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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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Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL₂(𝐙₃) can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over 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 𝐐(ζ₃)-rational points on a certain smooth plane quartic curve X′_H (arising as a quotient of the modular curve Xₙₛ^+(27)) defined over 𝐐(ζ₃) 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⊗𝐐ₚ) 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(𝐐(ζ₃)). This allows us to compute the set Xₙₛ^+(27)(𝐐), finishing the classification of 3-adic images of Galois.
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