Paper
Unit-generated orders of real quadratic fields I. Class number bounds
Authors
Gene S. Kopp, Jeffrey C. Lagarias
Abstract
Unit-generated orders of a quadratic field are orders of the form , where is a unit in the quadratic field. If the order is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud--Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as having quadratic discriminants (for ) and (for ). We show the (wide or narrow) class numbers of unit-generated orders satisfy as , using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is -torsion. We classify all unit-generated real quadratic maximal orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are -torsion for , for both wide and narrow class groups, which are conjecturally complete.