Arithmetic of algebraic groups over 2-dimensional fields

对于场上定义的连通线性代数群，近年来对p进域上曲线函数场上齐次空间的研究取得了相当大的进展。Parimala和Suresh最近证明了p进曲线函数域上至少有9个变量的每一个二次型都有一个非平凡零。更进一步，Harbater-Hartmann-Krashen发展的补片技术提供了工具来研究在假设代数群是有理的情况下齐次空间上有理点存在的某些局部-全局原理。研究了局部和全局域上曲线函数场上线性代数群下的齐次空间。对全局场的函数场的研究将会产生巨大的影响，并可能导致这样一个事实，即在函数场或曲线的函数场上，在全虚数场上，每个变量足够多的二次型都有一个非平凡的零。研究了齐次空间上连通线性代数群上有理点存在的Hasse原理问题。在单连通群的背景下，主齐次空间的Rost不变量是研究这些空间的有力工具。研究曲线函数场的Rost不变量在局部场和全局场上的注入性的障碍。线性代数群下齐次空间的研究包括对几个有趣的代数对象的研究，如二次型、中心简单代数、Octonian代数和Albert代数。类场理论技术丰富了对数场的研究。关于数域的一个经典的Hasse- mass - schiling定理是以下的Hasse原理：数域中的一个元素是一个中心简单代数的约简范数当且仅当它在所有实补上都是正的，其中除法代数是分枝的。二维域的简化范数判据是这一经典结果对数域的深远推广。我们提出用这些场的纯同调性质和相关算术曲面的几何来代替类场论，以研究局部场和全局场上曲线函数场的一般集合中齐次空间的性质。

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.