Bielliptic and hyperelliptic modular curves $X(N)$ and the group $\mathrm {Aut}(X(N))$
Volume 161 / 2013
Acta Arithmetica 161 (2013), 283-299
MSC: Primary 11G18; Secondary 11G30.
DOI: 10.4064/aa161-3-6
Abstract
We determine all modular curves $X(N)$ (with $N\geq 7$) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of $X(N)$ is $\operatorname {PSL}_2(\mathbb {Z}/N\mathbb {Z})$, whence it coincides with the normalizer of $\varGamma (N)$ in $\operatorname {PSL}_2(\mathbb {R})$ modulo $\pm \varGamma (N)$.
