Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917892244635648 |
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| author | Balakrishnan, Jennifer S. Betts, L. Alexander Hast, Daniel Rayor Jha, Aashraya Müller, J. Steffen |
| author_facet | Balakrishnan, Jennifer S. Betts, L. Alexander Hast, Daniel Rayor Jha, Aashraya Müller, J. Steffen |
| contents | Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve.
Our proof proceeds via computing the $\mathbf{Q}(ζ_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(ζ_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(ζ_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields Balakrishnan, Jennifer S. Betts, L. Alexander Hast, Daniel Rayor Jha, Aashraya Müller, J. Steffen Number Theory Algebraic Geometry Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve. Our proof proceeds via computing the $\mathbf{Q}(ζ_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(ζ_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(ζ_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois. |
| title | Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2501.07833 |