Integral points on cubic twists of Mordell curves
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909315314483200 |
|---|---|
| author | Chan, Stephanie |
| author_facet | Chan, Stephanie |
| contents | Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N)^{-\frac{1}{2}+ε})$. In particular, this implies that the number of positive integers $B\leq N$ such that $-3kB^2$ is the discriminant of an elliptic curve over $\mathbb{Q}$ is $o(N)$. The proof involves a discriminant-lowering procedure on integral binary cubic forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11366 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Integral points on cubic twists of Mordell curves Chan, Stephanie Number Theory 11D45, 11G05, 11D25 Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N)^{-\frac{1}{2}+ε})$. In particular, this implies that the number of positive integers $B\leq N$ such that $-3kB^2$ is the discriminant of an elliptic curve over $\mathbb{Q}$ is $o(N)$. The proof involves a discriminant-lowering procedure on integral binary cubic forms. |
| title | Integral points on cubic twists of Mordell curves |
| topic | Number Theory 11D45, 11G05, 11D25 |
| url | https://arxiv.org/abs/2203.11366 |