Effective bounds for common torsion points of elliptic curves

椭圆曲线公共扭转点的有效界

• 批准号: 2595074
• 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: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2595074
• 关键词: Effective; bounds; common; torsion; points

Given an elliptic curve E over a number field K we can consider the torsion points of E with respect to the group law of the curve. Passing to the algebraic closure, we see that there is an infinite number of such torsion points. In a paper titled 'Algebraic Varieties over Small Fields' Bogomolov and Tschinkel show that given two distinct elliptic curves E and E' with choices of 2:1 covers of the projective line; the intersection of the images of their torsion points gives a finite set. In a further paper with Fu titled 'Torsion of Elliptic Curves and Unlikely Intersections' they conjecture that this intersection is not only finite but uniformly bounded for all elliptic curves. Poineau was able to settle the conjecture with the caveat that the uniform bound is not effective.The approach of showing the original boundedness result by Bogomolov et al utilises the Manin-Mumford conjecture which shows that given an integral curve in an abelian variety of genus greater than 2, the number of torsion points on the curve must be finite. The conjecture was originally proved by Raynaud and in his paper, he claims that the bounds can be made effective and calculable with some assumptions on the abelian variety in question.Using the results of Raynaud and the assumptions he requires I have been able to obtain effective bounds for the Bogomolov-Fu-Tschinkel conjecture in the case of good reduction of the curves at a fixed small unramified prime . The aim of the project is to extend the proof to the multiplicative reduction case. Then, if we are able to extend the techniques to the case of small ramification degree, combining the multiplicative result with semistability one would be able to prove an effective version of the conjecture in full generality.The analogue of the problem for algebraic tori asks to study the intersection of roots of unity in the projective line after applying a projective transformation. With results of Beukers-Smyth I have been able to prove the effective analogue of the conjecture for the torus.The analogue of an elliptic curve's 2:1 cover of the projective line for abelian surfaces is the Kummer surface. I am also working on related arithmetic properties of Kummer surfaces attached to abelian surfaces with Alexei Skorobogatov at Imperial College London.

给定数域K上的一条椭圆曲线E，我们可以考虑E关于该曲线的群律的扭点。通过代数闭包，我们看到有无限数量的这样的扭点。Bogomolov和Tschinkel在一篇题为《小域上的代数变体》的论文中证明，给定两条截然不同的椭圆曲线E和E，并选择射影直线的2：1覆盖，它们的扭点的像的交集就是一个有限集。在另一篇题为《椭圆曲线的扭转和不太可能的交集》的论文中，他们猜想这个交集对所有的椭圆曲线来说不仅是有限的，而且是一致有界的。Poineau用Bogomolov等人的原始有界性结果证明了这一猜想是无效的，该方法利用了Manin-Mumford猜想，该猜想表明，给定一个大于2的交换变种中的一条积分曲线，该曲线上的扭点的个数一定是有限的.这个猜想最初是由Raynaud证明的，在他的论文中，他声称这个界是有效的和可计算的，利用Raynaud的结果和他所要求的假设，我得到了Bogomolov-Fu-Tschinkel猜想的有效界，在曲线在固定的小未分支素数处良好约化的情况下。该项目的目的是将证明推广到乘法约简的情况。然后，如果我们能够将技术扩展到小分支度的情况，将乘法结果与半稳定性相结合，就能够证明该猜想的一个完整的有效版本。代数环面问题的类比要求在应用射影变换后研究投影线上单位根的交集。利用Beukers-Smyth的结果，我证明了环面猜想的有效类比：椭圆曲线对阿贝尔曲面射影直线的2：1覆盖的类比是Kummer曲面。我还与伦敦帝国理工学院的Alexei Skorobogatov一起研究附着在阿贝尔曲面上的Kummer曲面的相关算术性质。