Equidistribution of Torus Orbits

数论的一个中心问题是找到多项式方程的整数解。有些方程组有大量的整数解。本文研究了实解连续区内整数解的分布。广泛的猜想表明，在许多情况下，整数解的分布模拟了环境空间中随机生成的集合。具有足够多的对称组的方程组被称为齐次方程组，它们在这个项目中起着中心作用。这些对称性可以用来将不同的整数解彼此联系起来，并有助于将动力系统理论的方法引入该研究领域。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关于积分点在球面上的均匀分布的结果，以及马古利斯关于整数点上的无理数二次形式的值的奥本海姆猜想的解。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。齐次变量上的积分点是点稳定器周期轨道的对偶。点稳定器作用于算术齐次空间。这个项目的主要焦点是环面稳定器。PI将研究周期环面轨道在算术齐次空间上的渐近分布。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。周期环面轨道除了在某些变体上的积分点外，还推广了模曲线及其高阶变体上的Heegner点和封闭测地线的概念，如Shimura变体上特殊点的伽罗瓦轨道。周期环面轨道与某些自同构l函数密切相关的周期公式，如Waldspurger公式和Hecke的爱森斯坦级数公式。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开放问题。一个根本的困难是缺乏对大多数动力学方法至关重要的单幂函数。PI打算利用齐次动力学、自同构形式、算术几何和乘法数论等方法的结合来取得进展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.