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 $X_{\mathrm { s}}(13)$, completing the classification of non-CM elliptic curves over $\mathbf {Q} $ with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.
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