财政年份: 2010
• 资助国家: 日本
• 起止时间: 2010 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22654003/
• 关键词: 代数体; 特異点; 底式; 定義方程式; 実射影平面; ２次形式; 判別式; 単位元形式; 特異点集合; 代数的整数論; 特異点解消; ゼータ関数

The defining equation of a number field is not unique and has in general many singular and infinitely near singular points. One can read some number theoretic properties, e.g. prime ideal decomposition, from these singularities, but we cannot find any properties of singutlarities, which distinguishes real and imaginary extensions. On the other hand I was able to prove the singular locus of the fundamental equation of a number field is contained in the zero points of a homogeneous form, which Hilbert called the Einheitsform in his Zahlbericht.

数字场的定义方程并非唯一，并且通常在奇异点附近具有许多单数和无限的奇异方程。可以读取一些数字理论属性，例如从这些奇异性来看，主要理想的分解，但我们找不到单一的任何特性，可以区分真实和虚构的扩展。另一方面，我能够证明一个数字基本方程的奇异轨迹包含在同质形式的零点中，希尔伯特在他的zahlbericht中称其为einheitsform。