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

这个项目涉及数论的研究，以研究整数方程的某些性质。本研究的目的是利用代数几何和数论中先进的现代方法，对某些丢芬图方程的解的个数给出一般的界限。丢番图方程的研究包括找到多项式等式的整数解，比如当两个五次方的和还是五次方的时候。对这类方程的研究可以追溯到近2000年前，它是所有数学中最困难的问题之一，这一点从45年前才确立的事实中可以看出，不可能设计出一个具有有限数量运算的一般过程来决定丢芬图方程是否有解。本研究项目中正在开发的边界仅取决于方程的程度（即指数的大小），将推进这一基础数学领域的知识。除了支持本科和研究生教育外，该项目还包括一个初中和高中充实计划。在这个项目中，研究者试图使用p进分析和Chabauty-Coleman方法，以及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量的统一界限（就属而言），改进先前的结果。这些条件通常涉及对莫德尔-韦尔秩的约束。利用相关的方法，他还将尝试证明一致的Manin—Mumford猜想，该猜想给出了特征为零的代数闭域上双曲曲线上扭包大小的一致界（再次以属的形式）。理想情况下，这个结果是无条件的；作为第一步，首席研究员将处理芒福德曲线和曲线与紧凑型减少。

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.