{"id":"6792c1d0-20bb-405f-9c7c-e055289b8185","arxiv_id":"2601.05406","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A finite-slope p-adic regulator formula for Asai–Flach classes is proved without finite-polynomial cohomology, and diagonal classes are reconstructed as pullback extension classes.","lead":"This paper gives new proofs of p-adic regulator formulas for Asai–Flach and diagonal cohomology classes, extending the Asai formula to finite-slope Hilbert modular forms and recasting diagonal classes as pullbacks. The methods avoid finite-polynomial cohomology and may generalize to settings where forms are ramified at p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.36 carries an admitted k1!k2! mismatch with [GLZ25, Thm 9.6.4] (Remark 5.37); until this constant-factor discrepancy is resolved, the central finite-slope regulator formula is not established.","rationale":"The reader's weakest_assumption emphasized the functorial comparison in Theorem 5.5 and the unverified identification in Remark 5.6(3). That is a legitimate structural risk, but the paper explicitly sidesteps the Hyodo–Kato identification by working with Grosse-Klonne rigid cohomology, where the Frobenius is known to match the crystalline Frobenius. The more concrete and directly load-bearing gap is the admitted k1!k2! factor mismatch in the final formula, flagged in Remark 5.37. Since Theorem A is an equality with an explicit constant, an unresolved mismatch with the established ordinary-level formula means the central theorem is not yet fully supported as stated. This is consistent with the reader's CONDITIONAL verdict: the issue is addressable by a direct computation, but the current text does not resolve it. I therefore recommend no change to the reader's verdict, but I would shift the emphasis of the concern from the comparison theorem to the factorial discrepancy.","tokens_in":43061,"tokens_out":5558,"duration_ms":58488,"concrete_test":"Specialize Theorem 5.36 to the ordinary case by taking α1 to be the unit root and compare the resulting constant term-by-term with [GLZ25, Theorem 9.6.4], using the L-function normalization of [KL25, Definition 7.1]. If the two formulas differ by k1!k2!, recompute Lemma 5.35 from the explicit Clebsch–Gordan formula in [GLZ25, §9.3] to locate the missing factor; this determines whether the discrepancy is a harmless convention change or a genuine error in the stated regulator formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem A (Theorem 5.36) is an equality with a specific rational factor. Remark 5.37 states that this factor differs from the established [GLZ25, Theorem 9.6.4] formula by exactly k1!k2!, and that the authors “do not completely understand” the accounting in GLZ25 Remark 9.5.6. This is not a cosmetic normalization issue: every prior step (Theorem 5.30, Corollary 5.34, Lemma 5.35) is an equality between explicit cohomology classes, so a missing factorial propagates directly into the final constant. The discrepancy must be located in one of: the normalization of η_1-ord / \\tilde{η}_1-ord in Definition 5.12/§5.6, the q-expansion in Lemma 5.35, the p-depletion Lemma 5.32, or the identification of the Eisenstein series in the [KL25] p-adic L-function. Since the finite-slope statement should specialise to the ordinary statement when α1 is chosen to be the unit root, the mismatch in the ordinary limit is directly checkable. The functorial comparison concern in Remark 5.6(3) is real but less decisive: the paper explicitly does not use Hyodo–Kato = log-crystalline, but rigid cohomology via Grosse-Klonne dagger spaces, whose Frobenius is known to match the crystalline Frobenius. The factorial discrepancy, by contrast, is an admitted unresolved gap inside the main theorem's computation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes new proofs and some generalizations of p-adic regulator formulae for Asai–Flach classes and diagonal classes. For the Asai case, working over a real quadratic field with a split prime p, the authors prove a finite-slope regulator formula (Theorem A / Theorem 5.36) relating the Bloch–Kato logarithm of the Asai–Flach class, paired with a de Rham class η_dR, to the imprimitive p-adic Asai L-function of [KL25]. The proof avoids finite-polynomial cohomology by using cone/mapping-fibre constructions, derived de Rham comparison results of [DN18], and reductions to the (partial) ordinary dagger locus. For diagonal classes, the paper gives a pullback reconstruction of the diagonal extension class (Proposition 6.2) and uses it to give a new derivation of the known diagonal regulator formula (Theorem 6.4). The novelty is explicitly claimed to lie in the proofs and in the finite-slope Asai generalization.","tokens_in":43488,"tokens_out":4240,"duration_ms":46097,"significance":"If fully correct, the paper would be a valuable contribution: it removes finite-polynomial cohomology from a technically heavy regulator computation, extends the Asai regulator formula beyond the ordinary case, and gives a clean categorical explanation of the diagonal class via mapping fibres. The homological Lemmas 2.1–2.2 and the reduction to coherent pairings on dagger ordinary loci are elegant and likely to be reusable. However, the central Asai formula contains an admitted factorial discrepancy with the previously established ordinary formula, and this must be resolved before the main theorem can be regarded as proved. The paper is therefore significant but currently not fully sound.","major_comments":[{"comment":"The final Asai regulator formula is stated with a specific rational factor, but Remark 5.37 admits an unresolved factor-k1!k2! discrepancy with [GLZ25, Thm 9.6.4]. This is not cosmetic: every intermediate identity (Theorem 5.30, Corollary 5.34, Lemma 5.35) is an equality of explicit cohomology classes, so an extra factorial in the final constant has to enter somewhere in the normalization of η_{1-ord} (Definition 5.12), the q-expansion computation in Lemma 5.35, the p-depletion Lemma 5.32, or the [KL25] p-adic L-function. Since the finite-slope formula should specialise to the ordinary formula as α1 approaches the unit root, this is checkable in the ordinary limit. As written, the central claim of Theorem A is not established.","section":"Remark 5.37; Theorem 5.36"},{"comment":"The proof of the syntomic identification in Theorem 5.11 relies on comparison functoriality of pro-étale, de Rham, and rigid/Hyodo–Kato cohomology. Remark 5.6(3) states that the Hyodo–Kato cohomologies 'should coincide' with log-crystalline cohomology but that this has not been verified. The paper's actual use of Grosse-Klonne rigid cohomology with a Frobenius known to match crystalline Frobenius makes this plausible, but the manuscript should either verify the required comparison or state a precise weaker assumption. Because all S(φ) and P(φ) manipulations feed into the regulator formula, this assumption should be explicit and justified rather than left as a remark.","section":"Remark 5.6(3); Theorem 5.5; Theorem 5.11"},{"comment":"The diagonal regulator proof is presented as a new method, but the final step is outsourced: 'one can proceed exactly as explained in [BSV20, pp. 1023-1024]' after identifying CG_{r-k} with the determinant element. The claimed equality of pairings is where the constant, including the Euler factor E_p(f,g,h) and the Petersson inner-product ratio, is determined. The manuscript should spell out, or at least verify in detail, that the BSV20 computation applies unchanged in the present setup; otherwise Theorem 6.4 rests on an unexamined black box.","section":"§6.2, final paragraph after Eq. (6.12)"}],"minor_comments":[{"comment":"The abstract and introduction state Theorem A only as 'can be expressed in terms of the p-adic Asai L-value'; the precise rational factor, including the factorial ambiguity, appears only in Theorem 5.36. Stating the exact factor in the introduction would help readers see the claimed normalization.","section":"Introduction, Theorem A vs Theorem 5.36"},{"comment":"The use of 'G' both for the reductive group Res_{F/Q} GL_{2,F} and for the functor υ_{proét,*} is flagged by the authors but remains confusing; suggest renaming one of them.","section":"Proposition 5.4 proof"},{"comment":"The text cites '[Mar2026]', but the reference list contains [Mar24] and [Mar25]; please reconcile the citation key and give the precise reference.","section":"Remark 6.5 and References"},{"comment":"The notation N(n) in the imprimitive Asai L-series is not defined; the later explanation about omitting Euler factors at primes dividing N is not sufficient to make the summands unambiguous for all n.","section":"Definition 5.1"},{"comment":"There are several minor typographical issues, e.g., 'Münster J. Math.' is abbreviated inconsistently, and in Remark 6.3 'The fact that this linear functional' is missing a 'that'. These are harmless but should be cleaned up.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on work by overlapping authors — [KL25], [GLZ25], [Hua24], [HK25], [SS] — for the p-adic L-function, the ordinary Asai formula, and the comparison results. This is not itself a defect, but it makes the unresolved k1!k2! discrepancy in Remark 5.37 especially important: an external verification of the factorial in the ordinary limit would substantially increase confidence. I would recommend asking for that check in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper has a real new result in view—the finite-slope Asai regulator formula—but the main theorem is not yet established because of an admitted k1!k2! factor mismatch with the ordinary-level formula [GLZ25, Thm 9.6.4]. The paper is worth refereeing, not desk-rejecting.\n\nThe genuinely new stuff: Theorem 5.36 extends the Asai regulator formula to small-slope p1-stabilisations, and the proof method (derived de Rham comparison plus homological algebra, avoiding fp-cohomology and partial compact support) is a credible simplification. The pullback reconstruction of the diagonal class in Prop 6.2 is a neat construction; it reproduces the known diagonal regulator formula with a much shorter argument, though the final computation is deferred to [BSV20].\n\nThe soft spots, in proportion:\n\n1. Remark 5.37 is the load-bearing one. The authors state that their finite-slope formula is k1!k2! times the GLZ25 formula and that they don't fully understand the GLZ25 accounting. That is not cosmetic. Every intermediate equality in Section 5 is an explicit cohomology identity, so a missing factorial propagates into the final constant. The finite-slope result should specialise to the ordinary one when α1 is the unit root, so the mismatch is directly checkable. Until it is resolved, Theorem A is conditional.\n\n2. Remark 5.6(3): the unverified identification of Hyodo–Kato cohomology with (log-)crystalline cohomology. The stress-test note is right that this is less decisive, because they work with Grosse-Klonne rigid cohomology and the Frobenius matching is expected, but the paper still relies on a comparison whose details are not spelled out, and Theorem 5.11 depends on it. Needs to be written down.\n\n3. The diagonal part: Prop 6.2 is good, but Theorem 6.4 is billed as a new proof while the final cup-product computation is 'exactly as explained in [BSV20, pp. 1023-1024]'—so it is not self-contained. For a paper advertising simplification, the reader has to fetch the old paper to verify the core computation.\n\nThese are addressable, not fatal. The homological algebra in Sections 2–3 is clean and likely reusable; the paper is honest about its gaps, which I appreciate.\n\nWho it's for: people working on p-adic regulator formulas, Euler systems for Hilbert modular forms, and the Asai L-function. A serious referee should be engaged. My recommendation: send to peer review, but the referee should push the authors to resolve the factorial mismatch in Remark 5.37 and to either verify Remark 5.6(3) or state it as a formal assumption. Only after that is the finite-slope Asai formula claim trustworthy.","headline":"The finite-slope Asai regulator formula is a genuine new result, but the admitted k1!k2! factorial mismatch with the ordinary case means the main theorem is not yet established as stated.","tokens_in":43946,"tokens_out":2458,"would_cite":false,"duration_ms":27753,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F41","11F67","11F80","11F85","14F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the p-adic Asai regulator formula extends from ordinary to finite-slope Hilbert modular forms, and that the same pullback construction recovers the diagonal regulator formula, by replacing finite-polynomial cohomology","keywords":["p-adic regulator formulae","Asai-Flach classes","diagonal classes","p-adic L-functions","finite slope Hilbert modular forms","Bloch-Kato logarithm","syntomic cohomology","Clebsch-Gordan pullback"],"falsifier":"Compute the two Frobenius actions explicitly on a Hilbert modular surface at a split prime (or on its dagger ordinary locus) and exhibit a class where the Hyodo-Kato Frobenius and the crystalline Frobenius differ; Theorem 5.11 and hence Theorem A would collapse, since all S(φ) manipulations assume this identification. A less drastic test is to verify the identification in Remark 5.6(3) directly for the good-reduction case.","tokens_in":42947,"feed_emoji":"🧮","tokens_out":4629,"duration_ms":47828,"temperature":0.7,"pith_summary":"The paper aims to prove p-adic regulator formulae—identities that express the Bloch-Kato logarithm of a special cohomology class as an explicit multiple of a p-adic L-value—using a new, lighter method. Its main result (Theorem A) establishes such a formula for Asai-Flach classes attached to a finite-slope stabilisation of a Hilbert modular form over a real quadratic field, removing the ordinarity assumption of earlier work. On the diagonal-class side, it shows that the diagonal class arises as an extension class from the mapping fibre of a Clebsch-Gordan pullback, which yields a simplified proof of the known diagonal regulator formula. The proofs avoid finite-polynomial cohomology and its cup-product computations, relying instead on elementary homological algebra and derived de Rham comparison theorems.","feed_headline":"Asai regulator formula extends to finite slope","feed_subtitle":"Pullback proof links Asai-Flach and diagonal Bloch-Kato logarithms straight to p-adic L-values.","key_machinery":"The central mechanism is the mapping fibre (mapping cone) of the Clebsch-Gordan pullback. For a representation V_l ⊗ V_m over Y^2, the Clebsch-Gordan map CG^*_{r-k}: V_l ⊗ V_m → V_k(k-r) induces a map of cohomology complexes, and the Asai-Flach and diagonal classes appear as extension classes in the long exact sequence attached to the triangle completing this map. The key technical lemmas are Lemma 2.2, a snake-lemma comparison identifying a connecting map with minus the triangle differential, and Proposition 3.7, an abstract formula for the Bloch-Kato logarithm of an extension in terms of a Frobenius polynomial P(φ) and a rigid primitive. Together they let the logarithm of the pushed-forwar","core_discovery":"The paper's central claim is that two regulator formulae—the Asai regulator formula for Hilbert modular forms and the diagonal/triple-product regulator formula—can both be proved by a single cone-and-pullback construction. For a Hilbert modular form Π over a real quadratic field with a small-slope p1-stabilisation α1, Theorem A constructs a de Rham class η_dR in Fil^0 D_dR(V_{p,As}(1+j)) such that the pairing of η_dR with the Bloch-Kato logarithm of the Asai-Flach class AF^j_ét(Π) equals an explicit rational factor times the p-adic Asai L-value L^imp_{p,As}(Π)(1+j). Proposition 6.2 realizes the diagonal class as an extension class coming from the long exact sequence of the Clebsch-Gordan pul","pith_inferences":["If the Hyodo-Kato/crystalline comparison flagged in Remark 5.6(3) is verified, the same mapping-fibre formalism should produce regulator formulae for other spherical pairs, including the balanced Asai case over a cubic field, where the paper identifies the relevant coherent cohomology group for a future p-adic L-function.","The paper's shift away from finite-polynomial cohomology suggests that regulator formulae for other Euler-system classes, such as Beilinson-Flach or GSp(4) classes, could be reproved by the same snake-lemma comparison whenever the relevant de Rham Eisenstein class satisfies the same Frobenius-killing condition.","The unverified comparison in Remark 5.6(3) is the natural target for a follow-up: a direct proof of that identification, or an explicit counterexample, would settle the status of Theorem A."],"forward_implications":["If Theorem A is correct, the Asai regulator formula holds for finite-slope (not necessarily ordinary) Hilbert modular forms, removing the ordinarity assumption of earlier work.","The same argument shows the Bloch-Kato logarithm of the Asai-Flach class is computed by the syntomic pushforward of the de Rham Eisenstein class, with no finite-polynomial cohomology or partial compact support rigid cohomology.","The pullback reconstruction of the diagonal class gives a new proof of the known diagonal regulator formula, reducing the computation to a rigid primitive over the ordinary locus of a modular curve.","The authors state the method should extend to other settings—such as twisted triple product/Hirzebruch-Zagier classes and possibly ramified-at-p representations—whenever the p-adic L-function is built from a coherent class on a good-reduction dagger affinoid."],"fun_headline_variants":["Cone-and-pullback unifies Asai and diagonal regulators","One construction proves both p-adic regulator formulae","Pullback proof links Asai-Flach and diagonal classes","Unified proof for two regulator formulae via cone","Single pullback yields Asai and diagonal regulator results"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof relies on the unverified identification, noted in Remark 5.6(3), that the Hyodo-Kato cohomology used in the comparison theorem coincides with usual log-crystalline cohomology in their good-reduction setting; if that Frobenius comparison fails, the regulator formula would compute a different quantity.","fun_headline_variants_meta":{"raw":{"variants":["Cone-and-pullback unifies Asai and diagonal regulators","One construction proves both p-adic regulator formulae","Pullback proof links Asai-Flach and diagonal classes","Unified proof for two regulator formulae via cone","Single pullback yields Asai and diagonal regulator results"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1114,"prompt_tokens":580,"completion_tokens":534,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":324,"completion_tokens_details":{"reasoning_tokens":456}},"tokens_in":324,"tokens_out":534,"duration_ms":5372,"temperature":1.0,"reasoning_tokens":456,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T11:37:23.407326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two Frobenius actions explicitly on a Hilbert modular surface at a split prime (or on its dagger ordinary locus) and exhibit a class where the Hyodo-Kato Frobenius and the crystalline Frobenius differ; Theorem 5.11 and hence Theorem A would collapse, since all S(φ) manipulations assume this identification. A less drastic test is to verify the identification in Remark 5.6(3) directly for the good-reduction case.","supporting_citations":[],"review_version":1}