A positive density of elliptic curves are diophantine stable in certain Galois extensions

TL;DR

This paper demonstrates that a positive density of elliptic curves over rationals remain diophantine stable in certain cyclic p-extensions, with the result being effective and applicable to specific primes 3 and 5.

Contribution

It establishes the existence of a positive density of elliptic curves that are diophantine stable in cyclic p-extensions for p in {3, 5}, providing an effective measure.

Findings

Positive density of elliptic curves diophantine stable in L

Effective bounds for the density

Applicable to cyclic p-extensions with p in {3, 5}

Abstract

Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.