On Larsen's conjecture on the ranks of Elliptic Curves

TL;DR

This paper proves Larsen's conjecture that the Mordell-Weil rank of elliptic curves over certain fixed fields of Galois groups is infinite, specifically for elements in particular infinite families of the Galois group.

Contribution

The paper confirms Larsen's conjecture for cases where the Galois group elements belong to specific infinite families, advancing understanding of elliptic curve ranks over fixed fields.

Findings

Proves Larsen's conjecture for certain Galois group elements.

Establishes infinite rank of elliptic curves over fixed fields in these cases.

Extends previous partial results to broader classes of Galois elements.

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langle\sigma_1, \dots, \sigma_n\rangle$ 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 $\sigma_i$ for each $i=1, \dots, n$ is an element of some infinite families of elements of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$.