Random integral matrices and the Cohen-Lenstra heuristics

We prove that given any ǫ > 0, random integral n×n matrices with independent entries that lie in any residue class modulo a prime with probability at most 1 -ǫ have cokernels asymptotically (as n → ∞) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture as the distribution for class groups of imaginary quadratic fields.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 × (n + u) matrices.