Distribution of rational points and automorphic forms

有理点的分布和自守形式

• 批准号: 0701753
• 负责人: Ramin Takloo-Bighash
• 金额: $ 11.99万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701753&HistoricalAwards=false
• 关键词: Distribution; rational; points; automorphic; forms

The PI proposes four research projects, the first three of which are related to the study of rational points of bounded height on certain classes of algebraic varieties in the context of Manin's conjecture for Fano varieties. In continuation of his joint work with Shalika and Tschinkel, the PI plans to study the distribution of rational points of bounded height on spherical varieties, and also compactifications of certain non-reductive algebraic groups. The fourth project, joint with a student, will be concerned with the distribution of orders in number fields. The proposed research, especially in the first three projects, is part of a broader program of bringing recent advances in the theory of automorphic forms to bear on the questions of arithmetic interest. The research theme applies methods from the theory of automorphic forms and ideas from arithmetic geometry to the study of rational points on homogeneous varieties. The PI believes that this research will advance knowledge and understanding of the arithmetic of higher dimensional varieties.Diophantine equations have been of interest since the antiquities. Often times a fundamental question of interest is whether a given Diophantine equation has solutions, or, if does, how many. In this research we propose to study certain classes of Diophantine equations with large groups of symmetries. The type of Diophantine equations we consider in this research a priori have an infinite number of solutions, so one desires a better understanding of the distribution of solutions. We propose to give approximate formulae for the number of solutions with bounded "height" - where here, "height" is a convenient measure of arithmetic complexity. We have also included an educational program in the proposal. Our previous work on the subject has produced a large number of concrete research problems accessible to undergraduate and graduate students. It has also led to the writing of a textbook joint with Steven J. Miller. We plan to develop these pedagogical programs further in the form of writing an advanced graduate textbook..

PI提出了四个研究项目，其中前三个与在马宁法诺簇猜想的背景下研究某些代数簇上的有界高度有理点有关。在继续与 Shalika 和 Tschinkel 合作的过程中，PI 计划研究球簇上有界高度有理点的分布，以及某些非还原代数群的紧化。第四个项目与一名学生合作，将涉及数字字段中的订单分配。拟议的研究，特别是前三个项目，是一个更广泛计划的一部分，该计划将自守形式理论的最新进展应用于算术兴趣问题。该研究主题运用自守形式理论的方法和算术几何的思想来研究同质簇上的有理点。 PI 相信这项研究将增进对高维簇算术的认识和理解。丢番图方程自古以来就引起了人们的兴趣。很多时候，一个令人感兴趣的基本问题是给定的丢番图方程是否有解，或者如果有，有多少个。在这项研究中，我们建议研究具有大对称群的某些类别的丢番图方程。 我们在这项研究中先验考虑的丢番图方程类型有无限多个解，因此人们希望更好地理解解的分布。 我们建议给出具有有限“高度”的解的数量的近似公式 - 其中，“高度”是算术复杂性的便捷度量。我们还在提案中纳入了教育计划。我们之前在该主题上的工作已经产生了大量本科生和研究生可以理解的具体研究问题。它还导致与史蒂文·J·米勒联合撰写了一本教科书。我们计划以编写高级研究生教科书的形式进一步开发这些教学计划。