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

Let Y-ns(+)(n) be the open non-cuspidal locus of the modular curve X-ns(+)(n) 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 Y-ns(+)(n) for suitably chosen n. In this note, we give a genus formula for the modular curves X-ns(+)(n) and we give three new solutions to the class number one problem using the modular curves X-ns(+)(n) for n = 16,20,21. These are the only such modular curves of genus <= 2 that had not yet been exploited. (C) 2010 Elsevier Inc. All rights reserved.