{"id":"a1ee9654-98a3-4ef9-9a4d-f4e174da5239","arxiv_id":"2510.10303","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Central derivative values of Rankin–Selberg L-functions are written as arithmetic heights of Hirzebruch–Zagier divisors on X_0(N)×X_0(N) or as geodesic Green's-function sums, by reorganizing Bruinier–Yang and AGHMP theorems.","lead":"A number-theory preprint rewrites derivative values of Rankin–Selberg L-functions—detectors of rank-one elliptic curves—as arithmetic heights of Hirzebruch–Zagier divisors on a product of modular curves, and as sums of Green's functions along real geodesics. The headline “two distinct proofs of Gross–Zagier” is not fully delivered: the imaginary-quadratic part repackages theorems of Bruinier–Yang and Andreatta–Goren–Howard–Madapusi Pera; the real-quadratic geodesic computatio","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Height formula in Theorem 7.8/Corollary 7.9 is proven only for d_k odd and maximal C_0(L_0); abstract and Theorem 1.4 claim it for all imaginary quadratic k, so central claim is overbroad.","rationale":"The reader's weakest assumption correctly identifies the load-bearing role of Theorem 7.5. I confirm that the central derivative height formula is not proven under the full hypotheses advertised in the abstract. The paper's own limitation statements (§3.3, Remark 7.6) are honest and should be credited; they weigh against ACCEPT but not against careful reading. The real-quadratic analogue in Theorem 1.3(ii) and Corollary 6.8(ii) does not claim an arithmetic height interpretation, so it is less affected by this concern. The L-function bridge (Prop 6.6) is standard and largely a matter of unpacking definitions; I do not see a concrete error there, though it is worth checking. The 'distinct proof of Gross-Zagier' claim is overstated: both proofs ultimately rest on Bruinier-Yang/AGHMP, but this is a novelty/emphasis issue rather than a correctness flaw. Overall, the paper remains valuable as a synthesis, and the correct verdict is CONDITIONAL, requiring the main theorems to be restated with the hypotheses under which they are actually proven. Since the reader already reached CONDITIONAL, I recommend UNCHANGED.","tokens_in":82625,"tokens_out":8021,"duration_ms":72226,"concrete_test":"Independently re-derive Theorem 7.5 from [13, Theorem 1.2] and [1, Theorem A], tracking every occurrence of the assumptions 'd_k odd' and 'C_0(L_0) maximal'. Determine the first step at which the proof fails if d_k is even or C_0(L_0) is nonmaximal; in particular, verify whether the integral model of the CM cycle Z(V_0) over Spec(O_{k}) is smooth only under those hypotheses. If the height formula indeed requires these hypotheses, then the abstract's unqualified claim for all imaginary quadratic k is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 7.8 / Corollary 7.9) equates Λ′(1/2, φ×θ(χ)) with a χ-averaged arithmetic height of Hirzebruch–Zagier divisors on Y_0(N)×Y_0(N). The proof passes through Theorem 7.5, imported from Bruinier–Yang [13] and AGHMP [1], which is stated only when d_{k(V_0)} is odd and C_0(L_0) is the maximal order O_{k(V_0)}. The body itself flags this: §3.3 says extending to nonmaximal orders 'remains an open problem,' and Remark 7.6 says Conjecture 7.4 (the general height formula) is not established. Yet the abstract and Theorem 1.4 state the height formula for arbitrary imaginary quadratic k with (d_k,N)=1 and η_k(−N)=−1, omitting the odd-discriminant and maximal-order hypotheses. If those hypotheses are essential, then Corollary 7.9 gives no proof for, e.g., k=Q(i) (d_k=−4) or for nonmaximal orders, and the advertised central claim is unsupported in those cases. This is not an internal inconsistency of the proof, but a mismatch between the proven statement and the headline claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new proof and refinement of the Gross-Zagier formula via regularized theta lifts and arithmetic Hirzebruch-Zagier divisors on Hilbert modular surfaces. For an elliptic curve E/Q of conductor N and a quadratic field k with (d_k,N)=1, it asserts that the central derivative Λ'(E/k,χ,1)=Λ'(1/2,φ×θ(χ)) equals a χ-twisted average of arithmetic heights of Hirzebruch-Zagier divisors on Y_0(N)×Y_0(N) when k is imaginary quadratic, and a sum of Green-function values along geodesic cycles when k is real quadratic. The proof is built on three ingredients: a sign-corrected derivation of the Bruinier-Yang summation formula for regularized theta lifts along CM cycles and a new geodesic analogue (Theorems 5.12 and 5.14); an L-function comparison linking those Rankin-Selberg integrals to Λ(s−1/2,φ×θ(χ)) via vector-valued lifts of φ (Proposition 6.6); and the arithmetic height formula imported from Bruinier-Yang and Andreatta-Goren-Howard-Madapusi Pera (Theorem 7.5). The paper also discusses consequences for BSD constants and periods, and for Fourier coefficients of half-integral weight forms.","tokens_in":82787,"tokens_out":6244,"duration_ms":57343,"significance":"If the central claims hold, the paper gives a genuinely new route to Gross-Zagier through the Kudla programme on Hilbert modular surfaces, and a real-quadratic analogue expressed through geodesic Green functions. The sign-corrected proof of the Bruinier-Yang summation formula and the treatment of the geodesic case are useful contributions. However, the main arithmetic-height formula is not established by a self-contained proof: it rests on the imported Theorem 7.5, whose hypotheses are explicitly flagged in the paper as not covering the general case stated in the abstract and Theorem 1.4. The L-function bridge in Proposition 6.6 also contains an asserted coefficient identity that is load-bearing and only justified as 'easy to see'. The paper is therefore more a conditional synthesis of existing theorems than a complete proof of the advertised headline, although the components are substantial enough that a repaired version could be valuable.","major_comments":[{"comment":"The headline formula is stated for every imaginary quadratic field k with (d_k,N)=1 and η_k(−N)=−1. The proof passes through Theorem 7.5, which is stated only when d_{k(V_0)} is odd and C_0(L_0) is the maximal order O_{k(V_0)}. The paper itself notes in §3.3 that extension to nonmaximal orders 'remains an open problem', and Remark 7.6 says Conjecture 7.4 is not established in general. Thus for k=Q(i), or for lattices whose even Clifford algebra is a nonmaximal order, the asserted equality in Theorem 1.4/Corollary 7.9 is unsupported. The abstract and Theorem 1.4 must either include the missing hypotheses or be accompanied by a proof of the height formula in the missing cases.","section":"Theorem 1.4 / §7.2, Theorem 7.8 and Corollary 7.9"},{"comment":"Theorem 7.5, as stated in Eq. (72), gives h[ Z(f),Z(V_0) ] = −deg(Z(V_0))/2 · (c_f^+(0,0) κ_{L_0}(0,0) + L'(0,ξ_{1−n/2}(f),θ_{L_0^⊥})). The proof of Theorem 7.8 uses only [ Z_A(f_{0,A}) : Z(V_{A,0}) ] = −h_k/w_k L'(0,g_{φ,A}×θ_{L_{A,0}^⊥}), silently omitting the c_f^+(0,0)κ term. No hypothesis on c_{f_{0,A}}^+(0,0) is stated. Unless one proves the existence of a choice of f_{0,A} with vanishing constant Fourier coefficient (or shows κ(0,0)=0 in this situation), the numerical constants in Theorem 7.8 and Corollary 7.9 are not justified.","section":"§7.1.2, Eq. (72), and §7.2 proof of Theorem 7.8"},{"comment":"The central equivalence L^*(2s−2,g_{φ,A}×θ_{L_{A,0}^⊥}) = Λ(s−1/2,φ×θ_A) rests on the scalar identity ⟨⟨g_{φ,A}, θ_{L_{A,0}}⊗E_{L_{A,0}}(·,s;1)⟩⟩ = φ(τ)θ_A(τ)E_A(τ,s;1), asserted just before the display with 'It is easy to see'. This identity is not proved; it requires a careful Hecke/Atkin-Lehner comparison of the Fourier coefficients c_{φ,A}(μ,m) with c_φ(m) and r_A(m), and a normalization statement for the Eisenstein series E_A. A normalization error at this point would propagate into every displayed constant in Theorems 6.7 and 7.8. Please supply a detailed proof or a precise reference for this identity.","section":"§6.1.5, proof of Proposition 6.6"},{"comment":"Theorem 7.5 and Conjecture 7.4 require the holomorphic part of f to have integral Fourier coefficients so that Z(f) is an arithmetic divisor. The form f_{0,A} produced by Corollary 6.4 is not shown to have this integrality property. If c_{f_{0,A}}^+(μ,m) are not integral, the height pairing in Theorem 7.8 is not defined on the arithmetic Chow group as stated. This is a technical but load-bearing condition: either prove integrality of the chosen f_{0,A}, or formulate the theorem for arithmetic Chow groups with rational coefficients and state the necessary compatibility.","section":"Theorem 7.5 and §7.2, definition of bZ_A(f_{0,A})"}],"minor_comments":[{"comment":"The identity after Eq. (55) is referred to as '(??)', indicating an unresolved cross-reference. Please fix.","section":"§6.1.3"},{"comment":"Several typos remain: 'Bruiner' for Bruinier in the proof of Theorem 4.1, 'discrimnant' in §6.1.2, and 'Strömberg' is cited inconsistently in §1 versus §6.1.4.","section":"Throughout"},{"comment":"The real-quadratic summation formula is claimed to be new, but the proof is only described as a 'minor generalization' of Theorem 5.12. It would help readers if the differences (especially the treatment of the nonholomorphic theta series θ_{L_W^⊥}) were spelled out more explicitly.","section":"§1.1.1, Theorem 1.2(ii)"},{"comment":"The derivation of the BSD constant relies on a long list of deep external results (Iwasawa main conjectures, Euler characteristic calculations) without precise statements of the hypotheses needed for each implication. A short appendix or a precise theorem block listing the exact inputs would improve verifiability.","section":"§7.3, proof of Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper has real value in its careful analytic computations and its real-quadratic summation formula, but the advertised central theorem is stated beyond the hypotheses of the imported height formula, and the proof of Proposition 6.6 omits a key coefficient identity. These are fixable in principle, but not by cosmetic changes: the statements in Theorem 1.4 and Corollary 7.9 must be restricted to the cases covered by Theorem 7.5, or the missing height formula must be supplied, and the L-function comparison needs a complete proof. I would be happy to reconsider after a substantive revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version. The paper is a careful, mostly honest reorganization of Bruinier–Yang and AGHMP into a Kudla-program-shaped formula for Gross–Zagier on X_0(N)×X_0(N), plus a plausibly new real-quadratic analogue via geodesic sums. It deserves a serious referee, but only after the authors restate what is proven and what is borrowed.\n\nWhat is actually good: Theorem 5.12 is a sign-corrected proof of the Bruinier–Yang summation formula, and that correction is real—useful for anyone who has struggled with the signs in [13]. Theorem 5.14, the geodesic summation, is explicitly flagged as new and is a reasonable extension of [55]; the real-quadratic derivative formula in Corollary 6.8(ii) is new if the proof holds together. The body is also honest: it says Theorem 1.2(i) reproves Bruinier–Yang/Schofer, and it flags the nonmaximal-order problem in §3.3 and Remark 7.6. That honesty is more than many papers manage.\n\nNow the soft spots. First, the stress-test note is on target: Theorem 7.8/Corollary 7.9 is proven only for d_k odd and maximal C_0(L_0), yet the abstract and Theorem 1.4 present the height formula for arbitrary imaginary quadratic k with (d_k,N)=1. That is a direct mismatch between proof and headline claim. It is not fatal—the paper states enough for a careful reader to see the restriction—but the advertised central claim is unsupported for k=\\mathbb{Q}(i) and for nonmaximal orders.\n\nSecond, Proposition 6.6 carries the L-function bridge, and one step in it is the scalar identity “It is easy to see.” Every constant in the final formulas runs through that normalization. It is probably fixable, but it needs to be written out before I trust the displayed constants.\n\nThird, the “two distinct proofs of Gross–Zagier” language oversells. The signature-(2,2) route derives the formula from the imported Bruinier–Yang/AGHMP height theorem plus L-function bookkeeping, and the comparison in §7.2 uses the Gross–Zagier formula as an input. That is a derivation, not an independent proof. It is fine as a contribution, but the abstract should say “reformulation” or “new derivation,” not “distinct proof.”\n\nFinally, Theorem 5.14, one of the genuinely new results, is proven largely by delegation to [55]. For a theorem the paper itself calls new, that is too thin.\n\nBottom line: a serious researcher in the Kudla program will want this around, especially for the sign correction and the real-quadratic formulas. It is refereeable if the claims are narrowed to the proven hypotheses and the L-function identity is filled in. I would send it to review, and I would tell the authors to revise the abstract and Theorem 1.4 first.","headline":"A genuinely useful recombination of Bruinier–Yang/AGHMP with a solid sign-correction and a new real-quadratic analogue, but the advertised height formula outruns its hypotheses and the 'two proofs of Gross–Zagier' claim is overstated.","tokens_in":83494,"tokens_out":2736,"would_cite":true,"duration_ms":26111,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11G18","11F41","11G50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Central L-derivatives become heights of Hirzebruch–Zagier divisors","keywords":["Rankin–Selberg L-functions","central derivative values","Hirzebruch–Zagier divisors","arithmetic heights","Hilbert modular surfaces","regularized theta lifts","Heegner divisors","elliptic curves"],"falsifier":"Compute both sides numerically for one elliptic curve and one imaginary quadratic field with odd discriminant and the sign condition η_k(−N) = −1; a mismatch in even the first digit would refute the theorem. More narrowly, inspect a single Fourier coefficient of the identity behind the L-function bridge—any mismatch there breaks the chain before heights enter.","tokens_in":82263,"feed_emoji":"🧮","tokens_out":7431,"duration_ms":62416,"temperature":0.7,"pith_summary":"The paper proves that the central derivative of the Rankin–Selberg L-function attached to an elliptic curve, a quadratic field, and a class-group character can be computed as a χ-weighted sum of arithmetic heights of Hirzebruch–Zagier divisors on the Hilbert modular surface Y0(N)×Y0(N) — in the imaginary-quadratic case — and as a χ-weighted sum of Green's functions along geodesic cycles in the real-quadratic case. This places rank-one leading terms of elliptic curve L-functions inside arithmetic intersection theory. It also gives two distinct derivations of the classical rank-one formula for Heegner divisors, one through the Hilbert modular surface and one through the modular curve itself, and extracts consequences for the refined leading-term conjecture and for half-integral weight Fourier coefficients.","feed_headline":"Central L-derivatives become heights of Hirzebruch–Zagier divisors","feed_subtitle":"Rank-one elliptic curve L-values are computed by arithmetic intersection theory on Y0(N)×Y0(N).","key_machinery":"Quadratic spaces (V_A, Q_A) of signature (2,2) attached to ideal classes A of k; their spin groups identify with GL2×GL2, so the associated Shimura varieties are Hilbert modular surfaces Y0(N)×Y0(N). The argument runs through: regularized theta lifts, which are automorphic Green's functions for special (Hirzebruch–Zagier) divisors; summation formulae expressing the value of these Green's functions along a CM cycle or geodesic set as a constant term plus a Rankin–Selberg L-derivative; an L-function bridge identifying those Rankin–Selberg L-functions with Λ(s−1/2, φ×θ_A); and an arithmetic height formula — imported under hypotheses of odd discriminant and maximal CM order — converting the cons","core_discovery":"On the paper's own terms, the central discovery is an identity of the form Λ′(1/2, φ×θ(χ)) = −2π Σ_{A∈C(O_k)} χ(A) [Ẑ_A(f_{0,A}) : Z(V_{A,0})], where the bracket is an arithmetic height of a Hirzebruch–Zagier divisor on the integral model of Y0(N)×Y0(N) against a CM cycle. The same machinery, with geodesic sets replacing CM cycles, yields an analogous formula for real quadratic k in terms of Green's function sums. A corollary is that the classical rank-one formula for Heegner divisors is recovered from heights on a two-dimensional Shimura surface rather than on the modular curve.","pith_inferences":["If the odd-discriminant/maximal-order hypotheses on the imported height formula can be lifted, the main identity would cover all quadratic twists and give a uniform arithmetic interpretation of rank-one leading terms.","The real-quadratic geodesic formula suggests a p-adic analogue: sums of Green's functions along real cycles may interpolate p-adically, connecting to anticyclotomic Iwasawa invariants.","A single numerical check on an elliptic curve and a quadratic field with even discriminant would delimit how far the height interpretation extends beyond the currently proven cases.","The quadratic-summation machinery is already sketched for higher-dimensional spin Shimura varieties, so the same reduction likely yields higher Gross–Zagier identities for CM cycles in any dimension."],"forward_implications":["Rank-one leading terms of elliptic curve L-functions are computed by arithmetic intersection theory on the Hilbert modular surface Y0(N)×Y0(N).","The classical rank-one Heegner-divisor formula follows from a height identity on this surface, giving a second proof.","Real-quadratic twists admit an analogous formula, with geodesic Green's function sums replacing CM-cycle heights.","Heights of Heegner divisors on X0(N) and heights of Hirzebruch–Zagier divisors on X0(N)×X0(N) are explicitly related.","The central derivative values are shown to lie in the ring of periods, conditional on the refined leading-term conjecture up to powers of 2 and 3."],"fun_headline_variants":["L-derivative identity via heights on X0(N)×X0(N)","Hirzebruch-Zagier divisors encode L-value derivatives","Rankin-Selberg derivatives as intersections on X0(N) x X0(N)","Central L-derivative equals height of a Hirzebruch-Zagier divisor","Distinct proof of Gross-Zagier via surface heights"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the imported arithmetic height formula, established only for odd discriminants and maximal CM orders; outside those hypotheses the height interpretation of the central derivative is not yet available.","fun_headline_variants_meta":{"raw":{"variants":["L-derivative identity via heights on X0(N)×X0(N)","Hirzebruch-Zagier divisors encode L-value derivatives","Rankin-Selberg derivatives as intersections on X0(N) x X0(N)","Central L-derivative equals height of a Hirzebruch-Zagier divisor","Distinct proof of Gross-Zagier via surface heights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002011,"raw_usage":{"total_tokens":7742,"prompt_tokens":870,"completion_tokens":6872,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":6787}},"tokens_in":614,"tokens_out":6872,"duration_ms":45333,"temperature":1.0,"reasoning_tokens":6787,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:18:57.738190+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides numerically for one elliptic curve and one imaginary quadratic field with odd discriminant and the sign condition η_k(−N) = −1; a mismatch in even the first digit would refute the theorem. More narrowly, inspect a single Fourier coefficient of the identity behind the L-function bridge—any mismatch there breaks the chain before heights enter.","supporting_citations":[],"review_version":1}