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

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arxiv 2502.18761 v1 pith:6FADSKJ6 submitted 2025-02-26 math.NT

classification math.NT
keywords mathbbrankanalyticellipticoverlineconjecturecurveslarsen
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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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  1. Elliptic curves and finitely generated Galois groups

    math.NT 2025-10 conditional novelty 7.0 of 10

    If the Galois group of a field is finitely generated, every elliptic curve over it has infinite Mordell–Weil rank.

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