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Quadratic points on modular curves with infinite Mordell–Weil group
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by Josha Box;

Math. Comp. 90 (2021), 321-343

DOI: https://doi.org/10.1090/mcom/3547

Published electronically: August 13, 2020

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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.

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