Study of homogeneous spaces under linear algebraic groups

设G是定义在场F上的连通线性代数群，X是定义在G下齐次空间上的连通线性代数群。我们的主要目的是研究X的算术性质，如在二维场上定义的r -等价类的有限性、弱逼近和Hasse原理：实场和代数闭场上曲面的数场和函数场是这类场的例子。在数字字段的情况下，由于Harder， Kneser, Borovoi, Sansuc, Colliot Th ' el 'ene等人的结果，这个主题很好理解。研究这一普遍性的主要动力来自Serre关于上同维2域上半单连通线性代数群下主齐次空间上有理点的存在性的一个猜想。由于P. Gille的结果，如果证明了素度除法代数在代数闭域上的循环性，就得到了E_8型群在代数闭域上的函数场上的猜想。素次代数的循环性是一个广泛的开放性问题。我们提出研究曲面函数场上的除法代数的结构，以期理解循环性。在寻找主齐次空间上的有理点时，人们会遇到一个较弱的问题，即寻找1次的零循环。一次零环的存在是否意味着主齐次空间上有理点的存在是一个有待解决的问题。这个问题对于数字域有一个肯定的答案，我们建议对一般域研究这个问题，特别参考类算术域。线性代数群下齐次空间的研究包含了对有趣的代数结构的研究——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。这些结构的研究渗透到数论、表示论和代数几何等数学领域。二次型——2次齐次多项式——的研究有着悠久而丰富的历史。经典的哈塞-闵可夫斯基定理将这种多项式的非平凡零的存在化约为若干同余模素数的解。我们的目标是研究这些与数域具有某些“上同调性质”的域上的代数结构，例如实数或复数表面的函数域。在这类领域中，我们建议研究诸如二次型和对合除法代数等代数结构的算术性质，在这些领域中，诸如类场理论和互易律等算术技术是不可用的。

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.