Study of homogeneous spaces under linear algebraic groups

线性代数群下齐次空间的研究

• 批准号: 0653382
• 负责人: Parimala Raman
• 金额: $ 21万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653382&HistoricalAwards=false
• 关键词: Study; homogeneous; spaces; under; linear

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.

令G为在g下定义的连接的线性代数组，在G下定义了均匀的空间。我们的主要目标是研究x的算术属性，例如r等于类别的有限性，弱近似值和x定义的x的hasse原理。 - 维字段： - 在真实和代数封闭的字段上表面的数字字段和功能字段是此类字段的示例。在数字字段的情况下，这要归功于Hard，Kneser，Borovoi，Borovoi，Sansuc，Colliot Th \'El \'Ene等的结果。该一般性研究的主要动力来自于serre的猜想，即在半岛上的主要同质空间上存在合理点，而半简单的同类阶段相互连接的线性代数群体在同一个学维度2的领域上。对于E_8类型的组，如果在此类函数字段上证明了Prime Legrive Division代数的循环性，则会在代数闭合字段上进行表面的功能字段。元素代数的循环性是通用地面的广泛开放问题。我们建议在表面的功能场上研究分裂代数的结构，以理解循环性。在寻找主要均质空间上的理性点时，人们遇到了一个较弱的问题，即发现零周期的第一学位。这是一个悬而未决的问题，一个学位的零周期是否意味着在主要同质空间上存在理性点。这个问题对数字字段有肯定的答案，我们建议对这个问题进行调查，以特别提及算术。像字段一样。对线性代数组下的均质空间的研究涵盖了有趣的代数结构的研究 - 与经典群体相关的二次形式和涉及分区代数，以及与特殊群体相关的Cayley和Albert代数。这些结构的研究通过数学理论，表示理论和代数几何形状等数学领域渗透。二次形式的研究 - 2度的均匀多项式 - 悠久而丰富的历史。 Hasse-Minkowski的经典定理减少了这种多项式的非平凡零对某些一致性modulo Primes的解决方案的存在。 ，在实际或复数上表面的功能字段。我们建议在此类的领域上研究代数结构（例如二次形式和涉及分区代数）的算术特性，在这些领域中，算术技术等算术技术和互惠定律等算术技术尚无。