2018/12/31 by Gilyoung Cheong, Yifeng Huang, Cheong, Gilyoung +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1812.11728
openalex publication_date 2018/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (R, \mathfrakm) be a complete discrete valuation ring with the finite residue field R/\mathfrakm = \mathbbFq. Given a monic polynomial P(t) ∈ R[t] whose reduction modulo \mathfrakm gives an irreducible polynomial P(t) ∈ \mathbbFq[t], we initiate the investigation of the distribution of coker(P(A)), where A ∈ Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n × n R-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the P-part of a random matrix A ∈ Matn(\mathbbFq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., ℤp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of ℚ with a probability inversely proportional to |Autℤ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between Matn(R) and Matn(\mathbbFq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(\mathbbFq) with respect to the uniform distribution.