Bielliptic modular curves $X_0^{\ast }(N)$ with square-free levels

Abstract

Let $N\ge 1$ be a square-free integer such that the modular curve $X_0^{\ast }(N)$ has genus $\ge 2$. We prove that $X_0^{\ast }(N)$ 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 $Aut(X_0^{\ast }(N))$ when the genus of $X_0^{\ast }(N)$ is $\ge 3$. Moreover, we prove that the set of all quadratic points over $\mathbb{Q}$ for the modular curve $X_0^{\ast }(N)$ with genus $\ge 2$ and $N$ square-free is not finite exactly for $51$ values of $N$.

References in corpus (1)

• Bielliptic intermediate modular curves