Equidistribution of Torus Orbits

数论中的一个中心问题是求多项式方程的整数解。有些方程组有大量的整数解。本文研究的是整数解在真实的解连续体中的分布。一个大范围的分布暗示着在许多情况下，整数解的分布模拟了周围空间中随机生成的集合。具有足够大的对称群的方程组被称为齐次方程组，它们在这个项目中起着核心作用。这些对称性可以用来将不同的整数解相互联系起来，并有助于将动力系统理论的方法引入这一研究领域。动力学方法在解决数论中长期存在的问题方面取得了巨大的成功。早期的突破包括林尼克的结果均匀分布的积分点的领域和马古利斯的解决方案的奥本海姆猜想的价值所达到的非理性二次形式在整数点。PI将整合动力学和数论的方法来研究这些技术本身无法解决的问题。齐次簇上的整点是点稳定子的周期轨道的对偶。点稳定器作用于算术齐性空间。该项目的主要重点是环面稳定器的情况。PI将研究算术齐性空间上周期环面轨道的渐近分布。周期环面轨道将数论和齐次动力学中的几个对象统一到一个框架中。除了某些簇上的整点外，周期环面轨道还推广了模曲线上的Heegner点和闭测地线的概念及其高阶变体，例如Shimura簇上特殊点的伽罗瓦轨道。周期环面轨道与一些自守L-函数通过周期公式如Waldspurger公式和Hecke的Eisenstein级数公式密切相关。齐次空间上高阶环面作用的所有轨道的完整描述是一个长期存在的公开问题。一个基本的困难是缺乏幂等元，这是至关重要的大多数动力学方法。PI旨在通过齐次动力学、自守形式、算术几何和乘法数论等方法的组合来取得进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

A central problem in Number Theory is finding integer solutions to polynomial equations. Some systems of equations have plenty of integer solutions. This research concerns the distribution of the integer solutions inside the continuum of real solutions. A wide-range of conjectures imply that in many cases the distribution of integer solutions mimics a randomly generated set in the ambient space. Systems of equation that have a large enough group of symmetries are called homogeneous and they play a central role in this project. These symmetries can be used to relate different integer solutions to each other and facilitate the introduction of methods from the theory of dynamical systems to this research area. The dynamical methods have been immensely successful in solving long standing problems in number theory. Early breakthroughs include Linnik's results about the equidistribution of integral points on the sphere and Margulis's solution of the Oppenheim conjecture regarding the values attained by an irrational quadratic form at integer points. The PI will integrate methods from dynamics and number theory to study questions which could not be solved by either of these techniques by itself. The integral points on a homogeneous variety are dual to periodic orbits of the point stabilizer. The point stabilizer acts on an arithmetic homogeneous space. The main focus of this project is the case of torus stabilizers. The PI will study the asymptotic distribution of periodic torus orbits on arithmetic homogeneous spaces. Periodic torus orbits unify several objects in number theory and homogeneous dynamics into a single framework. In addition to integral points on some varieties, periodic torus orbits also generalize the notion of Heegner points and closed geodesics on the modular curve and their higher rank variants, e.g. Galois orbits of special points on Shimura varieties. Periodic torus orbits are closely related to some automorphic L-functions by period formulae like the Waldspurger formula and Hecke's formula for Eisenstein series. A complete description of all the orbits for a higher rank torus action on a homogeneous space is a long standing open problem. A fundamental difficulty is the lack of unipotents which are crucial to most of the dynamical methods. The PI intends to make progress using a combination of methods from homogeneous dynamics, automorphic forms, arithmetic geometry and multiplicative number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.