Cohen-Lenstra heuristics and the Spiegelungssatz; function fields

In this paper, we study the compatibility of Cohen-Lenstra heuristics with Leopoldt's Spiegelungssatz (= the reflection theorem) in the case of cyclic function fields. First, we prove a group-theoretical version of the Spiegelungssatz for cyclic function fields. Then we show the internal consistency of the function field analogue of Cohen-Lenstra heuristics in the case of cyclic function fields. It thus supports the validity of Cohen-Lenstra heuristics in function fields.