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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.22895 |
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| _version_ | 1866911463614971904 |
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| author | Cerchia, Michael Rakvi |
| author_facet | Cerchia, Michael Rakvi |
| contents | We completely determine the $1085$ open subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level that satisfy $-I \in H$ and $\operatorname{det}(H)=\widehat{\mathbb{Z}}^{\times}$ for which the corresponding modular curve $X_H$ has infinitely many quadratic points. When $g(X_H)\geq 2$ this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level for which there are infinitely many elliptic curves $E/K$ over quadratic extensions such that $ρ_E(G_k)$ is conjugate to a subgroup of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular curves of prime-power level with infinitely many quadratic points Cerchia, Michael Rakvi Number Theory We completely determine the $1085$ open subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level that satisfy $-I \in H$ and $\operatorname{det}(H)=\widehat{\mathbb{Z}}^{\times}$ for which the corresponding modular curve $X_H$ has infinitely many quadratic points. When $g(X_H)\geq 2$ this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level for which there are infinitely many elliptic curves $E/K$ over quadratic extensions such that $ρ_E(G_k)$ is conjugate to a subgroup of $H$. |
| title | Modular curves of prime-power level with infinitely many quadratic points |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.22895 |