Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain their normalizers. prove that if $n$ is large enough, all are induced by ramified covering complex upper half-plane. get new results for non-split prime level $p\ge 13$: curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas $X_{\text{ns}}(p)$ exactly one automorphism. Moreover, as an immediate consequence our we compute automorphism group $X_0^*(n):=X_0(n)/W$, where $W$ generated Atkin-Lehner involutions $X_0(n)$ a enough square.