Intersecting the torsion of elliptic curves

TL;DR

This paper generalizes a conjecture about the finiteness of intersections of torsion point images on elliptic curves, using advanced methods to establish bounds and implications for elliptic curve ranks.

Contribution

It extends the Bogomolov-Fu-Tschinkel conjecture to rational functions of bounded degree, broadening the scope of previous results.

Findings

Proves a generalized conjecture on intersections of torsion point images.

Establishes lower bounds for ranks of elliptic curves over number fields.

Utilizes Nevanlinna theory combined with the Uniform Manin-Mumford conjecture.

Abstract

In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi_j\colon E_j\to \mathbb{P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi_j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. As it has been observed in the literature, the recent proof of the Uniform Manin-Mumford conjecture implies a full solution of the Bogomolov-Fu-Tschinkel conjecture. In this work we prove a generalization of the Bogomolov-Fu-Tschinkel conjecture where instead of even degree-$2$ maps one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the Uniform Manin-Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.