RANDOM INTEGRAL MATRICES AND THE COHEN-LENSTRA HEURISTICS

摘要

We prove that given any epsilon > 0, random integral n x n matrices with independent entries that lie in any residue class modulo a prime with probability at most 1-epsilon have cokernels asymptotically (as n -> infinity) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture to be the distribution for class groups of imaginary quadratic fields. This shows the Cohen-Lenstra distribution is universal for finite abelian groups given by generators and random relations-that the distribution of quotients does not depend on the way in which we choose (sufficiently nice) relations. This is a refinement of a result on the distribution of ranks of random matrices with independent entries in Z/pZ. This is interesting especially in light of the fact that these class groups are naturally cokernels of square matrices. We also prove the analogue for n x (n + u) matrices.