New Algebraic Points on Curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915700593917952 |
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| author | Khawaja, Maleeha Siksek, Samir |
| author_facet | Khawaja, Maleeha Siksek, Samir |
| contents | Let $C$ be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field $L$, we define the set of $L$-new points on $C$ to be $C(L)_{new} = \{P \in C(L) : \mathbb{Q}(P)=L\}$; this is the set of points on $C$ defined over $L$ but not any strictly smaller field. Let $n$ be at least 2. We conjecture that $C(L)_{new}$ is empty for 100 percent of degree $n$ number fields $L$ when ordered by absolute discriminant. For degrees $n=2$, $3$, we give sufficient criteria for our conjecture to hold in terms of an explicit model for $C$. For general $n$ we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for $n=2$ and $C=X_0(N)$ for the $18$ values $N \ne 37$ such that $X_0(N)$ is hyperelliptic, and also for $n=3$ and $C=X_0(23)$, $X_0(29)$, $X_0(31)$, $X_0(64)$. Moreover, we prove the analogue of our conjecture for the unit equation, again with $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Algebraic Points on Curves Khawaja, Maleeha Siksek, Samir Number Theory 11G30, 11G18 Let $C$ be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field $L$, we define the set of $L$-new points on $C$ to be $C(L)_{new} = \{P \in C(L) : \mathbb{Q}(P)=L\}$; this is the set of points on $C$ defined over $L$ but not any strictly smaller field. Let $n$ be at least 2. We conjecture that $C(L)_{new}$ is empty for 100 percent of degree $n$ number fields $L$ when ordered by absolute discriminant. For degrees $n=2$, $3$, we give sufficient criteria for our conjecture to hold in terms of an explicit model for $C$. For general $n$ we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for $n=2$ and $C=X_0(N)$ for the $18$ values $N \ne 37$ such that $X_0(N)$ is hyperelliptic, and also for $n=3$ and $C=X_0(23)$, $X_0(29)$, $X_0(31)$, $X_0(64)$. Moreover, we prove the analogue of our conjecture for the unit equation, again with $n=3$. |
| title | New Algebraic Points on Curves |
| topic | Number Theory 11G30, 11G18 |
| url | https://arxiv.org/abs/2511.15635 |