Rational points on homogeneous spaces, quadractic forms and Brauer groups

在数学中，寻找方程组的解和几何问题之间的相互作用是非常有成效的。PI的研究符合对这两个主题的这种思考方式。对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。将经典的关于数域的研究，例如有理数，扩展到函数域，其中包括PI建议的一部分的简单函数，如多项式，从几何的角度来看，这是有用的。让研究生参与与代数群和齐次空间有关的主题，这一领域有很多问题，学生可以通过研讨会和讲习班获得，这将是该项目执行期间PI活动的一部分。数域和p-ady域上函数域上的二次型及其零点的研究是本建议期间研究的对象的一个例子。PI计划调查与齐次空间研究有关的问题，特别是二次型和Brauer群。PI应研究数域上曲线的Brauer函数域的周期指数问题，以期给这些域的u-不变量定界。全虚数域上曲线的函数域上的足够多变量的二次型是否有一个非平凡的零，其条件结果取决于数域上曲线上扭曲的模空间的Hasse原理，这是一个悬而未决的问题；PI将调查这类空间对Hasse原理的阻碍。使用Bloch-Ogus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。

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.