Rank stability makes rings of integers diophantine
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908606184554496 |
|---|---|
| author | Poonen, Bjorn |
| author_facet | Poonen, Bjorn |
| contents | The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) = \operatorname{rank} A(L)$, then $\mathcal{O}_K$ is $\mathcal{O}_L$-diophantine. We present an alternative proof of this theorem and review how it is used. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rank stability makes rings of integers diophantine Poonen, Bjorn Number Theory Algebraic Geometry 11G10 (Primary) 11R04, 11U05, 14K15 (Secondary) The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) = \operatorname{rank} A(L)$, then $\mathcal{O}_K$ is $\mathcal{O}_L$-diophantine. We present an alternative proof of this theorem and review how it is used. |
| title | Rank stability makes rings of integers diophantine |
| topic | Number Theory Algebraic Geometry 11G10 (Primary) 11R04, 11U05, 14K15 (Secondary) |
| url | https://arxiv.org/abs/2510.19553 |