Modern Approaches for Classical Diophantine Problems

这项研究项目涉及丢番图逼近和数的几何。丢番图近似处理用有理数逼近实数，以及将给定的数分类为无理、代数或超越的问题。丢番图方程是具有整数系数的代数方程，可求其整数解。数的几何学研究用几何概念来解决数论中的问题，通常是通过求解整数中的方程。数学的基本问题之一是求一个给定的多项式方程的解。这类问题的难易程度取决于多项式的形状，更取决于要寻找的解的种类。找到作为多项式方程的解的复数是一项相对容易的任务。然而，寻找多项式方程的整数解是一个更微妙和更深入的问题，也是数论的焦点。本课题旨在拓宽和深化这一基本领域的知识，其目的是研究丢番图方程整数解的计数及其应用的一般问题。PI将通过将经典分析的技术与解析数论和算术几何的现代应用相结合来发展方法论的进步，以有效和明确地解决一些重要的和长期存在的丢番图问题。研究了丢番图方程的积分解的分布。预计该项目的结果将大大提高对某些算术对象(如椭圆曲线)的几何和分析性质的理解。

This research project concerns Diophantine approximation and the geometry of numbers. Diophantine approximation deals with approximation of real numbers by rational numbers and with questions of classification of given numbers as irrational, algebraic, or transcendental. Diophantine equations are algebraic equations with integer coefficients, for which integer solutions are sought. The geometry of numbers deals with the use of geometric notions to solve problems in number theory, usually via the solutions of equations in integers. One of the basic problems of mathematics is to find the solutions of a given polynomial equation. The level of difficulty of such problems depends on the shape of the polynomial and more so on what kind of solutions one is looking for. Finding complex numbers that are solutions to a polynomial equation is a relatively easy task. However, finding integer solutions to polynomial equations is a problem of much greater subtlety and depth, and is the focus of number theory. This project aims to broaden and deepen knowledge in this fundamental area.The purpose of this research project is to study the general problem of counting integral solutions of Diophantine equations and its applications. The PI will develop methodological advances by combining techniques from classical analysis with modern applications of analytic number theory and arithmetic geometry to address some important and long standing Diophantine problems effectively and explicitly. The research is also concerned with studying the distribution of integral solutions to Diophantine equations. Results of the project are anticipated to lead to substantially better understanding of the geometric and analytic properties of certain arithmetic objects, such as elliptic curves.