Refining the Chabauty--Coleman method for modular curves

改进模曲线的 Chabauty--Coleman 方法

• 批准号: 2441146
• 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: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2441146
• 关键词: Refining; Chabauty; Coleman; method; modular

Robin Visser's PhD project lies in the area of number theory, developing techniques to bound the number of rational solutions to a given equation. In particular, he will focus on finding upper bounds for the number of points on modular curves (certain specific algebraic curves related to the arithmetic of modular forms) which are defined over number fields of small degree. This problem has recently been intensively studied by Siksek, Visser's proposed supervisor, motivated by applications to modularity problems for elliptic curves over totally-real quadratic and cubic fields. Siksek used a classical technique due to Chabauty and Coleman to find all rational points on the d-th symmetric power of the curve for small d, which is equivalent to finding all points on the original curve over all degree d number fields simultaneously. At present, the bounds onthe set of solutions given by Chabauty--Coleman are far from optimal, which poses difficulties in applying this method to concrete problems arising in modularity theory. The goal of Visser's project is to refine the Chabauty-- Coleman method for modular curves by making use of the fact that the system of equations that arises is heavily over-determined, a property which has not been systematically exploited in previous work. This should allow much more precise bounds to be obtained, greatly strengthening the potential applications of the method to modularity of elliptic curves and other classical problems. This project addresses the EPSRC research area "Number theory", within the"Mathematical Sciences" theme.

Robin Visser 的博士项目属于数论领域，开发将给定方程的有理解的数量限制起来的技术。特别是，他将专注于寻找模曲线（与模形式算术相关的某些特定代数曲线）上点数的上限，这些曲线是在小阶数域上定义的。 Visser 提议的导师 Siksek 最近对这个问题进行了深入研究，其动机是应用于全实二次和三次域上的椭圆曲线的模块化问题。 Siksek 使用 Chabauty 和 Coleman 的经典技术来查找小 d 曲线的 d 次对称幂上的所有有理点，这相当于同时查找原始曲线上所有 d 次数域上的所有点。目前，Chabauty-Coleman 给出的解集的界限还远非最优，这给将该方法应用于模块化理论中出现的具体问题带来了困难。 Visser 项目的目标是通过利用所出现的方程组严重超定这一事实（这一特性在之前的工作中尚未系统地利用）来改进模曲线的 Chabauty-Coleman 方法。这应该可以获得更精确的界限，大大加强该方法在椭圆曲线模块化和其他经典问题上的潜在应用。该项目涉及 EPSRC 研究领域“数论”，属于“数学科学”主题。