{"id":"237612bc-5856-4c1a-b03c-1be644806063","arxiv_id":"2510.10277","paper_version":2,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The central L-derivative of an elliptic curve over a real quadratic field is expressed as a χ-twisted sum of regularized theta lifts (automorphic Green's functions) on X0(N)×X0(N).","lead":"This paper derives a new formula for the first derivative of L-functions of elliptic curves over real quadratic fields, expressing it as sums of automorphic Green's functions along geodesics. The result is a proposed real-quadratic analogue of the Gross-Zagier formula, with connections to the Birch-Swinnerton-Dyer conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central formula inherits an invalid classical input: the 'weight-zero' theta series θ_A of a real quadratic field is not a convergent modular form (negative-norm terms diverge), so Corollary 4.16's identification of the Rankin-Selberg L-derivative is unsupported.","rationale":"The reader's weakest_assumption (non-unique f0_A) is not the deepest issue: adding a weakly holomorphic h to f0_A changes both the CT term and Φ, but Theorem 4.15's relation CT + (vol/2)Φ = −L′(0,ξ0(f0),V2) would make the combination invariant under such changes. So that objection is answered by the paper's own algebra, modulo correctness of Theorem 4.15. A more serious, less easily repaired gap is the classical theta-series input. The proof of Theorem 4.17 is explicitly 'Formally, this is a consequence...' and leans on Corollary 4.16, which compares the vector-valued Rankin-Selberg L-function to the classical L(s,f×θ_A). But θ_A as defined in §4.9 is not a well-defined function on the upper half-plane for real quadratic fields: the norm form is indefinite, negative-norm units produce negative powers of q, and the series diverges. The paper calls θ_A weight 0, but the convergent real-quadratic theta series (after imposing total positivity) is weight 1, which would alter the Rankin-Selberg Γ-factors and invalidate the identity in Corollary 4.16. Until this is fixed—either by a proper adelic/idele theta kernel or by a coherent classical reference—the central derivative formula is unsupported. The reader's REJECT verdict therefore stands, but for a different load-bearing reason than the one stated.","tokens_in":52870,"tokens_out":20344,"duration_ms":192721,"concrete_test":"For K=Q(√2), take a=O_K and a⋆ a fundamental domain for the action of ε=1+√2 (Nε=−1). Expand the formal q-series Σ_{λ∈a⋆} e(Nλ·τ). Since λ=1−√2 has Nλ=−1, the coefficient of q^{−1} is nonzero, so θ_A contains divergent negative-frequency terms. Then repeat the calculation with λ restricted to totally positive elements (λ≫0, Nλ>0) and check whether Corollary 4.16's identity with the weight-2 Eisenstein series E_{L2}(τ,s;2) still holds for this weight-1 theta series. If the Γ-factor or level changes, the proof of Theorem 4.17 has no valid classical Rankin-Selberg input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.9 defines θ_A(τ)=1/w_K Σ_{λ∈a⋆} e(N_{K/Q}(λ)/Na·τ) and asserts it is a modular form of weight 0, level Γ0(d_K), character η. For a real quadratic K the norm form has signature (1,1): if the fundamental unit ε_K has norm −1 (e.g. K=Q(√2), Q(√5)), any fundamental domain a⋆ for ⟨ε_K⟩ contains elements with negative norm. The corresponding terms e(mτ) with m<0 diverge exponentially as v=ℑτ→∞, so the series does not define a function on H. Even if Nε_K=+1, convergence requires restricting to a totally positive ray class; the unrestricted sum has both positive and negative powers of q. The paper's subsequent use of θ_A as a weight-0 form in (52)–(53) and in Corollary 4.16 (L⋆(2s−2,g,V2)=2Λ(s,f×θ_A)) is therefore not a valid analytic identity. The standard convergent real-quadratic theta series has weight 1, not 0, and its Rankin-Selberg Γ-factors differ. This is independent of the f0_A ambiguity: Theorem 4.15 would make the CT+Φ combination depend only on the shadow if valid, but Theorem 4.17's proof is explicitly 'formal' and relies on Corollary 4.16, which is the point that breaks.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an integral formula for central derivative values Λ'(1/2, Π⊗χ) of elliptic curves over Q base-changed to a real quadratic field K and twisted by a ring class character χ, under a hypothesis making the central value vanish. The construction uses regularized theta lifts on quadratic spaces of signature (2,2), identifies the associated spin Shimura variety with Y0(N)×Y0(N), evaluates the lifts along geodesic subsets associated to ideal classes of K, and expresses the resulting sum in terms of automorphic Green's functions for Hirzebruch–Zagier-like divisors. The final theorem (Theorem 4.17 / Corollary 4.18) is then related to Birch–Swinnerton-Dyer constants in Section 5. The paper is organized around a Bruinier–Yang-style calculation: an identity for a regularized theta lift (Theorem 4.15), a bridge to the Rankin–Selberg L-function of the base-change representation (Corollary 4.16), and a twisted summation over the class group using the analytic class number formula.","tokens_in":53262,"tokens_out":8861,"duration_ms":90890,"significance":"If correct, the main theorem would give a real-quadratic analogue of the Gross–Zagier formula, expressing a central derivative of an L-function of an elliptic curve twisted by a real-quadratic ring class character as a χ-twisted sum of automorphic Green's functions evaluated on real geodesic cycles. Such a formula would be a notable advance and would connect the (still conjectural) arithmetic of elliptic curves over real quadratic fields to established theta-lifting machinery. The paper has credible structural strengths: it uses the standard apparatus of regularized theta lifts, the Bruinier–Funke ξ-operator, Siegel–Weil formulas, Stokes's theorem, and the analytic class number formula, and it gives an explicit vector-valued lift g_{f,A} of the elliptic curve newform in Theorem 4.6. The proposed formula is also sufficiently explicit to be numerically testable in principle. However, the central analytic bridge from the automorphic objects to the base-change L-function is not rigorously established, and the main theorem depends on an invalid classical theta-series assertion.","major_comments":[{"comment":"The theta series θ_A(τ) = (1/w_K) Σ_{λ∈a⋆} e(N_{K/Q}(λ)/N_a·τ) is asserted to be a holomorphic modular form of weight 0 for Γ0(d_K) with character η on the strength of 'a classical theorem of Hecke.' This is not correct as stated. For a real quadratic K, the norm form has signature (1,1), so a fundamental domain for the unit action contains elements with negative norm. The corresponding terms e(mτ) with m<0 grow exponentially as v=Im τ→∞, so the series does not converge on the upper half-plane. Even when N(ε_K)=+1, the unrestricted fundamental-domain sum contains both positive and negative norm terms. Thus θ_A is not a modular form on H, and the Rankin–Selberg identity Λ(s,f×θ_A) and the derivative identity in Corollary 4.16 are unsupported. The standard convergent real-quadratic theta series has weight 1, not weight 0, and its Rankin–Selberg Gamma factors differ. Since Theorem 4.17 and","section":"§4.9, Eqs. (52)–(53), and Corollary 4.16"},{"comment":"The harmonic weak Maass form f_{0,A} is specified only by its shadow: ξ0(f_{0,A}) = g_{f,A}. The kernel of ξ0 is the infinite-dimensional space of weakly holomorphic forms, so f_{0,A} is far from unique. Theorem 4.15, if valid, would make the combination CT⟨f_{0,A}^+, θ_{L_{A,1}}^+⊗E_{L_{A,2}}⟩ + L'(0,ξ0(f_{0,A}),V_{A,2}) independent of the choice of preimage, but no such invariance is proved. The two terms are not separately invariant under f_{0,A} ↦ f_{0,A}+h with ξ0(h)=0; the proof of Theorem 4.15 does not address the h-dependence. Unless a canonical choice of f_{0,A} is specified (e.g., by growth conditions at the cusps) or invariance of the full expression is proved, the right-hand side of Theorem 4.17 is not well-defined.","section":"§4.5 and Theorem 4.17"},{"comment":"The proof of Theorem 4.17 is explicitly labelled 'Formally, this is a consequence' and delegates the key analytic identification to Corollary 4.16, which is the step that breaks because of the invalid θ_A series. The proof of Corollary 4.16 itself relies on formal Dirichlet-series manipulation 'cf. [23, §IV (0.1)]' without checking the convergence of the indefinite theta series or the Gamma factors. A rigorous treatment would need to replace θ_A by a convergent object (such as the nonholomorphic Siegel theta series or a weight-one Hecke theta series) and re-derive the relation between L(s,g_{f,A},V_{A,2}) and Λ(s,Π⊗χ). As written, the central equality Λ′(1/2,Π⊗χ) = ... in Theorem 4.17 is therefore not established.","section":"Proof of Theorem 4.17 and Corollary 4.16"}],"minor_comments":[{"comment":"There are numerous typos and inconsistencies: 'Bruiner' for Bruinier, 'signture', 'disciminant', 'nontrivial automorphisms' (plural), and inconsistent capitalization. These should be corrected.","section":"Throughout"},{"comment":"The statement says 'inert level N^+' but the hypothesis is on N^-, the product of inert primes. This appears to be a typo.","section":"Corollary 4.18"},{"comment":"The theorem is labelled '(Theorem 4.17, Corollary 4.5)' but the referenced result is Corollary 4.18; there is no Corollary 4.5 in the text. Please correct the cross-reference.","section":"Introduction, Theorem 1.2"},{"comment":"The notation f_{0,A,µ}, θ_{L_{A,1},µ_1}, and E_{L_{A,2},µ_2} is introduced abruptly; the congruence condition µ_1+µ_2≡µ mod L_A is not fully defined here, and the reader must reconstruct it from earlier lattice dual-group notation.","section":"Eq. (54) and surrounding text"}],"recommendation":"reject","confidential_remarks":"The paper has an appealing architecture and the proposed formula is plausible as a research problem, but the main theorem as proven depends on a false analytic assertion about indefinite theta series. The f_{0,A} non-uniqueness is an additional unresolved well-definedness issue. These are not superficial presentation problems: they concern the central identification of the L-function. I do not see a local fix within the current framework; a substantive reworking of §4.9 and the proof of Corollary 4.16 would be required. For this reason I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper has a real new idea: a real-quadratic analogue of Gross–Zagier via regularized theta lifts and geodesic cycles. That would be genuinely new if it worked. Second, as it stands the main theorem is not well-defined: the weight-zero theta series θ_A is not convergent for a real quadratic field, and the harmonic weak Maass form f0_A is fixed only by its shadow, so the right-hand side of the central formula is not a well-defined number.\n\nWhat the paper does well: the Spin(2,2) setup, the Siegel–Weil and Stokes arguments in Theorem 4.15, and the careful adaptation of Bruinier–Yang’s machinery are real work. The author also flags two sign errors in the literature, which is good scholarly practice. If the analytic issues were fixed, the result would open a new line on BSD for real quadratic fields.\n\nNow the soft spots, and they are load-bearing. First, the stress-test note is right. In §4.9, θ_A(τ) is defined as a sum over a fundamental domain for the unit action, and for K=Q(√2) or Q(√5), elements with negative norm make e(mτ), m<0, diverge as v→∞. This is not a modular form of weight 0 or any weight, and the Rankin–Selberg identity in Corollary 4.16 is unsupported. Second, in §4.5, f0_A is chosen so that ξ0(f0_A)=g_{f,A}; the kernel of ξ0 is infinite-dimensional, and the paper never proves the RHS is independent of that choice. The proof of Theorem 4.17 is explicitly 'formal' at exactly the point where these issues would have to be resolved. Third, Proposition 2.2 gets the base-change cuspidality criterion backwards: cuspidality fails when Π≅Π^τ, not follows from it.\n\nNone of this is speculative. The reviewer is right to be skeptical, though the confidence should stay low rather than high: the paper’s architecture is plausible, and each gap is in principle fixable. But as written, the central theorem is not a theorem, and the BSD consequences in Section 5 inherit the same problems.\n\nThis paper is for a specialist in theta lifts and L-values. It deserves a serious referee—not a desk reject—because the program is important and the intermediate arguments are worth checking. But the expected outcome of that review should be major revision or rejection in present form. I would not cite it yet.","headline":"A plausible real-quadratic analogue of Gross–Zagier, but the main formula rests on an undefined theta series and an unspecified Maass form preimage.","tokens_in":53728,"tokens_out":3787,"would_cite":false,"duration_ms":40501,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F27","11F41","11G40","11F32","11F46","11G05","11G18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims an exact formula for the central derivative of a base-changed elliptic curve L-function over a real quadratic field, expressed as a twisted sum of constant terms and automorphic Green's functions evaluated along geodesics.","keywords":["L-functions","elliptic curves","real quadratic fields","ring class characters","regularized theta liftings","automorphic Green's functions","Hirzebruch-Zagier divisors","Gross-Zagier formula"],"falsifier":"Compute both sides of Theorem 4.17 for a small explicit example: take a semistable elliptic curve E with conductor N such that N^- is the squarefree product of an odd number of primes, a real quadratic field K of small discriminant with (N, d_K)=1, and a ring class character χ; numerically evaluate the χ-twisted finite sum of constant terms and geodesic Green's sums and compare it with a directly computed Λ′(1/2, Π⊗χ). A mismatch, or a variation of the right-hand side when f_{0,A} is replaced by another harmonic weak Maass form with the same shadow, would disprove the stated identity.","tokens_in":52732,"feed_emoji":"🔢","tokens_out":6643,"duration_ms":55450,"temperature":0.7,"pith_summary":"This paper tries to prove an integral formula for the central derivative Λ′(1/2, Π⊗χ) of the L-function of an elliptic curve base-changed to a real quadratic field and twisted by a ring class character. When the functional equation forces the central value to vanish, the derivative is written as a χ-twisted sum of two pieces: a constant term of a theta product and a sum of automorphic Green's functions along geodesics on X0(N)×X0(N). If the formula holds, it supplies a real-quadratic analogue of the Gross-Zagier formula, where central derivatives are captured by geodesic Green's sums rather than by CM point heights. The paper also derives consequences for the Birch-Swinnerton-Dyer conjecture, including a conditional description of Tate-Shafarevich and regulator terms and an unconditional BSD-related identity for E and its quadratic twist.","feed_headline":"Elliptic curve L-derivatives expressed as geodesic Green sums","feed_subtitle":"Birch-Swinnerton-Dyer data over real quadratic fields become geodesic sums of Green's functions.","key_machinery":"The load-bearing mechanism is the regularized theta lift Φ(f_{0,A}, ·) of a harmonic weak Maass form f_{0,A} whose shadow under the ξ0-operator is a canonical vector-valued lift of the eigenform f. Through the exceptional isomorphism GSpin(V_A) ≅ GL_2^2, the ambient spin Shimura variety is identified with Y0(N) × Y0(N), and the evaluation of the theta lift along the anisotropic geodesic subspace V_{A,2} reduces, via the Siegel-Weil formula and the derivative Eisenstein series E′_{L_2}(τ,0;2), to the displayed Green's sums. The factor √d_K/(log ε_K h_K) comes from Dirichlet's analytic class number formula for L(1, η).","core_discovery":"The paper's central claim (Theorem 4.17, Corollary 4.18) is an exact identity: for an elliptic curve E/Q with associated newform f, a real quadratic field K of discriminant d_K prime to the conductor N, and a ring class character χ of conductor c, the completed central derivative Λ′(1/2, Π⊗χ) — equivalently Λ′(E/K, χ, 1) — equals −√d_K/(log ε_K h_K) times half the χ-twisted sum, over ideal classes A of Pic(O_c), of CT⟨⟨f⁺_{0,A}(τ), θ⁺_{L_{A,1}}⊗E_{L_{A,2}}(τ)⟩⟩ plus (vol(U_{A,2})/2) times the automorphic Green's function G_{Z(f_{0,A})} evaluated along the geodesic set G(V_{A,2}). The proof adapts the Bruinier-Yang calculation, replacing the holomorphic projection used in Gross-Zagier by the","pith_inferences":["The formula's right-hand side depends on the choice of harmonic weak Maass form f_{0,A} through its holomorphic part; because the kernel of ξ0 is infinite-dimensional, a canonical normalization (e.g., minimal principal part or a specific preimage) is needed. Testing whether the displayed combination is actually independent of this choice is a natural next step.","The geodesic sets G(V_{A,2}) embed into boundary components of Borel-Serre compactifications of Siegel threefolds; the paper suggests this may illuminate the provenance of Stark-Heegner points for real quadratic fields, which could be probed by constructing p-adic analogues of the Green's sums.","Since the constant term (3) is algebraic while the Green's function values are typically periods, the formula predicts a clean separation between algebraic and transcendental parts of Λ′(1/2, Π⊗χ), a property that could be checked numerically for small conductors and discriminants.","The same framework might extend to imaginary quadratic fields or higher-weight newforms, potentially yielding a unified treatment of the Gross-Zagier formula over both types of quadratic fields via boundary components of Shimura varieties — though this is not worked out in the paper."],"forward_implications":["If the identity is correct, central derivatives in the real-quadratic forced-vanishing case become explicit finite sums over the class group, making numerical tests of the Birch-Swinnerton-Dyer conjecture feasible for ranks one.","The formula gives a real-quadratic analogue of Gross-Zagier: the leading Taylor coefficient at s=1/2 is encoded by automorphic Green's functions along geodesics, suggesting a height-like interpretation for rank-one curves over ring class fields of real quadratic fields, where no Heegner point construction is known.","Under the ersatz Heegner hypothesis (N^- the squarefree product of an odd number of primes), the paper's formula applies and produces non-vanishing expressions for Λ′(1/2, Π⊗χ), which can be compared against BSD regulator terms.","The unconditional Theorem 5.1 relates the product of BSD constants of E and its quadratic twist to the same geodesic Green's sums up to powers of 2 and 3, giving a concrete arithmetic identity independent of the rank-one conjecture."],"fun_headline_variants":["L-derivative identity via geodesic Green sums over class groups","Elliptic L-values from automorphic Green's sums on X0(N)^2","Central L-derivatives equal twisted geodesic Green's sums","Green's functions compute L-derivatives over real quadratic fields","New identity: L'(E,χ,1) equals twisted Green's sums"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the harmonic weak Maass forms f_{0,A} are well-defined enough that the combination of the constant term and the Green's function sum does not depend on the choice of preimage under ξ0; only the shadow ξ0(f_{0,A}) = g_{f,A} is fixed, and the kernel of ξ0 is infinite-dimensional.","fun_headline_variants_meta":{"raw":{"variants":["L-derivative identity via geodesic Green sums over class groups","Elliptic L-values from automorphic Green's sums on X0(N)^2","Central L-derivatives equal twisted geodesic Green's sums","Green's functions compute L-derivatives over real quadratic fields","New identity: L'(E,χ,1) equals twisted Green's sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00128,"raw_usage":{"total_tokens":5040,"prompt_tokens":686,"completion_tokens":4354,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":4260}},"tokens_in":430,"tokens_out":4354,"duration_ms":26972,"temperature":1.0,"reasoning_tokens":4260,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:18:59.998133+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of Theorem 4.17 for a small explicit example: take a semistable elliptic curve E with conductor N such that N^- is the squarefree product of an odd number of primes, a real quadratic field K of small discriminant with (N, d_K)=1, and a ring class character χ; numerically evaluate the χ-twisted finite sum of constant terms and geodesic Green's sums and compare it with a directly computed Λ′(1/2, Π⊗χ). A mismatch, or a variation of the right-hand side when f_{0,A} is replaced by another harmonic weak Maass form with the same shadow, would disprove the stated identity.","supporting_citations":[],"review_version":1}