Intersecting the torsion of elliptic curves
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916487410745344 |
|---|---|
| author | Garcia-Fritz, Natalia Pasten, Hector |
| author_facet | Garcia-Fritz, Natalia Pasten, Hector |
| contents | In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $π_j\colon E_j\to \mathbb{P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $π_j$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05708 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intersecting the torsion of elliptic curves Garcia-Fritz, Natalia Pasten, Hector Number Theory Primary: 11G05, Secondary: 14G25, 14H52 In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $π_j\colon E_j\to \mathbb{P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $π_j$ 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. |
| title | Intersecting the torsion of elliptic curves |
| topic | Number Theory Primary: 11G05, Secondary: 14G25, 14H52 |
| url | https://arxiv.org/abs/2308.05708 |