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
05.2024
Publikationsstatus
Veröffentlicht
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)