{"id":"4f671280-90bb-4926-a398-7e4d2a152af1","arxiv_id":"2604.13854","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper proves a p-adic Gross–Zagier-type formula via Beilinson–Flach elements and a wall-crossing strategy, but the stated constant A/B conflicts with the paper's own Corollary 6.13.","lead":"This number theory paper claims a new proof of the p-adic Gross–Zagier formula, connecting derivatives of p-adic L-functions to heights of Heegner cycles, and extends it to some non-ordinary higher-weight cases. The result matters for Iwasawa theory, but the main formula appears to contain an internal inconsistency in its constant factor.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.14/Theorem A's constant A/B is the reciprocal of what Corollary 6.13 and definition (6.5) actually yield; the stated formula is not derived unless A/B=1.","rationale":"The reader's weakest_assumption was Proposition 6.9 (rank-one Selmer via one-line external citation), which is indeed a genuine load-bearing dependency. But the single most decisive problem is the internal algebraic mismatch between Corollary 6.13, definition (6.5), and Theorem 6.14: the constant in the final formula is the reciprocal of what the preceding equations produce. This does not rely on external literature or on subtle Selmer-rank hypotheses; it can be checked by direct substitution. Granting Proposition 6.9 entirely, the stated Theorem A still does not follow. The conditional proof that A/B=1 under (NV) also raises concerns: (NV) is essentially a nonvanishing instance of the theorem itself, and the proof of Theorem 6.14 appears to conflate the generic condition NV1 with the elliptic-curve condition NV3. On a charitable reading the A/B versus B/A discrepancy might be a typo, and the rank-one assertion might be fillable from [KO20] and [Pot13]; but as the manuscript currently stands, the central claim is not supported by its own derivation. Hence I do not see grounds to change the reader's REJECT verdict, though the most load-bearing stated reason is the constant mismatch rather than Proposition 6.9.","tokens_in":38691,"tokens_out":11301,"duration_ms":92088,"concrete_test":"Perform the formal substitution from (6.3) into (6.5): write dL_p/ds = ♣·dL_p^{(f)}/ds and replace dL_p^{(f)}/ds by h_f/(4|D_K|)^{k/2} / (⋆·A/B). The result is dL_p/ds = (♣/⋆)(B/A)·h_f/(4|D_K|)^{k/2}, i.e. (1−p^{k/2}/α)^4(B/A) for p∤N and 4(B/A) for p||N. Compare this with (6.6): if the coefficient is B/A rather than A/B, the theorem as stated is not a consequence of the proof. Independently, check the NV1/NV3 switch in the proof of Theorem 6.14 by reading the statement of NV1 and the sentence 'we let h=f_{E,α}'; these do not match unless NV1 is intended to be an elliptic-curve condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula (1.1)/(6.6) does not follow from the paper's own derivation. Corollary 6.13 gives, for p∤N, h_f/(4|D_K|)^{k/2} = ⋆·(A/B)·dL_p^{(f)}/ds with ⋆ = α E(f_α)E*(f_α)(1−p^{k/2}/α)^{-4}. Definition (6.5) sets L_p = ♣·L_p^{(f)} with ♣ = α E(f_α)E*(f_α). Substituting gives dL_p/ds = (♣/⋆)(B/A)·h_f/(4|D_K|)^{k/2} = (1−p^{k/2}/α)^4·(B/A)·h_f/(4|D_K|)^{k/2}, not (A/B) times the same factor. The p||N case is identical: ⋆=α/4, ♣=α, so dL_p/ds = 4(B/A)h_f/(4|D_K|)^{k/2}, whereas (6.6) claims 4(A/B)h_f/(4|D_K|)^{k/2}. Thus the displayed theorem is algebraically incompatible with Corollary 6.13 and (6.5) unless A/B=B/A, i.e. A/B=1 for positive constants. The paper's conditional proof that A/B=1 under (NV) cannot repair this: as written, (NV1) is a generic p-old ordinary form condition, yet the proof of Theorem 6.14 switches to a form f_{E,α} attached to an elliptic curve, blurring NV1 and NV3. Moreover, comparing one known Gross–Zagier formula only normalizes constants if all Heegner-class and L_p normalisations match, which is not established. So the exact constant in the main theorem is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":39187,"tokens_out":8077,"duration_ms":68750,"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":[{"comment":"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.","section":"§6.2.4, Eq. (6.5)–(6.6) and Cor. 6.13; Theorem A, Eq. (1.1)"},{"comment":"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.","section":"Proposition 6.9"},{"comment":"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.","section":"Theorem 6.14 proof / (NV)"}],"minor_comments":[{"comment":"'p-sabilisation' should be 'p-stabilisation'.","section":"§5, opening"},{"comment":"'Propsition' is a typo; also the freeness/faithfulness needed to invert the scalar λ in the localized module should be stated explicitly.","section":"Proposition 6.10"},{"comment":"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.","section":"§2/§4"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the reciprocal-constant error is severe but appears mechanically fixable; I would not reject solely on that point if the authors can correct it. The rank-one citation in Prop. 6.9 and the (NV) normalization are more serious: they need real arguments, not just references. If the authors provide them, the paper may be suitable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: the paper's main formula has a reciprocal-constant error, and until that is fixed, Theorem A as stated is not derived from the paper's own equations. The approach is genuinely new and the paper is technically serious; this is not a crank document. Corollary 6.13 gives h_f/(4|D_K|)^{k/2} = (star)*(A/B)*dL_p^{(f)}/ds, and (6.5) sets L_p = (club)*L_p^{(f)} with club/star = (1 - p^{k/2}/\\alpha)^4 when p\\nmid N, and 4 when p\\parallel N. Substituting gives dL_p/ds = (B/A) times the displayed factors, not (A/B). So unless A/B=1, the displayed (1.1)/(6.6) is the reciprocal of what the derivation yields. The proof that A/B=1 under (NV) is itself thin: it compares to known Gross–Zagier formulae without checking that the Heegner-class and L_p normalisations match, and NV1/NV3 are blurred when choosing the auxiliary form. If (NV) fails, Remark 1.1 says the formula reduces to 0=0, but that does not repair the constant in the general statement; the unconditional formula as written is unsupported.\n\nWhat is genuinely good: the wall-crossing proof strategy via Beilinson–Flach elements and the BDP formula is different from the kernel-comparison approach; the treatment of Coleman families, triangulations, and local conditions is careful; and the paper honestly states what it does not do (the a_p = p^{k/2} semi-stable case, and the dependence on (NV)). The rank-one Selmer assertion in Proposition 6.9 is a load-bearing citation to [KO20]+[Pot13] without full hypothesis verification — a real concern, but the kind of thing a referee can check, not an obvious contradiction.\n\nNet: this paper deserves a serious referee report, but the central constant error means it should not be accepted as is. The authors need to either change (6.6) to B/A, or rework the definition of L_p so the factor is A/B, and then re-examine the (NV) constant-fixing argument.","headline":"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.","tokens_in":39681,"tokens_out":2495,"would_cite":false,"duration_ms":22169,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G40","11G18","11F67","11R23","11F33"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["p-adic Gross–Zagier formula","Heegner cycles","Beilinson–Flach elements","BDP formula","p-adic L-functions","Coleman families","non-ordinary primes","Selmer complexes"],"falsifier":"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.","tokens_in":38558,"feed_emoji":"🧮","tokens_out":13487,"duration_ms":111338,"temperature":0.7,"pith_summary":"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.","feed_headline":"Proves p-adic Gross–Zagier for ordinary and non-ordinary forms","feed_subtitle":"Derivative of the p-adic L-function equals the Heegner height, including new non-ordinary weight > 2 cases.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["New proof of p-adic Gross–Zagier via wall-crossing","Wall-crossing proves p-adic Gross–Zagier for all weights","Derivative equals height: new p-adic Gross–Zagier proof","Beilinson–Flach and wall-crossing settle p-adic Gross–Zagier","P-adic Gross–Zagier proven via wall-crossing, covers non-ordinary"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New proof of p-adic Gross–Zagier via wall-crossing","Wall-crossing proves p-adic Gross–Zagier for all weights","Derivative equals height: new p-adic Gross–Zagier proof","Beilinson–Flach and wall-crossing settle p-adic Gross–Zagier","P-adic Gross–Zagier proven via wall-crossing, covers non-ordinary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000672,"raw_usage":{"total_tokens":2906,"prompt_tokens":759,"completion_tokens":2147,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":2049}},"tokens_in":503,"tokens_out":2147,"duration_ms":13806,"temperature":1.0,"reasoning_tokens":2049,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T16:16:49.789742+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}