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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
MathSciNet review: 3957901
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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
MR Author ID: 955558
Email: ekin.ozman@boun.edu.tr

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

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