Diophantine stability for elliptic curves on average

Abstract: Let $K$ be a number field and $\ell\geq 5$ be a prime number. Mazur and Rubin introduced the notion of \emph{diophantine stability} for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves $E_{/\mathbb{Q}}$ of rank $1$, such that $E_{/K}$ is diophantine stable at $\ell$. This result has implications to Hilbert's tenth problem for number rings.