Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Quadratic points on modular curves with infinite Mordell-Weil group


Author: Josha Box
Journal: Math. Comp. 90 (2021), 321-343
MSC (2010): Primary 11G05, 14G05, 11G18
DOI: https://doi.org/10.1090/mcom/3547
Published electronically: August 13, 2020
MathSciNet review: 4166463
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578-602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461-2484] have recently determined the quadratic points on each modular curve $ X_0(N)$ of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the $ X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of $ N$ are 37, 43, 53, 61, 57, 65, 67, and 73.

The main tool used is a relative symmetric Chabauty method, in combination with the Mordell-Weil sieve. Often the quadratic points are not finite, as the degree 2 map $ X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $ X_0(N)^+$, and we specify the exceptional quadratic points on $ X_0(N)$ not coming from $ X_0(N)^+$. In particular, we determine the $ j$-invariants of the corresponding elliptic curves and whether they are $ {\mathbb{Q}}$-curves or have complex multiplication.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11G05, 14G05, 11G18

Retrieve articles in all journals with MSC (2010): 11G05, 14G05, 11G18


Additional Information

Josha Box
Affiliation: Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom
Email: j.box@warwick.ac.uk

DOI: https://doi.org/10.1090/mcom/3547
Keywords: Modular curves, quadratic points, Mordell--Weil, Jacobian, Chabauty
Received by editor(s): June 27, 2019
Received by editor(s) in revised form: February 3, 2020, and March 4, 2020
Published electronically: August 13, 2020
Additional Notes: During the work on this article, the author was supported by an EPSRC DTP studentship.
Article copyright: © Copyright 2020 American Mathematical Society