{"id":"3ab5188e-ea26-4ba9-89e4-2189ae60d220","arxiv_id":"2509.09446","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Higher weight rigid meromorphic cocycles are constructed and their periods define p-adic higher Green's functions on real quadratic points, with principal values appearing as p-adic logarithms of algebraic numbers.","lead":"This paper generalizes Darmon-Vonk's weight-2 rigid meromorphic cocycles to arbitrary even weight and uses them to define p-adic higher Green's functions attached to real-quadratic points. Computed values for principal cycles appear to be p-adic logarithms of algebraic numbers, supporting conjectures about Stark-Heegner cycles.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Existence of J_{k,D} for D with non-vanishing Deg is asserted but not proved: boundedness and Lemma 2.24 stabilization of Φ_{k,D}+χ are never verified, and all numerics avoid this case.","rationale":"The strongest claim depends on the existence of the canonical cocycle J_{k,D} for every inert RM-divisor. The reader's weakest assumption identifies the unverified stabilization hypothesis of Lemma 2.24 as the key risk. My reading agrees and sharpens it: the proof of Theorem 3.30(2) also needs boundedness of the corrective term χ before the Schneider–Teitelbaum lift can even be applied, and neither boundedness nor stabilization is established. The co-restriction surjectivity assertion is plausibly true via the standard index argument for corestriction, so I do not rest the objection there. The numerical examples are high-precision and genuinely support the algebraicity conjectures in the principal, strong-degree-zero case, but they deliberately avoid the difficult χ-correction by requiring Deg_{4,D1}=0. For that reason the central construction remains conditional on a missing technical lemma. This is exactly the kind of addressable gap that warrants a CONDITIONAL rather than ACCEPT verdict, and it does not change the reader's assessment. No objection to the author's integrity or to the overall program is intended; the issue is purely that a load-bearing existence proof is asserted rather than verified.","tokens_in":33116,"tokens_out":35823,"duration_ms":445568,"concrete_test":"Choose the smallest case where Deg_{k,D}≠0, e.g. p=3, k=4, D=[φ]. Compute Deg_{4,D}{0,∞}(v) for vertices of T_p up to level L, and define the tree-Laplacian solution χ_L(e)=Σ_{v beyond e} Deg_{4,D}{0,∞}(v) for edges oriented away from v_0. Test: (i) sup_{e∈T_p, level(e)≤L} |χ_L(e)| remains bounded as L→∞; (ii) for each path in P(v,K) starting at a T_p-vertex and truncated at level L, the values (Φ_{4,D}+χ_L)(e) stabilize to 0, or to the expected residue polynomial at the finitely many support points, as L grows. If either (i) or (ii) fails, Theorem 3.30(2) and Lemma 2.24 fail for this D, so Lemma 4.6 and Definition 4.22 are invalid in general. If both hold, the concern is not disproved for all D, but the missing boundedness/stabilization lemma should still be supplied analytically from Prop. 3.26 and the finiteness in Lemma 3.14/3.20.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central object is the canonical cocycle J_{k,D1} of Lemma 4.6, used to define G_k(D1,D2) in Definition 4.22 and to state Conjecture 1.1. For J_{k,D1} to exist for every inert RM-divisor, Theorem 3.30(2) must produce a harmonic boundary function Φ_{k,D}+χ whose Schneider–Teitelbaum lift is bounded and extends to a meromorphic function on H_p with divisor Div_{k,D}. The proof of Theorem 3.30(2) only asserts that χ∈MS^Γ(C^1_p(P_n)) exists with ∇χ=−Deg_{k,D}, justified by identifying ∇ with a co-restriction map. Even granting that surjectivity, the proof never shows that χ can be chosen bounded, so that ST applies, nor that Φ+χ satisfies the eventual stabilization hypothesis of Lemma 2.24. This is not a cosmetic gap: Φ_{k,D} vanishes on T_p by construction, so for any D with Deg_{k,D}≠0 the entire correction χ is responsible for harmonicity, and no bound or stabilization argument is supplied. Lemma 3.20 guarantees finiteness of Σ∩U_e for each fixed edge, but Lemma 2.24 requires a simultaneous statement about all sufficiently deep edges along each path in P(v,K), with limits equal to the expected residue polynomials at the finitely many support points. The paper asserts this rather than proving it. Moreover, the numerical evidence in §5.2 (Examples 5.9 and 5.10) uses only D1 with Deg_{4,D1} trivial, so χ=0 and the hard part of Theorem 3.30(2) is never exercised. If some D with Deg_{k,D}≠0 admits no bounded, stabilizing χ, then J_{k,D} is not a rigid meromorphic cocycle on H_p with the asserted divisor, and G_k is undefined for that D1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a higher-weight generalization of Darmon–Vonk's rigid meromorphic cocycles for the Ihara group Γ=SL₂(Z[1/p]). For an inert RM-divisor D, the author constructs a boundary function Φ_{k,D} on the Bruhat–Tits tree of the unramified quadratic extension K/Q_p, and then, after adding a correction χ to make it harmonic, applies the Schneider–Teitelbaum lift and Lemma 2.24 to obtain a weight-k rigid meromorphic cocycle J_{k,D} with prescribed divisor. The higher Green's function is defined by evaluating J_{k,D₁} on a homology class y^♯_{k,D₂} attached to a second RM-divisor (Definition 4.22). Conjectures 1.1 and 5.11 assert that these values are linear combinations of p-adic logarithms of algebraic numbers in the compositum of the relevant narrow class fields. Numerical examples for p=3, k=4, with deg-zero divisors D₁, exhibit such logarithms to high p-adic precision.","tokens_in":33689,"tokens_out":3715,"duration_ms":44528,"significance":"If the construction is completed, this gives a new real-quadratic analogue of higher Green's functions and Stark–Heegner cycles, with explicit algebraicity predictions and a conjectural factorization formula. The paper is commendably concrete: it provides an algorithm (Algorithm 5.7), reports high-precision numerical evidence, and does not fit parameters to obtain the alleged algebraic numbers; the α_i are outputs of the computation. The claimed link to Negrini's weight-k cocycles and to Rotger–Seveso's Darmon cycles is a natural and potentially important step. However, the load-bearing existence theorem for J_{k,D} is not fully proved, and the numerical evidence intentionally avoids the difficult case. The central claim is therefore conditional in its present form.","major_comments":[{"comment":"The existence of J_{k,D} for arbitrary inert RM-divisors D is asserted but not established. The proof of Theorem 3.30(2) reduces the problem to finding χ∈MS_Γ(C_p¹(P_n)) with ∇χ = −Deg_{k,D}, says ∇ is surjective by identifying it with a co-restriction map, and then applies ST to Φ_{k,D}+χ. Three load-bearing points are missing. First, no proof or reference is given that Deg_{k,D} lies in the image of the co-restriction map from MS_{Γ_0(p)}(P_n) to (MS_{SL₂(Z)}(P_n))²; the statement 'which is known to be surjective' is not enough. Second, even if a χ exists, the Schneider–Teitelbaum lift applies only to bounded harmonic functions; no boundedness of χ or of Φ_{k,D}+χ is shown. Third, Lemma 2.24 requires an eventual stabilization hypothesis on all paths in P(v,K), whereas Lemma 3.20 only establishes finiteness of Σ∩U_e for each fixed edge e. Since Φ_{k,D} vanishes on T_p by construction, f","section":"§3.4, Theorem 3.30(2) and Lemma 2.24"},{"comment":"The numerical evidence only exercises the case Deg_{k,D}=0, for which χ=0 and the difficult part of Theorem 3.30(2) disappears. Both examples state 'Deg_{4,D₁} is trivial' and use the direct ST lift of Φ_{k,D}. Thus they do not test the existence of a bounded, stabilizing correction χ, which is precisely the gap in the central construction. Since the main conjecture is formulated for 'principal' divisors (Obs₀(J_{k,D})=0), and the paper leaves open whether such divisors can have Deg_{k,D}≠0, the evidence is consistent with a version of the theory restricted to strong degree zero, but it does not support the full theorem as stated.","section":"§5.2, Examples 5.9 and 5.10"},{"comment":"Lemma 4.6 defines the canonical cocycle J_{k,D} by subtracting a bounded analytic cohomology class from an arbitrary lift J′_{k,D} with the correct divisor. This relies on the identifications in Proposition 3.10, but the proof of that proposition invokes Lemma 3.9 and Corollary 3.7, which in turn use the Hecke-module isomorphism of Corollary 3.7 only for k>2. For k=2 the argument is not covered, and for k>2 it is still contingent on the boundedness asserted in Proposition 3.4. The manuscript should make explicit that Lemma 4.6 is conditional on the missing parts of Theorem 3.30(2). As written, the uniqueness claim 'there exists a unique cohomology class' is too strong relative to what is proved.","section":"§4.2, Definition 4.22 and Lemma 4.6"}],"minor_comments":[{"comment":"In the second paragraph of the proof, the symbols are interchanged: the text writes '∇χ{r,s}=−Div_{k,D}{r,s}' and then 'Div_{k,D} is in the image of MS_Γ(C_p¹(P_n))→MS_Γ(C_p⁰(P_n))'. Since Div_{k,D} takes values in divisors, not in Cₚ⁰(P_n), the condition should involve Deg_{k,D}. This is confusing and should be corrected.","section":"§3.4, proof of Theorem 3.30(2)"},{"comment":"The hypothesis reads 'for every vertex v∈T_p¹', but T_p¹ is the set of edges; the intended object is presumably a vertex of T_p or a path beginning at a vertex of T_p. Also, 'eventually stabilizes to 0' is weaker than what the proof uses: it needs stabilization to the prescribed residue polynomials P_j at the finitely many support points of the exceptional paths. The statement should be aligned with the proof.","section":"Lemma 2.24"},{"comment":"The object y_{k,σ}=γ_σ⊗(σ)⊗((T−σ)^{n/2}(T−σ̄)^{n/2}/(σ−σ̄)^{n/2}) is called a 'Stark–Heegner cycle', but it is a homology class in H₁(Γ, Div(H_p)⊗P_n), not an algebraic cycle. The terminology is evocative but should be flagged as conjectural, especially since the introduction distinguishes the open problem of constructing Stark–Heegner cycles.","section":"Notation, Definition 4.11"},{"comment":"The text says the code is available on the author's website. For reproducibility, an archival version with version-control metadata and a precise list of SageMath dependencies would be preferable. This is not a scientific issue, but it would help readers verify the 145-digit claims.","section":"§5.1, Algorithm 5.7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a promising contribution to the p-adic RM program, and the numerical evidence is suggestive. The main obstruction is not disagreement with consensus but a genuine unproved existence step. If the author can supply the missing boundedness and stabilization arguments, or carefully restrict the main theorem and subsequent definitions to the strong-degree-zero case where those arguments are available, the paper would be publishable. I would not recommend rejection, because the gap appears local and potentially fixable within the manuscript's scope. The current overstatement of Theorem 3.30(2) and Lemma 4.6 is, however, substantial enough to require a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I've read the paper carefully. My take: the higher weight generalization of Darmon-Vonk is real and the numerical algebraicity is striking, but the main existence theorem has a load-bearing gap that the paper does not acknowledge, and the computations avoid it entirely.\n\nWhat's new: the construction of J_{k,D} as ST(Phi_{k,D}+chi) and the higher Green's pairing G_k(D1,D2) are original, and the explicit examples (e.g., equations (21) and (23)) give high-precision evidence for Conjecture 1.1. The paper also correctly positions itself relative to Negrini's cocycles and Rotger-Seveso's Stark-Heegner cycle conjecture. That's worth a serious look.\n\nThe soft spot is Theorem 3.30(2). The proof says: find chi with grad(chi) = -Div_{k,D}, using the surjectivity of the co-restriction map; then ST(Phi_{k,D}+chi) is a rigid analytic function, and Lemma 2.24 gives the meromorphic extension. But two things are never checked: (i) the co-restriction map's surjectivity is asserted, not proved; (ii) even granting it, there is no argument that chi can be chosen bounded (so ST applies), nor that Phi+chi satisfies the eventual stabilization hypothesis of Lemma 2.24. For D with Deg_{k,D}=0, chi=0 and everything works; for Deg_{k,D} != 0, the entire harmonicity correction comes from chi, and no proof is given. The numerics in Section 5.2 only use divisors with Deg_{4,D1} trivial, so they never test this case. That's not a minor technicality: G_k is defined for all inert RM-divisors, but the existence of J_{k,D} for the nontrivial case is the one the paper needs.\n\nI don't think this is a fatal flaw. The conjectures are framed conditionally, and the construction is plausible on its face. But the paper as written overstates what is proved. I would send it to a serious referee, with a request to focus on exactly this gap--either supply the missing boundedness/stabilization argument or state the needed condition as an assumption. As it stands, it's a promising preprint, not an accepted result.","headline":"A genuinely new higher-weight construction with attractive numerics, but the existence proof for J_{k,D} skips the hard case (Deg != 0) and the numerics never test it.","tokens_in":34105,"tokens_out":5926,"would_cite":true,"duration_ms":64665,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F85","11G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a higher-weight p-adic Green's pairing between real-quadratic divisors and conjectures that principal values are p-adic logarithms of algebraic numbers in predicted abelian extensions.","keywords":["rigid meromorphic cocycles","higher Green's functions","p-adic upper half-plane","RM-divisors","real quadratic fields","modular symbols","Bruhat-Tits tree","class-field logarithms"],"falsifier":"Run the paper's algorithm for a new principal RM-divisor, say at p=5, k=4, and compare the first 150 5-adic digits of J^0_{k,D1}[σ] against the predicted Q-linear combination of p-adic logarithms in the stated compositum; any disagreement would refute Conjecture 5.11. Alternatively, compute the corrected boundary measure Φ_{k,D}+χ on the tree of the unramified quadratic extension for some strong-degree-zero D and check whether it satisfies the stabilization condition of Lemma 2.24; a single infinite path with non-zero stable value would block the existence of J_{k,D} as a meromorphic cocycle.","tokens_in":32976,"feed_emoji":"🔢","tokens_out":13347,"duration_ms":132835,"temperature":0.7,"pith_summary":"This paper extends the weight-2 theory of rigid meromorphic cocycles to higher even weights, producing a p-adic higher Green's pairing G_k(D1,D2) between real-quadratic divisors on the p-adic upper half-plane. For each inert RM-divisor D, the author builds a canonical weight-k cocycle J_{k,D} whose divisor is supported on the Γ-orbits of D, and pairs it with a second divisor through a boundary duality. The central conjecture is that when D1 is principal, G_k(D1,σ) is a linear combination of p-adic logarithms of algebraic numbers in the compositum of the narrow class fields attached to the two divisors. The paper reports a computer implementation and numerical evidence, including p=3, k=4 calculations matching the conjecture to over 140 digits of p-adic accuracy. If correct, this gives the real-quadratic analogue of higher Green's functions and indirect evidence for the existence of algebraic analogues of Heegner cycles in the real-quadratic setting.","feed_headline":"Calculated p-adic Green's values match logarithms of algebraic numbers","feed_subtitle":"A higher-weight pairing of real-quadratic divisors generalizes singular moduli and predicts class-field logarithms.","key_machinery":"The load-bearing object is the boundary-measure modular symbol Φ_{k,D}, defined on edges of the Bruhat-Tits tree of the unramified quadratic extension of Q_p: on edges outside the Q_p-tree it records, for each RM-point w in the Γ-orbit, the intersection number of the geodesics (w,w̄) and (r,s) times the degree-(k-2) polynomial in T determined by w. This function is almost harmonic: its divergence is the degree modular symbol Deg_{k,D}. Adding a correction χ on the Q_p-tree makes it harmonic, after which a distribution-theoretic lift from bounded harmonic measures to rigid analytic functions produces a function on the unramified extension, and an eventual-stabilization step extends it to a ri","core_discovery":"The paper's central discovery is a canonical construction, for any inert RM-divisor D of strong degree zero, of a higher-weight rigid meromorphic cocycle J_{k,D} whose divisor is the prescribed divisor Div_{k,D} supported on Γ-orbits of real-quadratic points. The Green's function is defined as the period pairing G_k(D1,D2)=J_{k,D1}[y^#_{k,D2}] between this cocycle and a canonically chosen homology class attached to D2. The paper proves the resulting pairing recovers the weight-two logarithm of the multiplicative cocycle when k=2, and that for k>2 the construction is governed by the classical correspondence between modular symbols and p-new eigenforms. It then conjectures that for principal D","pith_inferences":["One could test the conjecture beyond the examples by computing the obstruction Obs_L for small discriminants and using it to search systematically for non-trivial principal combinations; the paper gives the framework but does not carry out such a search.","The same boundary-measure recipe might extend to ramified RM-points or to composite-level congruence subgroups; the author indicates the ramified case is deferred to a thesis, and if the stabilization lemma holds there, the range of testable examples would broaden considerably.","If the conjecture is established, the resulting logarithms could be assembled into a p-adic regulator on the relevant higher-dimensional cycles, giving a concrete shadow of the conjectural real-quadratic Heegner cycles; this is an implication the paper leaves implicit.","The prime-factorization conjecture suggests a reciprocity law for these p-adic logarithms: the q-adic valuations are zero unless q is inert in both orders, and otherwise equal weighted intersection numbers; this could be verified computationally by factoring the algebraic numbers in the numerical examples to high precision."],"forward_implications":["At k=2 the new pairing reduces to log_p of the known real-quadratic singular moduli, so the higher-weight theory is a direct generalization and can be probed against existing computations.","For principal divisors, the conjecture predicts explicit algebraic numbers in specific class fields, turning the Green's values into a source of computable p-adic logarithms; the p=3,k=4 examples already realize this.","The k>2 cohomology is spanned by p-new eigenforms, so the higher Green's functions carry modular-form information and could appear in p-adic special-value formulas for L-functions of the relevant higher-dimensional cycles.","The conjectural prime-factorization formula would give a purely quaternionic description of the q-adic valuations of the algebraic numbers, linking RM-values to optimal embeddings and geodesic intersection numbers.","If algebraicity holds, the values provide an analytic construction of elements of narrow class fields of real quadratic fields, complementing the CM case where singular moduli generate class fields."],"fun_headline_variants":["Higher-weight p-adic Green's pairing yields algebraic logarithms","Generalizing singular moduli: higher-weight p-adic Green's functions","Evidence for Stark-Heegner cycles from higher Green's values","Real-quadratic algebraicity via higher-weight p-adic pairings"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction depends on the eventual-stabilization condition (Lemma 2.24): the corrected boundary measure on the Bruhat-Tits tree of the unramified quadratic extension must be bounded and must stabilize to zero along all but finitely many infinite paths, so that the lifted function extends from rigid analytic on the quadratic extension to genuinely meromorphic on H_p; the paper asserts this for the corrected measures Φ_{k,D}+χ rather than proving it in full generality.","fun_headline_variants_meta":{"raw":{"variants":["Higher-weight p-adic Green's pairing yields algebraic logarithms","Generalizing singular moduli: higher-weight p-adic Green's functions","Evidence for Stark-Heegner cycles from higher Green's values","Real-quadratic algebraicity via higher-weight p-adic pairings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1661,"prompt_tokens":742,"completion_tokens":919,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":844}},"tokens_in":486,"tokens_out":919,"duration_ms":9185,"temperature":1.0,"reasoning_tokens":844,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:07:45.817161+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's algorithm for a new principal RM-divisor, say at p=5, k=4, and compare the first 150 5-adic digits of J^0_{k,D1}[σ] against the predicted Q-linear combination of p-adic logarithms in the stated compositum; any disagreement would refute Conjecture 5.11. Alternatively, compute the corrected boundary measure Φ_{k,D}+χ on the tree of the unramified quadratic extension for some strong-degree-zero D and check whether it satisfies the stabilization condition of Lemma 2.24; a single infinite path with non-zero stable value would block the existence of J_{k,D} as a meromorphic cocycle.","supporting_citations":[],"review_version":1}