A note on Larsen's conjecture and ranks of elliptic curves

TL;DR

This paper proves Larsen's conjecture for elliptic curves over Q regarding unbounded ranks in certain Galois extensions, and discusses conditions under which the conjecture follows from standard conjectures.

Contribution

It establishes the conjecture over Q for analytic and Selmer ranks and links the general case to standard conjectures under specific conditions.

Findings

Larsen's conjecture holds over Q for analytic and p-infinity Selmer ranks.

The conjecture is shown to follow from standard conjectures under certain conditions.

The paper connects the conjecture to broader conjectures in elliptic curve theory.

Abstract

Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant.