Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distribution of the cokernels of determinantal row-sparse matrices

Jungin Lee, Myungjun Yu

Abstract

We study the distribution of the cokernels of random row-sparse integral matrices $A_n$ according to the determinantal measure from a structured matrix $B_n$ with a parameter $k_n \ge 3$. Under a mild assumption on the growth rate of $k_n$, we prove that the distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ converges to that of Cohen--Lenstra for every prime $p$. Our result extends the work of A. Mészáros which established convergence to the Cohen--Lenstra distribution when $p \ge 5$ and $k_n=3$ for all positive integers $n$.

Distribution of the cokernels of determinantal row-sparse matrices

Abstract

We study the distribution of the cokernels of random row-sparse integral matrices according to the determinantal measure from a structured matrix with a parameter . Under a mild assumption on the growth rate of , we prove that the distribution of the -Sylow subgroup of the cokernel of converges to that of Cohen--Lenstra for every prime . Our result extends the work of A. Mészáros which established convergence to the Cohen--Lenstra distribution when and for all positive integers .
Paper Structure (7 sections, 34 theorems, 235 equations)

This paper contains 7 sections, 34 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Decomposition of the moments into sums of probabilities over nearly-uniform balls
  3. Computing the moments: sum over B(n,G)
  4. Computing the moments: bounding the sum over B1(n,H)
  5. Computing the moments: bounding the sum over B2(n,H)
  6. Convergence of moments and convergence to the Cohen--Lenstra distribution
  7. The limiting distribution of cok(An) is not Cohen--Lenstra when kn is a constant

Key Result

Theorem 1.1

(Theorem thm: cokernel distribution theorem 1) Let $G$ be a finite abelian group and $\mathcal{P}$ be a finite set of primes including those dividing $|G|$. Assume that a sequence $(k_n)_{n=1}^{\infty}$ satisfies the following: Then

Theorems & Definitions (65)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 55 more