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Publikationsdetails

Computing Quadratic Points on Modular Curves \(X_0(N)\)

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 \(X_0(N)\) 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 \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on \(X_0(N)\) for \(N\in \mathcal{L}\) are CM points, except for one pair of Galois conjugate points on \(X_0(103)\) defined over \(\mathbb{Q}(\sqrt{2885})\). 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
Typ
Preprint
Publikationsdatum
22.03.2023
Publikationsstatus
Elektronisch veröffentlicht (E-Pub)
Elektronische Version(en)
https://doi.org/10.48550/arXiv.2303.12566 (Zugang: Offen)