Cohen-Lenstra sums over \(\mathbb{Z}_p[C_p]\)

Abstract

The Cohen-Lenstra heuristics predict that the \(p\)-part of the class group of a random imaginary quadratic field is isomorphic to an abelian \(p\)-group \(A\) with probability proportional to \(\frac{1}{|\text{Aut} A|}\). This probability distribution has since arisen in a myriad of different settings, and serves as a natural model for a random abelian group. More generally, the Cohen-Lenstra-Martinet heuristics deal with sums of the form \(\sum_M \frac{1}{|\text{Aut} M|}\) over a collection of \(\mathbb{Z}_p[G]\)-modules \(M\), where \(G\) is a group such that \(p \nmid |G|\). When \(p \mid |G|\), however, things become more complicated, and much less is known. We consider the simplest such case, when \(G\) is the cyclic group \(C_p\) of order \(p\). We study modules over \(\mathbb{Z}_p[C_p]\) and evaluate \(\sum_M \frac{1}{|\text{Aut} M|}\) for various collections of \(\mathbb{Z}_p[C_p]\)-modules \(M\).