Coincidences of Division Fields of an elliptic curve defined over a number field
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916786196185088 |
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| author | Yvon, Zoé |
| author_facet | Yvon, Zoé |
| contents | For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the surjectivity of this representation are either local (i.e. at a prime), or due to nonsurjectivity on the product of local Galois images. In this article, we study an extreme case: the coincidence i.e. the equality of $n$-division fields, generated by the $n$-torsion points, attached to different positive integers $n$. We give necessary conditions for coincidences, dealing separately with vertical coincidences, at a given prime, and horizontal coincidences, across multiple primes, in particular when the Galois group on the $n$-torsion contains the special linear group. We also give a non-trivial construction for coincidences not occurring over $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14370 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coincidences of Division Fields of an elliptic curve defined over a number field Yvon, Zoé Number Theory Group Theory Primary 11G05, 11F80, Secondary 11R32 For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the surjectivity of this representation are either local (i.e. at a prime), or due to nonsurjectivity on the product of local Galois images. In this article, we study an extreme case: the coincidence i.e. the equality of $n$-division fields, generated by the $n$-torsion points, attached to different positive integers $n$. We give necessary conditions for coincidences, dealing separately with vertical coincidences, at a given prime, and horizontal coincidences, across multiple primes, in particular when the Galois group on the $n$-torsion contains the special linear group. We also give a non-trivial construction for coincidences not occurring over $\mathbb{Q}$. |
| title | Coincidences of Division Fields of an elliptic curve defined over a number field |
| topic | Number Theory Group Theory Primary 11G05, 11F80, Secondary 11R32 |
| url | https://arxiv.org/abs/2407.14370 |