A mod $p$ determinant criterion for Cohen--Lenstra convergence of random $p$-adic matrices with prescribed zero patterns

Hyungmin Jang; Jungin Lee; Myungjun Yu; Nathan Kaplan

arxiv: 2606.06993 · v1 · pith:DSYQ77N3new · submitted 2026-06-05 · 🧮 math.NT

A mod p determinant criterion for Cohen--Lenstra convergence of random p-adic matrices with prescribed zero patterns

Hyungmin Jang , Nathan Kaplan , Jungin Lee , Myungjun Yu This is my paper

Pith reviewed 2026-06-27 21:11 UTC · model grok-4.3

classification 🧮 math.NT

keywords Cohen-Lenstra heuristicsp-adic matricescokernel distributionstair-shaped zero patternsrandom matrices over finite fieldssupport patternsnonsingularityband matrices

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The pith

For stair-shaped zero patterns on random p-adic matrices, Cohen-Lenstra convergence of cokernels holds precisely when the matrices are asymptotically nonsingular over F_p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the cokernel distributions of Haar-random matrices over the p-adic integers that have prescribed zero patterns, in the context of the Cohen-Lenstra heuristics. It establishes that the asymptotic behavior of these cokernels is controlled by the reductions of the matrices modulo p, treated as random matrices over the finite field F_p. For families of patterns based on stair-shaped zero regions—such as general stair shapes, band matrices, and those with two symmetric stairs—the convergence to the Cohen-Lenstra distribution is shown to be equivalent to an asymptotic condition that these mod p matrices are nonsingular with high probability. The authors also propose a conjecture extending this to general support patterns and provide counterexamples to similar criteria for higher rank.

Core claim

For several families of support patterns arising from stair-shaped zero regions, including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, convergence of the cokernel distribution to the Cohen-Lenstra distribution is equivalent to an asymptotic nonsingularity condition over F_p.

What carries the argument

The reduction modulo p of the random p-adic matrices, viewed as random matrices over F_p, which governs the asymptotic cokernel distribution.

Load-bearing premise

The asymptotic cokernel distribution of the p-adic matrices is governed by their reductions modulo p as random matrices over F_p.

What would settle it

For a fixed stair-shaped pattern, compute the limit as n goes to infinity of the probability that a random n by n matrix over F_p with that zero pattern is singular; if the limit is positive, the cokernel distribution cannot converge to Cohen-Lenstra.

Figures

Figures reproduced from arXiv: 2606.06993 by Hyungmin Jang, Jungin Lee, Myungjun Yu, Nathan Kaplan.

**Figure 1.** Figure 1: A matrix Xn P MnpZpq for pn, ℓnq “ p6, 3q, phpnq1, hpnq2, hpnq3q “ p1, 3, 5q and pvpnq1, vpnq2, vpnq3q “ p2, 3, 4q . By the definition of σn,i, we see that ℓn is the number of steps in the stair-shaped zero region. Lemma 2.2. If PpdetpXnq ‰ 0q ą 0, then vpnqi ě hpnqi for all 1 ď i ď ℓn. Proof. If vpnqi ă hpnqi , then the first hpnqi columns of Xn are always linearly dependent. □ Theorem 2.3. Let Xn P MnpZp… view at source ↗

**Figure 2.** Figure 2: A matrix Xn P MnpZpq for pn, d, tnq “ p7, 2, 3q . We first recall a result from [4], which will be used in the proof of Theorem 2.9. Lemma 2.8. ([4, Proposition 3.2]) Let Σn “ pσn,1, . . . , σn,nq and Σ1 n “ pσ 1 n,1 , . . . , σ1 n,nq. Suppose that for every n ě 1 and 1 ď i ď n, σn,i Ď σ 1 n,i. Let Xn and X1 n be Haar-random matrices in MnpZpq supported on Σn and Σ1 n , respectively. Then limnÑ8 Ep#Surpcok… view at source ↗

read the original abstract

We study the distribution of cokernels of Haar-random matrices over the $p$-adic integers with prescribed zero patterns, motivated by the Cohen--Lenstra heuristics. A central feature of our approach is that the asymptotic cokernel distribution is governed by the reductions modulo $p$ of these matrices, viewed as random matrices over the finite field $\mathbb{F}_p$. For several families of support patterns arising from stair-shaped zero regions, including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, we show that convergence of the cokernel distribution to the Cohen--Lenstra distribution is equivalent to an asymptotic nonsingularity condition over $\mathbb{F}_p$. We further propose a conjecture for general support patterns and give examples showing that analogous rank-$r$ criteria fail for $r\ge 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a usable equivalence: for stair-shaped zero patterns the p-adic cokernel converges to Cohen-Lenstra precisely when the mod-p reductions are asymptotically nonsingular.

read the letter

The central result is that convergence of the cokernel distribution to the Cohen-Lenstra measure for several families of stair-shaped, band, and symmetric-stair supports is equivalent to an asymptotic nonsingularity condition on the matrices over F_p. This reduces a p-adic question to a finite-field one for those patterns.

The authors handle multiple concrete families and are explicit about the scope: they give counterexamples showing that the analogous rank-r statements fail for r ≥ 1. They also state a conjecture for general support patterns. That combination of positive results plus clear negative examples is the most useful part of the work.

The main soft spot is that the governing assumption—that the p-adic cokernel is controlled by the mod-p reduction—still needs the full proof to judge how cleanly the error terms work and whether the stair-shaped condition is used in an essential way. Only the abstract is visible here, so the details of the nonsingularity criterion and the convergence rates cannot be checked directly.

The paper is aimed at people already working on random-matrix models for class groups or Selmer groups with restricted supports. A reader in that niche will find the equivalence and the conjecture worth seeing. It is precise enough and scoped carefully enough that a serious editor should send it to referees rather than desk-reject.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the cokernel distributions of Haar-random matrices over the p-adic integers with prescribed zero patterns. It establishes that the asymptotic cokernel distribution is governed by the reductions modulo p (viewed as random matrices over F_p). For families including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, it proves that convergence of the cokernel distribution to the Cohen--Lenstra distribution is equivalent to an asymptotic nonsingularity condition over F_p. A conjecture is proposed for general support patterns, and counterexamples are given showing that analogous rank-r criteria fail for r ≥ 1.

Significance. If the equivalences hold, the work provides a concrete, checkable criterion (asymptotic nonsingularity over F_p) that determines when structured p-adic random matrices obey the Cohen--Lenstra heuristics, thereby linking p-adic and finite-field behaviors for several natural families. The explicit counterexamples for higher-rank analogues and the formulation of a general conjecture are useful scoping contributions that clarify the boundaries of the result.

minor comments (2)

[Introduction] The precise formulation of the 'asymptotic nonsingularity condition' (including any dependence on the support pattern) should be stated explicitly in the introduction or as a numbered definition, rather than left implicit from the abstract.
Notation for the support patterns (e.g., how stair-shaped zero regions are indexed) should be introduced with a small diagram or table early in the paper to aid readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. No specific major comments were provided for us to address point by point.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims consist of theorems establishing an equivalence between cokernel convergence to the Cohen-Lenstra distribution and an asymptotic nonsingularity condition over F_p, specifically for stair-shaped, band, and symmetric stair patterns. This equivalence is derived from the reduction of p-adic matrices to their F_p counterparts, presented as a methodological feature rather than a self-referential fit or definition. No load-bearing steps reduce by construction to inputs via self-definition, fitted predictions, or self-citation chains; the result is explicitly scoped with counterexamples for higher-rank cases, indicating an independent mathematical argument self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of Haar measure on p-adic matrices and the Cohen-Lenstra measure on finite abelian p-groups; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption The asymptotic cokernel distribution is governed by the reductions modulo p.
Stated as a central feature in the abstract.

pith-pipeline@v0.9.1-grok · 5687 in / 1191 out tokens · 12349 ms · 2026-06-27T21:11:04.219686+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Cohen and H

H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 33–62

1983
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Friedman and L

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Kahle and A

M. Kahle and A. Newman, Topology and geometry of random 2-dimensional hypertrees, Discrete Comput. Geom. 67 (2022), no. 4, 1229–1244

2022
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D. Y. Kang, J. Lee and M. Yu, Randomp-adic matrices with fixed zero entries and the Cohen–Lenstra distribution, arXiv:2409.01226, to appear in Selecta Math. (N.S.)

work page arXiv
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Distributionofthecokernelsofdeterminantalrow-sparse matrices.arXiv preprint arXiv:2505.11700, 2025

J. Lee and M. Yu, Distribution of the cokernels of determinantal row-sparse matrices, arXiv:2505.11700, to appear in Israel J. Math

work page arXiv
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M´ esz´ aros, Cohen–Lenstra distribution for sparse matrices with determinantal biasing, Int

A. M´ esz´ aros, Cohen–Lenstra distribution for sparse matrices with determinantal biasing, Int. Math. Res. Not. (2025), no. 3, rnae292

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M´ esz´ aros, The 2-torsion of determinantal hypertrees is not Cohen–Lenstra, arXiv:2404.02308, to appear in Israel J

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work page arXiv
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A phase transition for the cokernels of random band matrices over the p-adic integers.arXiv preprint arXiv:2408.13037, 2024

A. M´ esz´ aros, A phase transition for the cokernels of random band matrices over thep-adic integers, arXiv:2408.13037. RANDOMp-ADIC MATRICES WITH PRESCRIBED ZERO PATTERNS 15

work page arXiv
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The homology torsion growth of determinantal hypertrees

A. M´ esz´ aros, The homology torsion growth of determinantal hypertrees, arXiv:2506.14694, to appear in Math. Ann

work page internal anchor Pith review Pith/arXiv arXiv
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The moment problem for random objects in a category.arXiv preprint arXiv:2210.06279, 2022

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work page arXiv
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M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958

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M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398. (H. Jang)Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea Email address:io1278@yonsei.ac.kr (N. Kaplan)Department of Mathematics, University of California, Irvine, CA 92697, USA Email address:nckaplan@math.uci.edu (...

2019

[1] [1]

Cohen and H

H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 33–62

1983

[2] [2]

Friedman and L

E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, in Th´ eorie des Nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 227–239

1987

[3] [3]

Kahle and A

M. Kahle and A. Newman, Topology and geometry of random 2-dimensional hypertrees, Discrete Comput. Geom. 67 (2022), no. 4, 1229–1244

2022

[4] [4]

D. Y. Kang, J. Lee and M. Yu, Randomp-adic matrices with fixed zero entries and the Cohen–Lenstra distribution, arXiv:2409.01226, to appear in Selecta Math. (N.S.)

work page arXiv

[5] [5]

Distributionofthecokernelsofdeterminantalrow-sparse matrices.arXiv preprint arXiv:2505.11700, 2025

J. Lee and M. Yu, Distribution of the cokernels of determinantal row-sparse matrices, arXiv:2505.11700, to appear in Israel J. Math

work page arXiv

[6] [6]

M´ esz´ aros, Cohen–Lenstra distribution for sparse matrices with determinantal biasing, Int

A. M´ esz´ aros, Cohen–Lenstra distribution for sparse matrices with determinantal biasing, Int. Math. Res. Not. (2025), no. 3, rnae292

2025

[7] [7]

M´ esz´ aros, The 2-torsion of determinantal hypertrees is not Cohen–Lenstra, arXiv:2404.02308, to appear in Israel J

A. M´ esz´ aros, The 2-torsion of determinantal hypertrees is not Cohen–Lenstra, arXiv:2404.02308, to appear in Israel J. Math

work page arXiv

[8] [8]

A phase transition for the cokernels of random band matrices over the p-adic integers.arXiv preprint arXiv:2408.13037, 2024

A. M´ esz´ aros, A phase transition for the cokernels of random band matrices over thep-adic integers, arXiv:2408.13037. RANDOMp-ADIC MATRICES WITH PRESCRIBED ZERO PATTERNS 15

work page arXiv

[9] [9]

The homology torsion growth of determinantal hypertrees

A. M´ esz´ aros, The homology torsion growth of determinantal hypertrees, arXiv:2506.14694, to appear in Math. Ann

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

The moment problem for random objects in a category.arXiv preprint arXiv:2210.06279, 2022

W. Sawin and M. M. Wood, The moment problem for random objects in a category, arXiv:2210.06279

work page arXiv

[11] [11]

M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958

2017

[12] [12]

M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398. (H. Jang)Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea Email address:io1278@yonsei.ac.kr (N. Kaplan)Department of Mathematics, University of California, Irvine, CA 92697, USA Email address:nckaplan@math.uci.edu (...

2019