The Cohen-Lenstra heuristics, postulated in the early 80s, conceptually explained numerous phenomena in the behaviour of ideal class groups of number fields that had puzzled mathematicians for decades, by proposing a probabilistic model: the probability that the class group of an imaginary quadratic field is isomorphic to a given group $A$ is inverse proportional to $\#\text{Aut}(A)$. This is a very natural model for random algebraic objects. But the probability weights for more general number fields, while agreeing well with experiments, look rather mysterious. I will explain how to recover the original heuristic in a very conceptual way by phrasing it in terms of Arakelov class groups instead. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. We build up a theory of commensurability of modules, of groups, and of rings, in order to remove this obstacle. This is joint work with Hendrik Lenstra.
