Geodesic
Explicit Chabauty–Kim for the split Cartan modular curve of level 13
Jennifer S. Balakrishnan1 ; Netan Dogra2 ; J. Steffen Müller3 ; Jan Tuitman4 ; Jan Vonk5
1 Department of Mathematics and Statistics, Boston University, Boston, MA
2 Jesus College, University of Oxford, Oxford, UK
3 Bernoulli Institute, University of Groningen, Groningen, The Netherlands
4 Departement Wiskunde, KU Leuven, Leuven, Belgium
5 Mathematical Institute, University of Oxford, Oxford, UK
Annals of mathematics, Tome 189 (2019) no. 3, pp. 885-944

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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 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.

DOI : 10.4007/annals.2019.189.3.6

Jennifer S. Balakrishnan 1 ; Netan Dogra 2 ; J. Steffen Müller 3 ; Jan Tuitman 4 ; Jan Vonk 5

1 Department of Mathematics and Statistics, Boston University, Boston, MA
2 Jesus College, University of Oxford, Oxford, UK
3 Bernoulli Institute, University of Groningen, Groningen, The Netherlands
4 Departement Wiskunde, KU Leuven, Leuven, Belgium
5 Mathematical Institute, University of Oxford, Oxford, UK 
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Jennifer S. Balakrishnan; Netan Dogra; J. Steffen Müller; Jan Tuitman; Jan Vonk. Explicit Chabauty–Kim for the split Cartan modular curve of level 13. Annals of mathematics, Tome 189 (2019) no. 3, pp. 885-944. doi: 10.4007/annals.2019.189.3.6

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