Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of GL2(Z/N) and certain subgroups of their normalizers. We prove that if N is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for nonsplit curves of prime level p greater or equal to 13: the curve X_ns+(p) has no nontrivial automorphisms, whereas the curve Xns(p) has exactly one nontrivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of X_0*(n):=X_0(n)/W, where W is the group generated by the Atkin–Lehner involutions of X_0(n) and n is a large enough square.