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arXiv · 2501.07833

Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields

Jennifer S. Balakrishnan·L. Alexander Betts·Daniel Rayor Hast·Aashraya Jha·J. Steffen Müller
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Abstract

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 $\mathbf{Q}(\zeta_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(\zeta_3)$ 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 $X'_H(\mathbf{Q}(\zeta_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.

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Jennifer S. Balakrishnan, L. Alexander Betts, Daniel Rayor Hast, Aashraya Jha, J. Steffen Müller. 2025-01-14. Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields. https://arxiv.org/abs/2501.07833

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