Rational points on homogeneous spaces, quadractic forms and Brauer groups

齐次空间、二次型和布劳尔群上的有理点

• 批准号: 1401319
• 负责人: Parimala Raman
• 金额: $ 24.61万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401319&HistoricalAwards=false
• 关键词: Rational; points; homogeneous; spaces; quadractic

The interplay between finding solutions to sets of equations and questions in geometry has been very fruitful in Mathematics. The PI's research fits into this way of thinking about both topics. The study of linear algebraic groups and homogeneous spaces provides a unified plank to understanding distinct interesting objects in algebra, geometry and number theory. Extending the classical study over number fields, for example the rational numbers, to function fields, which includes the class of simple functions such as polynomials, which is part of the PI's proposal,is useful from the geometric perspective. Engaging graduate students on topics related to algebraic groups and homogeneous spaces, an area ripe with questions accessible to students, via seminars and workshops, will be part of the activities of the PI during this project execution.The study of quadratic forms and their zeros over function fields over number fields and p-adic fields is an example of the objects studied during this proposal period. The PI plans to investigate questions related to the study of homogeneous spaces with special reference to quadratic forms and Brauer groups. The PI shall study the period-index questions for the Brauer group of function fields of curves over number fields with a view to bounding the u-invariant of such fields. It is an open question whether quadratic forms in sufficiently many variables over function fields of curves over totally imaginary number fields have a nontrivial zero with conditional results dependent on the Hasse principle for twisted moduli spaces over curves over number fields; the PI will investigate the obstruction to the Hasse principle for such spaces. Higher reciprocity obstructions using the Bloch-Ogus theory will be used to study the existence of rational points on homogeneous spaces over function fields. The PI also proposes to study G-trace forms, via construction of invariants, towards answering realisability questions.

在数学上找到方程组和问题的解决方案之间的相互作用在数学上非常有效。 PI的研究符合这种思考这两个主题的方式。对线性代数组和均匀空间的研究提供了一个统一的木板，以理解代数，几何和数理论中不同有趣的对象。 从几何角度来看，将经典研究扩展到数字字段（例如有理数字段）上，其中包括简单函数（例如多项式）等简单函数的类别，这是PI建议的一部分，从几何角度来看是有用的。 吸引研究生参与与代数群体和同质空间相关的主题，该区域成熟，学生可以通过研讨会和研讨会来访问该项目，这将成为PI在此项目执行过程中的活动的一部分。数字字段和P-ADIC字段上的功能字段是在此提案期间研究的对象的一个​​示例。 PI计划调查与研究统一空间有关的问题，并特别参考了二次形式和Brauer群体。 PI应研究数字字段上曲线的Brauer功能字段的周期索引问题，以期界定此类领域的U不变。这是一个开放的问题，是否在完全虚构数字字段上的曲线功能字段上有足够多的变量中的二次形式具有非平凡的零，其条件结果取决于曲线上曲线上的扭曲模量空间的hasse原理； PI将研究此类空间的Hasse原理的障碍。使用Bloch-Ogus理论将使用较高的互惠障碍来研究功能场上均匀空间上的理性点的存在。 PI还建议通过构建不变性来研究G-Trace形式，以回答可实现的问题。