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 #AutA, 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 #AutA” 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.