# Effective bounds for common torsion points of elliptic curves

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

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.

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.

### Organisations

## People |
## ORCID iD |

Gavin Brown (Primary Supervisor) | |

David Hubbard (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/W523793/1 | 30/09/2021 | 29/09/2025 | |||

2595074 | Studentship | EP/W523793/1 | 03/10/2021 | 29/09/2025 | David Hubbard |