Arithmetic of Homogeneous Spaces under Linear Algebraic Groups

线性代数群下齐次空间的算术

基本信息

批准号：
1801951
负责人：
Parimala Raman
金额：
$ 27万
依托单位：
Emory University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801951&HistoricalAwards=false
关键词：
Arithmetic Homogeneous Spaces under Linear

项目摘要

This award concerns work in Arithmetic Geometry, an important area of mathematics at the intersection of algebraic geometry and number theory. Recent progress on the topic of the study of homogeneous spaces has set forth new questions and conjectures whose study is part of the proposal. There are related problems accessible to graduate students and they will be engaged via topical graduate courses, seminars and workshops. The PI and Co-PI plan to organize workshops in related areas and disseminate the outcome of research to mathematical community through lectures and seminars. Encouraging and engaging women graduate students will be part of the PI's mission.The PI and Co-PI plan to investigate questions related to the study of homogeneous spaces under connected linear algebraic groups with special reference to function fields of curves over local and global fields. This study, with reference to classical groups, is closely related to the study of the Brauer group of the field and the period-index bounds as well as the study of quadratic forms and the u-invariant. The PI and Co-PI plan to investigate the obstruction to the Hasse principle for homogeneous spaces via the Brauer-Manin like obstructions as well as higher reciprocity obstructions which have already been constructed. They will investigate the Brauer-Manin obstruction for moduli spaces of twisted sheaves over curves over totally imaginary number fields to get a hold on the u-invariant of function fields of such curves whose finiteness is a big open question. They will also study Serre/Totaro question on the existence of closed points of degree d on principal homogeneous spaces under absolutely simple simply connected groups admitting a zero cycle of degree d.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项涉及算术几何形状的工作，这是代数几何学和数理论的交集中数学的重要领域。关于同质空间研究的主题的最新进展提出了新的问题和猜想，这些问题和猜想是该提案的一部分。研究生有相关的问题，它们将通过局部研究生课程，研讨会和讲习班参与。 PI和CO-PI计划在相关领域组织研讨会，并通过讲座和研讨会将研究结果传播给数学社区。鼓励和引人入胜的女性研究生将成为PI任务的一部分。PI和Co-Pi计划调查与在连接的线性代数群体下研究同质空间有关的问题，并特别参考了本地和全球领域的曲线功能领域。这项研究与古典群体有关，与该领域的Brauer群体和周期索引边界以及二次形式和U不变的研究密切相关。 PI和CO-PI计划通过Brauer-Manin（例如障碍物）以及已经构建的较高的互惠障碍物来调查均匀空间的Hasse原理的障碍。他们将研究曲线上曲线的模量空间的Brauer-manin障碍物，在完全虚构的数字场上，以掌握此类曲线的功能场的U-unimitiant，其有限的曲线是一个很大的悬而未决的问题。他们还将研究Serre/Totaro的问题，即在绝对简单的简单连接组下，在主要均质空间上存在D学位的封闭点，承认该学位的零周期。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来获得支持的。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Third Galois cohomology group of function fields of curves over number fields

数域曲线函数域的第三伽罗瓦上同调群

DOI：
10.2140/ant.2020.14.701
发表时间：
2020
期刊：
Algebra & Number Theory
影响因子：
1.3
作者：
Suresh, Venapally
通讯作者：
Suresh, Venapally

Local-global principles for tori over arithmetic curves

算术曲线上环面的局部全局原理

DOI：
10.14231/ag-2020-022
发表时间：
2020
期刊：
Algebraic Geometry
影响因子：
1.5
作者：
Colliot-Thélène, Jean-Louis;Harbater, David;Hartmann, Julia;Krashen, Daniel;Parimala, Raman;Suresh, Venapally
通讯作者：
Suresh, Venapally

Embeddings of Maximal Tori in Classical Groups, Odd Degree Descent and Hasse Principles

最大托里在经典群中的嵌入、奇次下降和哈塞原理