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arXiv · 1711.05846

Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13

Jennifer S. Balakrishnan·Netan Dogra·J. Steffen Müller·Jan Tuitman·Jan Vonk
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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 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.

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Jennifer S. Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman, Jan Vonk. 2017-11-15. Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13. https://arxiv.org/abs/1711.05846

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