{"id":"50534eb8-8fff-46ec-90a0-1b19d6ab27f6","arxiv_id":"2509.01378","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct a weak Maass form omega_{k+1,D} related to the divisor of Zagier's f_{k,D} and show its generating function yields a new theta lift.","lead":"This paper defines a new function built from Zagier's hyperbolic Poincare series and shows it is a weak Maass form whose generating function is modular, producing a new theta lift. A smart generalist might care because theta lifts are a standard bridge between modular forms and L-values, so a new one gives new tools for arithmetic problems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theta lift in Thm 1.3 pairs the kernel at −z instead of −\\bar z; with the sesquilinear inner product of (2.7), the proof’s un-conjugated integrand yields ω(\\bar z), not ω(z).","rationale":"The reader’s stated weakest assumption—the SAGE eigenfunction check and C^2 regularity of p—is a legitimate but likely repairable concern: for even k>2, D_+^{k−1/2} is C^{k−1}, so C^2 across D=0 should hold, and the symbolic identity can be verified independently. The more serious issue is the sesquilinearity of the Petersson inner product in the proof of Theorem 1.3. The displayed computation omits the conjugation required by (2.7). Once restored, the lift with argument −z yields ω(\\bar z), not ω(z), and these differ for generic upper-half-plane z, as shown by the identity ω=i f'/k+f/y from Theorem 1.1(ii). The fix is to replace −z by −\\bar z in the definition of the theta lift (and in the analogous Kohnen–Zagier identity), which is a small but substantive correction to the stated central theorem. Therefore the paper should be accepted only conditionally, pending this correction; the mathematical idea appears sound, but Theorem 1.3 as written is not.","tokens_in":10392,"tokens_out":35423,"duration_ms":400624,"concrete_test":"Re-derive the displayed inner product in the proof of Theorem 1.3 keeping the complex conjugation from (2.7): compute Λ_k(τ,−z)̄ and perform the u-integral. The selected coefficient is ω_{k+1,D}(−z)̄=ω_{k+1,D}(z̄). Then compare ω_{k+1,D}(z) with ω_{k+1,D}(z̄) for a generic point such as z=1+i, using the explicit formula ω_{k+1,D}(z)=i f_{k,D}'(z)/k+f_{k,D}(z)/y for a small discriminant (e.g., D=1, k=4) with the Fourier expansion of f_{k,D} truncated; the values differ, confirming that the −z version is wrong and the corrected lift uses −z̄.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Definition (2.7) is sesquilinear: ⟨f,g⟩ contains g(z)̄. In the proof of Theorem 1.3 the displayed integrand is P(τ)Λ_k(τ,−z)v^{k+1/2}dμ, i.e. the conjugation has been dropped. Restoring it, the coefficient of e^{−2πidτ̄} in the conjugate kernel is ω_{k+1,d}(−z)̄. By the paper’s own b↦−b symmetry, ω_{k+1,d}(−z)=ω_{k+1,d}(z), and termwise Q_z∈R and Q(z,1)̄=Q(z̄,1) give ω(z)̄=ω(z̄). Thus the u-integral selects ω_{k+1,D}(z̄), not ω_{k+1,D}(z). These are not equal in general: from Theorem 1.1(ii), ω=i f_{k,D}'/k+f_{k,D}/y, so ω(z)̄ = −i f_{k,D}'(z̄)/k + f_{k,D}(z̄)/y while ω(z̄)=i f_{k,D}'(z̄)/k+f_{k,D}(z̄)/y. The theorem is therefore false as stated for generic z; the lift should be defined with Λ_k(·,−z̄), and the analogous identity (1.3) needs the same correction. This is load-bearing because Theorem 1.3 is the central new theta lift, not a peripheral gap.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the divisor modular form attached to Zagier's hyperbolic Poincaré series f_{k,D}. It constructs a weak Maaß form ω_{k+1,D} and proves three theorems: (1.1) ω_{k+1,D} is a weak Maaß form of weight 2k+2 whose quotient by f_{k,D} is related to the divisor modular form; (1.2) the generating function Λ_k(τ,z) of the ω_{k+1,D} is modular of weight k+1/2 for Γ0(4) in Kohnen's plus space, via Vignéras' indefinite theta theorem; (1.3) a Petersson-inner-product theta lift of Λ_k(·,−z) against the plus-space Poincaré series P^+_{k+1/2,D} reproduces ω_{k+1,D}(z).","tokens_in":10789,"tokens_out":23623,"duration_ms":253559,"significance":"If correct, the connection between Zagier's f_{k,D} and the new weak Maaß form ω_{k+1,D}, together with the modularity of its generating function, would be a valuable addition to the literature on indefinite theta series and theta lifts. The paper is well organized, gives clear preliminary material, and includes a reproducible Sage computation in an appendix. The main theorem, however, contains a load-bearing conjugation error that makes the statement false as written; the surrounding framework is otherwise sound and appears fixable.","major_comments":[{"comment":"The Petersson inner product (2.7) is sesquilinear: ⟨f,g⟩ = (1/6)∫ f(z) \\overline{g(z)} y^κ dμ. The displayed line after 'Hence, we obtain' drops the conjugation on Λ_k(τ,−z). Restoring it, the unfolded integral selects \\overline{ω_{k+1,D}(−z)}. Using the stated b↦−b symmetry, ω_{k+1,D}(−z)=ω_{k+1,D}(z), and since \\overline{ω(z)}=ω(\\bar z), the selected coefficient is ω_{k+1,D}(\\bar z), not ω_{k+1,D}(z). These are not equal in general (for example D=1, k=4 gives ω(z)=2x/(y z^5)). Thus Theorem 1.3 is false as stated; the proof establishes instead the value for ⟨Λ_k(·,−z), f⟩. The lift should be defined with Λ_k(·,−\\bar z), or the conclusion changed to ω_{k+1,D}(\\bar z).","section":"Section 4, proof of Theorem 1.3; Eq. (2.7)"},{"comment":"Vignéras' theorem requires p ∈ C²(R³) and the eigenfunction identity (E − Δ/(4π))p = (k−1)p on all of R³. The proof asserts C² at D=0 without a detailed argument, and the eigenfunction identity is delegated to the Sage script in Appendix A. Since Theorem 1.3 depends on this modularity, please provide a short written verification or at least justify why the piecewise-defined p is C² across the discriminant-zero surface for k>2 and why the formal symbolic identity in Sage covers the actual p (which vanishes for D≤0).","section":"Section 4, proof of Theorem 1.2"}],"minor_comments":[{"comment":"Once the conjugation is restored, the factor written as e^{2πiDτ}e^{-2πidτ} should be e^{2πiDτ}e^{-2πid\\barτ}; the displayed v-integral is convergent only with the conjugated exponential.","section":"Section 4, proof of Theorem 1.3"},{"comment":"The phrase 'p is twice continuously differentiable at D=0' deserves a one-line expansion: for k>2 the exponent k−1/2 exceeds 2, so D_+^{k−1/2} is C² across the cone. Also, the Sage code uses symbolic complex powers for D<0; clarify that the verification applies to the D>0 branch and is extended by continuity.","section":"Section 4, proof of Theorem 1.2"},{"comment":"There are a few typographical issues: 'iff satsifies' in Definition 2.4 and inconsistent use of τ vs. z in some integrals. These do not affect the mathematics.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid and load-bearing: Theorem 1.3 as stated is false because of the missing conjugate in the Petersson inner product. The paper's central claim can likely be repaired by redefining the lift with Λ_k(·,−\\bar z) or by adjusting the statement and diagram accordingly, but the current version requires substantive correction before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper makes a natural move: define ω_{k+1,D} from Zagier's f_{k,D}, bundle them into a generating function Λ_k, and show it has the same Γ0(4) plus-space modularity as the Kohnen–Zagier kernel. Theorem 1.1 is solid: the splitting in (ii) and the divisor formula in (iii) are clean. Theorem 1.2 is also in decent shape, though the Vignéras eigenvalue check is only a SAGE computation and the C^2 regularity across the discriminant-zero surface is asserted rather than proved. Those are minor referee requests, not real flaws.\n\nThe serious problem is Theorem 1.3. The Petersson inner product in (2.7) is printed without a complex conjugate; I think that is a typo, because the unfolding argument only works with a sesquilinear pairing. Restore the conjugate. Then the coefficient selected by the u-integral is not ω_{k+1,D}(z) but \\overline{ω_{k+1,D}(−z)}, which by the paper's own b↦−b symmetry equals \\overline{ω_{k+1,D}(z)} = ω_{k+1,D}(\\bar z). These are different from ω_{k+1,D}(z) in general: from Theorem 1.1(ii), ω = i f'/k + f/y, so switching z to \\bar z flips the sign of the derivative term. So the theorem as stated is false for generic z. The fix is to change the kernel to Λ_k(·,−\\bar z), and the same correction applies to identity (1.3). This is load-bearing — the theta lift is the advertised new result.\n\nThe construction is well-motivated and the paper is otherwise competent. The error is precise and repairable, so this deserves a serious referee rather than a desk rejection. My own verdict: major revision; I would not cite it in its current form.","headline":"Nice construction, but Theorem 1.3 has a missing conjugate and is false as stated; the fix (kernel at −\\bar z) likely saves it.","tokens_in":11233,"tokens_out":10177,"would_cite":false,"duration_ms":101345,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11E16","11F27"],"pacs":[],"model":"deepseek-v4-flash","headline":"A theta lift sends half-integral weight cusp forms to the weak Maass form ω_{k+1,D}, which encodes the divisor modular form of f_{k,D}.","keywords":["Indefinite theta series","Weak Maass forms","Theta lifts","Divisor modular forms","Hyperbolic Poincaré series","Half-integral weight modular forms","Plus space","Binary quadratic forms"],"falsifier":"Take the function p(a,b,c) defined in Section 4 and compute its second mixed partial derivatives in a neighborhood that crosses the cone b^2 = 4ac; a discontinuity there would invalidate the application of the theta-series theorem. An independent high-precision evaluation of (E − Δ/4π)p − (k−1)p at random parameter triples would verify or refute the paper's computer algebra check.","tokens_in":10333,"feed_emoji":"📐","tokens_out":9736,"duration_ms":95417,"temperature":0.7,"pith_summary":"The paper sets up a new theta lift in the theory of modular forms. Its central object is a function ω_{k+1,D}(z) built by summing, over all integral binary quadratic forms of discriminant D, the quantity Q_z/Q(z,1)^{k+1}; this sum has no poles on the upper half-plane and is a weak Maass form of weight 2k+2 with eigenvalue 2k. The authors prove that the generating function Λ_k(τ,z) of these functions, taken over positive discriminants D, transforms like a half-integral weight modular form in the plus space, exactly as the classical kernel that generates the hyperbolic Poincaré series f_{k,D} does. They then compute the Petersson inner product of Λ_k with the plus-space Poincaré series of index D and obtain ω_{k+1,D} up to an explicit gamma constant. Because ω_{k+1,D} also appears in a formula for the divisor modular form of f_{k,D}, the lift connects half-integral weight cusp forms directly to the zeros and poles of that series.","feed_headline":"Theta lift pairs cusp forms with divisor modular forms","feed_subtitle":"The new kernel Λ_k mirrors the classical half-integral weight generating function and recovers the divisor form.","key_machinery":"The engine of the argument is the new function ω_{k+1,D}(z)=∑_{Q∈Q_D} Q_z/Q(z,1)^{k+1}, where Q_z=(a|z|^2+bx+c)/y encodes the hyperbolic geodesic attached to Q. Each summand is a weak Maass form piece: the function is modular of weight 2k+2, has eigenvalue 2k under the hyperbolic Laplacian, and splits into a holomorphic derivative part plus y^{-1}f_{k,D}(z). The generating function Λ_k(τ,z)=∑_{D>0} D^{k−1/2}ω_{k+1,D}(z)e^{2πiDτ} is shown, via a standard indefinite theta series criterion, to transform like a weight k+1/2 modular form in the plus space. The Petersson coefficient formula then turns a Fourier coefficient evaluation into the closed-form lift.","core_discovery":"The paper's main result is an exact evaluation: the Petersson inner product of the plus-space Poincaré series P^+_{k+1/2,D} with Λ_k(·,−z) equals Γ(k−1/2)/(6(4π)^{k−1/2}) ω_{k+1,D}(z). Since the P^+ series generate the plus space, this is a theta lift realizing ω_{k+1,D} as the image of the index-D series. The supporting results are modularity of Λ_k in weight k+1/2 for Γ0(4) and a formula expressing the divisor modular form of f_{k,D} as k/(2π) ω_{k+1,D}/f_{k,D} plus k/6 E_2^*. The lift therefore connects half-integral weight cusp forms to the zeros and poles of the hyperbolic Poincaré series.","pith_inferences":["Editorial inference: extending the lift to the full plus space by linearity yields a natural map from half-integral weight cusp forms to a space spanned by the ω_{k+1,D}, potentially giving a new analog of the classical correspondence between half-integral and integral weight forms.","Editorial inference: the explicit divisor formula suggests that coefficients of ω_{k+1,D} could be translated into information about vanishing behavior of f_{k,D}; this could be tested numerically by comparing the lift values with known divisor data for small k and D.","Editorial inference: the same construction applied to negative discriminants, where f_{k,D} has poles at CM points rather than cusps, might produce a hyperbolic analog with singularities at CM points, yielding a different but related theta lift."],"forward_implications":["Because the plus-space Poincaré series generate the entire plus space, the theta lift extends by linearity to a map defined on all half-integral weight cusp forms satisfying the plus-space condition.","Theorem 1.1(iii) gives an explicit formula for the divisor modular form of f_{k,D}: normalized by its first nonzero coefficient, it equals k/(2π) ω_{k+1,D}/f_{k,D} plus k/6 E_2^*.","Since Λ_k has the same modularity type as the classical kernel Ω_k, modularity-based constructions that work for Ω_k—such as coefficient extraction and Petersson inner products—have direct counterparts for Λ_k.","The lift pairs half-integral weight cusp forms with real-analytic modular objects that carry divisor data, making the zeros and poles of the hyperbolic Poincaré series accessible through theta-lift methods."],"supporting_citations":[{"why":"Introduces the hyperbolic Poincaré series f_{k,D} and records the first nonzero Fourier coefficient used in Theorem 1.1(iii).","marker":"[26]"},{"why":"Supplies the general identity relating the derivative of a modular form to its divisor modular form, used in the proof of Theorem 1.1(iii).","marker":"[10]"},{"why":"Introduce the generating function Ω_k of the f_{k,D} and establish its modularity and Petersson coefficient formula, which Λ_k is built to parallel.","marker":"[16,17]"},{"why":"Gives the indefinite theta series criterion that proves modularity of Λ_k in Theorem 1.2.","marker":"[25]"},{"why":"Introduces the plus-space projection and the plus-space Poincaré series used to evaluate the lift.","marker":"[15]"},{"why":"States the Hermitian property of the plus-space projection and sets up the theta lift convention used in Theorem 1.3.","marker":"[5]"},{"why":"Provides the computer algebra verification of the eigenfunction identity required by the theta series criterion.","marker":"[30]"}],"fun_headline_variants":["Theta lift connects divisor form to weak Maass form","Exact theta lift yields weak Maass form from cusp forms","Petersson product gives new theta lift for divisor form","Kohnen-Zagier kernel inspires theta lift to Maass form"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument treats a computer algebra check and a smoothness claim on a three-dimensional auxiliary function as established; the modularity theorem collapses if the check or the smoothness is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Theta lift connects divisor form to weak Maass form","Exact theta lift yields weak Maass form from cusp forms","Petersson product gives new theta lift for divisor form","Kohnen-Zagier kernel inspires theta lift to Maass form"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":2845,"prompt_tokens":647,"completion_tokens":2198,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":2127}},"tokens_in":391,"tokens_out":2198,"duration_ms":19349,"temperature":1.0,"reasoning_tokens":2127,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:36:00.352246+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the function p(a,b,c) defined in Section 4 and compute its second mixed partial derivatives in a neighborhood that crosses the cone b^2 = 4ac; a discontinuity there would invalidate the application of the theta-series theorem. An independent high-precision evaluation of (E − Δ/4π)p − (k−1)p at random parameter triples would verify or refute the paper's computer algebra check.","supporting_citations":[{"cited_title":"Zagier,Modular forms associated to real quadratic fields, Invent","cited_arxiv_id":null,"evidence_quote":"Introduces the hyperbolic Poincaré series f_{k,D} and records the first nonzero Fourier coefficient used in Theorem 1.1(iii)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the general identity relating the derivative of a modular form to its divisor modular form, used in the proof of Theorem 1.1(iii)."},{"cited_title":"Lecture Notes in Mathematics (J.-P","cited_arxiv_id":null,"evidence_quote":"Gives the indefinite theta series criterion that proves modularity of Λ_k in Theorem 1.2."},{"cited_title":"Kohnen,Fourier coefficients of modular forms of half-integral weight, Math","cited_arxiv_id":null,"evidence_quote":"Introduces the plus-space projection and the plus-space Poincaré series used to evaluate the lift."},{"cited_title":"Bringmann, B","cited_arxiv_id":null,"evidence_quote":"States the Hermitian property of the plus-space projection and sets up the theta lift convention used in Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the computer algebra verification of the eigenfunction identity required by the theta series criterion."}],"review_version":1}