Diophantine equations and modular curves

丢番图方程和模曲线

• 批准号: 2274692
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2274692
• 关键词: Diophantine; equations; modular; curves

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.

丢番图方程是一种寻求整数解的方程。尽管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。自从怀尔斯在 20 世纪末著名地证明了费马大定理以来，一种被称为“模方法”的求解丢番图方程的策略得到了显着的发展。该项目的总体目标是通过引入抽象和计算新技术来进一步发展该方法，然后使用这些技术来研究丢番图方程的各个族，例如“广义费马”和“勒贝格-纳格尔”方程。重点关注模曲线所扮演的角色，这些复杂的几何对象通常与丢番图方程相关。特别是，将开发新的几何技术，通过利用这些曲线的对称性来研究这些曲线。

项目成果

Quadratic points on non-split Cartan modular curves

非分裂嘉当模曲线上的二次点

• DOI: 10.1142/s1793042122500178
• 发表时间: 2021
• 期刊: International Journal of Number Theory
• 影响因子: 0.7
• 作者: Michaud-Rodgers P

On elliptic curves with p -isogenies over quadratic fields

二次域上具有 p 同基因的椭圆曲线

• DOI: 10.4153/s0008414x22000244
• 发表时间: 2022
• 期刊: Canadian Journal of Mathematics
• 作者: Michaud-Jacobs P