# Bielliptic modular curves $X_0^*(N)$ with square-free levels

## TL;DR

This paper classifies bielliptic properties and automorphism groups of certain modular curves with square-free levels, and analyzes the finiteness of quadratic points over rationals for these curves.

## Contribution

It identifies exactly 19 square-free levels where $X_0^*(N)$ is bielliptic and determines their automorphism groups, providing new examples for higher genus cases.

## Key findings

- Exactly 19 levels where $X_0^*(N)$ is bielliptic.
- First examples of nontrivial automorphism groups for genus ≥ 3.
- Finiteness of quadratic points over $\\mathbb{Q}$ for these curves holds for 51 levels.

## Abstract

Let $N\geq 1$ be a square-free integer such that the modular curve $X_0^*(N)$ has genus $\geq 2$. We prove that $X_0^*(N)$ is bielliptic exactly for $19$ values of $N$, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial $Aut(X_0^*(N))$ when the genus of $X_0^*(N)$ is $\geq 3$. Moreover, we prove that the set of all quadratic points over $\mathbb{Q}$ for the modular curve $X_0^*(N)$ with genus $\geq 2$ and $N$ square-free is not finite exactly for $51$ values of $N$.