{"id":"26a765d9-d923-4c6a-9f60-cf9e473084b0","arxiv_id":"2411.16381","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For GL_3 base change to a real quadratic field, new middle-degree automorphic periods satisfy a Hida-type congruence formula and yield a p-adic divisibility of periods, conditional on the Calegari-Geraghty hypotheses and up to an uncomputed constant.","lead":"The paper proves that for certain 3-by-3 matrix automorphic forms over the rationals, the product of the two classical period invariants divides a newly defined 'middle degree' period of the base change to a real quadratic field, up to a constant.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's central divisibility depends on an uncomputed local constant u2,ram; unless u2,ram is shown to be a p-adic unit, the advertised clean relation Ω2(π)Ω3(π) | Ω5(Π,ε,−) is not actually established.","rationale":"I read the paper in good faith as a conditional theorem: under the stated hypotheses (Split), (CG), (LGC), residual absolute irreducibility, and the p-integrality conditions, the announced divisibility is derived from Theorem D and the Jacquet-Ye period computation. The structure is coherent and the paper is unusually explicit about its limitations. The reader's weakest_assumption emphasizes the Calegari-Geraghty conjectures and local-global compatibility at bad places; those are genuine hypotheses, but they are declared assumptions, so I do not treat them as a hidden gap in the argument. The more load-bearing concern is the uncomputed factor u2,ram: it appears in the conclusion of the main theorem, and its p-adic valuation directly controls whether the advertised period relation holds in the clean form. This matches part of the reader's rationale, which already flags the uncomputed constant as a reason for a conditional verdict. My concrete check would settle whether this concern lands: if u2,ram is in fact a p-adic unit, the theorem has the intended content; if not, the abstract and Theorem A overstate the result. Because the reader already returned CONDITIONAL and my concern reinforces rather than overturns that verdict, I recommend no change.","tokens_in":72204,"tokens_out":13161,"duration_ms":136120,"concrete_test":"Compute u2,ram explicitly by repeating the local zeta-integral computation in the proof of Theorem C at v = 2 and at primes of E ramified over Q, using the Matringe/Miyauchi formulas for essential vectors already used in §3.2.2 and §4.3.1, and verify that the resulting product lies in O^× whenever p ∤ 6N_E/Q(n)h_E(n)D_E. If u2,ram is a p-adic unit, Theorem A becomes the clean divisibility; if not, the statement and the abstract must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem, stated as Corollary 5.1 via Theorem C, asserts Ω2(π)·Ω3(π) | Ω5(Π,ε,−)·u2,ram, where u2,ram is a nonzero complex factor depending on local components above 2 and at primes ramified in E. The paper explicitly says that u is expected to be trivial but is hard to compute. This is not cosmetic: the divisibility sign means the quotient lies in O after the fixed identification of C with Q_p, and the p-adic valuation of u2,ram is unknown. If v_p(u2,ram) < 0, the right-hand side loses integrality and the claimed period divisibility can fail; if v_p(u2,ram) > 0, the proved relation is weaker than the clean divisibility advertised in the abstract. Since the whole point is to prove the integral period relation (0.0.2), an uncomputed factor inside the conclusion leaves the central claim not fully quantified. The Calegari-Geraghty hypotheses and residual absolute irreducibility are declared assumptions rather than internal gaps, but the uncomputed u is a missing computation inside the theorem itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies p-adic integral period relations for two base-change transfers to GL_3 over a real quadratic field E: the Arthur–Clozel base change from GL_3(Q) and the Rogawski–Mok stable base change from the quasi-split unitary group U_E. Since the middle cohomology of GL_3(E) has two-dimensional isotypic parts, the author introduces new 'base-change periods' Ω_5(Π,ι,±) attached to the ±-eigenspaces of the Galois involution σ (for self-conjugate Π) or the conjugation-duality involution ε (for conjugate self-dual Π). Theorem D (Theorem 3.1) establishes an à la Hida formula: the ±-part of the congruence number η_{λ_Π}(M)[±] equals Λ_imp(Π,Ad,1)/(Ω_5(Π,ι,±)Ω_5(Π∨,ι,∓)) up to O-units. Theorems B and C (Theorems 4.1 and 5.1) then give one-sided divisibilities relating the base-change congruence numbers to twisted adjoint L-values, using a cohomological interpretation of the Flicker–Rallis and Jacquet–Ye periods. The headline Theorem A (Corollary 5.1) deduces the divisibility Ω_2(π)Ω_3(π) | Ω_5(Π,ε,−) up to an uncomputed factor u_{2,ram}. The paper is conditional on the Calegari–Geraghty running hypotheses (Galm), (LGC_m), (Van_m), and (CG), and on residual absolute irreducibility of the associated Galois representation.","tokens_in":107,"tokens_out":8580,"duration_ms":279585,"significance":"If the core computations are correct, this is a substantive advance: it provides the first middle-degree Hida-style adjoint formula for GL_3 over a real quadratic field and a workable definition of middle-degree periods in a setting where the isotypic cohomology is two-dimensional. The structure is not circular: the periods are fixed by canonically normalized Eichler–Shimura maps and integral O-structures, and the congruence numbers η_{λ_Π}(M)[±] are defined independently of the periods, so Theorem D is a genuine identity rather than a fitted one. The local computations are explicit, using Miyauchi–Matringe essential vectors and Chen's archimedean generators, and the congruence-number formalism is carefully adapted to semi-linear involutions. The main caveat is that the advertised clean period relation (0.0.2) is not actually proved: the theorem contains the uncomputed local factor u_{2,ram}, whose p-adic valuation is not controlled. The theorem is therefore best regarded as a divisibility with a local error term, and the headline claim needs to be rephrased or completed.","major_comments":[{"comment":"The central theorem advertised in the introduction as 0.0.2 is not what Corollary 5.1 proves. The proved relation is Ω_2(π)Ω_3(π) | Ω_5(Π,ε,−)·u_{2,ram}, where u_{2,ram} is an uncomputed nonzero complex factor depending on local components above 2 and at primes ramified in E. Since divisibility is defined after identifying C with Q_p via the fixed isomorphism j_p, the p-adic valuation of u_{2,ram} is load-bearing: if v_p(u_{2,ram})<0, then Ω_5(Π,ε,−)·u_{2,ram} is not integral and the claimed divisibility of periods can fail; if v_p(u_{2,ram})>0, the proved statement is strictly weaker than the clean relation 0.0.2. The text says only that u is 'expected to be trivial', and expectation is not a proof. To make Theorem A a theorem about (0.0.2), the author must either compute u_{2,ram}, prove that it is a p-adic unit, or state the clean divisibility as an explicit conditional consequence.","section":"§5.1 (Theorem C and Corollary 5.1)"},{"comment":"Theorem A is conditional on the Calegari–Geraghty hypotheses (Galm), (LGC_{m_π}), and (CG), and the paper itself notes in §2.2.4 that no local-global compatibility at bad places is known for the representation of [CGJ23] when p is completely split in E. Since Theorem A assumes p split in E, the freeness result of Theorem 2.3, and hence the divisibility argument, relies on a conjectural compatibility that is currently open in exactly the relevant case. The hypotheses are declared, so this is not an internal inconsistency, but the abstract and introduction should not present Theorem A as an unconditional proof of a period divisibility; it is a result conditional on those running conjectures. If the local-global compatibility fails, the divisibility argument collapses, and the reader should be told this limitation in the statement of the main theorem as well as in a remark.","section":"§2.2.4 and §2.3.3 (Theorem 2.3 and Theorem A)"}],"minor_comments":[{"comment":"The differential operator defining the kernel L_n(K) is written as ∂²/(∂X∂A)+∂²/(∂Y∂B)+∂²/(∂X∂A); the third term should presumably be ∂²/(∂Z∂C).","section":"§2.1"},{"comment":"There are several typos and grammatical slips: 'beetween', 'cus pidal', 'is a ap-adic', and 'when Limp(Π×Π′,s) has not' in §1.2.1. These should be corrected in a final revision.","section":"Abstract and §1.1.2"},{"comment":"The introduction contains two unresolved cross-references ('see ?? and ??') when describing the stable base change case; these should be replaced by precise theorem or section numbers.","section":"Introduction"},{"comment":"The notation n is used both for the mirahoric level and for the cohomological weight, which can be confusing; for example, the phrase 'mirahoric level n' and 'cohomological weight n' appear close together in Theorem 3.1. A notational distinction would improve readability.","section":"§3.1 and §5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is long and technically demanding, and much of its correctness depends on very recent and partly conjectural input (Calegari–Geraghty theory, Mok's base change, Scholze's Galois representations). I would recommend that the editor seek a second opinion from an expert in p-adic automorphic periods, particularly on the local computations in Sections 3–5. The main advertised period divisibility is not fully proved because of the uncomputed factor u_{2,ram}; this is fixable in principle, but it changes the central statement. If the author can prove the p-adic unitarity of u_{2,ram}, or clearly separate the clean conditional relation from the proved one, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The takeaway: this is a substantial paper and it deserves a real referee. What is genuinely new is the construction of middle-degree periods on the 5th cuspidal cohomology of GL3(E) via σ- and ε-involutions. That construction is the load-bearing idea, and Theorem D — the middle-degree à la Hida formula — is carefully derived, with explicit ramified computations using Miyauchi–Matringe essential vectors and archimedean computations modeled on Chen. I checked the setup for circularity: the periods are fixed before any L-value comparison, via canonically normalized Eichler–Shimura maps and integral O-structures, and the congruence numbers are defined independently. Theorem D looks like a genuine identity, not a fitted one.\n\nThe soft spot is exactly where the stress-test lands. The headline divisibility in Theorem A / Corollary 5.1 is not Ω2(π)Ω3(π) | Ω5(Π,ε,−), but Ω2(π)Ω3(π) | Ω5(Π,ε,−)·u2,ram, with u2,ram an uncomputed local constant. The paper says u is expected to be trivial but hard to compute. That is not cosmetic: the p-adic valuation of u is unknown, and if v_p(u) is negative the divisibility can fail, while if it is positive the proved statement is weaker than what the abstract advertises. So the clean relation (0.0.2) is not actually established. I do not, however, see this as a fatal flaw. The paper is honest about the gap, and the structural core — Theorem D and the period construction — is unaffected. The theorems also carry the declared Calegari–Geraghty hypotheses (CG), (LGCm), and residual absolute irreducibility; the paper even notes that local-global compatibility at bad places is not known for the [CGJ23] representation when p is split. Those are assumptions, not hidden gaps. The stable base change part yields only partial results because no adjoint à la Hida formula exists for the unitary group, which the paper states clearly.\n\nWho is this for? People working on integral period relations, Hida theory, and base change for GLn. It is technically demanding but clearly written, and the computations are explicit enough for a referee to check. My recommendation: send it to peer review. A referee should press for the u2,ram computation and for a verification or weakening of the residual irreducibility assumption, but the paper's central new construction and Theorem D justify a thorough refereeing process.","headline":"A serious, carefully-built paper that defines new middle-degree base-change periods and proves an à la Hida formula for GL3; the advertised divisibility is real but carries an uncomputed local constant u2,ram that keeps the clean headline relation conditional.","tokens_in":73030,"tokens_out":2326,"would_cite":true,"duration_ms":24682,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F70","11F75","11R42","22E55"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new class of middle-degree automorphic periods carries a p-adic divisibility between the periods of a GL3(Q) representation and those of its base change to a real quadratic field.","keywords":["p-adic periods","automorphic periods","base change","GL3","real quadratic field","middle-degree cohomology","congruence numbers","adjoint L-values"],"falsifier":"Compute, for an explicit triple $(\\pi,E,p)$ satisfying the hypotheses, the $p$-adic valuations of the transfer congruence number $\\eta^\\#_{\\lambda_\\Pi}(M^*)[+]$ and of $\\Lambda^{\\mathrm{imp}}(\\pi,\\mathrm{Ad}\\otimes\\chi_E,1)/(\\Omega_5(\\Pi,\\sigma,+)\\Omega_5(\\Pi^\\vee,\\sigma,-))$; the claimed divisibility is false if the valuation of the congruence number exceeds that of the quotient. A more direct check of the proof's gate is to test the assumed local-global compatibility at a ramified place above $2$ or above a prime ramified in $E$ for the Galois representation of [CGJ23] when $p$ is completely split in $E$, since the freeness conclusion of Theorem 2.3 is the only bridge from the Hecke algebra to the divisibility.","tokens_in":71809,"feed_emoji":"➗","tokens_out":18115,"duration_ms":151616,"temperature":0.7,"pith_summary":"This paper proves a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\\mathrm{GL}_3(\\mathbb{Q})$ and the periods of its classical base change to a real quadratic field $E$. The new ingredient is a class of periods attached to the middle degree, degree $5$, of the cuspidal cohomology of $\\mathrm{GL}_3(E)$, where the $\\Pi$-isotypic part is two-dimensional and is split into two one-dimensional eigenspaces by the conjugation or conjugation-duality involution. Under the running hypotheses (Split), (CG), and (LGC$_{m_\\pi}$), and with $p$ outside a finite exceptional set, the product $\\Omega_2(\\pi)\\Omega_3(\\pi)$ divides $\\Omega_5(\\Pi,\\varepsilon,-)\\cdot u$, where $u=u_{2,\\mathrm{ram}}$ is an uncomputed local constant. This generalizes the period relations known for $\\mathrm{GL}_2$ to $\\mathrm{GL}_3$ and gives an automorphic shadow of the conjectural Bloch--Kato period relation for the twisted adjoint motive; a parallel middle-degree divisibility is proved for the stable base change from the unitary group $U_E$.","feed_headline":"New periods prove p-adic divisibility for GL3 base change","feed_subtitle":"The periods of a GL3(Q) representation divide the middle-degree periods of its base change to a real quadratic field.","key_machinery":"The central mechanism is the degree-$5$ cuspidal cohomology $H^5_{\\mathrm{cusp}}(Y_E(K_f), L_{\\mathfrak{n}}(\\mathcal{O}))_{m_\\Pi}$, localized at the maximal ideal attached to $\\Pi$, viewed as a Hecke module with a semi-linear involution $\\iota$ ($\\sigma$ for self-conjugate $\\Pi$, $\\varepsilon$ for conjugate self-dual $\\Pi$). The involution splits the two-dimensional $\\Pi$-isotypic part into one-dimensional $\\pm$-eigenspaces carrying canonical $\\mathcal{O}$-structures, which makes possible the canonically normalized Eichler--Shimura maps $\\delta^\\pm_\\iota$ and hence the periods $\\Omega_5(\\Pi,\\iota,\\pm)$. The proof of the divisibility uses the congruence-number formalism for Hecke modules with semi-linear involution, the transfer congruence number $\\eta^\\#_{\\lambda_\\Pi}(M^*)[\\pm]$, and cohomological interpretations of the period integrals that detect classical and stable base changes, with ramified factors computed from essential-vector formulas and archimedean factors from explicit generators of the relevant relative Lie algebra cohomology.","core_discovery":"The central claim is Theorem A: if $\\pi$ is a self-dual cohomological cuspidal automorphic representation of $\\mathrm{GL}_3(\\mathbb{A}_\\mathbb{Q})$, not isomorphic to $\\pi\\otimes\\chi_E$, ramified only at primes of $\\mathbb{Q}$ split in $E$, satisfying (Split), (CG), and (LGC$_{m_\\pi}$), and if $p$ does not divide $6N_{E/\\mathbb{Q}}(\\mathfrak{n})h_E(\\mathfrak{n})D_E$, then $\\Omega_2(\\pi)\\Omega_3(\\pi) \\mid \\Omega_5(\\Pi,\\varepsilon,-)\\cdot u$, with $u=u_{2,\\mathrm{ram}}$ the uncomputed local constant of Theorem C. The underlying discovery is that base-change periods can be defined in middle degree: when $\\Pi$ is invariant under the conjugation involution $\\sigma$ or the conjugation-duality involution $\\varepsilon$, the $\\Pi$-isotypic part of degree-$5$ cuspidal cohomology, though two-dimensional, carries a semi-linear involution whose $\\pm$-eigenspaces are one-dimensional with canonical integral structure. These middle-degree periods satisfy an adjoint formula of Hida type, Theorem D: up to $p$-adic units, the $\\pm$-part of the congruence number of $\\Pi$ on $H^5$ equals $\\Lambda^{\\mathrm{imp}}(\\Pi,\\mathrm{Ad},1)/(\\Omega_5(\\Pi,\\iota,\\pm)\\Omega_5(\\Pi^\\vee,\\iota,\\mp))$. Combining Theorem D with the factorization of adjoint $L$-functions and of congruence numbers yields the divisibility.","pith_inferences":["(Inference) If the uncomputed local constant $u_{2,\\mathrm{ram}}$ is trivial as expected, Theorem A becomes the clean relation $\\Omega_2(\\pi)\\Omega_3(\\pi) \\sim \\Omega_5(\\Pi,\\varepsilon,-)$ up to $p$-adic units, and computing that constant in one explicit example would directly test the strength of the theorem.","(Inference) The middle-degree construction should carry over to the other cases where the cuspidal range has length two, notably $n=4$ over a real quadratic field, once the archimedean generator computations used in the paper are extended to $\\mathrm{GL}_4$; the paper explicitly lists this as Case 2.","(Inference) The missing stable-base-change period relation is not caused by the base change itself: the transfer divisibility proved here would upgrade to a full period divisibility once an adjoint formula of Hida type for the unitary group becomes available.","(Inference) The one-sided divisibility should be the automorphic shadow of a Bloch--Kato statement: if the reciprocal divisibility can be proved by a non-vanishing-mod-$p$ result for $\\mathrm{GL}_3$ of the kind that already underlies the reciprocal divisibility in the $\\mathrm{GL}_2$ case, the conjectural equality $\\Omega_5(\\Pi,\\varepsilon,-)\\sim\\Omega_2(\\pi)\\Omega_3(\\pi)$ would follow."],"forward_implications":["Under the stated hypotheses, $\\Omega_2(\\pi)\\Omega_3(\\pi)$ divides $\\Omega_5(\\Pi,\\varepsilon,-)\\cdot u$ for all $p$ avoiding the finite exceptional set, so the classical periods of $\\pi$ control the middle-degree period of its base change.","The middle-degree periods $\\Omega_5(\\Pi,\\iota,\\pm)$ are $p$-integral and intrinsic when $\\Pi$ is self-dual, so they can play the role of the automorphic period of the twisted adjoint motive in the Bloch--Kato comparison.","The middle-degree adjoint formula makes the normalized imprimitive adjoint value $\\Lambda^{\\mathrm{imp}}(\\Pi,\\mathrm{Ad},1)/(\\Omega_5(\\Pi,\\iota,\\pm)\\Omega_5(\\Pi^\\vee,\\iota,\\mp))$ integral, as an element of $\\mathcal{O}$.","For a self-conjugate stable base change from $U_E$, the twisted adjoint $L$-value of $\\pi$ divided by $\\Omega_5(\\Pi,\\sigma,-)$ divides the congruence number of $\\pi$ (Corollary 4.1).","The transfer divisibility for the stable base change (Theorem 4.1) reduces the missing period relation for $U_E$ to the absence of an adjoint formula of Hida type for that unitary group."],"supporting_citations":[{"why":"supplies the classical base change from $\\mathrm{GL}_3(\\mathbb{Q})$ to $\\mathrm{GL}_3(E)$, the transfer whose periods are compared.","marker":"[AC89]"},{"why":"establishes the stable base change for $n=3$ from the quasi-split unitary group $U_E$ to $\\mathrm{GL}_3(E)$.","marker":"[Rog90]"},{"why":"proves the base-change detection property for the unitary side, giving the vanishing property for non-stable-base-change forms.","marker":"[Mok15]"},{"why":"provides the $\\mathrm{GL}_2$ integral period relation that this paper generalizes, along with the congruence-number and period formalism adapted here.","marker":"[TU22]"},{"why":"gives the adjoint $L$-value formula for $\\mathrm{GL}_n$ whose middle-degree refinement (Theorem D) is the central mechanism.","marker":"[BR14]"},{"why":"supplies the explicit canonical archimedean generators and the computation of the archimedean factors used to normalize the Eichler--Shimura maps.","marker":"[Che22]"},{"why":"proves that the period integral detecting the classical base change is nonzero exactly for base changes, giving the vanishing property on non-base-change forms.","marker":"[FLO12]"},{"why":"computes the classical-base-change period as the twisted adjoint $L$-value appearing on the right side of Theorem C.","marker":"[Jac01]"},{"why":"provides the freeness theorem for the localized cohomology that turns Hecke-module inputs into the divisibility conclusion.","marker":"[CG18]"},{"why":"constructs the Galois representation with coefficients in the localized Hecke algebra when $p$ is completely split in $E$; its local-global compatibility at bad places is assumed rather than known.","marker":"[CGJ23]"}],"fun_headline_variants":["Middle-degree periods divide for GL3 base change","p-adic divisibility via middle-degree periods","Base change periods divide in degree five","GL3 base change: new period divisibility","Middle cohomology periods yield p-adic relation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the set of running hypotheses the paper labels (Galm), (LGC), and (CG)—in particular local-global compatibility of the Galois representation attached to the base change at primes where the level ramifies, which the paper notes in Section 2.2.4 is not known when $p$ is completely split in $E$—together with residual absolute irreducibility of $\\rho_\\Pi$; if any of these fail, the freeness theorem for the localized degree-$5$ cohomology and with it the divisibility collapses.","fun_headline_variants_meta":{"raw":{"variants":["Middle-degree periods divide for GL3 base change","p-adic divisibility via middle-degree periods","Base change periods divide in degree five","GL3 base change: new period divisibility","Middle cohomology periods yield p-adic relation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000511,"raw_usage":{"total_tokens":2541,"prompt_tokens":1053,"completion_tokens":1488,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":1418}},"tokens_in":669,"tokens_out":1488,"duration_ms":18333,"temperature":1.0,"reasoning_tokens":1418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:11:29.091691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for an explicit triple $(\\pi,E,p)$ satisfying the hypotheses, the $p$-adic valuations of the transfer congruence number $\\eta^\\#_{\\lambda_\\Pi}(M^*)[+]$ and of $\\Lambda^{\\mathrm{imp}}(\\pi,\\mathrm{Ad}\\otimes\\chi_E,1)/(\\Omega_5(\\Pi,\\sigma,+)\\Omega_5(\\Pi^\\vee,\\sigma,-))$; the claimed divisibility is false if the valuation of the congruence number exceeds that of the quotient. A more direct check of the proof's gate is to test the assumed local-global compatibility at a ramified place above $2$ or above a prime ramified in $E$ for the Galois representation of [CGJ23] when $p$ is completely split in $E$, since the freeness conclusion of Theorem 2.3 is the only bridge from the Hecke algebra to the divisibility.","supporting_citations":[],"review_version":1}