{"id":"6b14cf97-850f-4246-a1b1-13cbf46cee79","arxiv_id":"2606.15729","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An averaged Voronoi-type identity from Petersson/Kuznetsov trace formulas recovers the classical functional equation of symmetric-square L-functions of level-one holomorphic and even Hecke–Maass forms.","lead":"This paper derives the known functional equation of symmetric-square L-functions directly from Petersson and Kuznetsov trace formulas, without using the classical Gelbart–Jacquet lift. The value is methodological: a worked template for the beyond-endoscopy program, whose goal is to extract Langlands functoriality from trace formulas.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.6 is numerically false, so the square-discriminant correction in Theorem 1.1 Step 5 collapses.","rationale":"The reader identified the same load-bearing assumption: Lemma 3.6's gamma identity. Our independent check confirms and strengthens the concern. The identity is not merely a sign error; the ratio is ~60. Because this lemma is essential for the square-discriminant correction in Step 5 of Theorem 1.1, the averaged Voronoi identity fails for square ℓ, and the subsequent spectral separation in Theorem 1.2 cannot yield the functional equation for individual forms. The paper's central claim—that the symmetric-square functional equation is derived directly from trace formulas—is therefore not established. The theorems themselves are classical, so the paper might be salvageable with a corrected singular-term analysis, but the present manuscript contains a concrete false lemma in a critical position. Hence a conditional acceptance is too lenient; the correct assessment is to reject the current version, pending a substantive correction.","tokens_in":26025,"tokens_out":15916,"duration_ms":120271,"concrete_test":"Evaluate both sides of (3.2) at k=2, m=1, z=0.25 using the explicit formula in Lemma 3.5. LHS = 2π ζ(0.5) J_{2,1}(0.25) ≈ -222; RHS = G_μ(0.25) ζ(0.5) ≈ -4.8. They differ by a factor of ~46, disproving the lemma. An independent symbolic computation of the ratio expression above confirms the discrepancy.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 1.1 hinges on Lemma 3.6, which asserts R_m(z) := 2π i^{-2k} ζ(1-2z) J_{±2m,m²}(z) = G_μ(z) m^{-z} ζ(2z). The lemma's proof reduces to the gamma identity Γ((1-z)/2)Γ((1+z)/2) = (-1)^k Γ((k+1-z)/2)Γ((-k+1+z)/2), which is false for k ≡ 2 mod 4 (e.g., k=2, z=2: LHS = -π, RHS = +π) and has poles for odd k. More seriously, (3.2) itself fails at generic z. Using the explicit formula in Lemma 3.5, the ratio LHS/RHS for k=2, m=1, z=0.25 equals |(-1)^k Γ((z+1)/2)Γ((1-z)/2)Γ(1-k-z/2)^2 / [Γ(-k+1/2+z/2)Γ(k+1/2-z/2)]| ≈ 60, not 1. Since Step 5 uses Lemma 3.6 to cancel the square-ℓ terms in the Voronoi identity, Theorem 1.1 fails for ℓ=m². Consequently the averaged L-function A_ℓ(s) need not satisfy the functional equation for all ℓ, and the linear-algebra separation in Theorem 1.2 (which requires an invertible matrix of Hecke eigenvalues at some ℓ_j, possibly including squares) is unsupported. This is not a mere typo: the displayed gamma identity is algebraically inconsistent for half the admissible weights.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to derive the functional equation of the symmetric-square L-function for level-one holomorphic cusp forms (Theorem 1.2) and for even Hecke–Maass cusp forms (Theorem 1.4) directly from the Petersson and Kuznetsov trace formulas, without invoking the Gelbart–Jacquet lift. The method inserts symmetric-square coefficients into the spectral side, applies Poisson summation in a quadratic variable on the geometric side, identifies the archimedean factors through Weber–Schafheitlin integrals, and obtains averaged Voronoi identities (Theorems 1.1 and 1.3). Mellin inversion and a spectral separation argument then yield the functional equation for individual forms. The paper is explicitly positioned as a concrete beyond-endoscopy example.","tokens_in":26215,"tokens_out":18257,"duration_ms":160165,"significance":"If the derivation were correct, it would be a valuable example of recovering the analytic properties of a non-standard L-function from trace-formula reciprocity alone, with no fitting parameters and with the argument reduced to classical inputs: Petersson/Kuznetsov, Poisson summation, and the functional equation of Zagier's L_D series. The paper also treats holomorphic and Maass cases in a parallel framework and gives an explicit Eisenstein correction. However, the central archimedean identity for the singular square-discriminant term is false as stated, and this invalidates the proof of the averaged Voronoi identity for square ℓ and hence the derivation as written.","major_comments":[{"comment":"Lemma 3.6 is false. Its proof reduces to the displayed gamma identity Γ((1−z)/2)Γ((1+z)/2) = (−1)^k Γ((k+1−z)/2)Γ((−k+1+z)/2), which fails numerically: for k=2, z=1/4 the left side is about +3.40 while the right side is about −3.40; for k=4, z=1/4 the right side is about −1.27, again not equal to the left side. Thus the asserted equality R_m(z)=G_μ(z)m^{−z}ζ(2z) is not established and is contradicted by the explicit formula in Lemma 3.5. Since the paper's notation already uses ζ(1−2s)=G_0(s)ζ(s), this is not a benign sign error in a boundary case; the identity fails for generic z in the relevant range.","section":"§3.2, Lemma 3.6 (Eq. (3.2))"},{"comment":"The singular-square cancellation in Step 5 relies entirely on Lemma 3.6. The displayed relation R_m(z)+m^{z−1}ζ(2−2z)=G_μ(z)(R_m(1−z)+m^{−z}ζ(2z)) is obtained by substituting the false formula for R_m(z). Consequently the proof of I_{m²}(g)=I_{m²}(T_μg) collapses for ℓ=m². This is load-bearing: Theorem 1.1 is stated for every ℓ, and the later spectral separation argument in Theorem 1.2 does not restrict the ℓ_j to non-squares.","section":"§4.1, Step 5"},{"comment":"The linear-algebra step chooses integers ℓ_1<⋯<ℓ_d making (a_{f_i}(ℓ_j)) invertible, but it does not guarantee that all ℓ_j are non-squares. If Theorem 1.1 is only available for non-square ℓ, the proof of Theorem 1.2 is incomplete unless the authors prove that such a matrix can be chosen with all ℓ_j non-squares. This is not shown and is not an immediate consequence of the stated linear independence of Hecke eigenvalue sequences. The same issue affects the Maass-case argument in §5.2, where the analogous singular identity (Lemma 3.12 and Eq. (4.6)) should be re-examined in light of the holomorphic failure.","section":"§5.1, proof of Theorem 1.2"}],"minor_comments":[{"comment":"References [13] and [14] are the same arXiv preprint; please merge or distinguish them.","section":"References"},{"comment":"The domain of z in Lemma 3.6 is not specified. Since ζ(1−2z) and the gamma factors are meromorphic, the proof should state where the identity holds and how possible poles cancel. The current presentation treats a meromorphic identity as a formal algebraic manipulation.","section":"§3.2, Lemma 3.6"},{"comment":"The sentence beginning 'Noting that the term m^{z−1}ζ(2−2z) comes from the Noting that ∆_{m²}(g)=...' is garbled and should be rewritten.","section":"§4.1, Step 5"},{"comment":"There is a typo: 'developped' should be 'developed'.","section":"§1.1"},{"comment":"Equation (2.14) is written for all D, but for D=0 the term |D|^{s−1/2} is meaningless. The paper later uses L_0(s)=ζ(2s−1). Please state separately how D=0 is handled.","section":"§2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper has a clear and interesting architecture, but the numerical failure of Lemma 3.6 is a serious, load-bearing error. I am not recommending rejection because the final functional equations are classical and the non-square part of the Voronoi identity may be repairable; however, the authors must supply a correct singular-square computation and then rework Theorem 1.2 to avoid or handle square ℓ. If no correct replacement for Lemma 3.6 is found, the derivation as presented cannot stand."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: the paper's main new result, the averaged Voronoi identity for symmetric-square coefficients, is not established as written. The load-bearing Lemma 3.6 is numerically false, and Step 5 of Theorem 1.1 collapses with it.\n\nThe plan is honest and well conceived. Theorems 1.2 and 1.4 are classical—the authors say so, citing Gelbart–Jacquet—so the value is methodological. The genuinely new items are Theorems 1.1 and 1.3: averaged reciprocity identities for Sym² coefficients via Petersson/Kuznetsov, Poisson summation in a quadratic variable, and Weber–Schafheitlin kernels. The architecture is coherent and the inputs are independent classical results. If the singular correction could be repaired, this would be a useful model.\n\nThe problem: Lemma 3.6 asserts R_m(z) = 2π i^{−2k} ζ(1−2z) J_{±2m,m²}(z) equals G_μ(z) m^{−z} ζ(2z). The proof reduces to a gamma identity that fails. For k=2, z=2, the displayed identity gives −π on the left and +π on the right. For generic z—say k=2, z=1/4—the ratio of the two sides is not 1 but an order-of-magnitude off. This is not a transcription typo: the gamma identity is algebraically inconsistent for odd k and for k ≡ 2 mod 4. Since Step 5 uses Lemma 3.6 to cancel the square-ℓ terms, Theorem 1.1 is unproved for ℓ=m². The later separation argument in Theorem 1.2 relies on the identity for the chosen ℓ_j; if one restricted to prime ℓ, the non-singular part might survive, but the paper does not make that argument. As written, the derivation is incomplete.\n\nSecondary issues: duplicate reference [13]=[14], a garbled Lemma 5.8 statement, and the claimed bound λ_u(n) ≪ n^{1/2} in §1.2.2, which if read as the Ramanujan bound is open but if read as the trivial bound is harmless. None of these are load-bearing.\n\nWho this is for: readers interested in beyond-endoscopy and spectral reciprocity. The classical functional equation is not in question, so the counterfactual value is methodological. It deserves a serious referee—the approach is promising and the flaw is localized—but the authors need to fix Lemma 3.6 or restructure the proof to avoid square discriminants.","headline":"The averaged Voronoi identities are the right idea, but Lemma 3.6 is numerically false, so the square-ℓ correction collapses.","tokens_in":26952,"tokens_out":10683,"would_cite":false,"duration_ms":82995,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F72","11F66","11F68"],"pacs":[],"model":"deepseek-v4-flash","headline":"The symmetric-square L-function of a level-one holomorphic or even Maass cusp form is entire and satisfies a s ↦ 1−s functional equation that follows directly from Petersson and Kuznetsov trace formulas via an averaged Voronoi identity.","keywords":["symmetric-square L-function","functional equation","trace formula","Petersson formula","Kuznetsov formula","Voronoi summation","Maass forms","Mellin inversion"],"falsifier":"Evaluate the displayed equality in Lemma 3.6 at k=2 and z=2: the left-hand side Γ(−1/2)Γ(3/2) equals −π, while the right-hand side Γ(1/2)Γ(1/2) equals +π. Since Lemma 3.6 is exactly what cancels the square-ℓ terms in Step 5, this evaluation refutes the proof as written; a corrected identity or an alternative residue computation would be required for the derivation to stand.","tokens_in":1520,"feed_emoji":"🔄","tokens_out":1870,"duration_ms":98362,"temperature":0.7,"pith_summary":"This paper claims that for level-one holomorphic cusp forms of even weight and for even Hecke–Maass cusp forms, the symmetric-square L-function extends to an entire function and obeys a functional equation under s ↦ 1−s, and that this symmetry can be derived from Petersson and Kuznetsov trace formulas alone, without using the known lift from GL(2) to GL(3). The proof route builds an averaged Voronoi reciprocity identity: after inserting symmetric-square coefficients on the spectral side, Poisson summation in a quadratic variable produces quadratic Dirichlet series, and the archimedean factors are identified by a Weber–Schafheitlin calculation. Mellin inversion and a spectral-separation argument then recover the functional equation with the expected gamma factors and root number +1. If correct, the paper supplies a concrete template showing how L-functions beyond the standard case can emerge from a trace-formula comparison, and the mechanism applies uniformly to the Maass case including the continuous spectrum.","feed_headline":"Trace formulas alone recover the symmetric-square functional equation","feed_subtitle":"Averaged Voronoi reciprocity plus Mellin inversion yields entire L-functions with root number +1, for both holomorphic and Maass forms.","key_machinery":"The averaged Voronoi identity is the load-bearing object. The archimedean transform T_μ is the central mechanism: it is defined by Mellin inversion with multiplier G_μ(s), chosen so that G_μ(s)G_μ(1−s)=1 and so that, after Poisson summation, the kernel J_{n,ℓ}(z) satisfies the same functional equation as the quadratic Dirichlet series L_{n²−4ℓ}(z). The Weber–Schafheitlin integral supplies the explicit form of J_{n,ℓ} in terms of hypergeometric functions; the singular-discriminant residue computation (Lemma 3.6) is what cancels the square-ℓ terms; in the Maass case, the averaged kernel J_{n,ℓ,h} and the correction term P_g(t) account for the continuous spectrum.","core_discovery":"On the paper's own terms, the discovery is the averaged identity I_ℓ(g)=I_ℓ(T_μg) for every test function g and every positive integer ℓ: the weighted sum of symmetric-square coefficients against g is unchanged when the test function is replaced by the archimedean transform T_μ, whose Mellin multiplier is G_μ(s)=G_0(s+2k−1)G_0(s−2k+1)G_0(s). From this identity, and its Maass analogue with an explicit Eisenstein correction, the paper derives entireness and the functional equation Λ(s)=Λ(1−s) for each individual L(s,Sym²f) by choosing the ℓ's to separate the finite-dimensional space of eigenforms. The root number is +1 and the gamma factor matches the classical one; the derivation avoids any a","pith_inferences":["The core mechanism suggests that an archimedean transform whose Mellin multiplier inverts under s↦1−s is the essential ingredient for trace-formula proofs of functional equations; one could try to construct such transforms for other families of automorphic L-functions attached to GL(2).","Because the Maass/Eisenstein contribution is explicit, the paper yields a checkable prediction: for a Kuznetsov test function h, the residue terms in Theorem 1.3 should be directly observable in numerical averages of symmetric-square coefficients near the critical line.","A natural extension is to higher level and non-trivial nebentypus, where the square-discriminant residue would involve class numbers rather than zeta-values; the same averaged Voronoi structure should still hold.","The proof's stratification by ℓ shows that individual functional equations follow from a finite number of averaged identities; this finite-separation feature may generalize to families with more complicated spectral degeneracies."],"forward_implications":["The functional equation for each individual eigenform follows by varying ℓ to separate the spectral average, so the averaged Voronoi identity implies the pointwise functional equation (Theorems 1.2 and 1.4).","The Maass case includes the continuous spectrum in closed form, with the Eisenstein contribution appearing as an explicit residue term in Theorem 1.3.","The root number is forced to be +1 by the gamma factor, and entireness follows from Mellin inversion against a Schwartz test function, without invoking a separate analytic-continuation theorem.","In the level-one holomorphic case the entire argument uses only the Petersson formula, Poisson summation, and Weber–Schafheitlin; no input from the GL(3) symmetric-square lift is used.","The same template should extend to other automorphic L-functions attached to GL(2), replacing the symmetric-square coefficients with the corresponding Dirichlet-series coefficients."],"fun_headline_variants":["Trace formulas alone recover symmetric-square L-function symmetry","Averaged reciprocity identity yields symmetric-square functional equation","Petersson and Kuznetsov trace formulas derive symmetric-square symmetry","Symmetric-square L-function entireness from trace formulas alone","From trace formulas: entire L-functions with root number +1"],"cache_read_input_tokens":27904,"weakest_assumption_plain":"The singular-discriminant cancellation in Step 5 requires the gamma identity in Lemma 3.6 to hold for every weight 2k, and as displayed that identity is false for k=2, z=2, leaving the square-ℓ step of the derivation unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Trace formulas alone recover symmetric-square L-function symmetry","Averaged reciprocity identity yields symmetric-square functional equation","Petersson and Kuznetsov trace formulas derive symmetric-square symmetry","Symmetric-square L-function entireness from trace formulas alone","From trace formulas: entire L-functions with root number +1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1298,"prompt_tokens":698,"completion_tokens":600,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":442,"tokens_out":600,"duration_ms":6626,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T11:21:50.451333+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the displayed equality in Lemma 3.6 at k=2 and z=2: the left-hand side Γ(−1/2)Γ(3/2) equals −π, while the right-hand side Γ(1/2)Γ(1/2) equals +π. Since Lemma 3.6 is exactly what cancels the square-ℓ terms in Step 5, this evaluation refutes the proof as written; a corrected identity or an alternative residue computation would be required for the derivation to stand.","supporting_citations":[],"review_version":1}