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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.11996 |
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| _version_ | 1866914199941152768 |
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| author | Calger, Chris |
| author_facet | Calger, Chris |
| contents | An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of $j$-invariants, in extensions of bounded degree, that arise as the $j$-invariant of an isolated point on a modular curve. We obtain finiteness results on these sets for families of modular curves with prime-power level. This is related to recent work of Bourdon and Ejder, who classified rational $j$-invariants of isolated points on the families $X_1(n)$ and $X_0(n)$, for $n$ a prime power. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isolated points on modular curves of prime-power level Calger, Chris Number Theory An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of $j$-invariants, in extensions of bounded degree, that arise as the $j$-invariant of an isolated point on a modular curve. We obtain finiteness results on these sets for families of modular curves with prime-power level. This is related to recent work of Bourdon and Ejder, who classified rational $j$-invariants of isolated points on the families $X_1(n)$ and $X_0(n)$, for $n$ a prime power. |
| title | Isolated points on modular curves of prime-power level |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.11996 |