Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

Seokhyun Choi; Bo-Hae Im

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arXiv · 2502.18761

Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

Seokhyun Choi·Bo-Hae Im

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Abstract

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.

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Seokhyun Choi, Bo-Hae Im. 2025-02-26. Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. https://arxiv.org/abs/2502.18761

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