Differential Galois theory and linear algebraic groups over algebraic function fields

代数函数域上的微分伽罗瓦理论和线性代数群

• 批准号: 279644768
• 负责人: Dr. Annette Bachmayr
• 金额: --
• 依托单位: Arbeitsgruppe Arithmetische Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/279644768?language=en
• 关键词: Differential; Galois; theory; linear; algebraic

Differential Galois theory assigns a matrix group, called differential Galois group, to each linear differential equation. The differential Galois group contains information on the differential field generated by the solutions of the equation. It also measures the algebraic relations among the solutions. The differential Galois group of an ordinary linear differential equation is a linear algebraic group. The inverse differential Galois problem asks which linear algebraic groups occur as differential Galois groups over a given differential field. Over rational function fields k(x) over algebraically closed fields k it has been known for almost 20 years that every linear algebraic group occurs. The first part of the project is to show that in addition, the absolute differential Galois group of k(x) is a free proalgebraic group. The approach is based on field patching methods and on the study of differential embedding problems.For a differential equation depending on an additional discrete parameter, one can define its sigma-differential Galois group, which is a subgroup of the differential Galois group. It can be regarded as a refined version of the latter, as it measures difference-algebraic relations among the solutions. Sigma-differential Galois groups are linear differential algebraic groups. In the second part of this project, we will study the corresponding inverse problem over C(x).The third part of this project is devoted to the study of torsors under linear algebraic groups: Using the field patching method, we aim to prove a local-global principle for reduced norms.

微分伽罗瓦理论赋予每个线性微分方程一个矩阵群，称为微分伽罗瓦群。微分伽罗瓦群包含由方程的解产生的微分场的信息。它还测量解之间的代数关系。一个常线性微分方程的微分伽罗瓦群是一个线性代数群。逆微分伽罗瓦问题（英语：Inverse differential Galois problem）是问在一个给定的微分域上，哪些线性代数群是微分伽罗瓦群。在代数闭域k上的有理函数域k（x）上，几乎20年前就知道每个线性代数群都存在。该项目的第一部分是证明，此外，绝对微分伽罗瓦群的k（x）是一个自由proalgebraic群。该方法基于域修补方法和微分嵌入问题的研究，对于依赖于一个附加离散参数的微分方程，可以定义它的σ-微分Galois群，它是微分Galois群的一个子群.它可以被视为后者的改进版本，因为它测量解之间的差分代数关系。Σ-微分伽罗瓦群是线性微分代数群。第二部分研究C（x）上相应的反问题，第三部分研究线性代数群下的扭子：利用域修补方法，证明了一个约化范数的局部-整体原理。