Hilbert's tenth problem in Anticyclotomic towers of number fields
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911835117060096 |
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| author | Ray, Anwesh Weston, Tom |
| author_facet | Ray, Anwesh Weston, Tom |
| contents | Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb{Z}_p$-extension $K_{anti}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in $K_{anti}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04157 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hilbert's tenth problem in Anticyclotomic towers of number fields Ray, Anwesh Weston, Tom Number Theory Logic 11R23, 11U05 Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb{Z}_p$-extension $K_{anti}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in $K_{anti}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$. |
| title | Hilbert's tenth problem in Anticyclotomic towers of number fields |
| topic | Number Theory Logic 11R23, 11U05 |
| url | https://arxiv.org/abs/2302.04157 |