Cohen-Lenstra heuristics and the Spiegelungssatz: Number fields

In this paper we study the compatibility of Cohen-Lenstra heuristics with Leopoldt's Spiegelungssatz (the reflection theorem). We generalize Dutarte's ([1983, in "Théorie des nombres, Besançon, 1983-1984"]) work to every prime number p: He proved the compatibility of the Cohen-Lenstra conjectures with the Spiegelungssatz in the case p = 3. We also show that the Spiegelungssatz is compatible with the conjectural probabilities on the p-rank of some subgroups of the class group of a cyclic extension of degree q over Q, where q is a prime number dividing p-1.