Hilbert's tenth problem via additive combinatorics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911281344151552 |
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| author | Koymans, Peter Pagano, Carlo |
| author_facet | Koymans, Peter Pagano, Carlo |
| contents | For all infinite rings $R$ that are finitely generated over $\mathbb{Z}$, we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic curves $E$ without rank growth in certain quadratic extensions $L/K$. To achieve such a result unconditionally, our key innovation is to use elliptic curves $E$ with full rational $2$-torsion which allows us to combine techniques from additive combinatorics with $2$-descent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert's tenth problem via additive combinatorics Koymans, Peter Pagano, Carlo Number Theory Logic For all infinite rings $R$ that are finitely generated over $\mathbb{Z}$, we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic curves $E$ without rank growth in certain quadratic extensions $L/K$. To achieve such a result unconditionally, our key innovation is to use elliptic curves $E$ with full rational $2$-torsion which allows us to combine techniques from additive combinatorics with $2$-descent. |
| title | Hilbert's tenth problem via additive combinatorics |
| topic | Number Theory Logic |
| url | https://arxiv.org/abs/2412.01768 |