2025/01/14 by Balakrishnan, Jennifer S., Betts, L. Alexander, Hast, Daniel Rayor +2 · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.07833
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\otimesQp) 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.