{"id":"69e7fee5-2580-4bf0-abe8-60d79ac0443e","arxiv_id":"2412.12410","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"L(1/2, f⊗g) ≤ p^{1/2 - 1/524 + ε} for newforms f of prime level p with any nebentypus and fixed eigenforms g, improving the prior δ = 1/1413.","lead":"This paper proves a sharper subconvexity bound for Rankin-Selberg L-functions of a newform of prime level p and a fixed level-1 form, improving the exponent to p^{1/2 - 1/524}. It uses the delta method together with trilinear Kloosterman sum bounds, avoiding GL2 spectral machinery and extending to weight-1 forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (4.3) is not an identity for non-quadratic χ: the amplifier lower bound is unjustified, so Lemma 4.1's division by A(L) is unsupported for the full range of central characters claimed in Theorem 1.1.","rationale":"The reader's weakest assumption points at the amplifier lower bound (4.3) and the absence of an unconditional lower bound for a prime character sum. My independent reading confirms that (4.3) is the key step: Lemma 4.1 divides by A(L), and every subsequent estimate inherits that normalization. However, the more precise defect is algebraic, not merely analytic: for non-quadratic χ, the identity asserted in (4.3) is false under the paper's own definitions. This is a stronger statement than 'the character sum is not known to be large'; as written, the expression is not even the character sum the paper claims. The fix is likely simple, namely conjugate the χ-weight in the second amplified term so that χ(ν)^{-1}(λ_f(ν)^2-λ_f(ν^2))=1. There is a reasonable chance the rest of the proof goes through unchanged, since all weights have modulus 1 and only character twists enter the later Kloosterman-sum estimates, but the manuscript currently does not provide that verification. I agree with the reader that the appropriate verdict is conditional acceptance pending a correct amplifier lemma. I do not see a second comparably load-bearing defect: the Voronoi, Poisson, zero-frequency, and Bettin-Chandee trilinear steps are carefully laid out, use only allowed norms, and the exponent computation around N=p^{1+ε}, L=p^{1/151}, σ=1/20 gives 1/524 consistently. Credit is due for the clean use of Theorem 1.8 and for building the argument around GL1 harmonic analysis, but the amplifier gap blocks the claimed full uniformity over all central characters as written.","tokens_in":30913,"tokens_out":18875,"duration_ms":171152,"concrete_test":"Independently re-derive (4.3) from the definitions in §4.1, relation (3.6), and the Hecke relation λ_f(ν^2)=λ_f(ν)^2-χ(ν), using χ a primitive character of order 3 modulo a prime. A symbolic computation with λ_f(ν)=χ(ν)^{1/2} shows the left-hand side equals Σγν[1+(χ(ν)^{-1}-χ(ν))λ_f(ν)^2], not Σγν. Then test the suggested repair: replace a_2(ν) by -overline{χ}(ν), verify that A(L)=Σγν, and check that the bounds (4.2), Lemma 4.7, Lemma 4.9, and Lemma 4.19 remain valid with b_2(ℓ)=overline{χ}(√ℓ); since |χ|=1, the coefficient bounds are unchanged, but the character sums in Lemma 4.17 must be rechecked for the conjugated weight.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing defect is in §4.1, equation (4.3). With A_j defined as A_j(L) = Σ γν a_j(ν) λ_f(ν^j), a_1(ν)=λ_f(ν), a_2(ν)=-χ(ν), we have A(L) = Σ γν(λ_f(ν)^2 - χ(ν)λ_f(ν^2)). The Hecke relation (3.5) gives λ_f(ν^2) = λ_f(ν)^2 - χ(ν), so A(L) = Σ γν[(1-χ(ν))λ_f(ν)^2 + χ(ν)^2]. The paper instead asserts, via (3.6), that A(L) = Σ γν χ(ν)(λ_f(ν)^2 - λ_f(ν^2)) = Σ γν. For non-quadratic χ this is algebraically false: (3.6) says λ_f(ν)=χ(ν)overline{λ_f(ν)}, so λ_f(ν)^2 = χ(ν)|λ_f(ν)|^2, hence |λ_f(ν)|^2 = χ(ν)^{-1}λ_f(ν)^2, not χ(ν)λ_f(ν)^2. If the intended first form was |λ_f(ν)|^2 - χ(ν)λ_f(ν^2), the same relations give Σ γν[1 + (χ(ν)^{-1}-χ(ν))λ_f(ν)^2], again not a prime count. Thus for a primitive character of order >2 the claimed lower bound A(L) ≫ L^{1-ε} does not follow from the prime number theorem, and no unconditional lower bound of that strength is supplied. Since Lemma 4.1 requires A(L) in the denominator, and the subsequent Cauchy-Schwarz and trilinear-Kloosterman steps all use the resulting normalization, the proof as written does not establish uniformity over all central characters. The defect appears repairable by replacing the second weight a_2(ν)=-χ(ν) with -overline{χ}(ν) (or equivalently -χ(ν)^{-1}), in which case the identity becomes Σ γν, but this repair must be traced through §4.5–§4.11 because the b_j weights enter Lemma 4.7, Lemma 4.9, and Lemma 4.19.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a level-aspect subconvexity bound L(1/2, f \\otimes g) \\ll p^{1/2 - 1/524 + \\varepsilon} for a Hecke newform f of prime level p with arbitrary central character \\chi, and a fixed level-one eigenform g that may be holomorphic, Maass, or Eisenstein. The method uses the DFI delta symbol, Voronoi summation, Cauchy-Schwarz amplification, Poisson summation, and the Bettin-Chandee trilinear Kloosterman-fraction bound. The claimed exponent improves Harcos's \\delta = 1/1413 and is stated uniformly in the nebentypus, in the type of g, and without recourse to the Ramanujan conjecture.","tokens_in":31395,"tokens_out":9804,"duration_ms":86001,"significance":"If correct, the result is a significant advance: it is the first delta-method treatment of the level-aspect GL(2) x GL(2) subconvexity problem that is uniform over all nebentypi and over cuspidal/Eisenstein choices of g, and it avoids the spectral machinery used in earlier moment-method proofs. The paper is careful and detailed: the exponent arithmetic is checkable, the use of external results (DFI delta lemma, Bettin-Chandee, Voronoi, stationary phase) is explicit, and the argument does not circularly assume the target bound. The main technical innovation, the exploitation of cancellation in correlations of Ramanujan sums, is clearly explained and is independent of the automorphic forms involved. However, the unconditional amplifier lower bound in Section 4.1 is load-bearing and is not justified for non-quadratic nebentypi; this must be repaired before the claimed uniformity over all central characters is established.","major_comments":[{"comment":"The lower bound A(L) \\gg L^{1-\\varepsilon} is not proved for non-quadratic \\chi and is in fact algebraically incorrect as written. From (3.6), \\lambda_f(\\nu) = \\chi(\\nu)\\overline{\\lambda_f(\\nu)}, so with \\alpha = \\lambda_f(\\nu) we have |\\alpha|^2 = \\overline{\\chi(\\nu)} \\alpha^2 = \\chi(\\nu)^{-1}\\alpha^2, not \\chi(\\nu)\\alpha^2. Substituting the Hecke relation \\lambda_f(\\nu^2)=\\alpha^2-\\chi(\\nu) into A(L)=\\sum_\\nu \\gamma_\\nu(|\\lambda_f(\\nu)|^2-\\chi(\\nu)\\lambda_f(\\nu^2)) gives \\sum_\\nu \\gamma_\\nu[(\\chi(\\nu)^{-1}-\\chi(\\nu))\\alpha^2 + \\chi(\\nu)^2], not \\sum_\\nu\\gamma_\\nu. Even the intermediate line \\sum_\\nu\\gamma_\\nu\\chi(\\nu)(\\lambda_f(\\nu)^2-\\lambda_f(\\nu^2)) equals \\sum_\\nu\\gamma_\\nu\\chi(\\nu)^2, which has no prime-number-theorem lower bound for a character of order greater than 2. Consequently Lemma 4.1's division by A(L) is unsupported for arbitrary \\chi, and Proposition 4.2 and Theorem 1.1 are not established in the stated generality. A likely repair is to replace a_2(\\nu) = -\\chi(\\nu) by -\\chi(\\nu)^{-1} (equivalently -\\overline{\\chi}(\\nu)), which makes the identity exact, but this change must be traced through the b_j weights used in Lemmas 4.7, 4.9, and 4.19.","section":"§4.1, Eq. (4.3)"},{"comment":"There is an internal inconsistency in the amplifier definition: a_2(\\nu) is defined as -\\chi(\\nu), but the alternative expression A_j(L)=\\sum_{\\ell\\in L_j} b_j(\\ell)\\lambda_f(\\ell) sets b_2(\\ell)=\\chi(\\sqrt{\\ell}), which has the opposite sign. The later estimates mostly use |b_j(\\ell)|, so the sign may be harmless once absolute values are taken, but the two displayed definitions of A_2(L) are not equal as written and should be reconciled.","section":"§4.1, definition of b_2"},{"comment":"The asserted bound S_0(N,A_1) \\ll p^{-2024} is not a defined order of magnitude; the paper's convention of 'very small' means \\ll_M p^{-M} for every M, and the constant 2024 appears to be a placeholder. The proof also asserts that the n-sum 'is essentially empty' for j=1 or c/c'=\\nu^2 without displaying the truncation argument; this should be stated as a precise 'very small' estimate with the relevant ranges from (4.13).","section":"Lemma 4.7"}],"minor_comments":[{"comment":"In the decomposition of T'(N,A_j), the displayed condition 'n_1\\ell_2 \\neq n_1\\ell_1' should read 'n_1\\ell_2 \\neq n_2\\ell_1'.","section":"§4.6, after Eq. (4.19)"},{"comment":"In the prime-power analysis, the notation m_0' = m_0/q_{10} is not defined globally; it should be m_0' = m_0/(c_{10},c_{20}) or the local prime-power version should be introduced explicitly.","section":"§4.10, Lemma 4.17"},{"comment":"The improved exponent under assumption (1.3) is helpful for orientation, but the reader would benefit from a one-sentence reminder that (1.3) is not used in the unconditional proof and that the unconditional amplifier is designed precisely to avoid it.","section":"§1.1, Remark 1.4"}],"recommendation":"major_revision","confidential_remarks":"The amplifier defect in §4.1 is localized and appears repairable by changing a_2 to -\\chi^{-1}(\\nu) and re-checking the later lemmas that use b_2. Because the rest of the argument is detailed and the claimed result would be a strong and welcom, I recommend major revision rather than rejection. The authors should also clarify the sign convention in the definition of b_2 and the 'very small' statement in Lemma 4.7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a serious and largely convincing extension of the delta method to GL2×GL2 level-aspect subconvexity. The genuinely new ingredient is a Poisson-summation treatment of correlations of Ramanujan sums inside the amplified delta-method setup, used with the Bettin–Chandee trilinear Kloosterman-fraction bound. That is a real innovation, and it buys a better exponent (1/524 vs. Harcos's 1/1413) plus uniformity over cuspidal/Eisenstein and holomorphic/Maass cases, with no dependence on the Ramanujan theta. The exponent arithmetic checks out: sigma=1/20 gives delta=1/524. The paper is careful, and the one self-citation [Leu24] is a standard delta-symbol lemma, not the target result, so no circularity.\n\nThe load-bearing problem is equation (4.3). The amplifier is A(L) = sum_nu gamma_nu (|lambda_f(nu)|^2 - chi(nu) lambda_f(nu^2)). The Hecke relation gives lambda_f(nu^2) = lambda_f(nu)^2 - chi(nu), and (3.6) gives lambda_f(nu)^2 = chi(nu)|lambda_f(nu)|^2. Substituting, A(L) = sum_nu gamma_nu [(1 - chi(nu)^2)|lambda_f(nu)|^2 + chi(nu)^2], not sum_nu gamma_nu unless chi^2 = 1. The paper's displayed identity A(L) = sum gamma_nu is algebraically wrong for non-quadratic characters. Since Lemma 4.1 divides by A(L), the claimed uniformity over all central characters is not established. The proof as written does cover trivial and quadratic chi. The fix is likely simple: take a_2(nu) = -chi(nu)^{-1} instead of -chi(nu); then the identity becomes sum gamma_nu. But that change must be traced through the b_j weights in Lemmas 4.7, 4.9, and 4.19, which the authors have not done. So my verdict is conditional: the method is sound and the gap is repairable, but Theorem 1.1 as stated is currently unsupported for characters of order greater than 2.\n\nThis paper deserves a serious referee. It is not desk-reject material; it is the kind of result the field wants, with one identifiable technical defect. I would send it to review and ask the referee to push specifically on the amplifier and the trace-through of the repaired weights. I would not cite Theorem 1.1 in its current form.","headline":"Sharp new method and exponent for GL2×GL2 level-aspect subconvexity, but the amplifier lower bound (4.3) is algebraically wrong for non-quadratic characters, so the stated uniformity is not yet proven.","tokens_in":31959,"tokens_out":3933,"would_cite":false,"duration_ms":33219,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F66","11F67","11L05","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"For each newform $f$ of prime level $p$, the paper establishes $L(1/2, f\\otimes g)\\ll p^{1/2-1/524+\\varepsilon}$ for every fixed level-one eigenform $g$.","keywords":["level aspect","subconvexity","Rankin–Selberg L-functions","delta method","Kloosterman fractions","Dirichlet characters","automorphic L-functions"],"falsifier":"Compute, for a primitive non-quadratic character $\\chi\\pmod p$ and $L=p^{1/151}$, the prime character sum $\\sum_{L/2<\\nu\\le L}\\chi(\\nu)^2$; a single prime $p$ for which this sum is $o(L^{1-\\varepsilon})$ would invalidate equation (4.3) and with it the amplifier lemma (Lemma 4.1) on which the proof rests.","tokens_in":30692,"feed_emoji":"🔢","tokens_out":15247,"duration_ms":124809,"temperature":0.7,"pith_summary":"The paper proves a subconvexity bound for the central value of the Rankin–Selberg $L$-function $L(1/2, f\\otimes g)$ as the level of $f$ runs over primes $p$: for every $\\varepsilon>0$, $L(1/2, f\\otimes g)\\ll p^{1/2-1/524+\\varepsilon}$, uniformly in the nebentypus of $f$ and with $g$ any fixed level-one cusp form or Eisenstein series, with $f$ holomorphic or Maass. This improves the previously best exponent $1/1413$ and, because the proof uses only $\\mathrm{GL}(1)$ harmonic analysis, it covers weight-one forms whose $L$-functions include class-group and Artin $L$-functions. The argument obtains the saving by applying an amplifier, Voronoi summation, Poisson summation, and a trilinear bound on Kloosterman fractions; it does not solve shifted convolution problems and is independent of any approximation to the Ramanujan conjecture.","feed_headline":"Subconvex bound for GL(2)xGL(2) L-functions hits 1/524","feed_subtitle":"Uniform in nebentypus and in g cuspidal or Eisenstein; the proof needs only GL(1) harmonic analysis.","key_machinery":"The engine is a delta-symbol method, a smooth-weight version of the circle method, applied to an amplified short sum of Hecke eigenvalues $S(N)=\\sum_{n\\asymp N}\\lambda_f(n)\\lambda_g(n)$ with amplifier length $L=p^{1/151}$. An amplifier $A(L)=\\sum_{\\nu\\le L}\\gamma_\\nu(|\\lambda_f(\\nu)|^2-\\chi(\\nu)\\lambda_f(\\nu^2))$ is meant to recover the size of the moments; then Voronoi summation converts the opening exponential sums into sums of Ramanujan sums, and a Poisson summation step evaluates the correlation of two Ramanujan sums. That correlation produces a trilinear exponential sum whose kernel is a Kloosterman fraction $e(am/n)$; the final cancellation comes from a known bilinear bound on such fractions, refined by an extra savings when a third variable is averaged. This trilinear refinement is what supplies the specific exponent $1/524$.","core_discovery":"The main theorem states that for prime $p$, $f$ a Hecke newform of level $p$ with any central character $\\chi\\pmod p$, and $g$ a fixed Hecke eigenform of level $1$, one has $L(1/2, f\\otimes g)\\ll p^{1/2-1/524+\\varepsilon}$, with the implied constant depending on $g$, $\\varepsilon$, and the archimedean parameter of $f$. The proof is uniform: $g$ may be cuspidal or Eisenstein, $f$ holomorphic or Maass, and $\\chi$ primitive or trivial, and the same estimate extends to $L(1/2+it, f\\otimes g)$ with polynomial dependence on $t$. Unlike earlier moment-method proofs, the argument never solves a shifted convolution problem and is free of any dependence on the Ramanujan-conjecture parameter $\\theta$, because all cancellation is extracted from arithmetic sums—Ramanujan sums and Kloosterman fractions—after Voronoi and Poisson summation.","pith_inferences":["Inference: if the obstructing character-sum lower bound becomes available, the same proof immediately upgrades the exponent to $1/302$.","Inference: the Ramanujan-sum correlation step is independent of the particular forms $f$ and $g$, so this Poisson-summation device should transfer to other delta-method level-aspect subconvexity problems.","Inference: because the route avoids $\\mathrm{GL}(2)$ spectral theory, it suggests that weight-one subconvexity can be pursued without developing the full spectral machinery; a testable next step is a hybrid aspect bound along these lines."],"forward_implications":["Weight-one holomorphic cusp forms are covered, so the result applies to $L$-functions attached to odd two-dimensional Artin representations and to class-group $L$-functions of imaginary quadratic fields.","If the conjectural mean-square lower bound $\\sum_{\\nu\\le L}\\gamma_\\nu|\\lambda_f(\\nu)|^2\\gg L^{1-\\varepsilon}$ is assumed, the same argument sharpens the exponent from $1/524$ to $1/302$.","The estimate extends to $L(1/2+it, f\\otimes g)$ with polynomial dependence on $t$ and on the archimedean parameter of $f$.","The method is expected to work for general levels and arbitrary central characters, not only prime level.","The bound does not depend on $\\theta$, so it is unaffected by the quality of the available approximation to the Ramanujan conjecture."],"supporting_citations":[{"why":"supplies the bilinear bound for Kloosterman fractions that is the base case for the final cancellation.","marker":"[DFI97]"},{"why":"provides the three-variable refinement that yields the improved exponent $1/524$.","marker":"[BC18]"},{"why":"gives the smooth delta-symbol identity used to open the convolution.","marker":"[Leu24]"},{"why":"furnishes the Voronoi summation formula and bounding tools for Fourier coefficients.","marker":"[KMV02]"},{"why":"supplies the functional equation and approximate functional equation that reduce the $L$-value to the sum $S(N)$.","marker":"[IK04]"},{"why":"is the previous moment-method benchmark for primitive central character with Maass $g$.","marker":"[HM06]"},{"why":"holds the previous best exponent $1/1413$, which Theorem 1.1 improves.","marker":"[Har11]"},{"why":"is the mean-square estimate for Hecke eigenvalues used throughout the bounding.","marker":"[Iwa92]"},{"why":"introduces the standard amplifier workaround that the paper adapts in Section 4.1.","marker":"[DFI94a]"}],"fun_headline_variants":["Subconvexity for GL(2)×GL(2) L-functions with exponent 1/524","Level-aspect subconvex bound improves to 1/524","Subconvexity via Kloosterman fractions: exponent 1/524","Uniform subconvex bound for GL(2)×GL(2) L-functions","New subconvex estimate beats Harcos-Michel exponent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the amplifier sum $A(L)=\\sum_{\\nu\\le L}\\gamma_\\nu(|\\lambda_f(\\nu)|^2-\\chi(\\nu)\\lambda_f(\\nu^2))$ to be $\\gg L^{1-\\varepsilon}$, and the paper's derivation of that lower bound reduces to a prime sum weighted by $\\chi(\\nu)^2$; for non-quadratic characters no unconditional lower bound of this strength is known, and without it the amplification lemma fails.","fun_headline_variants_meta":{"raw":{"variants":["Subconvexity for GL(2)×GL(2) L-functions with exponent 1/524","Level-aspect subconvex bound improves to 1/524","Subconvexity via Kloosterman fractions: exponent 1/524","Uniform subconvex bound for GL(2)×GL(2) L-functions","New subconvex estimate beats Harcos-Michel exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001167,"raw_usage":{"total_tokens":4827,"prompt_tokens":945,"completion_tokens":3882,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":3777}},"tokens_in":561,"tokens_out":3882,"duration_ms":25501,"temperature":1.0,"reasoning_tokens":3777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:11:58.993230+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a primitive non-quadratic character $\\chi\\pmod p$ and $L=p^{1/151}$, the prime character sum $\\sum_{L/2<\\nu\\le L}\\chi(\\nu)^2$; a single prime $p$ for which this sum is $o(L^{1-\\varepsilon})$ would invalidate equation (4.3) and with it the amplifier lemma (Lemma 4.1) on which the proof rests.","supporting_citations":[],"review_version":1}