Diophantine equations and modular curves

丢番图方程是一个寻求整数解的方程。虽然丢番图方程的研究可以追溯到古代，但它们在现代数论中仍然发挥着核心作用。自20世纪末Wiles著名的费马大定理证明以来，一种被称为“模法”的求解丢番图方程的策略得到了长足的发展。这个项目的主要目的是通过引入抽象和计算的新技术来进一步发展这种方法，然后使用这些技术来研究各种族的丢番图方程，例如“广义费马”和“勒贝格-纳格尔”方程。一个关键的焦点将是模曲线所起的作用，模曲线是经常与丢番图方程联系在一起的复杂几何对象。特别是，将开发新的几何技术来研究这些曲线，利用它们的对称性。

A Diophantine equation is an equation for which one seeks whole number solutions. Although the study of Diophantine equations dates back to antiquity, they still play a central role in modern number theory. Since Wiles' celebrated proof of Fermat's Last Theorem at the end of the 20th century, a strategy for solving Diophantine equations known as the "modular method" has seen significant development. The broad aim of this project is to further develop this method by introducing new techniques, both abstract and computational, and to then use these to study various families of Diophantine equations, such as the "generalised Fermat" and "Lebesgue-Nagell" equations. A key focus will be on the role played by modular curves, complex geometric objects often associated with Diophantine equations. In particular, new geometric techniques will be developed to study these curves by exploiting their symmetries.