Cohen-Lenstra sums over local rings

Abstract

We study series of the form ∑ M |Aut R (M)| -1 |M| -u , where R is a commutative local ring, u is a non-negative integer, and the summation extends over all finite R-modules M, up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If R has additional properties, we will relate the above sum to a limit of zeta functions of the free modules R n , where these zeta functions count R-submodules of finite index in R n . In particular we will show that this is the case for the group ring ℤ p [C p k ] of a cyclic group of order p k over the p-adic integers. Thereby we are able to prove a conjecture from [5], stating that the above sum corresponding to R=ℤ p [C p k ] and u=0 converges. Moreover we consider refined sums, where M runs through all modules satisfying additional cohomological conditions.