Quadratic points on modular curves with infinite Mordell–Weil group

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 ) 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 ) X_0(N) of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of N 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 ) → X 0 ( N ) + 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 ) + X_0(N)^+ , and we specify the exceptional quadratic points on X 0 ( N ) X_0(N) not coming from X 0 ( N ) + X_0(N)^+ . In particular, we determine the j j -invariants of the corresponding elliptic curves and whether they are Q {\mathbb {Q}} -curves or have complex multiplication.