ON THE AUTOMORPHISMS OF THE NONSPLIT CARTAN MODULAR CURVES OF PRIME LEVEL

We study the automorphisms of the nonsplit Cartan modular curves $X_{\text{ns}}(p)$ of prime level $p$. We prove that if $p\geqslant 29$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1~\text{mod}~12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$. We also prove that for every $p\geqslant 29$ the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{\text{ns}}^{+}(p)$ associated to the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$.