Diophantine stability for elliptic curves on average

Let $K$ be a number field and $\mathrm{\ell }\ge 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{\mathrm{/}K}$ at a prime $\mathrm{\ell }$. We show that there is a positive density set of elliptic curves $E_{\mathrm{/}\mathbb{Q}}$ of rank $1$ such that $E_{\mathrm{/}K}$ is diophantine stable at $\mathrm{\ell }$. This has implications for Hilbert’s Tenth Problem over ${\mathcal{O}}_K$. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of Diophantine equations over ${\mathcal{O}}_K$ has a solution.