Some Algorithmic Questions Related to the Mordell Conjecture

数论是研究整数性质的数学的一个基本分支。自古以来，人们一直对方程的整数解感兴趣，称为丢番图方程；毕达哥拉斯方程、佩尔方程和费马最后定理都是这类著名的问题。在现代，研究集中在解决这类问题的一般技术上。Faltings证明的一个突破性结果表明，某些类型的方程只有有限多个整数解。然而，通常还没有已知的算法来寻找这些解；丢番图方程的算法方法是一个活跃的研究领域。令人惊讶的是，这个问题与数学中的许多其他领域有关，包括代数几何、复分析、伽罗瓦理论等。为了计算丢番图问题的解，我们需要开发一些方法来执行这些其他领域的计算。这个项目包括研究其中几个困难的计算问题。这个项目的目标是研究与数论中的有限定理有关的各种计算和算法主题。例如，Faltings定理保证了某一类型的多项式方程只能有有限多个有理解。(在几何语言中，亏格至少为两个的曲线只能有有限多个有理点。)所有已知的法林斯定理的证明都是无效的：人们知道解在数量上是有限的，但证明没有提供任何方法来判断一个人是否已经找到了所有的解。最近的工作，包括PI和Venkatesh对Faltings定理的新证明，开辟了通往算法解决方案的有希望的新途径。PI将学习代数几何和数论中的算法和计算问题，动机是福林斯定理和代数和算术几何中的其他问题。PI希望研究Shafarevich猜想(确定固定数域上的阿贝尔簇，并从固定的有限组坏素数得到良好的约化)和Riemann-Hilbert对应(找到具有给定单行表示的代数微分方程式)的算法方法。反过来，Riemann-Hilbert对应应该适用于寻找曲线的分支覆盖的问题，以及研究给定基上的Hodge结构的变化的问题。这两个问题都与法林斯的理论有关。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory is a fundamental branch of mathematics that deals with properties of the integers. Since ancient times, people have been interested in integer solutions to equations, called Diophantine equations; the Pythagorean equation, the Pell equation, and Fermat's Last Theorem are well-known problems of this type. In modern times, research has focused on general techniques for this sort of problem. A breakthrough result, proven by Faltings, shows that certain types of equations have only finitely many integer solutions. However, there is no known algorithm to find these solutions in general; algorithmic approaches to Diophantine equations are an active area of research. Surprisingly, this problem relates to many other areas in math, including algebraic geometry, complex analysis, Galois theory, and more. In order to compute solutions to Diophantine problems, we need to develop methods to perform some calculations in these other fields. This project involves studying several of these difficult computational problems.The goal of this project is to investigate various computational and algorithmic topics related to finiteness theorems in number theory. For example, a theorem of Faltings guarantees a polynomial equation of a certain type can have only finitely many rational solutions. (In geometric language, a curve of genus at least two can have only finitely many rational points.) All known proofs of Faltings's theorem are ineffective: one knows that the solutions are finite in number, but the proof provides no way to tell whether one has found them all. Recent work, including a new proof of Faltings's theorem by the PI and Venkatesh, has opened up promising new avenues toward an algorithmic solution. The PI will study algorithmic and computational questions in algebraic geometry and number theory, motivated by Faltings's theorem and other problems in algebraic and arithmetic geometry. The PI hopes to study algorithmic approaches to the Shafarevich conjecture (determining abelian varieties over a fixed number field, with good reduction away from a fixed finite set of bad primes) and the Riemann-Hilbert correspondence (finding algebraic differential equations with given monodromy representations). The Riemann-Hilbert correspondence, in turn, should be applicable to the problem of finding branched covers of curves, as well as the problem of studying variations of Hodge structure over a given base. Both these problems relate to Faltings's theorem.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.