Effective bounds for common torsion points of elliptic curves

给定一条椭圆曲线E在一个数域K上，我们可以根据曲线的群律来考虑E的扭转点。通过代数闭包，我们看到有无限个这样的扭转点。Bogomolov和Tschinkel在一篇名为“小域上的代数变化”的论文中表明，给定两条不同的椭圆曲线E和E'，其投影线的覆盖选择为2:1；它们的扭转点像的交点给出了一个有限集合。在另一篇题为“椭圆曲线的扭转和不可能相交”的论文中，他们推测这种相交不仅是有限的，而且对所有椭圆曲线都是一致有界的。波涅奥能够解决这个猜想，但警告说，统一边界是无效的。Bogomolov等人证明原始有界性结果的方法利用了Manin-Mumford猜想，该猜想表明，给定一个大于2的阿贝尔变属的积分曲线，曲线上的扭转点的个数必须是有限的。这个猜想最初是由雷诺证明的，在他的论文中，他声称可以通过对所讨论的阿贝尔变化的一些假设使边界有效和可计算。利用雷诺的结果和他所要求的假设，我已经能够得到Bogomolov-Fu-Tschinkel猜想在曲线在一个固定的小的未分叉素数处良好约简的情况下的有效界。该项目的目的是将证明推广到乘法约简的情况。然后，如果我们能够将该技术推广到小分支度的情况下，将相乘结果与半稳定性相结合，就能够证明该猜想的有效版本具有完全的普遍性。与代数环面问题类似，要求研究在进行射影变换后，单位根在射影线上的交点。利用beukes - smyth的结果，我已经能够证明环面猜想的有效类比。椭圆曲线对阿贝尔曲面的投影线的2:1覆盖的类比是Kummer曲面。我也在伦敦帝国理工学院与Alexei Skorobogatov一起研究与阿贝尔曲面相关的Kummer曲面的算术性质。

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.