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arxiv: 2605.23526 · v1 · pith:VIXI6JHAnew · submitted 2026-05-22 · 🧮 math.AT · math.NT

Top-dimensional rational cohomology of the congruence subgroup Gamma_(0,n)^+(p)

Tatiana Abdelnaim This is my paper

Pith reviewed 2026-05-25 02:32 UTC · model grok-4.3

classification 🧮 math.AT math.NT
keywords congruence subgrouprational cohomologySL_n(Z)vanishing theoremtop-dimensional cohomologyarithmetic grouplevel p
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The pith

The top rational cohomology of Γ_{0,n}^+(p) vanishes in degree binom(n,2) for p in {2,3,5,7,13} when n≥3 and for all p≤6n-14.

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

This paper studies the top-dimensional rational cohomology of the congruence subgroup Γ_{0,n}^+(p) inside SL_n(Z). It proves vanishing of this group for the listed small primes when the matrix size n is at least three, and vanishing whenever the level p is at most 6n minus 14. The paper also establishes that the same cohomology group is nonzero when n equals two for any prime, and when n equals three except for those five small primes. A sympathetic reader would care because these results clarify the structure of cohomology for arithmetic groups in their highest possible degree.

Core claim

We prove that the top-dimensional cohomology group H^{binom(n,2)}(Γ_{0,n}^+(p);Q) vanishes for p in {2,3,5,7,13} if n≥3, as well as for p≤6n-14. Additionally, we prove a non-vanishing result, showing that this cohomology group is nonzero for n=2 for every prime p, and for n=3 for all primes p not in {2,3,5,7,13}.

What carries the argument

The level-p congruence subgroup Γ_{0,n}^+(p) inside SL_n(Z) whose first column is congruent to (*,0,...,0)^t mod p, whose top-degree rational cohomology is shown to vanish or not under the stated conditions on p and n.

If this is right

  • The top cohomology vanishes for every prime p at most 6n-14, regardless of whether p belongs to the listed set.
  • For n=2 the top cohomology group is nonzero for every prime p.
  • For n=3 the top cohomology group is nonzero precisely when p avoids the five small primes {2,3,5,7,13}.
  • Vanishing holds uniformly once n is large enough relative to a fixed p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar vanishing might hold for other congruence subgroups defined by conditions on different columns or rows.
  • The cutoff p≤6n-14 suggests a linear dependence between matrix size and level that could be tested for other arithmetic groups.
  • The exceptional primes {2,3,5,7,13} may mark a transition in representation-theoretic or modular-form behavior that appears only for n=3.

Load-bearing premise

The stated congruence condition on the first column defines a subgroup of SL_n(Z) whose top-dimensional rational cohomology can be analyzed by the methods of the paper.

What would settle it

An explicit computation of H^{binom(4,2)}(Γ_{0,4}^+(2);Q) that yields a nonzero group would falsify the vanishing claim for n=4 and p=2.

Figures

Figures reproduced from arXiv: 2605.23526 by Tatiana Abdelnaim.

Figure 1
Figure 1. Figure 1: The order complex whose poset is P± n (Fp)\T ± n (Fp) 5.2 Determinant-1 partial frames Let p be a prime. In this section, we will describe the actions of P± n (Fp) and Tk on MD(F n p )k for all n. We recall that Tk = Σk+1 ⋉ {−1, 1} k+1 and MD(F n p )k =  B = (v0| . . . |vk) [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The order complex whose poset is P± n (Fp)\T ± n (Fp), with the subposet Y enclosed in red. We observe that O(Y ) has a cone point at [(Un−1, ±ωn−1)]. Thus, O(Y ) is contractible. Moreover, Link (Vn−1, ±ω ′ n−1 ) [PITH_FULL_IMAGE:figures/full_fig_p089_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: P± n (Fp)\T ± n (Fp), with (P± n (Fp)\T ± n (Fp))>V enclosed in red. This implies that Hk [PITH_FULL_IMAGE:figures/full_fig_p093_3.png] view at source ↗
read the original abstract

Let $\Gamma_{0,n}^+(p)\subset \mathrm{SL}_n(\mathbb{Z})$ be the congruence subgroup of level-$p$ whose first column is of the form $(*,0,\dots,0)^t\bmod p$. We prove that the top-dimensional cohomology group $H^{\binom{n}{2}}(\Gamma_{0,n}^+(p);\mathbb{Q})$ vanishes for $p\in\{2,3,5,7,13\}$ if $n \geq 3$, as well as for $p \leq 6n-14$. Additionally, we prove a non-vanishing result, showing that this cohomology group is nonzero for $n = 2$ for every prime $p$, and for $n=3$ for all primes $p \notin \{2,3,5,7,13\}$.

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.

Referee Report

0 major / 2 minor

Summary. The paper defines the level-p congruence subgroup Γ_{0,n}^+(p) ⊂ SL_n(ℤ) by the condition that its first column is congruent to (*,0,…,0)^t mod p. It proves that the top-dimensional rational cohomology H^{binom(n,2)}(Γ_{0,n}^+(p);ℚ) vanishes when p ∈ {2,3,5,7,13} and n ≥ 3, as well as when p ≤ 6n−14. It also proves non-vanishing for n=2 and every prime p, and for n=3 and all primes p ∉ {2,3,5,7,13}.

Significance. If the stated vanishing and non-vanishing ranges hold, the results supply explicit arithmetic conditions under which the top rational cohomology of these congruence subgroups is zero or nonzero. Such concrete ranges are useful for testing conjectures on the cohomology of arithmetic groups and for relating the top-degree behavior to the virtual cohomological dimension of SL_n(ℤ).

minor comments (2)
  1. The abstract states the definition of Γ_{0,n}^+(p) but does not indicate whether this notation is standard or newly introduced; a brief comparison with the usual congruence subgroups Γ_0(p) or Γ_1(p) would clarify the construction.
  2. The exponent binom(n,2) is used without explicit justification that it equals the virtual cohomological dimension of the group in question; a short reference or sentence recalling this fact would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report. The summary accurately reflects the main theorems of the paper. No major comments are listed in the report, and the recommendation is uncertain without further elaboration on any concerns. We therefore have no specific points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states direct vanishing and non-vanishing theorems for the top-degree rational cohomology of an explicitly defined congruence subgroup Γ_{0,n}^+(p) of SL_n(Z). The subgroup is defined by a standard congruence condition on the first column, which is closed under the group law and independent of the cohomology results. No equations, parameters, or self-citations are shown reducing the claimed statements to the inputs by construction. The derivation chain consists of standard group-cohomology techniques applied to this fixed object, with no fitted inputs renamed as predictions or ansatzes smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; none can be identified.

pith-pipeline@v0.9.0 · 5672 in / 1183 out tokens · 23179 ms · 2026-05-25T02:32:04.020363+00:00 · methodology

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