On Larsen's conjecture on the ranks of Elliptic Curves
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915029556658176 |
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| author | Hadavand, A. |
| author_facet | Hadavand, A. |
| contents | Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langleσ_1, \dots, σ_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the Mordell-Weil group $E(\overline{\mathbb{Q}}^G)$ is infinite where ${\overline{\mathbb Q}}^G$ is the $G$-fixed sub-field of $\overline{\mathbb Q}$. In this paper we prove the conjecture for the case in which $σ_i$ for each $i=1, \dots, n$ is an element of some infinite families of elements of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14097 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Larsen's conjecture on the ranks of Elliptic Curves Hadavand, A. Number Theory Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langleσ_1, \dots, σ_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the Mordell-Weil group $E(\overline{\mathbb{Q}}^G)$ is infinite where ${\overline{\mathbb Q}}^G$ is the $G$-fixed sub-field of $\overline{\mathbb Q}$. In this paper we prove the conjecture for the case in which $σ_i$ for each $i=1, \dots, n$ is an element of some infinite families of elements of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. |
| title | On Larsen's conjecture on the ranks of Elliptic Curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.14097 |