Arithmetic of algebraic groups over 2-dimensional fields

对于域上的连通线性代数群，近年来关于p进域上曲线的函数域上的齐性空间的研究取得了很大的进展。Parimala和Suresh最近证明了p-adic曲线的函数域上的每一个至少九个变量的二次型都有一个非平凡的零点。此外，由Harbater-Hartmann-Krashen开发的修补技术提供了工具来研究某些局部-全局原理，这些原理在代数群是有理的假设下，在这样的域上的齐次空间上存在有理点。本文研究了局部域和整体域上曲线的函数域上线性代数群下的齐性空间。对整体域上函数域的研究将产生巨大的影响，并可能导致这样一个事实，即全虚数域上曲线的函数域上的每一个具有足够多变量的二次型都有一个非平凡的零。本文研究了齐性空间上连通线性代数群下有理点存在的Hasse原理问题。在单连通群的背景下，主齐性空间的Rost不变量是研究这类空间的有力工具。我们提出研究局部域和整体域上曲线的函数域的Rost不变量的内射性的障碍.线性代数群下的齐性空间的研究包括研究几个有趣的代数对象，如二次型，中心单代数，Octonian和Albert代数。类场论技术丰富了数域的研究。哈塞-马斯-席林关于数域的一个经典定理是以下哈塞原理：数域中的元素是中心单代数的约化范数当且仅当它在除代数是分歧的所有真实的完备化处都是正的。二维域的约化范数的一个类似的准则是数域的这一经典结果的一个意义深远的推广。我们建议取代类场理论的纯同调性质的这些领域和几何的相关算术表面研究性质的齐次空间的一般设置的函数场的曲线在当地和全球的领域。

For a connected linear algebraic group defined over field there has been considerable progress in recent years in the study of homogeneous spaces over function fields of curves over a p-adic field. It has been proved recently by Parimala and Suresh that every quadratic form in at least nine variables over the function field of a p-adic curve has a nontrivial zero. Further, patching techniques developed by Harbater-Hartmann-Krashen have provided tools to study certain local-global principle for existence of rational points on homogeneous spaces over such fields under the assumption that the algebraic group is rational. We propose to study homogeneous spaces under linear algebraic groups over function fields of curves over local and global fields. The study over function fields over global fields would have tremendous consequences and could lead to the fact that every quadratic form in large enough number of variables over function fields of curves over totally imaginary number fields has a nontrivial zero. We propose to study questions of Hasse principle for existence of rational points on homogeneous spaces under connected linear algebraic groups over such fields. In the context of simply connected groups, the Rost invariant for principal homogeneous spaces is a powerful tool to study these spaces. We propose to study obstruction to the injectivity of the Rost invariant for function fields of curves over local and global fields.The study of homogeneous spaces under linear algebraic groups includes study of several interesting algebraic objects like quadratic forms, central simple algebras, Octonian and Albert algebras. The study over number fields is enriched by class field theory techniques. A classical theorem of Hasse-Maass-Schiling for number fields is the following Hasse principle: an element in the number field is a reduced norm from a central simple algebra if and only if it is positive at all real completions where the division algebra is ramified. A similar criterion for reduced norms for 2-dimensional fields is a far-reaching extension of this classical result for number fields. We propose to replace class field theory by purely homological properties of these fields and the geometry of the associated arithmetic surfaces to study properties of homogeneous spaces in the general setting of function fields of curves over local and global fields.