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。该研究旨在使用代数几何和数论中复杂的现代方法来产生某些丢番图方程的解的数量的一般界限。 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2,000年前，是所有数学中最困难的问题之一，这一事实证明，它是在45年前才建立的，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番图方程是否有解。 本研究项目中正在开发的边界仅取决于方程的阶数（即，指数的大小），将推进数学这一基本领域的知识。该项目还涉及一个初中和高中充实计划，除了支持本科和研究生教育。在这个项目中，研究人员试图使用p-adic分析和Chabauty-Coleman方法，沿着热带和非阿基米德几何的想法，给出满足一定条件的双曲曲线上有理点的数量的统一界限（在亏格方面），改进早期的结果。这些条件通常涉及对Mordell-Weil秩的约束。使用相关的方法，他还将试图证明统一的Manin-芒福德猜想，这给了一个统一的界限（再次在条款的属）的大小扭转包的双曲曲线超过代数封闭领域的特征零。理想情况下，这一结果将是无条件的;作为第一步，主要研究者将处理芒福德曲线和具有紧凑型简化的曲线。

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.