Abstract

The original Cohen-Lenstra heuristic predicts frequencies with which a given abelian group appears as the class group of a quadratic field. Postulated just over 30 years ago, the heuristic helped explain in a very compelling and intuitive way various phenomena in the study of class groups that had been observed over the preceding decades and centuries. Roughly speaking, the probability of an abelian group A occurring as the class group of an imaginary quadratic field is inverse proportional to #Aut(A) – a random object tends to have few symmetries. Already for real quadratic fields, the weights are postulated to be different, and the heuristic explanation is less intuitive. Later, the heuristic was extended by Cohen and Martinet to more general number fields, this time without even an attempt at an intuitive explanation of why these should be the right weights. I will explain that in fact, the intuitive heuristic that the rarity of an algebraic object is proportional to the number of symmetries does explain distributions of class groups of arbitrary number fields and recovers all the above heuristics, but the object one has to look at is not the class group. This is still work in progress, jointly with Hendrik Lenstra.