Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929693056303104 |
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| author | Alpöge, Levent Bhargava, Manjul Ho, Wei Shnidman, Ari |
| author_facet | Alpöge, Levent Bhargava, Manjul Ho, Wei Shnidman, Ari |
| contents | We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field Alpöge, Levent Bhargava, Manjul Ho, Wei Shnidman, Ari Number Theory Logic 11U05, 14H40, 11G10, 11G30 We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$. |
| title | Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field |
| topic | Number Theory Logic 11U05, 14H40, 11G10, 11G30 |
| url | https://arxiv.org/abs/2501.18774 |