New perspectives on $p$-adic regulator formulae

Ananyo Kazi; Luca Marannino; Ting-Han Huang

REVIEW 3 major objections 5 minor 11 references

New perspectives on $p$-adic regulator formulae

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2601.05406 v2 pith:JMNLSEHZ submitted 2026-01-08 math.NT

Ting-Han Huang , Ananyo Kazi , Luca Marannino This is my paper · ORCID

classification math.NT MSC 11F4111F6711F8011F8514F30

keywords p-adicregulatorformulaeAsai-FlachclassesdiagonalL-functionsfiniteslopeHilbertmodularformsBloch-KatologarithmsyntomiccohomologyClebsch-Gordanpullback

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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].

The soft spots, in proportion:

1. 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.

2. 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.

3. 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.

These 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Remark 5.37; Theorem 5.36] 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.
[Remark 5.6(3); Theorem 5.5; Theorem 5.11] 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.
[§6.2, final paragraph after Eq. (6.12)] 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.

minor comments (5)

[Introduction, Theorem A vs Theorem 5.36] 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.
[Proposition 5.4 proof] 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.
[Remark 6.5 and References] The text cites '[Mar2026]', but the reference list contains [Mar24] and [Mar25]; please reconcile the citation key and give the precise reference.
[Definition 5.1] 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.
[General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the regulator identity relates independently defined quantities; cited overlapping-author works are external inputs, and the admitted factorial mismatch is a correctness gap, not a circular step.

full rationale

The central regulator identity (Theorem 5.36) relates two independently defined objects: the Bloch–Kato logarithm of the Asai–Flach class, constructed via pro-étale pushforward of an Eisenstein class (Proposition 5.4, Section 3), and the p-adic Asai L-function constructed in [KL25, Definition 7.1] as a coherent pairing with an Eisenstein family and required to interpolate critical L-values. The paper does not define the L-function as the regulator pairing; instead it reduces the regulator side to a coherent pairing (Theorem 5.30, Corollary 5.34) and then identifies the resulting p-depleted Eisenstein series with the one appearing in the L-function by explicit q-expansion (Lemma 5.35, with the note that 1/(k2−j)!∇^{−1−(k2−j)}F^{t+2,[p]}_{0,1/N} is the Eisenstein series in the L-function formula). This is a genuine computation, not a renaming or a definitional identity. The heavy use of works with overlapping authors ([KL25], [GLZ25], [Hua24], [HK25]) is external dependence: those results are stated with their own assumptions, are not fitted to the present conclusion, and do not reduce the argument to an unverified self-citation. Remark 5.37 explicitly acknowledges an unresolved factorial mismatch with [GLZ25, Theorem 9.6.4]; that is a correctness gap about the constant factor, not evidence that the claimed equality holds by construction. Likewise, Remark 5.6(3) flags an unverified Hyodo–Kato identification, but the paper says it instead uses Grosse-Klönne rigid cohomology, so the load-bearing comparison is not being smuggled in as a tautology. Overall, no step in the derivation chain is circular by the paper's own equations or by self-citation alone.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or forces. Its real external dependencies are the derived comparison theorems of [DN18], the p-adic Asai L-function of [KL25], the Asai-Flach class construction and Eisenstein q-expansions of [GLZ25], and the final diagonal pairing computation of [BSV20]. The unverified Hyodo-Kato/rigid Frobenius identification and the unresolved factorial normalization are the two points where the argument relies on assumptions not fully discharged in the text.

free parameters (3)

α1, β1 = roots of the p1-Hecke polynomial; v_p(α1) < k1+1
Chosen p1-stabilisation data of Π; the regulator formula and denominators depend on these values.
α2, β2 = roots of the p2-Hecke polynomial
Hecke eigenvalues at p2 appearing in the polynomial P(φ) that kills η_dR and in the final formula.
normalization of η_dR (and η_rig^{1-ord}) = unspecified
η_dR is chosen as a basis of a 1-dimensional L_P-module; the factorial discrepancy with [GLZ25] in Remark 5.37 may be absorbed in this choice, but the paper does not fix the normalization.

assumptions (5)

standard math Derived de Rham and semistable comparison theorems with coefficients (Theorem 5.5, [DN18, §4.15])
The whole syntomic-to-Bloch-Kato logarithm computation assumes these functorial quasi-isomorphisms.
domain assumption Hyodo-Kato cohomology coincides with log-crystalline/rigid cohomology with matching Frobenius (Remark 5.6(3))
The authors explicitly state they have not verified this coincidence; the Frobenius actions used in the regulator formula depend on it.
domain assumption Vanishing of the Hecke-localised target cohomology, H^3_ét(Y_G,...)^{G_Q} = 0 on the Π_f^∨-eigenspace ([GLZ25, Note 6.1.1])
The Asai-Flach extension class is obtained from an exact sequence only because this target vanishes; without it the class construction breaks.
standard math Spherical-pair multiplicity one for GL2 in GL2 × GL2 gives uniqueness of the Clebsch-Gordan map CG_{r-k} (Prop 6.2(1))
The pullback reconstruction of the diagonal class requires the map used to be the unique equivariant map up to scalar.
ad hoc to paper The final pairing computation in the diagonal regulator formula is exactly the one performed in [BSV20, pp. 1023-1024]
The new proof of Theorem 6.4 stops before computing the final Petersson pairing and delegates it to [BSV20]; the theorem as proved depends on that external computation.

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Pith. "Pith review of New perspectives on $p$-adic regulator formulae." pith.science (2026). https://pith.science/paper/JMNLSEHZ

@misc{pith2026260105406,
  author       = {Pith},
  title        = {Pith review of: New perspectives on $p$-adic regulator formulae},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMNLSEHZ}},
  note         = {Machine review of arXiv:2601.05406}
}

abstract

Inspired by the pullback method in the recent work of Sangiovanni-Vincentelli--Skinner, we reconstruct the diagonal class of Darmon--Rotger. Moreover, we reinterpret the computation of the $p$-adic regulator formula.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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