## Details

Original language | English |
---|---|

Pages (from-to) | 1371-1397 |

Number of pages | 27 |

Journal | Mathematics of Computation |

Volume | 93 |

Issue number | 347 |

Early online date | 3 Oct 2023 |

Publication status | Published - 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 X_{0}(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 X_{0}(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X_{0}(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

- Mathematics(all)
**Algebra and Number Theory**- Mathematics(all)
**Computational Mathematics**- Mathematics(all)
**Applied Mathematics**

## Cite this

- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS

**Computing Quadratic Points on Modular Curves X**/ Adžaga, Nikola; Keller, Timo; Michaud-Jacobs, Philippe et al.

_{0}(N).In: Mathematics of Computation, Vol. 93, No. 347, 05.2024, p. 1371-1397.

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

_{0}(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

_{0}(N).

*Mathematics of Computation*,

*93*(347), 1371-1397. https://doi.org/10.48550/arXiv.2303.12566, https://doi.org/10.1090/mcom/3902

_{0}(N). Mathematics of Computation. 2024 May;93(347):1371-1397. Epub 2023 Oct 3. doi: 10.48550/arXiv.2303.12566, 10.1090/mcom/3902

}

TY - JOUR

T1 - Computing Quadratic Points on Modular Curves X0(N)

AU - Adžaga, Nikola

AU - Keller, Timo

AU - Michaud-Jacobs, Philippe

AU - Najman, Filip

AU - Ozman, Ekin

AU - Vukorepa, Borna

N1 - Funding Information: The second author was supported by the Deutsche Forschungsgemeinschaft (DFG), Projektnummer STO 299/18-1, AOBJ: 667349 while working on this article. The third author was supported by an EPSRC studentship EP/R513374/1 and had previously used the surname Michaud- Rodgers. The fourth and sixth authors were supported by QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). The fifth author was partially supported by TUBITAK Project No 122F413.

PY - 2024/5

Y1 - 2024/5

N2 - 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.

AB - 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.

KW - elliptic curves

KW - Jacobians

KW - Modular curves

KW - Mordell–Weil sieve

KW - quadratic points

KW - symmetric Chabauty

UR - http://www.scopus.com/inward/record.url?scp=85187143570&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2303.12566

DO - 10.48550/arXiv.2303.12566

M3 - Article

VL - 93

SP - 1371

EP - 1397

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 347

ER -