Diophantine stability for elliptic curves on average
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918168159584256 |
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| author | Ray, Anwesh Weston, Tom |
| author_facet | Ray, Anwesh Weston, Tom |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Diophantine stability for elliptic curves on average Ray, Anwesh Weston, Tom Number Theory 11G05, 11U05, 11R45, 11R32 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. |
| title | Diophantine stability for elliptic curves on average |
| topic | Number Theory 11G05, 11U05, 11R45, 11R32 |
| url | https://arxiv.org/abs/2304.09742 |