Open image computations for elliptic curves over number fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916413150593024 |
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| author | Zywina, David |
| author_facet | Zywina, David |
| contents | For a non-CM elliptic curve $E$ defined over a number field $K$, the Galois action on its torsion points gives rise to a Galois representation $ρ_E: Gal(\overline{K}/K)\to GL_2(\widehat{\mathbb{Z}})$ 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 $GL_2(\widehat{\mathbb{Z}})$. In an earlier work of the author, an algorithm was given, and implemented, that computed the image of $ρ_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$ in the special case $K=\mathbb{Q}$. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields $K\neq \mathbb{Q}$. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre's theorem for a typical elliptic curve over a fixed number field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Open image computations for elliptic curves over number fields Zywina, David Number Theory 11G05 (Primary) 11F80 (Secondary) For a non-CM elliptic curve $E$ defined over a number field $K$, the Galois action on its torsion points gives rise to a Galois representation $ρ_E: Gal(\overline{K}/K)\to GL_2(\widehat{\mathbb{Z}})$ 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 $GL_2(\widehat{\mathbb{Z}})$. In an earlier work of the author, an algorithm was given, and implemented, that computed the image of $ρ_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$ in the special case $K=\mathbb{Q}$. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields $K\neq \mathbb{Q}$. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre's theorem for a typical elliptic curve over a fixed number field. |
| title | Open image computations for elliptic curves over number fields |
| topic | Number Theory 11G05 (Primary) 11F80 (Secondary) |
| url | https://arxiv.org/abs/2403.16147 |