Hilbert's 10th Problem via Mordell curves
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866912237260636160 |
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| author | Jha, Somnath Kundu, Debanjana Majumdar, Dipramit |
| author_facet | Jha, Somnath Kundu, Debanjana Majumdar, Dipramit |
| contents | We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(ζ_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(ζ_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert's 10th Problem via Mordell curves Jha, Somnath Kundu, Debanjana Majumdar, Dipramit Number Theory We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(ζ_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(ζ_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof. |
| title | Hilbert's 10th Problem via Mordell curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.04253 |