Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.
Daileda, R.C., Krishnamoorthy, R., & Malyshev, A. (2010). Maximal class numbers of CM number fields. Journal of Number Theory, 130(4), 936-943. doi: 10.1016/j.jnt.2009.09.013
Journal of Number Theory