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Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6

Volume 208 / 2023

Nikola Adžaga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, Timo Keller, Boya Wen Acta Arithmetica 208 (2023), 15-49 MSC: Primary 14G05; Secondary 11G30, 11G18. DOI: 10.4064/aa220110-7-3 Published online: 5 June 2023

Abstract

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus 4, 5, and 6. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus 5 and 6. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197, 211,223,269,271,359$}.

Authors

  • Nikola AdžagaDepartment of Mathematics
    Faculty of Civil Engineering
    University of Zagreb
    10000 Zagreb, Croatia
    e-mail
  • Vishal ArulDepartment of Mathematics
    University College London
    London WC1E 6BT, United Kingdom
    e-mail
  • Lea BeneishDepartment of Mathematics
    University of California at Berkeley
    Berkeley, CA 94720, USA
    e-mail
  • Mingjie ChenSchool of Computer Science
    University of Birmingham
    Birmingham B15 2TT, United Kingdom
    e-mail
  • Shiva ChidambaramDepartment of Mathematics
    Massachusetts Institute of Technology
    Cambridge, MA 02139-4307, USA
    e-mail
  • Timo KellerIAZD, Leibniz Universität Hannover
    30159 Hannover, Germany
    and
    Department of Mathematics
    Universität Bayreuth
    95440 Bayreuth, Germany
    e-mail
  • Boya WenDepartment of Mathematics
    University of Wisconsin at Madison
    Madison, WI 53706, USA
    and
    Simons Laufer Mathematical Sciences
    Institute (formerly MSRI)
    Berkeley, CA 94720-5070, USA
    e-mail

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