No general Riemann-Hurwitz formula for relative p-class groups

We disprove, by means of numerical examples and theoretical arguments, illustrated with p=3, the existence of a Riemann-Hurwitz formula for the p-ranks of relative class groups in a p-ramified p-extension K/k of number fields of CM-type containing μp in contradiction with a result published in 1996. In the cyclic case of degree p, under some assumptions on the p-class group of k and the decomposition of the p-places, we prove some results on the structure of the p-class group of K and justify that some theoretical structures do not exist in this particular situation. In this context, an analogue of Kida's formula is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is not always the case, as shown by a numerical table for p=3.