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REVIEW 3 major objections 3 minor 11 references

A proof of $p$-adic Gross--Zagier theorem via BDP formula

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Pith's one-line read A uniform proof of the p-adic Gross–Zagier formula: the derivative of the base-change p-adic L-function equals the p-adic height of the Heegner cycle, including non-ordinary weight > 2 cases.

desk verdict The new proof strategy is real, but the main formula has a reciprocal-constant error that is not derived from the paper's own equations unless A/B=1. read the letter →

arxiv 2604.13854 v2 pith:TMHRUEG6 submitted 2026-04-15 math.NT

Kâz{i}m Büyükboduk , Peter Neamti This is my paper · ORCID
classification math.NT MSC 11G4011G1811F6711R2311F33
keywords p-adicGross–ZagierformulaHeegnercyclesBeilinson–FlachelementsBDPL-functionsColemanfamiliesnon-ordinaryprimesSelmercomplexes
topics Birch and Swinnerton-Dyer
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 sets out to prove the p-adic Gross–Zagier formula in a uniform way: at the central point, the derivative of the p-adic L-function attached to a modular form over an imaginary quadratic field equals, up to an explicit algebraic factor, the p-adic height of the corresponding Heegner cycle. Earlier proofs compared two kernels and were tied to ordinary primes or to weight two; here the comparison is replaced by a wall-crossing argument centred on the BDP formula and Beilinson–Flach elements. The result covers ordinary and non-ordinary eigenforms of weight k ≥ 2, including the previously open case k > 2 with positive slope (ord_p(a_p(f)) > 0). If a non-vanishing derivative is known to exist (condition NV), the algebraic factor A/B is forced to be 1, making the formula exact; if not, the theorem is still consistent with earlier degenerate 0 = 0 cases. A sympathetic reader would care because the same mechanism now explains and extends the known formulae, rather than a case-by-case kernel computation.

What carries the argument

The argument runs through a wall-crossing comparison between two anticyclotomic classes: the big Beilinson–Flach element cBF^τ_{ac,f} and the big Heegner class Z^τ_∞, both lying in the same Selmer group for the self-dual twist V^{†,τ}_{f,ac}. The BDP formula — a p-adic Waldspurger limit formula — together with the first and second reciprocity laws for Beilinson–Flach elements identifies their images at p. An Euler-system argument, relying on a rank-one assertion, then upgrades the local comparison to a global equality Z^τ_∞ = λ · cBF^τ_{ac,f} in the Selmer group. Substituting this equality into a height-pairing computation (a cyclotomic-derivative height formula) yields the derivative of the

What would settle it

Compute the rank of H^1_f(G_K,S, V^{†,τ}_{f,ac}) over the anticyclotomic Iwasawa algebra for a Coleman family of positive slope through a non-ordinary eigenform of weight > 2 satisfying hypotheses (Disc), (Heeg), and (BI). If for any anticyclotomic character τ this rank exceeds one, the paper's Proposition 6.9 is false and the global comparison Z^τ_∞ = λ·cBF^τ_{ac,f} cannot be justified; a direct proof of rank one in that setting would supply the missing verification.

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Extended reading notes

Core claim

The central claim is Theorem A (equation (1.1)). For a normalised cuspidal eigen-newform f of weight k ≥ 2 whose base change to K satisfies the strong Heegner hypothesis, let fα be its p-stabilisation (with a_p(f) ≠ p^{k/2} when p|N_f). Then the derivative at s = k/2 + 1 of the p-adic L-function L_p(fα/K, s) equals A/B times (1 − p^{k/2}/α)^4 · h_f(z_f, z_f)/(4|D_K|)^{k/2} if p∤N_f, and A/B times 4 · h_f(z_f, z_f)/(4|D_K|)^{k/2} if p||N_f. Here z_f is the Heegner cycle attached to the base change of f, h_f is the p-adic height pairing, and A, B are explicit algebraic constants; under the non-vanishing condition (NV), A/B = 1. The paper proves this by describing the p-adic L-function through

Load-bearing premise

The proof needs a certain Galois-cohomology Selmer group, formed with local conditions along the anticyclotomic tower, to have exactly one generator; this rank-one fact is cited from earlier work, not verified for the general finite-slope Coleman families at hand, and the global comparison of Heegner and Beilinson–Flach classes collapses if the rank is larger than one.

Editorial extensions

If this is right

  • The formula holds uniformly for p-ordinary and non-ordinary forms, including weight k > 2 with positive slope, a range not covered by earlier proofs.
  • When (NV) holds, A/B = 1, so the derivative of the p-adic L-function is exactly the p-adic height of the Heegner cycle; when (NV) fails, the theorem remains consistent with the 0 = 0 conclusion forced by earlier degenerate cases.
  • Specialising the family-level formula yields p-adic Gross–Zagier for generalised Heegner cycles f/K × χ with χ^c = χ^{-1} and infinity type (−ℓ, ℓ), |ℓ| < k/2, reproving and extending earlier results in that setting.
  • The proof produces a new reciprocity statement: the derivative term L^f_{ac,1}(f, τ) is computed by the height of a Beilinson–Flach element, which is the analytic input behind the formula.
  • The case p||N_f with a_p(f) = p^{k/2} is the only finite-slope case left out; the theorem explicitly defers it as work in progress.

Reading between the lines

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

  • Because A/B is independent of the form f, a single non-zero instance of the formula for any eigenform satisfying the hypotheses would pin A/B = 1 without assuming (NV); the paper uses (NV) only to guarantee such an instance exists from earlier work.
  • The same wall-crossing route is a plausible template for the missing case p||N_f, a_p(f) = p^{k/2}; replacing the classical Heegner class by its generalised analogue is the natural first move toward that case.
  • If the rank-one Selmer assertion fails for some finite-slope family, the global comparison may still hold after localising or inverting a single element; a rank-two example would not refute the final derivative formula, only the specific global-comparison proof given here.
  • The reliance on Beilinson–Flach elements suggests that improvements to BDP-type formulas in higher weight or higher rank settings would automatically upgrade this p-adic Gross–Zagier theorem, since the BDP formula is the external input carrying the arithmetic of Heegner cycles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper claims a new proof of the p-adic Gross–Zagier formula for the p-adic L-function of the base change of a cuspidal eigenform f of weight k ≥ 2 to an imaginary quadratic field K, treating ordinary and non-ordinary primes uniformly, including k > 2 with ord_p(a_p(f)) > 0. The route uses the BDP formula, Beilinson–Flach elements, big Heegner classes, and a Rubin-style height formula to first prove a version up to algebraic factors (Cor. 6.13), then normalizes L_p by (6.5) to obtain a 'usual form' (Thm 6.14, Thm A) with a rational constant A/B, claimed to be 1 under hypothesis (NV). The paper is largely a synthetic argument over existing results; the new input is the comparison of Beilinson–Flach and Heegner classes in Selmer groups.

Significance. The proposed strategy is coherent and, if the constant issue were repaired, would give a genuinely uniform proof and extend known results to new non-ordinary higher-weight cases. The paper is honest about relying on major external theorems (BDP, reciprocity laws, Pottharst/Kobayashi–Ota). It contains no machine-checked proofs or reproducible code; its value depends on the correctness of the normalization and on the rank-one Selmer assertion. Currently the central displayed formula is not supported by the paper's own algebra, so the significance is prospective rather than established.

major comments (3)
  1. [§6.2.4, Eq. (6.5)–(6.6) and Cor. 6.13; Theorem A, Eq. (1.1)] Corollary 6.13 gives H/(4|D_K|)^{k/2} = ⋆·(A/B)·dL^{(f)}_p/ds, with ⋆ = αE(fα)E*(fα)(1−p^{k/2}/α)^{-4} (p∤N; the p||N case is analogous with ⋆=α/4). Since (6.5) defines L_p = ♣·L^{(f)}_p with ♣=αE(fα)E*(fα) (resp. ♣=α), substitution yields dL_p/ds = (♣/⋆)(B/A)·H/(4|D_K|)^{k/2} = (1−p^{k/2}/α)^4(B/A)·H/(4|D_K|)^{k/2} (resp. 4(B/A)H/(...)). This is the reciprocal of the constant in (6.6)/(1.1). Thus Theorem A does not follow from the paper's own equations unless B/A=A/B, which is neither proved nor implied by the (NV) argument; that argument first uses (6.6) and so cannot supply the missing identity.
  2. [Proposition 6.9] The assertion rank_{H_ac,f} H^1_f(G_K,S,V^{†,τ}_{f,ac})=1 is proved by a one-line citation to [KO20] and [Pot13, Thms 1.9, 1.16], after reducing to torsion of H^1_{∅,0}. No hypotheses of those theorems are verified for the arbitrary finite-slope Coleman family f under (BI): e.g. whether [KO20] applies in the stated non-ordinary/non-étale triangulation setting, for all τ∈Δ_ac, or with the chosen local conditions. Since Prop. 6.10 uses this rank-one property to write Z∞=λ·cBF and to identify λ by a reciprocity law, failure of the rank bound would invalidate Theorem 6.12. This is a load-bearing gap, not a presentation issue.
  3. [Theorem 6.14 proof / (NV)] The normalization step is not justified. (NV1) postulates existence of an arbitrary p-old ordinary h of weight m≥2, but the proof chooses f_{E,α} attached to an elliptic curve without showing this h arises as such or that (H) and the cited GZ theorems apply to it. Comparing (6.6) to [PR87, Nek95, Kob13, Kob14] also requires matching the normalizations of the Heegner class, of L_p, and of the height pairing; the paper only remarks on one factor (h_K·z_{h◦} in footnote 8). Without such matching, the conclusion A/B=1 is not established. Moreover, even with A/B=1, major comment 1 remains unless B/A=1.
minor comments (3)
  1. [§5, opening] 'p-sabilisation' should be 'p-stabilisation'.
  2. [Proposition 6.10] 'Propsition' is a typo; also the freeness/faithfulness needed to invert the scalar λ in the localized module should be stated explicitly.
  3. [§2/§4] The symbol H is used both for the standing hypothesis (H) and for Perrin–Riou's algebra H(Z_p^×); this can confuse readers and should be disambiguated.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 6.14/Theorem A's coefficient is not derived: substituting Definition (6.5) into Corollary 6.13 gives B/A, not A/B; the constant is imported from prior Gross–Zagier theorems.

  1. other [Corollary 6.13 (eq. 6.3), Definition 6.5 (eq. 6.5), Theorem 6.14 (eq. 6.6), Theorem A (eq. 1.1)]
    "Corollary 6.13: hf(zf,zf)/(4|DK|)^{k/2} = (⋆)· A/B · d/ds L^{(f)}_p(fα,1^(p), s)|_..., (⋆)= αE(fα)E*(fα)(1−p^{k/2}/α)^{−4} if p∤N, α/4 if p||N. On setting (6.5) Lp(fα/K,s):=(♣)·L^{(f)}_p(fα,1^(p),s), (♣)= αE(fα)E*(fα) if p∤N, α if p||N, we obtain (6.6) d/ds Lp(...)= A/B × [(1−p^{k/2}/α)^4·hf/(4|DK|)^{k/2} ...]."

    Definition (6.5) gives dL^{(f)}_p/ds = ♣^{−1} dLp/ds. Substituting this into (6.3) forces dLp/ds = (♣/⋆)(B/A)·hf/(4|DK|)^{k/2}. Since ⋆=♣(1−p^{k/2}/α)^{−4} when p∤N and (⋆,♣)=(α/4,α) when p||N, the factor is (1−p^{k/2}/α)^4(B/A) or 4(B/A), respectively — the reciprocal of the A/B displayed in (6.6)/(1.1). Thus the main formula's coefficient is not a consequence of the paper's own universal formula; it becomes true only when A/B=1, a value imported by comparing with the prior Gross–Zagier theorems [PR87, Nek95, Kob13, Kob14] rather than derived from the BDP/Beilinson–Flach chain.

full rationale

Apart from the normalization mismatch, the proof is a chain of genuine external theorems: BDP formula (Thm 5.7), reciprocity laws for Beilinson–Flach elements (Thm 4.10, Cor 4.11), big Heegner interpolation (Thm 5.1), and Rubin-style height formulas (Thm 3.2), the latter backed by [Nek06, Büy16, BB23] in addition to the authors' [BB26]. I found no load-bearing self-citation loop: the self-references used are either accompanied by independent published/peer-reviewed sources or are not the source of the central identity. The rank-one assertion in Proposition 6.9 is a serious unverified premise — it cites [KO20] and [Pot13] without checking their hypotheses for arbitrary finite-slope families — but it is an external-citation gap, not circularity. The reciprocal A/B vs B/A error in Theorem 6.14 is likewise not a self-referential loop; it is an algebraic flaw in the final specialization. However, it means the exact coefficient in the advertised formula is currently an input (fixed by comparison with earlier Gross–Zagier theorems) rather than an output of the new derivation, which is why I set the score at 4 rather than 0. Once the reciprocal is corrected or A/B is independently established, the remaining argument would be largely self-contained and the circularity score would drop to 0–2.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper relies on a large network of external deep theorems (BDP formula, Beilinson–Flach reciprocity, big Heegner classes, rank-one Selmer results). The only really ad hoc assumption is (NV), introduced to force the constant A/B=1. The free constant A/B is not derived from first principles and appears to be affected by an internal algebra error.

free parameters (1)
  • A/B = 1 if (NV) holds; otherwise undetermined (and possibly B/A in the internal derivation)
    The wall-crossing comparison yields the formula only up to the algebraic constant A/B. The paper proves A/B=1 by comparison with prior Gross–Zagier theorems under the auxiliary existence hypothesis (NV), not by a first-principles evaluation. Moreover, the paper's own Corollary 6.13 combined with definition (6.5) implies the coefficient should be B/A, so the value of A/B is effectively a free const
assumptions (6)
  • ad hoc to paper Hypothesis (NV): There exists a p-old p-ordinary or p-supersingular form, or a supersingular elliptic curve, with nonvanishing p-adic L-derivative satisfying (H).
    Introduced specifically to pin the constant A/B=1 (Theorem 6.14 proof). No proof of (NV) is given; Remark 1.1 suggests it may follow from improvements of Smith's work, but this is not established.
  • domain assumption Big image condition (BI) defined in §2.5.7.
    Assumed in Theorem A and used in Proposition 6.9 to apply Euler-system machinery. It is a standard Galois-representation hypothesis, not proved here.
  • domain assumption The rank-one Selmer assertion Proposition 6.9, imported from [KO20] and [Pot13].
    Used to prove the global comparison Z∞ = λ·cBF in Proposition 6.10. The paper does not verify the hypotheses of the cited theorems in detail.
  • domain assumption BDP / Waldspurger reciprocity law, Theorem 5.7 = [JLZ19, Thm 8.2.4].
    A deep external theorem interpolating Heegner classes; it is a key input for the wall-crossing comparison.
  • domain assumption Reciprocity laws for Beilinson–Flach elements, Theorem 4.10 = [LZ16, BDV22].
    Used to compute the logarithms of Beilinson–Flach elements in terms of p-adic L-functions.
  • standard math Rubin's formula, Theorem 3.2 = [Nek06, BB26].
    Used to express p-adic heights in terms of local duality pairings; treated as a known general theorem.

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Pith. "Pith review of A proof of $p$-adic Gross--Zagier theorem via BDP formula." pith.science (2026). https://pith.science/paper/TMHRUEG6

@misc{pith2026260413854,
  author       = {Pith},
  title        = {Pith review of: A proof of $p$-adic Gross--Zagier theorem via BDP formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMHRUEG6}},
  note         = {Machine review of arXiv:2604.13854}
}
abstract

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.

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