Quadratic points on bielliptic modular curves

TL;DR

This paper completes the classification of quadratic points on bielliptic modular curves of genus up to 11, using advanced methods to identify all exceptional points, which correspond to CM elliptic curves.

Contribution

It fully describes quadratic points on all bielliptic modular curves $X_0(n)$ for specific $n$, extending previous results to higher genus cases up to 11.

Findings

All exceptional points correspond to CM elliptic curves.

Complete classification for $X_0(n)$ with $n=60,62,69,79,83,89,92,94,95,101,119,131$.

Methods include Box's relative symmetric Chabauty and moduli descriptions of $ ext{Q}$-curves.

Abstract

Bruin and Najman, Ozman and Siksek, and Box 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 $\Q$-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.