Heuristics for Arakelov class groups

Alex Barthel (Warwick/MPIM Bonn)

In the 1980s, Cohen and Lenstra proposed a probabilistic model that explains the statistical properties of class groups of quadratic number fields. They postulated that a “random” algebraic object should be isomorphic to a given object $A$ with probability inverse proportional to $\#\mathop{\rm Aut} A$, and the (odd parts of) class groups of imaginary quadratic fields indeed appear to obey this rule. The Cohen-Lenstra heuristic has been extended to arbitrary number fields by Cohen-Martinet, but their model looks much more complicated. In this talk, I will explain how to restore the simple rule stated above in the case of class groups of arbitrary number fields by passing to Arakelov class groups. The main difficulty, which I will explain how to resolve, is that Arakelov class groups of number fields typically have infinite automorphism groups, so “inverse proportional to $\#\mathop{\rm Aut} A$” does not appear to make any sense in this context. On the other hand, I will also disprove the Cohen-Martinet conjecture, and will discuss possible ways of fixing it. This is ongoing joint work with Hendrik Lenstra.