Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910845600006144 |
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| author | Choi, Seokhyun Im, Bo-Hae |
| author_facet | Choi, Seokhyun Im, Bo-Hae |
| contents | We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the rank of $E$ over the fixed subfield $\overline{\mathbb{Q}}^G$ of $\overline{\mathbb{Q}}$ under $G$ is infinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$ Choi, Seokhyun Im, Bo-Hae Number Theory 11G05 We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the rank of $E$ over the fixed subfield $\overline{\mathbb{Q}}^G$ of $\overline{\mathbb{Q}}$ under $G$ is infinite. |
| title | Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$ |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2502.18761 |