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

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read 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.

desk verdict 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 →

arxiv 2606.06993 v1 pith:DSYQ77N3 submitted 2026-06-05 math.NT

Hyungmin Jang , Nathan Kaplan , Jungin Lee , Myungjun Yu This is my paper · ORCID
classification math.NT
keywords Cohen-Lenstraheuristicsp-adicmatricescokerneldistributionstair-shapedzeropatternsrandomoverfinitefieldssupportnonsingularityband
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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.

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.

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Extended reading notes

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.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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)
  1. [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.
  2. 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

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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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (1)
  • domain assumption The asymptotic cokernel distribution is governed by the reductions modulo p.
    Stated as a central feature in the abstract.

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Pith. "Pith review of A mod $p$ determinant criterion for Cohen--Lenstra convergence of random $p$-adic matrices with prescribed zero patterns." pith.science (2026). https://pith.science/paper/DSYQ77N3

@misc{pith2026260606993,
  author       = {Pith},
  title        = {Pith review of: A mod $p$ determinant criterion for Cohen--Lenstra convergence of random $p$-adic matrices with prescribed zero patterns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSYQ77N3}},
  note         = {Machine review of arXiv:2606.06993}
}
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 $r\ge 1$.

Figures

Figures reproduced from arXiv: 2606.06993 by the authors.

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 MnpZpq be the H… 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#SurpcokpXnq, Gqq … view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Universality for cokernels of partially random integral matrices

    math.PR 2026-07 conditional novelty 7.0 of 10

    The Cohen-Lenstra distribution for cokernels of random p-adic matrices persists when up to a linear fraction of entries per row and column are non-random, when within-column dependence is allowed, and for band matrice...

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Works this paper leans on

12 extracted references · 6 canonical work pages · cited by 1 Pith paper

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