Explicit Chabauty—Kim for the split Cartan modular curve of level 13

By Jennifer S. Balakrishnan, Netan Dogra, J. S. Muller, Jan Tuitman, Jan Vonk. Annals of Mathematics, 2017

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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 an algorithm to compute 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 algorithm is then applied to 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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