Quadratic points on bielliptic modular curves.

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2\leq g(X_0(n)) \leq 5. Since all the hyperelliptic curves X_0(n) are of genus \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus \leq 1, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves X_0(n) for which this has not been done already. The values of n that we deal with are n=60, 62, 69, 79, 83, 89, 92, 94, 95, 101, 119 and 131; the curves X_0(n) are of genus up to 11. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box's relative symmetric Chabauty method and an application of a moduli description of \mathbb {Q}-curves of degree d with an independent isogeny of degree m, which reduces the problem to finding the rational points on several quotients of modular curves. [ABSTRACT FROM AUTHOR]