Development and Applications of Non-Archimedean Analytic Geometry and Tropical Geometry

这个项目涉及数论的研究，以研究整数中方程的某些性质。这项研究的目的是利用代数几何和数论中复杂的现代方法，给出某些丢番图方程解的个数的一般界限。丢番图方程的研究涉及寻找多项式等式的整数解，例如当两个五次幂的和再次是五次幂时。对这类方程的研究可以追溯到近2000年前，是所有数学中最困难的问题之一，证明它是在45年前才建立起来的，不可能设计出一个具有有限运算次数的一般过程来确定一个丢番图方程是否有解。这个研究项目中正在开发的界限只取决于方程的阶数(即指数的大小)，将促进这一基本数学领域的知识。除了对本科生和研究生教育的支持外，该项目还包括一个初中和高中丰富项目。在这个项目中，研究人员试图使用p-进分析和Chabauty-Coleman方法，结合热带和非阿基米德几何的思想，给出满足某些条件的双曲曲线上有理点数目的统一界(关于亏格)，改进了以前的结果。这些条件通常涉及对Mordell-Weil等级的限制。使用相关方法，他还将尝试证明一致Manin-Mumford猜想，该猜想给出特征为零的代数闭域上的双曲曲线上的扭包的大小的一致界(同样是根据亏格)。理想情况下，这一结果是无条件的；作为第一步，首席研究员将用紧凑型简化处理Mumford曲线和曲线。

This project concerns research in number theory to study certain properties of equations in the whole numbers. The research aims to use sophisticated modern methods in algebraic geometry and number theory to produce general bounds on the number of solutions to certain Diophantine equations. The study of Diophantine equations involves finding whole number solutions to polynomial equalities, such as when the sum of two fifth powers is again a fifth power. The study of such equations dates back almost 2,000 years and is among the most difficult problems in all of mathematics, as evidenced by the fact that it was established only 45 years ago that it is not possible to devise a general process with a finite number of operations that can decide whether a Diophantine equation has a solution. The bounds under development in this research project, which depend only on the degree of the equation (i.e., the size of the exponents), will advance knowledge in this fundamental area of mathematics. The project also involves a middle- and high-school enrichment program, in addition to support for undergraduate and graduate education. In this project, the investigator seeks to use p-adic analysis and the Chabauty-Coleman method, along with ideas from tropical and non-Archimedean geometry, to give uniform bounds (in terms of the genus) on the number of rational points on hyperbolic curves satisfying certain conditions, refining earlier results. These conditions generally involve a constraint on the Mordell-Weil rank. Using related methods, he will also attempt to prove the uniform Manin--Mumford conjecture, which gives a uniform bound (again in terms of the genus) on the size of a torsion packet on a hyperbolic curve over an algebraically closed field of characteristic zero. Ideally this result would be unconditional; as a first step, the principal investigator will treat Mumford curves and curves with compact-type reduction.