Quadratic Chabauty for modular curves: algorithms and examples

Compositio Mathematica, vol. 159, pp. 1111–1152

Abstract

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$ . This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$ , the curve $X_{S_4}(13)$ , as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$ .

Authors 5

1. Jennifer S. Balakrishnan

   Boston University

   Affiliation as printed

   Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, MA 02215, USA 

   Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, MA 02215, USA

2. Netan Dogra

   King's College London

   Affiliation as printed

   Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK 

   Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK

3. J. Steffen Müller

   University of Groningen

   Affiliation as printed

   Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands 

   Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands

4. Jan Tuitman
   Affiliation as printed

   3053 Haasrode, Belgium 

   3053 Haasrode, Belgium

5. Jan Vonk Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands 

   Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands