On the automorphisms of the nonsplit cartan modular curves of prime level

We study the automorphisms of the nonsplit Cartan modular curves Xns(p) of prime level p. We prove that if p ≥ 29 all the automorphisms preserve the cusps. Furthermore, if p ≡ 1 mod 12 and p ≠13 , the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of GL2(Fp). We also prove that for every p ≥ 29 the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve X+ns(p) associated to the normalizer of a nonsplit Cartan subgroup of GL2(Fp).