Distribution of rational points and automorphic forms

PI提出了四个研究项目，其中前三个是在Manin的Fano猜想的背景下，研究一类代数变量上有界高度的有理点。为了继续他与Shalika和Tschinkel的合作，PI计划研究球面上有界高度的有理点的分布，以及某些非约代数群的紧化。第四个项目将与一名学生合作，研究数字字段中订单的分布。提出的研究，特别是前三个项目，是一个更广泛的计划的一部分，该计划将自同构形式理论的最新进展应用于算术兴趣问题。本课题将自同构形式理论的方法和等差几何的思想应用于齐次变异上有理点的研究。PI认为，这项研究将促进对高维数列算法的认识和理解。丢番图方程自古以来就备受关注。通常我们感兴趣的一个基本问题是给定的丢番图方程是否有解，或者如果有，有多少个解。在本研究中，我们提出研究一类具有大群对称性的丢番图方程。我们在本研究中先验地考虑的丢芬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界“高度”的解的数目的近似公式——在这里，“高度”是算术复杂度的方便度量。我们还在提案中列入了一项教育计划。我们以前在这个主题上的工作已经产生了大量的具体研究问题，可供本科生和研究生使用。它还导致了与史蒂文·j·米勒（Steven J. Miller）合作编写教科书。我们计划以编写高级研究生教材的形式进一步发展这些教学计划。

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..