Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

Let be the open non-cuspidal locus of the modular curve associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on for suitably chosen n. In this note, we give a genus formula for the modular curves and we give three new solutions to the class number one problem using the modular curves for n=16,20,21. These are the only such modular curves of genus ⩽2 that had not yet been exploited.