arXiv subjects

Gilyoung Cheong

Publications and source records attributed to Gilyoung Cheong.

8 recordsLinked to original sources

arXiv2023

The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of $p$-adic integers whose entries are independent. Previously, Wood showed that regardless of the distribution of $X_n$, as long as each entry of $X_n$ is not too concentrated on a single residue modulo $p$, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen--Lenstra distribution, as $n \rightarrow \infty$. In this paper, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$, where $B_n$ is a random $n \times n$ matrix over $\mathbb{Z}_p$. We show that for every fixed $n$ and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$, we can explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ when $B_n$ is a Haar-random matrix. Using this, we also show that for specific choices of $A_n$ a much wider class of random matrices $B_n$ gives the same distribution of $\mathrm{cok}(P(X_n))$. For the Haar-random $B_n$, we deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$.

math.NT↗

arXiv2023

The distribution of the cokernel of a polynomial evaluated at a random integral matrix

Given a prime $p$, let $P(t)$ be a non-constant monic polynomial in $t$ over the ring $\mathbb{Z}_{p}$ of $p$-adic integers. Let $X_{n}$ be an $n \times n$ random matrix over $\mathbb{Z}_{p}$ with independent entries that lie in any residue class modulo $p$ with probability at most $1 - ε$ for a fixed real number $0 < ε< 1$. We prove that as $n \rightarrow \infty$, the distribution of the cokernel $\mathrm{cok}(P(X_{n}))$ of $P(X_{n})$ converges to the distribution given by a finite product of some explicit measures that resemble Cohen--Lenstra measures. For example, the random matrix $X_{n}$ can be taken as a Haar-random matrix or a uniformly random $(0,1)$-matrix. We consider the distribution of $\mathrm{cok}(P(X_{n}))$ as a distribution of modules over $\mathbb{Z}_{p}[t]/(P(t))$, which gives us a clearer formulation in comparison to considering the distribution as that of abelian groups. For the proof, we first reduce our problem into a problem over $\mathbb{Z}/p^{k}\mathbb{Z}$, for large enough positive integer $k$, in place of $\mathbb{Z}_{p}$. Then we use a result of Sawin and Wood to reduce our problem into another problem of computing the limit of the expected number of surjective $(\mathbb{Z}/p^{k}\mathbb{Z})[t]/(P(t))$-linear maps from $\mathrm{cok}(P(X_{n}))$ modulo $p^{k}$ to a fixed finite size $(\mathbb{Z}/p^{k}\mathbb{Z})[t]/(P(t))$-module $G$. To estimate the expected number and compute the desired limit, we carefully adopt subtle techniques developed by Wood, which were originally used to compute the asymptotic distribution of the $p$-part of the sandpile group of a random graph.

math.NT↗

arXiv2023

Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization of counting solutions to $P(X) = 0$ as the distribution of the cokernel $\mathrm{cok}(P(X))$ of $P(X)$ up to isomorphisms, where $X$ is a uniform random matrix in $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$. This distribution involves an explicit formula when we fix the residue class of $X$ modulo $p$. We prove this conjecture for the special case when the image of $P(t)$ in $\mathbb{F}_{p}[t]$ modulo $p$ is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution. Our proof involves algebraic and combinatorial arguments in linear algebra over $\mathbb{Z}/p^{k}\mathbb{Z}$ and builds upon a previous work of Cheong and Kaplan.

math.CO↗

arXiv2022

Jordan--Landau theorem for matrices over finite fields

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors when $n$ is large. We also estimate the analogous probability that $f_{A}(t)$ has $r$ irreducible factors counting with multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and the error term $O((\log n)^{r-2}n^{-1})$, whose implied constant only depends on $r$ but not on $q$ nor $n$, coincide with the probability that a random permutation on $n$ letters is a product of $r$ disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree $n$ monic polynomial in $\mathbb{F}_{q}[t]$ is square-free with $r$ irreducible factors and the analogous probability that the polynomial has $r$ irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of $n \times n$ matrices with a fixed characteristic polynomial over $\mathbb{F}_{q}$.

math.CO↗

arXiv2022

Generalizations of results of Friedman and Washington on cokernels of random $p$-adic matrices

Let $p$ be prime and $X$ be a Haar-random $n \times n$ matrix over $\mathbb{Z}_{p}$, the ring of $p$-adic integers. Let $P_{1}(t), \dots, P_{l}(t) \in \mathbb{Z}_{p}[t]$ be monic polynomials of degree at most $2$ whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. For each $j$, let $G_{j}$ be a finite module over $\mathbb{Z}_{p}[t]/(P_{j}(t))$. We show that as $n$ goes to infinity, the probabilities that $\mathrm{cok}(P_{j}(X)) \simeq G_{j}$ are independent, and each probability can be described in terms of a Cohen-Lenstra distribution. We also show that for any fixed $n$, the probability that $\mathrm{cok}(P_{j}(X)) \simeq G_{j}$ for each $j$ is a constant multiple of the probability that that $\mathrm{cok}(P_{j}(\bar{X})) \simeq G_{j}/pG_{j}$ for each $j$, where $\bar{X}$ is an $n \times n$ uniformly random matrix over $\mathbb{F}_{p}$. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.

math.NT↗

arXiv2021

Betti and Hodge numbers of configuration spaces of a punctured elliptic curve from its zeta functions

Given an elliptic curve $E$ defined over $\mathbb{C}$, let $E^{\times}$ be an open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th Betti number of the unordered configuration space $\mathrm{Conf}^{n}(E^{\times})$ of $n$ points on $E^{\times}$ appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the $\mathbb{F}_{q}$-point counts of $\mathrm{Conf}^{n}(E^{\times})$, which can be obtained from the zeta function of $E$ over a finite field $\mathbb{F}_{q}$. We show that the mixed Hodge structure of the $i$-th singular cohomology group $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$ with complex coefficients is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$.

math.AG↗

arXiv2020

Pólya enumeration theorems in algebraic geometry

We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^{n}/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_{n}$ acting on $X^{n}$ by permuting coordinates. Our result is completely axiomatic: in a general setting, given an endomorphism on the cohomology $H^{\bullet}(X)$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $H^{\bullet}(X^{n})^{G}$ to that of the given endomorphism on $H^{\bullet}(X)$ in the presence of the Künneth formula with respect to a cup product. For example, when $X$ is a compact manifold, we take the Lefschetz series given by the singular cohomology with rational coefficients. On the other hand, when $X$ is a projective variety over a finite field $\mathbb{F}_{q}$, we use the $l$-adic étale cohomology with a suitable choice of prime number $l$. We also explain how our formula generalizes the Pólya enumeration theorem, a classical theorem in combinatorics that counts colorings of a graph up to given symmetries, where $X$ is taken to be a finite set of colors. When $X$ is a smooth projective variety over $\mathbb{C}$, our formula also generalizes a result of Cheah that relates the Hodge numbers of $X^{n}/G$ to those of $X$. We will also see that our result generalizes the following facts: 1. the generating function of the Poincaré polynomials of symmetric powers of a compact manifold $X$ is rational; 2. the generating function of the Hodge-Deligne polynomials of symmetric powers of a smooth projective variety $X$ over $\mathbb{C}$ is rational; 3. the zeta series of a projective variety $X$ over $\mathbb{F}_{q}$ is rational. We also prove analogous rationality results when we replace $S_{n}$ with $A_{n}$, alternating groups.

math.AG↗

arXiv2019

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

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 $n \times n$ $R$-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.

math.NT↗