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

TL;DR

This paper proves Larsen's conjecture for elliptic curves over the rationals with analytic rank at most one, showing that the rank over certain fixed fields is infinite, advancing understanding of Galois representations and ranks.

Contribution

It establishes Larsen's conjecture for elliptic curves over  with analytic rank  at most one, linking Galois groups and infinite ranks over fixed fields.

Findings

Proves Larsen's conjecture for rank   1.

Shows infinite rank over fixed subfields under Galois groups.

Extends understanding of elliptic curve ranks and Galois actions.

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.