Points on a curve with a power on a curve
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914636973998080 |
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| author | Boxall, Gareth |
| author_facet | Boxall, Gareth |
| contents | Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose $C_1$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $1$ and $C_1\cup C_2$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $2$. It is a conjecture that at most finitely many points $x\in C_1$ have the property that there is a positive integer $n$ such that $x^n\in C_2$ and $[n]C_1\nsubseteq C_2$, where $[n]C_1=\{x^n:x\in C_1\}$. We prove this in the case where at least one of the two curves is not defined over $\overline{\mathbb{Q}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Points on a curve with a power on a curve Boxall, Gareth Algebraic Geometry Logic Number Theory 11U09, 11U07, 03C64 Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose $C_1$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $1$ and $C_1\cup C_2$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $2$. It is a conjecture that at most finitely many points $x\in C_1$ have the property that there is a positive integer $n$ such that $x^n\in C_2$ and $[n]C_1\nsubseteq C_2$, where $[n]C_1=\{x^n:x\in C_1\}$. We prove this in the case where at least one of the two curves is not defined over $\overline{\mathbb{Q}}$. |
| title | Points on a curve with a power on a curve |
| topic | Algebraic Geometry Logic Number Theory 11U09, 11U07, 03C64 |
| url | https://arxiv.org/abs/2401.05024 |