Cohen–Lenstra Heuristics and the Spiegelungssatz: Number Fields Original Research Article

In this paper we study the compatibility of Cohen–Lenstra heuristics with Leopoldtʹs Spiegelungssatz (the reflection theorem). We generalize Dutarteʹs ([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.