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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
Published electronically: August 13, 2020
MathSciNet review: 4166463
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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.

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Additional Information

Josha Box
Affiliation: Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom

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