Computing Quadratic Points on Modular Curves X0(N)

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Nikola Adžaga
  • Timo Keller
  • Philippe Michaud-Jacobs
  • Filip Najman
  • Ekin Ozman
  • Borna Vukorepa

Research Organisations

External Research Organisations

  • University of Zagreb
  • University of Warwick
  • Bogazici University
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Details

Original languageEnglish
Pages (from-to)1371-1397
Number of pages27
JournalMathematics of Computation
Volume93
Issue number347
Early online date3 Oct 2023
Publication statusPublished - May 2024

Abstract

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}. We obtain that all the non-cuspidal quadratic points on X0(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X0(103) defined over Q(√2885). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

Keywords

    elliptic curves, Jacobians, Modular curves, Mordell–Weil sieve, quadratic points, symmetric Chabauty

ASJC Scopus subject areas

Cite this

Computing Quadratic Points on Modular Curves X0(N). / Adžaga, Nikola; Keller, Timo; Michaud-Jacobs, Philippe et al.
In: Mathematics of Computation, Vol. 93, No. 347, 05.2024, p. 1371-1397.

Research output: Contribution to journal › Article › Research › peer review

Adžaga, N, Keller, T, Michaud-Jacobs, P, Najman, F, Ozman, E & Vukorepa, B 2024, 'Computing Quadratic Points on Modular Curves X0(N)', Mathematics of Computation, vol. 93, no. 347, pp. 1371-1397. https://doi.org/10.48550/arXiv.2303.12566, https://doi.org/10.1090/mcom/3902
Adžaga, N., Keller, T., Michaud-Jacobs, P., Najman, F., Ozman, E., & Vukorepa, B. (2024). Computing Quadratic Points on Modular Curves X0(N). Mathematics of Computation, 93(347), 1371-1397. https://doi.org/10.48550/arXiv.2303.12566, https://doi.org/10.1090/mcom/3902
Adžaga N, Keller T, Michaud-Jacobs P, Najman F, Ozman E, Vukorepa B. Computing Quadratic Points on Modular Curves X0(N). Mathematics of Computation. 2024 May;93(347):1371-1397. Epub 2023 Oct 3. doi: 10.48550/arXiv.2303.12566, 10.1090/mcom/3902
Adžaga, Nikola ; Keller, Timo ; Michaud-Jacobs, Philippe et al. / Computing Quadratic Points on Modular Curves X0(N). In: Mathematics of Computation. 2024 ; Vol. 93, No. 347. pp. 1371-1397.
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abstract = "In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}. We obtain that all the non-cuspidal quadratic points on X0(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X0(103) defined over Q(√2885). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.",
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