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.

求解方程组和几何问题之间的相互作用在数学中是非常富有成效的。PI的研究符合对这两个主题的思考方式。线性代数群和齐次空间的研究为理解代数、几何和数论中不同的有趣对象提供了一个统一的平台。 将经典的研究扩展到数域，例如有理数，函数域，包括简单函数类，例如多项式，这是PI提案的一部分，从几何角度来看是有用的。 在本项目执行期间，PI将通过研讨会和讲习班，让研究生参与与代数群和齐次空间相关的主题，这是一个学生可以通过研讨会和讲习班获得的问题成熟的领域。在数域和p-adic域上的函数域上的二次型及其零点的研究是本项目期间研究对象的一个例子。PI计划研究与齐次空间研究有关的问题，特别是二次型和布劳尔群。PI将研究数域上曲线的函数域的Brauer群的周期指数问题，以期界定这些域的u-不变量。这是一个悬而未决的问题是否有足够多的变量二次型函数域的曲线在全虚数域有一个非平凡的零与条件的结果依赖于哈塞原则扭曲模空间的曲线在数域; PI将调查的障碍哈塞原则为这样的空间。使用布洛赫-奥格斯理论的高互易障碍将被用来研究函数域上齐次空间上有理点的存在性。PI还建议通过构造不变量来研究G-迹形式，以回答可实现性问题。