{"id":"f11df1b0-a5b2-425a-8be5-029b42be8dd6","arxiv_id":"2508.20847","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For cocompact arithmetic lattices of simplest type in PU(n,1), the cohomology of the profinite completion is nontrivial up to degree 2n for large primes.","lead":"This paper proves that the profinite completion of certain arithmetic lattices in complex hyperbolic space has nonzero cohomology in all degrees up to twice the complex dimension, for large primes. This is the first such nonvanishing result for ball quotients in dimensions at least two, and it improves the known lower bound on virtual cohomological dimension from 1 to 2n.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 assumes an integral Albanese lift of K_X that does not follow from the stated complex cup-product condition; Theorem 1.1 appears sound.","rationale":"I read the main nonvanishing theorem carefully. The strategy—produce a nonzero complex class in the cup-product image, lift to an infinite-order integral class, reduce mod p, and transfer to profinite cohomology via the Albanese map and goodness of Z^d—is internally coherent. The transfer argument justifies the passage to every finite-index subgroup. The only serious soft spot I found is in Theorem 1.3, not in Theorem 1.1. Since the abstract advertises the profinite fundamental class and canonical-class statement, the paper should be accepted only after that integral-lift step is repaired or clarified. This agrees with the reader that Theorem 2.1(1) is the delicate input, but my specific concern is an internal gap downstream of it rather than an independent check of the cited result.","tokens_in":10408,"tokens_out":32615,"duration_ms":336213,"concrete_test":"For an explicit Γ0 as in Corollary 2.2 (e.g., a congruence subgroup of a simplest-type lattice with b1 ≥ 2), compute the subgroup V = α^*_Z(H^2(Alb(X),Z)) in H^2(X,Z)/torsion and check whether [K_X] ∈ V. If not, Theorem 1.3 fails. If yes, supply the missing saturation argument. Alternatively, re-derive [28, Cor. 3.6] from [5, Thm. 9.3] and determine whether the containment proved there is integral or only complex.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.3 (and the abstract's claim that the canonical class is profinite modulo torsion) contains an unjustified integral-lift step. Corollary 2.2(1) only yields that the complex image of K_X lies in the image of α^*: H^2(Alb(X),C) -> H^2(X,C). The text then asserts there is ψ ∈ H^2(Alb(X),Z) with α^*_Z(ψ) = K_X + σ for a torsion class σ. But complex containment only places the free part of K_X in the rational span of the integral image α^*_Z(H^2(A,Z)); the integral image need not be saturated, so an integral class in that rational span need not be an integral pullback. No saturation or integrality argument is supplied. Thus Theorem 1.3 is not established as written. This does not affect Theorem 1.1: there one only needs some nonzero free class in Im(c^Z_j), and because α^*_Z is an isomorphism on H^1, any such class is automatically an Albanese pullback, so the profinite-restriction diagram applies.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies profinite completions \\hat{\\Gamma} of cocompact arithmetic lattices \\Gamma < PU(n,1) of simplest type. The main theorem (Thm. 1.1) asserts the existence of an open subgroup \\hat{\\Gamma}_0 \\le \\hat{\\Gamma} such that for every open subgroup \\hat{\\Delta} \\le \\hat{\\Gamma}_0 and every j \\le 2n, the cohomology group H^j(\\hat{\\Delta}, \\mathbb{F}_p) is nonzero for all sufficiently large primes p. Consequently, vcd(\\hat{\\Gamma}) \\ge 2n (Cor. 1.2). Two further results are claimed: Theorem 1.3 says that the canonical class and its exterior powers are \"profinite modulo torsion\" on finite covers, and Theorem 1.4 gives a low-degree virtual surjectivity statement for congruence lattices. The proofs reduce the statements to complex cohomology via a structure theorem (Thm. 2.1), use the Albanese map and goodness of abelian groups to transfer classes to profinite cohomology, and use theta-correspondence results for Part 3 of Theorem 2.1.","tokens_in":10669,"tokens_out":17673,"duration_ms":173159,"significance":"If correct, Theorem 1.1 is a significant advance: for n \\ge 2 it gives the first nonvanishing results for H^j(\\hat{\\Gamma}, \\mathbb{F}_p) in every degree up to the cohomological dimension 2n, improving the known lower bound vcd(\\hat{\\Gamma}) \\ge 1 to vcd(\\hat{\\Gamma}) \\ge 2n. The proof of Theorem 1.1 is coherent and largely sound: it exploits the Albanese map and the goodness of abelian groups, and the key input from [28] is an independent published result rather than circular. The main caveat is the integral-lift gap in Theorem 1.3 and, to a lesser extent, in Theorem 1.4; Theorem 1.1 is not affected because it only needs an infinite-order integral class in the image of the cup product. The reliance on deep external results is clearly stated, and the paper is well organized.","major_comments":[{"comment":"The step \"there is a class \\psi \\in H^2(Alb(X),\\mathbb{Z}) such that \\alpha_Z^*(\\psi)=K_X+\\sigma\" is not justified. Corollary 2.2(1) gives only \\alpha_C^*(\\psi_C)=K_X\\otimes\\mathbb{C}, i.e. complex containment. Because \\alpha_Z^*(H^2(A,\\mathbb{Z})) need not be saturated in H^2(X,\\mathbb{Z}), a rational preimage of K_X need not be integral up to torsion (e.g. multiplication-by-m maps). Thus Theorem 1.3 and the abstract's \"canonical class is profinite modulo torsion\" are not established as written. Theorem 1.1 is unaffected, since it only needs an infinite-order element of Im(c_j^\\mathbb{Z}).","section":"§2, Proof of Theorem 1.3"},{"comment":"Part 3 of Corollary 2.2 yields \\psi_i \\in H^j(Alb(X),\\mathbb{C}), not H^j(Alb(X),\\mathbb{Z}). The proof nevertheless \"fixes\" integral \\psi_i with \\alpha^*(\\psi_i)=\\phi_i. As in Theorem 1.3, only rational lifts follow from complex containment. This is repairable: one may choose integral multiples m_i\\psi_i^\\mathbb{Z} with \\alpha^*(m_i\\psi_i^\\mathbb{Z})=m_i\\phi_i; for primes p not dividing \\prod m_i, the \\mathbb{F}_p-span equals the span of the reductions of the \\phi_i. The manuscript should carry out this argument explicitly.","section":"§2, Proof of Theorem 1.4"}],"minor_comments":[{"comment":"The sentence \"Theorem 1.3 then leads to the following significant improvement\" before Corollary 1.2 appears to be a typo: Corollary 1.2 is proved from Theorem 1.1 and is stated before Theorem 1.3.","section":"Section 1"},{"comment":"The statement \"Betti numbers only grow under finite covers\" is imprecise; what is needed is that restriction on rational cohomology is injective (transfer argument), so an infinite-order integral class pulls back to an infinite-order class.","section":"§2, proof of Theorems 1.1 and 1.4"},{"comment":"The hard Lefschetz step as written covers degrees n through 2n-1; the top degree 2n follows from K_X^n being in Im(c_{2n}) and should be mentioned.","section":"§2, proof of Part 2 of Theorem 2.1"},{"comment":"Please clarify the passage from arbitrary arithmetic lattices of simplest type to congruence lattices via [6, Prop. 1.4], in particular whether \"contained in\" or commensurability is meant and how the finite-index subgroup is chosen.","section":"§2, proof of Part 2 of Theorem 2.1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The main theorem (Thm. 1.1) appears sound and is publishable once the presentation issues are fixed. The gap in Thm. 1.3 and Thm. 1.4 is real but likely repairable; the author should be asked to supply a saturation or denominator-clearing argument, or to adjust the statements. The dependence on [28] is legitimate and not circular. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is genuinely new and the proof is largely sound: Theorem 1.1 gives nontrivial H^j(Δ-hat, F_p) for all j ≤ 2n after passing to an open subgroup, for cocompact arithmetic lattices of simplest type. That improves the previous vcd lower bound from 1 to 2n, and it's the first nonvanishing in degrees j ≥ 2 for n ≥ 2. The strategy is clean: use the Albanese map, goodness of abelian groups, and hard Lefschetz to push low-degree classes up to the cohomological dimension. The paper is honest about what remains open and about the reliance on external results.\n\nThe main soft spot is the proof of Theorem 1.3. Corollary 2.2(1) only gives a complex class ψ_X on the Albanese pulling back to K_X in complex cohomology. The proof then asserts there is an integral class ψ with α^*_Z(ψ) = K_X + σ for torsion σ. Complex containment does not imply integral containment without a saturation argument, and none is supplied. So Theorem 1.3 and the abstract's claim that the canonical class is profinite modulo torsion are not established as written. This does not affect Theorem 1.1, because there you only need some infinite-order class in the image of the integral cup product, and such classes are automatically Albanese pullbacks since α^* is an isomorphism on H^1. It may be fixable by passing to a finite cover or by a separate integrality argument, but the current text has a gap.\n\nThe paper leans heavily on a prior self-cited result with Toledo ([28, Cor. 3.6]) for the key cup-product input. That is an independently published theorem with its own proof, so it's not a red flag by itself, but the referee should check it carefully. The Bergeron–Millson–Moeglin input for Theorem 1.4 also deserves scrutiny, though the reduction there looks plausible.\n\nWho is this for? People working on algebraic fundamental groups, profinite rigidity, and Serre goodness for complex hyperbolic lattices. It deserves a serious referee. The main theorem is important, likely correct, and the proof is intelligible. I'd send it to review with a request to fix or weaken Theorem 1.3.","headline":"New nonvanishing for profinite cohomology of ball quotients up to degree 2n, with a fixable gap in the canonical-class theorem.","tokens_in":11172,"tokens_out":7064,"would_cite":true,"duration_ms":64361,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F35","11F75","22E40","32Q45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The profinite completion of a ball-quotient lattice has cohomology in every degree up to 2n","keywords":["ball quotients","profinite completion","cohomological dimension","arithmetic lattices","simplest type","Albanese variety","theta correspondence","good groups"],"falsifier":"Take an explicit cocompact arithmetic lattice of simplest type, pass to a small open subgroup $\\hat{\\Delta}$ of its profinite completion, and compute $H^j(\\hat{\\Delta}, F_p)$ for some $j \\leq 2n$ for infinitely many primes $p$; a single prime $p$ with $H^j(\\hat{\\Delta}, F_p) = 0$ in that range would contradict Theorem 1.1. Alternatively, compute $\\mathrm{vcd}(\\hat{\\Gamma})$ directly; any value below $2n$ for any such lattice would contradict Corollary 1.2.","tokens_in":10301,"feed_emoji":"📐","tokens_out":9853,"duration_ms":94199,"temperature":0.7,"texified_at":"2026-08-05T20:17:34.101532+00:00","pith_summary":"This paper proves that a cocompact arithmetic lattice of simplest type in the isometry group of the complex n-ball has a profinite completion whose mod p cohomology is nonzero in every degree from 0 to 2n, once one passes to a suitable open subgroup. For $n \\geq 2$ this is the first nonvanishing result above degree 1, and it raises the known virtual cohomological dimension of the profinite completion from 1 to 2n. The proof pulls the relevant cohomology classes back from the Albanese variety of the ball quotient, where the fundamental group is an abelian group and therefore 'good' in Serre's sense, so discrete and profinite cohomology coincide. The same mechanism shows the canonical class is profinite modulo torsion, and gives virtual surjectivity statements for congruence lattices in low degrees. The results are about the algebraic fundamental group of these projective varieties, since for torsion-free lattices the profinite completion is exactly that group.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8259,"prompt_tokens":880,"completion_tokens":7379,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":880,"completion_tokens_details":{"reasoning_tokens":6499}},"feed_headline":"Ball-quotient lattices: profinite cohomology lives through degree 2n","feed_subtitle":"First nonvanishing above degree 1; virtual cohomological dimension jumps from 1 to 2n.","key_machinery":"The load-bearing mechanism is the Albanese map $\\alpha: X \\to \\mathrm{Alb}(X)$ for a compact ball quotient $X = \\Delta \\backslash B^n$, together with the fact that the fundamental group of $\\mathrm{Alb}(X)$ is a finitely generated free abelian group, hence good. Because $\\alpha^*$ identifies $H^1(\\mathrm{Alb}(X), C)$ with $H^1(X, C)$, every class in the cup-product span of $H^1(X, C)$ is pulled back from the Albanese variety, and integral classes pulled back this way can be lifted to profinite cohomology through the goodness isomorphism. The paper's Theorem 2.1, assembled from a result of Stover and Toledo ([28, Cor. 3.6]) plus Venkataramana's Hecke-correspondence theorem ([31, Thm. 8]) and hard Lefschetz, ensures that after a finite-index pass the canonical c","core_discovery":"The paper's central claim is Theorem 1.1: for any cocompact arithmetic lattice $\\Gamma < \\mathrm{PU}(n,1)$ of simplest type, there is an open subgroup $\\hat{\\Gamma}_0$ of the profinite completion $\\hat{\\Gamma}$ such that $H^j(\\hat{\\Delta}, F_p)$ is nontrivial for every open subgroup $\\hat{\\Delta} \\leq \\hat{\\Gamma}_0$, every $j \\leq 2n$, and all sufficiently large primes $p$. This is a statement about algebraic fundamental groups, because when $\\Gamma$ is torsion-free the profinite completion is the algebraic fundamental group of the projective variety $\\Gamma \\backslash B^n$. The paper also establishes Theorem 1.3, that the canonical class and its exterior powers are 'profinite modulo torsion': a fixed torsion class $\\sigma$ exists such that the reductions of $(K_X + \\sigma)^i$ lie in the image of the restrictio","pith_inferences":["If the cup-product generation of the canonical class could be established for a wider class of lattices in PU(n,1), the same Albanese-pullback argument would likely transfer the nonvanishing and vcd conclusions; the paper's remarks identify the missing ingredients for non-arithmetic lattices.","The 'profinite fundamental class' formulation suggests a strengthening one could test: if the torsion class σ in Theorem 1.3 can be chosen zero, then the profinite completion would carry a genuine profinite fundamental class, possibly giving profinite Poincare duality properties for Γ̂.","Theorem 1.4 isolates where a proof of goodness would need new input: it already supplies virtual surjectivity below degree (n+1)/2, so the open question is the range j ≥ (n+1)/2.","Explicit covers from the ball-quotient literature could be used to probe sharpness, e.g. testing whether H^{2n+1} of the profinite completion can be nonzero or whether vcd(Γ̂) is exactly 2n."],"forward_implications":["For every open subgroup beneath the chosen Γ̂_0, all mod p cohomology groups H^j(·, F_p) with j ≤ 2n are nonzero for all sufficiently large primes p, so the nonvanishing is stable under further finite covers.","The virtual cohomological dimension of Γ̂ is at least 2n, matching the virtual cohomological dimension of the discrete lattice Γ itself and far exceeding the previously known lower bound of 1.","The canonical class of any finite cover is represented, up to a fixed torsion class, by a class that survives the restriction map from profinite to discrete cohomology, and the same holds for every exterior power up to degree 2n.","For congruence lattices in degrees j < (n+1)/2, arbitrary finite families of integral classes become virtually contained in the image of the profinite restriction map after reduction modulo p, for all large p.","The restriction map H^j(Γ̂, F_p) → H^j(Γ, F_p) is virtually almost surjective in low degrees, giving concrete quantitative control on Serre goodness in the range the paper can reach."],"supporting_citations":[{"why":"Provides the starting fact that for simplest-type lattices the canonical class of X lies in the cup-product image of H^1(X, C).","marker":"[28, Cor. 3.6]"},{"why":"Supplies the Hecke-correspondence argument that nonzero classes in low degrees can be wedged to nonzero classes, allowing induction up to degree n.","marker":"[31, Thm. 8]"},{"why":"Shows low-degree cohomology of the Shimura variety is generated by theta lifts, the engine for Theorem 2.1(3) and Theorem 1.4.","marker":"[5, Cor. 7.3]"},{"why":"Hard Lefschetz theorem, used to amplify nontriviality from degrees up to n to all degrees up to 2n by wedging with powers of the canonical class.","marker":"[13, p. 122]"},{"why":"Gives the infinite H^1 for a finite-index subgroup that anchors the degree-one nonvanishing.","marker":"[17, Thm. 1]"},{"why":"Extends the congruence-lattice inputs to all arithmetic lattices of simplest type.","marker":"[6, Prop. 1.4]"}],"fun_headline_variants":["Profinite cohomology of ball quotients: nonzero up to degree 2n","Ball-quotient lattices: cohomology resists vanishing through 2n","Virtual cohomological dimension of ball quotients leaps to 2n","For ball quotients, profinite cohomology lives to degree 2n","Arithmetic lattices: profinite classes persist to degree 2n"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The entire proof depends on a finite-index subgroup of the lattice having the property that on every further finite cover, the first Chern class of the canonical bundle lies in the subspace of $H^2$ spanned by cup products of degree-one classes; without this, the Albanese pullback step and everything after it collapses.","fun_headline_variants_meta":{"raw":{"variants":["Profinite cohomology of ball quotients: nonzero up to degree 2n","Ball-quotient lattices: cohomology resists vanishing through 2n","Virtual cohomological dimension of ball quotients leaps to 2n","For ball quotients, profinite cohomology lives to degree 2n","Arithmetic lattices: profinite classes persist to degree 2n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1250,"prompt_tokens":823,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":323}},"tokens_in":567,"tokens_out":427,"duration_ms":4535,"temperature":1.0,"reasoning_tokens":323,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:45:58.129129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit cocompact arithmetic lattice of simplest type, pass to a small open subgroup $\\hat{\\Delta}$ of its profinite completion, and compute $H^j(\\hat{\\Delta}, F_p)$ for some $j \\leq 2n$ for infinitely many primes $p$; a single prime $p$ with $H^j(\\hat{\\Delta}, F_p) = 0$ in that range would contradict Theorem 1.1. Alternatively, compute $\\mathrm{vcd}(\\hat{\\Gamma})$ directly; any value below $2n$ for any such lattice would contradict Corollary 1.2.","supporting_citations":[],"review_version":1}