Rational points on homogeneous spaces, quadractic forms and Brauer groups

求解一组方程和几何问题之间的相互作用在数学上取得了丰硕的成果。PI的研究符合这种思考这两个主题的方式。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。从几何的角度来看，将经典的对有理数等数域的研究扩展到函数域，其中包括多项式等简单函数，这是PI提案的一部分，是有用的。通过研讨会和工作坊，让研究生参与与代数群和同质空间相关的主题，这是一个向学生提出问题的领域，这将是项目执行期间PI活动的一部分。二次型及其在函数域、数域和p进域上的零点的研究是本建议期间研究对象的一个例子。PI计划研究与齐次空间研究相关的问题，特别涉及二次型和Brauer群。PI将研究数域上曲线函数域的Brauer群的周期-指标问题，以限定这些域的u-不变量。在全虚数域上曲线的函数域上具有足够多变量的二次型是否具有非平凡零，其条件结果依赖于数域上曲线上扭模空间的Hasse原理，这是一个有待解决的问题；PI将调查对此类空间的Hasse原理的阻碍。利用Bloch-Ogus理论研究函数场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轨迹形式，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.