Iwasawa-Cohen-Lenstra heuristics

Abstract

In this note we propose an analog of the well-known Cohen-Lenstra heuristics for modules over the Iwasawa algebra $\Lambda$. It turns out that only the analog of the real-quadratic situation leads to a convergent series and hence to potential predictions. We determine the sum of this series, which runs over all isomorphism classes of finite $\Lambda$-modules, and we discuss the partial sum that arises by restricting to cyclic $\Lambda$-modules. We demonstrate that this subsum is almost as large as the total sum. No attempt is made to test the heuristics numerically.