Cohen–Lenstra heuristics and the Spiegelungssatz; function fields

In this paper, we study the compatibility of Cohen–Lenstra heuristics with Leopoldtʹs 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.