Automorphisms of Cartan modular curves of prime and composite level

Abstract

We study the automorphisms of modular curves associated to Cartan subgroups of GL(2,Z/nZ) 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 non-split curves of prime level p>12: the curve Xns+(p) has no non-trivial automorphisms, whereas the curve Xns(p) has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of X0*(n):=X0(n)/W, where W is the group generated by the Atkin-Lehner involutions of X0(n) and n is a large enough square.