REVIEW 5 major objections 5 minor 4 references
On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper proves that for odd p the E2 page of the LHS spectral sequence for natural Sylow p-subgroup extensions of GL_2(Z/p^n) and SL_2(Z/p^n) is independent of n>1, and gives an explicit stable-elements description of the cohomology…
desk verdict A promising approach to H^*(GL_2(Z/p^n),F_p) whose main theorem rests on an unproven powerfulness claim; still worth refereeing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery has two parts. First, the congruence kernels L_n and K_n are treated as pro-p groups and shown to be powerful and Ω-extendable; a classification theorem for powerful pro-p groups then yields H^*(K_n)=Λ(x_1,x_2,x_3)⊗F_p[y_1,y_2,y_3] and H^*(L_n)=Λ(x_1,x_2,x_3,x_4)⊗F_p[y_1,y_2,y_3,y_4] with |x_i|=1 and |y_i|=2. The second part is the Lyndon–Hochschild–Serre spectral sequence for the extensions whose quotient is C_p; the key mechanism is that the C_p-action on $H^{1}$ is the same for every n, so the exterior-symmetric shape of the cohomology is C_p-equivariantly independent of n. For the stable elements, the machinery is fusion systems: the paper reduces the inverse limit over all F-centric, p-radical subgroups to just the Sylow subgroup itself and the normal subgroup K_n (or L_n), using the fact that a normal subgroup of the fusion system lies in every such subgroup.
What would settle it
For p=3 and n=3, take h=1+9X in K_3 with X=[[0,1],[1,0]]. The proof's candidate cube root g=1+3X is not in K_3 because det(1+3X)≡19 mod 27, not 1, so one can exhaustively test whether any Y≡X mod 3 in M_2(Z/27) with det(1+3Y)≡1 mod 27 satisfies (1+3Y)^3≡h. The proof requires such a root for every X, so a single X with no such Y would refute Proposition 3.4 and, with it, the foundation of Theorem 4.2.
Extended reading notes
Core claim
The discovery is a stability statement for the second page of a spectral sequence. For an odd prime p and any n>1, write M=$H^{1}$(K_n,F_p) and N=$H^{1}$(L_n,F_p), where K_n and L_n are the kernels of reduction modulo p in SL_2(Z/p^n) and GL_2(Z/p^n). The paper argues that H^*(K_n,F_p)=S(M)⊗Λ(M) and H^*(L_n,F_p)=S(N)⊗Λ(N) as C_p-modules, with C_p acting by conjugation through the quotient S_p(1,·)≅C_p. From this it follows that the E2 page of the LHS spectral sequence attached to 1→K_n→S_p(n,SL)→C_p→1 is isomorphic to the one for n=2 for all n>2, and likewise for GL. The paper's Theorem 5.14 then describes the stable elements in H^*(S_p(n,SL),F_p) and H^*(S_p(n,GL),F_p) as the intersection of two invariant subrings, one from the Sylow normalizer and one from restriction to K_n or L_n; because only those two subgroups survive the F-centric, p-radical reduction, the description has the same shape for every n.
Load-bearing premise
The load-bearing step is the claim that K_n and L_n are powerful pro-p groups with the stated generator counts for all n>1, which the proof of Proposition 3.4 establishes by writing top-level elements as p-th powers whose determinants are checked only modulo p, not modulo p^n; if that step fails, the cohomology formula and the n-independence of the E2 page lose their foundation.
Editorial extensions
If this is right
- The E2 page of the LHS spectral sequence for the Sylow-p-subgroup extensions is the same for all n>2, so computations of that page for n=2 transfer verbatim to every n.
- H^*(K_n,F_p) and H^*(L_n,F_p) are exterior-algebra-tensor-symmetric-algebra rings generated in degrees 1 and 2, for every n>1, giving explicit generators for the E2 page.
- Stable elements in H^*(S_p(n,SL),F_p) are exactly the classes invariant under the Sylow normalizer quotient N_{SL_2(Z/p^n)}(S)/S whose restriction to K_n is invariant under N_{SL_2(Z/p^n)}(K_n)/K_n; the analogous statement holds for GL_2 with L_n.
- Only the Sylow subgroup and K_n (respectively L_n) need to be tested among F-centric p-radical subgroups, because K_n and L_n are normal in the fusion system.
- The paper does not claim that H^*(GL_2(Z/p^n),F_p) is isomorphic to H^*(GL_2(Z/p^{n+1}),F_p); the remaining n-dependence lives in differentials and extension problems of the spectral sequence.
Reading between the lines
- The paper leaves implicit that the computational bottleneck after its theorem is differentials and extension problems, not the E2 page; a natural next computation is the full spectral sequence for p=3, n=3 versus n=2.
- An extension the paper does not claim is that the same E2-page-independence strategy is plausible for GL_m and SL_m once analogous powerfulness and generator-count results for the congruence kernels are established.
- The determinant check in the powerfulness proof is a soft spot to test directly: prove or disprove surjectivity of the p-th power map from K_n to K_{n,n-1} for p=3, n=3.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the mod-p cohomology of GL2(Z/p^n) and SL2(Z/p^n) for odd primes p. It sets up Lyndon-Hochschild-Serre spectral sequences for the Sylow p-subgroup extensions with kernels L_n and K_n, and claims that the E2 page is independent of n (Theorem 4.2 and Corollary 4.3). This is intended to reduce the computation to the case n=2, together with a stable-elements description via fusion systems (Theorem 5.14). The main technical steps are: proving that K_n and L_n are powerful pro-p groups (Proposition 3.4), proving they are Ω-extendable (Lemma 3.6), applying the Minh-Symonds classification to obtain explicit cohomology rings (Theorem 3.8), and then identifying the C_p-module structure on these rings (Theorem 4.2). The stable-elements section then identifies the F-centric p-radical subgroups and gives an intersection formula for the stable elements.
Significance. If the E2-page independence result were established, it would be a substantial simplification of the computation of H*(GL2(Z/p^n),F_p) and H*(SL2(Z/p^n),F_p), reducing the spectral-sequence input to the n=2 case. The fusion-systems description of stable elements is a fresh approach to this problem. The paper also explicitly acknowledges the limitation that the result concerns only the E2 page and not convergence or lifting of generators, which is measured and appropriate. However, the central claims rest on proofs that currently contain serious gaps; the powerfulness and Ω-extendability arguments are not valid as written, and the C_p-module decomposition in Theorem 4.2 is not established. These issues are load-bearing for the main theorems.
major comments (5)
- [§3, Proposition 3.4] The proof that K_n is powerful contains a determinant miscalculation. To show K_{n,n-1} ⊆ K_n^p, the text sets h=1+p^{n-1}X and g=1+pY with Y=p^{n-3}A, and verifies only det(1+pY) ≡ 1 mod p. Membership in K_n requires det(1+pY)=1 in Z/p^n, i.e., equality modulo p^n. For p=3, n=3, X=diag(1,-1), the element h=1+9X lies in K_3 (det = -80 ≡ 1 mod 27), while the proposed g=1+3X has det = -8 ≡ 19 mod 27, so g∉K_3. Thus the displayed p-th-power computation does not establish K_{3,2} ⊆ K_3^3, and the induction proving [K_n,K_n] ⊆ K_n^p collapses. Additionally, the statement 'det(X)=1 implies tr(X)=2' is false; the condition det(1+p^{n-1}X)=1 mod p^n gives tr(X) ≡ 0 mod p. The powerfulness of K_n is therefore unproven.
- [§3, Theorem 3.8] The proof asserts m=d=k=3 (and 4 for L_n) without justification. The text states 'We use Ω ∼= K_{n,n-1}' and then 'This gives m=d=k=3', but m=d(K_n), d=d(Φ(K_n)), and k=d(Ω_1(K_n)) are independent numerical invariants that require proof. In particular, d(K_n) is the dimension of K_n/Φ(K_n) over F_p, and no computation of this quotient is supplied. The identification Ω_1(K_n) ≅ K_{n,n-1} is also asserted rather than proved. Since the applicability of the Minh-Symonds classification (Theorem 3.7) depends on these equalities, Theorem 3.8 is not established.
- [§3, Lemma 3.6] The lift B=1+p^{n-1}X into K_{n+1} is not shown to satisfy the defining determinant condition. For B to lie in K_{n+1} (or even in K_{n+1,n-1}), one needs det(B) ≡ 1 mod p^{n+1}. For arbitrary X∈M_2(Z/p^n), det(B) ≡ 1 + p^{n-1} tr(X) mod p^{n+1} (for n≥3 the remaining terms vanish), which is not generally congruent to 1. Hence B need not be a valid element of K_{n+1}, and the central-extension construction in Definition 3.5 is not realized. The binomial-coefficient estimate for B^{p^2} also contains an unproved claim about the exponent '3+nk-2k' being greater than n+1; this is false for small k (e.g., n=3, k=1 gives equality). Thus Ω-extendability is not proven.
- [§4, Theorem 4.2] The proof that H^*(K_n) ≅ S(M)⊗Λ(M) as C_p-modules is not convincing. The claim that H^1(K_2) ≅ H^1(K_n) as C_p-modules is justified only by the isomorphism K_{2,1} ≅ K_{n+1,n} of the bottom layers; however H^1(K_n) ≅ Hom(K_n/Φ(K_n), F_p), and no argument shows that the C_p-conjugation action on the Frattini quotient is independent of n. The treatment of H^2 relies on an injection Λ^2 H^1 ↪ H^2, cites [SW] for this, but then asserts without proof that Λ^2 H^1 'must survive onto the E_∞ page', that H^2 decomposes as Λ^2 H^1 ⊕ im(δ), and that S^1(y_1,y_2,y_3) ≅ im(δ). The injectivity argument for δ depends on the unproved assertion that H^1(K_n) ≅ H^1(K_{n+1}) as groups. Consequently Corollary 4.3, which is the paper's main E2-independence claim, rests on an unproved theorem.
- [§5, Theorem 5.14] In the statement for GL_2(Z/p^n), the formula for Ψ_n uses the normalizer N_{SL_2(Z/p^n)}(Q)/Q, but Q is a Sylow p-subgroup of GL_2(Z/p^n); the normalizer should be taken in GL_2(Z/p^n). In the proof, the claim 'A computation shows that S is not in the centraliser of K_n' is not carried out; this claim is needed to conclude C_S(K_n)=K_n, which is a key step in showing K_n is F-centric. These issues are local and can be repaired, but as written the stable-elements description is not fully rigorous.
minor comments (5)
- [Abstract and Section 1] The abstract contains a grammatical error ('it's Sylow p-subgroup' should be 'its Sylow p-subgroup'), and Section 1 has a typo ('seqeuences' for 'sequences').
- [§2, Proposition 2.1] The statement 'det(A)=1 implies tr(A)=2' is imprecise. For A=1+p^{n-1}X, the condition det(A)=1 mod p^n leads to tr(X) ≡ 0 mod p; the argument should be phrased in terms of X, not A, and the congruence should be stated explicitly.
- [§3, Lemma 3.6] The diagram in the proof is garbled and does not clearly display the claimed maps; it should be redrawn. Also, the notation 'E ∼= K_{n,n-1} = Ω_1(K_n)' conflates an elementary abelian subgroup with the full Ω_1-subgroup without proof that they coincide.
- [§4, proof of Theorem 4.2] The statement 'Since C_p ∼= (F_p)' is false as written; C_p is isomorphic to F_p as an abelian group or as an F_p-vector space, but not as a ring in any way that is used here. The intended meaning should be clarified.
- [§2, Definition of K_{n,m}] In the definition of K_{n,m}, the text writes 'Z/pnZ' where it should be 'Z/p^n Z'; the missing exponent makes the definition hard to read.
Circularity Check
No circularity: the central derivation rests on external published theorems and independent group-theoretic checks; no step reduces to its own input.
full rationale
The paper's derivation chain is not circular. Theorem 3.8 applies the Minh--Symonds classification ([MS, p. 239]) after Propositions 3.4 and Lemma 3.6 aim to verify powerfulness and Omega-extendability; these are external criteria, not the target cohomology. Theorem 4.2 then establishes the C_p-module structure of H^*(K_n) and H^*(L_n) using the ring description from Theorem 3.8 together with [SW, Thm 5.1.6] and an LHS spectral sequence argument; the C_p-module claim is genuinely additional content rather than an assumption. Corollary 4.3's E2-page independence follows from the module isomorphism and the definition of the LHS spectral sequence, not from a disguised restatement. Section 5 invokes fusion-system results from [BLO] and [C] and identifies the relevant F-centric p-radical subgroups by the normality and maximality of K_n; this is a computation within the fusion-system framework, not a circular definition. The heavy reliance on earlier work by the author's supervisor is notable, but the cited results are published external theorems with proofs, and the present paper's author is not an author of [MS] or [SW]; under the review rules this counts as independent support rather than circularity. The paper itself, in the closing Remark, explicitly disclaims the stronger statement that the full cohomology rings of GL_2(Z/p^n) are independent of n, which further indicates that the E2-page result is not being inflated into an unsupported conclusion. Separately, Proposition 3.4's determinant verification appears to check det(g) ≡ 1 mod p rather than the required mod p^n when proving K_{n,n-1} ⊆ K_n^p; this is a potential rigor or correctness gap in the powerfulness proof, but it is not a circularity, because Theorem 3.8 is quoted from an external source and the later arguments do not define their conclusion into their hypotheses. No circular step satisfying the quoted-reduction standard was found.
Assumptions & free parameters
assumptions (4)
- standard math Minh-Symonds classification of cohomology of powerful pro-p groups (Theorem 3.7)
- standard math Symonds-Weigel injectivity of Λ^2(H^1) into H^2 for powerful p-groups
- standard math Broto-Levi-Oliver equivalences for fusion systems and p-local groups
- standard math Craven's Proposition 5.13: a normal subgroup of F is contained in every F-centric p-radical subgroup
Cite this review
Pith. "Pith review of On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$." pith.science (2026). https://pith.science/paper/VGZCGMQJ
@misc{pith2026250604720,
author = {Pith},
title = {Pith review of: On the modular cohomology of $GL_2(\mathbbZ/p^n)$ and $SL_2(\mathbbZ/p^n)$},
year = {2026},
howpublished = {\url{https://pith.science/paper/VGZCGMQJ}},
note = {Machine review of arXiv:2506.04720}
}
abstract
Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on $n>1$. Finally, we use the theory of fusion systems to describe the ring of stable elements.
Reference graph
Works this paper leans on
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[1]
On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
On the modular cohomology ofGL(2, pn) andSL(2, pn) Anja Meyer Abstract Letpbe an odd prime. Denote a Sylowp-subgroup ofGL 2(Z/pn) andSL 2(Z/pn) byS p(n, GL) andSp(n, SL) respectively. The theory of stable elements tells us that the mod-pcohomology of a finite group is given by the stable elements of the mod-pcohomology of it’s Sylowp-subgroup. We prove th...
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[3]
Looking at theE 3-page we have thatE 2,0 3 ∼= H 2(Kn)/im(δ). AsK n+1 is powerful, Λ2(H 1(Kn,F p)) must survive onto theE ∞ page, and thus Λ 2(H 1(Kn,F p))∩im(δ) = 0.This gives that H 2(Kn,F p) ∼= Λ2(H 1(Kn,F p))⊕im(δ) asF p-vector spaces, and thusS 1(y1, y2, y3) ∼= im(δ). AsC p ∼= (Fp), andδis an injective map ofC p-modules,S 1(y1, y2, y3) is aC p-module....
work page 1956
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[282]
The proposition following the next definition prevents this
To require examination of every single subgroup to determine whether it isF-centric andp-radical would make our computations impossible. The proposition following the next definition prevents this. Definition 5.12.TakeFa fusion system over a p-group S, and fixQ≤S. Q is normal inF, writeQ ◁F, ifQ ◁ S, and for allP, R≤Sand for allϕ∈Hom F (P, R),ϕextends to ...
work page 1980
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[1972]
New Horizons in Pro-p Groups, du Sautoy, Segal and Shalev eds., Birkhauser (2000) 349-416
[SW]Peter SYMONDS, Thomas WEIGEL:Cohomology ofp-adic Analytic Groups. New Horizons in Pro-p Groups, du Sautoy, Segal and Shalev eds., Birkhauser (2000) 349-416 . 14
work page 2000
Reviewed August 7, 2026 · model on record in the stance chip above.
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