[Paper Review] Cohen-Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields

Gilyoung Cheong, Yifeng Huang|arXiv (Cornell University)|Dec 31, 2018

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper establishes a connection between Cohen-Lenstra distributions in number theory and random matrix theory over complete discrete valuation rings with finite residue fields. By analyzing the cokernel of random matrices over such rings using Haar measure, the authors generalize Friedman and Washington's results and prove that the limiting distribution of cokernels matches the Cohen-Lenstra heuristic, with probabilities inversely proportional to automorphism group sizes.

ABSTRACT

Let $(R, \mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/\mathfrak{m} = \mathbb{F}_{q}$. Given a monic polynomial $P(t) \in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial $\bar{P}(t) \in \mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $\mathrm{coker}(P(A))$, where $A \in \mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $\mathrm{Mat}_{n}(R)$ of $n imes n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $\bar{P}$-part of a random matrix $\bar{A} \in \mathrm{Mat}_{n}(\mathbb{F}_{q})$ 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., $\mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $\mathbb{Q}$ with a probability inversely proportional to $|\mathrm{Aut}_{\mathbb{Z}}(H)|$. 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 $\mathrm{Mat}_{n}(R)$ and $\mathrm{Mat}_{n}(\mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $\mathrm{Mat}_{n}(R)$ into problems about a random matrix in $\mathrm{Mat}_{n}(\mathbb{F}_{q})$ with respect to the uniform distribution.

Motivation & Objective

• To generalize Friedman and Washington's results on the distribution of cokernels of random matrices over $\mathbb{Z}_p$ to more general complete discrete valuation rings with finite residue fields.
• To relate the distribution of $\overline{P}$-parts of random matrices over $\mathbb{F}_q$ to the Cohen-Lenstra conjecture on class group distributions.
• To provide a new random matrix-theoretic framework that heuristically supports the Cohen-Lenstra conjecture via cokernel statistics over $R[[t]]$.
• To establish a universal limiting behavior of cokernel distributions as matrix size $n \to \infty$, matching Cohen-Lenstra probabilities.
• To clarify the discrepancy between cokernel distributions of matrices over $\mathbb{F}_q$ and over $\mathbb{F}_q[[t]]$, despite both converging to the same limiting distribution.

Proposed method

• Use Haar measure on $\mathrm{Mat}_n(R)$ for a complete discrete valuation ring $R$ with finite residue field $\mathbb{F}_q$, to define a probability distribution on $n \times n$ matrices.
• Leverage the projection $R \to \mathbb{F}_q$ to relate cokernels of matrices over $R$ to those over $\mathbb{F}_q$, translating Haar-random problems into uniform-random problems.
• Apply combinatorial techniques to compute the probability that $\mathrm{coker}(P(A))$ is isomorphic to a given finite $R$-module, particularly focusing on $P(t)$ irreducible modulo $\mathfrak{m}$.
• Use generating functions and zeta functions $\hat{\boldsymbol{Z}}(\{P_j\}, \mathbf{x}, 1)$ to compute limiting probabilities of cokernel structures.
• Prove that the limiting distribution of $\mathrm{coker}(A)$ for $A$ Haar-random in $\mathrm{Mat}_n(R)$ matches the Cohen-Lenstra measure, with probability $\frac{1}{|\mathrm{Aut}(H)|} \prod_{i=1}^\infty (1 - p^{-i})$.
• Compare two different cokernel probabilities: one for $\overline{A} \in \mathrm{Mat}_n(\mathbb{F}_q)$ via $\mathrm{coker}(\overline{A} - tI_n)$, and another for $A \in \mathrm{Mat}_n(\mathbb{F}_q[[t]])$, showing both converge to the same Cohen-Lenstra limit.

Experimental results

Research questions

• RQ1 How does the distribution of $\mathrm{coker}(P(A))$ for Haar-random $A \in \mathrm{Mat}_n(R)$ relate to the Cohen-Lenstra conjecture when $R$ is a complete DVR with finite residue field?
• RQ2 Can the classical Friedman-Washington result over $\mathbb{Z}_p$ be generalized to arbitrary complete DVRs with finite residue fields?
• RQ3 What is the relationship between the $\overline{P}$-part of a random matrix over $\mathbb{F}_q$ and the Cohen-Lenstra distribution of $p$-parts of class groups?
• RQ4 Why do two different random matrix models—over $\mathbb{F}_q$ and over $\mathbb{F}_q[[t]]$—yield the same limiting cokernel distribution despite differing finite-size probabilities?
• RQ5 How do the automorphism group sizes and zeta functions encode the probability of a given cokernel structure in the limit?

Key findings

• The limiting distribution of $\mathrm{coker}(A)$ for $A$ Haar-random in $\mathrm{Mat}_n(R)$, where $R$ is a complete DVR with residue field $\mathbb{F}_q$, matches the Cohen-Lenstra measure: $\mathrm{Prob}(\mathrm{coker}(A) \simeq H) = \frac{1}{|\mathrm{Aut}(H)|} \prod_{i=1}^\infty (1 - p^{-i})$ as $n \to \infty$.
• For $P(t) \in R[t]$ with irreducible reduction $\overline{P}(t) \in \mathbb{F}_q[t]$, the probability that $\mathrm{coker}(P(A)) \simeq H$ converges to $\frac{1}{|\mathrm{Aut}(H)|} \prod_{i=1}^\infty (1 - q^{-i \deg(P)})$ as $n \to \infty$, generalizing Friedman and Washington's $p$-adic result.
• The distribution of the $\overline{P}$-part of a uniformly random matrix $\overline{A} \in \mathrm{Mat}_n(\mathbb{F}_q)$ converges to the Cohen-Lenstra measure on $\mathbb{F}_q[t]$-modules, matching the heuristic for class groups of function fields.
• The authors prove that the limiting probability for $\mathrm{coker}(\overline{A} - tI_n) \simeq H$ over $\mathbb{F}_q$ is $\frac{1}{|\mathrm{Aut}_{\mathbb{F}_q[t]}(H)|} \prod_{i=1}^n (1 - q^{-i})$ when $n \geq \dim_{\mathbb{F}_q}(H)$, and zero otherwise.
• Despite differing finite-size probabilities, both the $\mathbb{F}_q$-matrix model and the $\mathbb{F}_q[[t]]$-matrix model yield the same limiting cokernel distribution, demonstrating a form of universality.
• The zeta function $\hat{\boldsymbol{Z}}(\{P_j\}, \mathbf{x}, 1)$ encodes the limiting probability of specified $P_j$-parts in the cokernel, and its product form confirms the independence of parts at distinct primes in the limit.