Quadratic Chabauty for modular curves and modular forms of rank one

Journal article

Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL_2(Z_3) 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 X+_ns(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 X+_ns(27)(Q), finishing the classification of 3-adic images of Galois

Authors: Dogra, N; Le Fourn, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Mathematische Annalen
• Volume: 380
• Issue: 1-2
• Pages: 393-448
• Publication date: 2020-11-19
• DOI: 10.1007/s00208-020-02112-3
• EISSN: 1432-1807
• ISSN: 0025-5831
• Language: English
• Keywords: Quotient; Rank (graph theory); Modular design; Applied mathematics; Dimension (graph theory); Geometry; Combinatorics; Modular form; Prime (order theory); Mathematics; Discrete mathematics; Modular curve; Quadratic equation; Jacobian matrix and determinant; Modular group; Pure mathematics