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

Abstract

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.