Computing Quadratic Points on Modular Curves X₀(N)

authored by

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

University of Zagreb
University of Warwick
Bogazici University

Type

Article

Journal

Mathematics of Computation

Volume

93

Pages

1371-1397

No. of pages

27

ISSN

0025-5718

Publication date

03.10.2023

Publication status

E-pub ahead of print

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory, Computational Mathematics, Applied Mathematics

Electronic version(s)

https://doi.org/10.48550/arXiv.2303.12566 (Access: Open)
https://doi.org/10.1090/mcom/3902 (Access: Closed)