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arXiv · 2304.09742

Diophantine stability for elliptic curves on average

Anwesh Ray·Tom Weston
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Abstract

Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of 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 has implications for Hilbert's Tenth Problem over $\mathscr{O}_K$. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of Diophantine equations over $\mathscr{O}_K$ has a solution.

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Anwesh Ray, Tom Weston. 2023-04-19. Diophantine stability for elliptic curves on average. https://doi.org/10.1007/s40879-025-00872-3

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