Abstract
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 $g \ge 2$ 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 $X_s(13)$, completing the classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan level structure due to Bilu--Parent and Bilu--Parent--Rebolledo.
Citation
Jennifer Balakrishnan. Netan Dogra. J. Steffen Müller. Jan Tuitman. Jan Vonk. "Explicit Chabauty–Kim for the split Cartan modular curve of level 13." Ann. of Math. (2) 189 (3) 885 - 944, May 2019. https://doi.org/10.4007/annals.2019.189.3.6
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