Explicit open images for elliptic curves over $\mathbb{Q}$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929285378342912 |
|---|---|
| author | Zywina, David |
| author_facet | Zywina, David |
| contents | For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $ρ_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $ρ_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $ρ_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ for non-CM elliptic curves $E/\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14959 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Explicit open images for elliptic curves over $\mathbb{Q}$ Zywina, David Number Theory 11G05 (Primary) 11F80 (Secondary) For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $ρ_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $ρ_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $ρ_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ for non-CM elliptic curves $E/\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series. |
| title | Explicit open images for elliptic curves over $\mathbb{Q}$ |
| topic | Number Theory 11G05 (Primary) 11F80 (Secondary) |
| url | https://arxiv.org/abs/2206.14959 |