Refining the Chabauty--Coleman method for modular curves

罗宾·维瑟的博士项目是数论领域，开发技术来限制给定方程的有理解的数量。特别是，他将着重于寻找在小次数域上定义的模曲线（与模形式的算术有关的某些特定代数曲线）上点的数目的上界。这个问题最近被Visser提议的导师Siksek深入研究，其动机是将其应用于全实数二次和三次场上椭圆曲线的模块化问题。Siksek使用了Chabauty和Coleman的经典技术，在小d的曲线的d次对称幂上找到所有有理点，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—Coleman给出的解集上的界远不是最优的，这给将该方法应用于模块化理论中的具体问题带来了困难。Visser项目的目标是通过利用出现的方程系统严重过度确定的事实来改进模曲线的Chabauty- Coleman方法，这一特性在以前的工作中没有被系统地利用。这将允许获得更精确的边界，极大地加强了该方法在椭圆曲线的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。

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.