RTG Seminar on Number Theory Seminar

In the famous 1983 paper, when studying the heuristic distribution of class groups of imaginary quadratic fields, Cohen and Lenstra considered the weighting of a finite abelian group G with a weight proportional to 1/#Aut(G). More generally, for a given Dedekind domain R, they studied the statistics of finite-cardinality R-modules under the 1/#Aut weighting. They defined a "zeta" function \sum_M 1/#Aut(M) |M|^{-s} summing over all finite-cardinality R-modules, and they showed that it is an infinite product involving the Dedekind zeta function of R. In this talk, we discuss this Cohen--Lenstra zeta function defined for other families of rings, where the known results are organized in terms of the Krull dimension. The "nodal singularity" R=Fq[u,v]/(uv) is a surprisingly interesting example that gives rise to a peculiar q-series, which we will describe in more detail. Speaker(s): Yifeng Huang (University of Michigan)