Computing Quadratic Points on Modular Curves X₀(N)

verfasst von

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

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

University of Zagreb
University of Warwick
Bogazici University

Typ

Artikel

Journal

Mathematics of Computation

Band

93

Seiten

1371-1397

Anzahl der Seiten

27

ISSN

0025-5718

Publikationsdatum

03.10.2023

Publikationsstatus

Elektronisch veröffentlicht (E-Pub)

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie, Computational Mathematics, Angewandte Mathematik

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2303.12566 (Zugang: Offen)
https://doi.org/10.1090/mcom/3902 (Zugang: Geschlossen)