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.

罗宾·维瑟的博士项目是在数论领域，开发技术来限制一个给定方程的有理解的数量。特别是，他将专注于寻找上限的一些点上的模块化曲线（某些特定的代数曲线有关的算术的模块化形式），这是定义了一些领域的小程度。这个问题最近被深入研究的Siksek，维瑟的建议主管，动机应用到模块化问题的椭圆曲线在全实二次和三次领域。Siksek使用了Chabauty和科尔曼的经典技术来找到曲线的d次对称幂上的所有有理点，这相当于同时在所有d次数域上找到原始曲线上的所有点。目前，Chabauty-科尔曼给出的解集上的界远不是最优的，这给将这种方法应用于模块化理论中的具体问题带来了困难.维瑟的项目的目标是完善的Chabauty-科尔曼方法的模块化曲线利用的事实，即系统的方程，产生的是严重超定，属性尚未系统地利用在以前的工作。这应该允许更精确的界限，以获得，大大加强了该方法的模块化的椭圆曲线和其他经典问题的潜在应用。该项目涉及EPSRC研究领域“数论”，在“数学科学”的主题。