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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Quadratic points on modular curves
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by Ekin Ozman and Samir Siksek HTML | PDF
Math. Comp. 88 (2019), 2461-2484 Request permission

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.

References
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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
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 2461-2484
  • MSC (2010): Primary 11G05, 14G05, 11G18
  • DOI: https://doi.org/10.1090/mcom/3407
  • MathSciNet review: 3957901