Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929207318151168 |
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| author | Gillibert, Jean Hallouin, Emmanuel Levin, Aaron |
| author_facet | Gillibert, Jean Hallouin, Emmanuel Levin, Aaron |
| contents | We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable assumptions on the factorization of $f$, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When $f$ has degree at most $6$, we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the $k$-curves corresponding to the $2$ and $3$-torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11353 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$ Gillibert, Jean Hallouin, Emmanuel Levin, Aaron Algebraic Geometry Number Theory 14J27 (Primary) 11G05, 14J20, 11D61 (Secondary) We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable assumptions on the factorization of $f$, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When $f$ has degree at most $6$, we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the $k$-curves corresponding to the $2$ and $3$-torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation. |
| title | Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$ |
| topic | Algebraic Geometry Number Theory 14J27 (Primary) 11G05, 14J20, 11D61 (Secondary) |
| url | https://arxiv.org/abs/2306.11353 |