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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2306.16983 |
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| _version_ | 1866918383841181696 |
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| author | Kai, Wataru |
| author_facet | Kai, Wataru |
| contents | We prove a number field analogue of the Green--Tao--Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational points for certain fibrations $X\to \mathbb{P}^1$ over number fields $K$ which had only been available over $\mathbb Q $ by Harpaz--Skorobogatov--Wittenberg, and construction of elliptic curves having some specified ranks due to Koymans--Pagano and Zywina. This latter family of results led to a negative answer to a generalized Hilbert Tenth Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16983 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear patterns of prime elements in number fields Kai, Wataru Number Theory Algebraic Geometry Combinatorics 14G12, 11N32, 11P32, 11B30, 11N37 We prove a number field analogue of the Green--Tao--Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational points for certain fibrations $X\to \mathbb{P}^1$ over number fields $K$ which had only been available over $\mathbb Q $ by Harpaz--Skorobogatov--Wittenberg, and construction of elliptic curves having some specified ranks due to Koymans--Pagano and Zywina. This latter family of results led to a negative answer to a generalized Hilbert Tenth Problem. |
| title | Linear patterns of prime elements in number fields |
| topic | Number Theory Algebraic Geometry Combinatorics 14G12, 11N32, 11P32, 11B30, 11N37 |
| url | https://arxiv.org/abs/2306.16983 |