Dihedral Artin representations and CM fields

Abstract

For a fixed CM field K K with maximal totally real subfield F F , we consider isomorphism classes of dihedral Artin representations of F F which are induced from K K , distinguishing between those which are “canonically” induced from K K and those which are “noncanonically” induced from K K . The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.