Bielliptic quotient modular curves of $X_0(N)$

TL;DR

This paper classifies certain bielliptic quotient modular curves derived from $X_0(N)$, identifying cases with genus at least two where the quotient is bielliptic or has infinitely many quadratic points over $Q$.

Contribution

It determines all pairs $(N,W_N)$ where the quotient $X_0(N)/W_N$ is bielliptic or has infinitely many quadratic points, extending understanding of modular curve quotients.

Findings

Identified all pairs $(N,W_N)$ with bielliptic quotient curves.

Classified pairs $(N,W_N)$ with infinitely many quadratic points over $Q$.

Provided explicit descriptions of these modular curve quotients.

Abstract

Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.