Refining the Chabauty--Coleman method for modular curves

Robin Visser的博士项目位于数论领域，开发了将有理解的数量限制在给定方程的数量的技术。特别是，他将专注于寻找定义在小次数数域上的模曲线(与模形式的算术相关的某些特定代数曲线)上的点数的上界。最近，Visser推荐的主管Siksek对这个问题进行了深入的研究，其动机是将其应用于全实二次和三次域上椭圆曲线的模性问题。Siksek利用Chabauty和Coleman提出的一种经典方法求出了小d的曲线的d次对称幂上的所有有理点，这相当于同时求出所有d次数域上原曲线上的所有点。目前，Chabauty-Coleman给出的解集的界远不是最优的，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。Visser项目的目标是通过利用所产生的方程组严重超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。这应该允许获得更精确的界，极大地加强了该方法在椭圆曲线的模性和其他经典问题中的潜在应用。本项目以“数学科学”为主题，探讨EPSRC的研究领域“数论”。

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.