Study of homogeneous spaces under linear algebraic groups

设G是定义在域F上的连通线性代数群，X是G下的齐次空间，我们的主要目的是研究二维域上定义的R-等价类的类X的有限性、弱逼近和Hasse原理的算术性质：-数域和实域和代数闭域上曲面的函数域。在数字字段的情况下，这个主题是很好理解的，这要归功于Harder、Knerer、Borovoi、Sansuc、Colliot Th‘El’ene和其他人的结果。这一一般性研究的主要推动力来自于Serre关于上同调维域上的半单单连通线性代数群在主齐性空间上存在有理点的猜想。由于P.Gille的一个结果，如果证明了这样的函数域上的素数次除代数的循环性，就可以得到代数闭域上曲面的函数域上的E_8型群的猜想。素数次代数的循环性在一般的基域中是一个广泛公开的问题。为了了解曲面的循环性，我们建议研究曲面函数域上的除法代数的结构。在寻找主齐性空间上的有理点时，人们遇到了一个较弱的问题，即寻找一次零循环。一次零圈的存在是否意味着主齐次空间上有理点的存在是一个悬而未决的问题。这个问题对于数域有一个肯定的答案，我们建议在一般域中研究这个问题，特别参考类算术域。线性代数群下的齐性空间的研究包括研究有趣的代数结构--与经典群相关的二次型和对合除法代数，以及与例外群相关的Cayley和Albert代数。对这些结构的研究渗透到数学的几个领域，如数论、表示论和代数几何。二次型--2次齐次多项式--的研究有着悠久而丰富的历史。Hasse-Minkowski的经典定理将这种多项式的非平凡零点的存在性归结为某些模素数同余的解，我们的目的是研究与数域相同的域上的这些代数结构，例如实数或复数上的曲面的函数域。我们建议在这类域上研究二次型和对合除代数等代数结构的算术性质，其中类场理论和互易定律等算术技巧是不可用的。

Let G be a connected linear algebraic group defined over a field F and X a homogeneous space under G. Our main goal is to study arithmetic properties of X like finiteness of R-equivalence classes, weak approximation and Hasse principle for X defined over a 2-dimensional field:- number fields and function fields of surfaces over real and algebraically closed fields are examples of this class of fields. In the case of number fields, the subject is well-understood, thanks to the results of Harder, Kneser, Borovoi, Sansuc, Colliot Th\`el\'ene, and others. The main impetus for the study in this generality came from a conjecture of Serre concerning the existence of rational points on principal homogeneous spaces under semisimple simply connected linear algebraic groups over fields of cohomological dimension 2. Thanks to a result of P. Gille, the conjecture for groups of type E_8 over function fields of surfaces over algebraically closed fields would follow if one proves cyclicity of prime degree division algebras over such function fields. Cyclicity of prime degree algebras is a wide open question for a general ground field. We propose to study the structure of division algebras over function fields of surfaces with a view to understanding cyclicity. While looking for rational points on principal homogeneous spaces, one comes across the weaker question of finding zero cycles of degree one. It is an open question whether the existence of zero cycles of degree one implies existence of rational points on principal homogeneous spaces.This question has an affirmative answer for number fields and we propose to investigate this question for a general field, with special reference to arithmetic like fields.The study of homogeneous spaces under linear algebraic groups encompasses the study of interesting algebraic structures--quadratic forms and involutorial division algebras which are associated to classical groups and Cayley and Albert algebras which are associated to exceptional groups. The study of these structures permeates through several areas of mathematics like Number Theory, Representation Theory and Algebraic Geometry. The study of quadratic forms--homogeneous polynomials of degree 2 --has a long and rich history. The classical theorem of Hasse-Minkowski reduces the existence of nontrivial zeros of such a polynomial to solutions of certain congruences modulo primes.Our objective is to study these algebraic structures over fields which share certain `cohomological properties' in common with number fields, for example, the function fields of surfaces over real or complex numbers. We propose to investigate arithmetic properties of algebraic structures like quadratic forms and involutorial division algebras over this class of fields where arithmetic techniques like class field theory and reciprocity laws are not available.