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Mathematics of Computation

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Quadratic points on modular curves


Authors: Ekin Ozman and Samir Siksek
Journal: Math. Comp. 88 (2019), 2461-2484
MSC (2010): Primary 11G05, 14G05, 11G18
DOI: https://doi.org/10.1090/mcom/3407
Published electronically: December 28, 2018
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Abstract: In this paper we determine the quadratic points on the modular curves $ X_0(N)$, where the curve is non-hyperelliptic, the genus is $ 3$, $ 4$, or $ 5$, and the Mordell-Weil group of $ J_0(N)$ is finite. The values of $ N$ are $ 34$, $ 38$, $ 42$, $ 44$, $ 45$, $ 51$, $ 52$, $ 54$, $ 55$, $ 56$, $ 63$, $ 64$, $ 72$, $ 75$, $ 81$.

As well as determining the non-cuspidal quadratic points, we give the $ j$-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic $ \mathbb{Q}$-curves.


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Additional Information

Ekin Ozman
Affiliation: Department of Mathematics, Bogazici University, Bebek, Istanbul, 34342, Turkey
Email: ekin.ozman@boun.edu.tr

Samir Siksek
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: samir.siksek@gmail.com

DOI: https://doi.org/10.1090/mcom/3407
Keywords: Modular curves, quadratic points, Mordell--Weil, Jacobian
Received by editor(s): June 21, 2018
Received by editor(s) in revised form: August 16, 2018, and October 2, 2018
Published electronically: December 28, 2018
Additional Notes: The first-named author was partially supported by Bogazici University Research Fund Grant Number 10842 and TUBITAK Research Grant 117F045.
The second-named author was supported by an EPSRC LMF: L-Functions and Modular Forms Programme Grant EP/K034383/1.
Article copyright: © Copyright 2018 American Mathematical Society