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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，在代数几何和数论的交叉数学的一个重要领域的工作。最近的进展专题的研究齐性空间提出了新的问题和proposals的研究是该提案的一部分。研究生可以接触到相关的问题，他们将通过专题研究生课程，研讨会和讲习班参与。PI和Co-PI计划组织相关领域的研讨会，并通过讲座和研讨会向数学界传播研究成果。鼓励和吸引女研究生将是PI的使命的一部分。PI和Co-PI计划调查与连接线性代数群下的齐次空间研究有关的问题，特别是关于局部和全局领域的曲线函数领域。本研究以经典群为参考，与域的Brauer群和周期指数界的研究以及二次型和u-不变量的研究密切相关。PI和Co-PI计划通过类似Brauer-Manin的障碍以及已经构建的更高的互易性障碍来研究均匀空间的Hasse原理的障碍。他们将研究全虚数域上曲线上扭曲层的模空间的Brauer-Manin阻塞，以掌握这种曲线的函数场的u-不变量，其有限性是一个很大的开放问题。他们还将研究塞尔/Totaro问题上存在的封闭点的程度d的主要齐性空间下的绝对简单的简单连通群承认零循环的程度d.这个奖项反映了NSF的法定使命，并已被认为是值得支持的通过评估使用基金会的智力价值和更广泛的影响审查标准。

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

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

DOI：
10.1007/s44007-021-00007-6
发表时间：
2022
期刊：
La Matematica
影响因子：
0
作者：
Bayer-Fluckiger, Eva;Lee, Ting-Yu;Parimala, Raman
通讯作者：
Parimala, Raman

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

Hasse principles for multinorm equations