TY - JOUR
T1 - Explicit Chabauty--Kim for the split Cartan modular curve of level 13
AU - Balakrishnan, Jennifer
AU - Dogra, Netan
AU - Müller, Jan Steffen
AU - Tuitman, Jan
AU - Vonk, Jan
PY - 2019/5/14
Y1 - 2019/5/14
N2 - We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in a method to compute a finite set of p-adic points, containing the rational points, on a curve of genus g > 2 over the rationals whose Jacobian has Mordell–Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This is then applied to determine the rational points of the modular curve Xs (13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.
AB - We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in a method to compute a finite set of p-adic points, containing the rational points, on a curve of genus g > 2 over the rationals whose Jacobian has Mordell–Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This is then applied to determine the rational points of the modular curve Xs (13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.
UR - https://arxiv.org/abs/1711.05846
U2 - 10.4007/annals.2019.189.3.6
DO - 10.4007/annals.2019.189.3.6
M3 - Article
SN - 0003-486X
VL - 189
SP - 885
EP - 944
JO - ANNALS OF MATHEMATICS
JF - ANNALS OF MATHEMATICS
IS - 3
ER -