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

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 $X_H^{\prime }$ (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 $X_H^{\prime }(\mathbf{Q}({\zeta }_3))$. This allows us to compute the set $X_{ns}^{+}(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.