Equidistribution of Torus Orbits

Northwestern University, Evanston IL

Investigators

Abstract

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.