{"id":"ce7c8314-dd66-471e-94a9-ee7d441d866f","arxiv_id":"2509.05968","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Burgess-type subconvexity bound for GL(2)xGL(2) L-functions holds uniformly in both spectral parameters when one form is dihedral or of level 1.","lead":"This paper proves new upper bounds for products of two automorphic L-functions, improving the trivial growth rate in the spectral aspect. The advance is uniformity in both frequency parameters at once, and it gives a new quantum-equidistribution result for a special family of Maass forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved level-q extension of [18, Prop 6.1] is load-bearing; without it, the C2 bound in §5.4.1 and hence Theorem 1.1 collapse.","rationale":"The paper’s strongest claim is a uniform Burgess-type subconvexity bound for GL(2)×GL(2) L-functions in the spectral aspect. The proof architecture is: Theorem 1.1 gives a second moment bound, and various L4-norm results convert the moment to an individual bound. The moment theorem itself is a long, self-contained analytic calculation, but in the mixed case t_g ≪ T^{1+ε}/H it explicitly depends on an external proposition that is only stated for level 1. The footnote “It is clear to see...” is an assertion, not a proof, and the proposition is used at a critical quantitative step: bounding C2, which controls the discrete-spectrum contribution. If the level-q1 version of [18, Prop 6.1] has different exponents or additional level factors, the bound C2 ≼ T/H could fail, and the entire second moment bound (1.2) would not follow. This would undermine all of the corollaries, since both the dihedral and level-1 Burgess bounds are derived from Theorem 1.1. I do not see an internal contradiction in the rest of the argument; the Bessel and stationary phase steps appear carefully executed, and the use of the L4 norm of g is legitimate given the existing dihedral bound (1.3). The L4 norm bound of Ki is a separate announced result and is a genuine dependency for Corollary 1.3 (1.8); but the most fundamental missing proof is the level extension, because it is needed for the main theorem itself. The reader identified exactly this point, and I agree. The paper remains a plausible substantial contribution, conditional on a complete proof of Lemma 5.1 (and the other sketched extensions).","tokens_in":32827,"tokens_out":11959,"duration_ms":123849,"concrete_test":"Write out the proof of Lemma 5.1 for squarefree q1 by following the proof of [18, Prop 6.1] (approximate functional equation, large sieve Lemma 2.1, Li’s Theorem 2), explicitly computing the conductor of L(1/2, φ×ad g) for φ∈B*(q1), g∈B*(q), and tracking all q1-dependence in the large sieve and in Li’s bound. Verify that the first inequality still reads ≪ t_g^{2+ε} M for M≤2t_g and ≪ M^{3+ε} for M≥2t_g, and the second ≪ M^{1+ε} U, with implied constants independent of M,U,t_g (q1-dependence is acceptable). If any extra power of M or t_g appears, substitute the corrected bounds into (5.42) and test whether (5.43) remains ≤ N T^{3/2+ε}+N T^{1+ε}H; if not, Theorem 1.1(1.2) and the corollaries fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.1 reproduces [18, Proposition 6.1], a second-moment bound for L(1/2, φ×ad g) and L(1/2, φ) at level q1. The original proposition is stated for q=1; the paper’s footnote asserts that “the tools there … work well in our case” but supplies no proof. This is the single most load-bearing step: in Section 5.4.1, for the case t_g ≪ T^{1+ε}/H, the bound on C2 is reduced via (5.33) and Cauchy–Schwarz to C3^{1/2}C4^{1/2}, and then Lemma 5.1 is invoked to conclude C2 ≼ T/H, leading to (5.43) and hence to the second assertion (1.2) of Theorem 1.1. Both Corollaries 1.2 and 1.3 rely on Theorem 1.1, so a failure of the level extension would invalidate the paper’s advertised Burgess bounds. The gap is not an internal contradiction; the adaptation may be routine, but it is a nontrivial spectral/large-sieve statement whose proof is entirely absent. Secondary but related unproved assertions are the extension of the Watson–Ichino formula to non-primitive ψ (footnote before Corollary 1.3) and the use of Ki’s announced L4 norm bound (1.5) for Corollary 1.3’s level-1 Burgess bound; these also need full proofs, but they are less fundamental than Lemma 5.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a second moment of GL(2)xGL(2) Rankin--Selberg L-functions L(1/2, f x g) in the spectral aspect, with a uniform treatment of both spectral parameters t_f and t_g. The main technical result, Theorem 1.1, bounds a spectral average of |L(1/2, f x g)|^2 over f in a short interval around T, in terms of the L4-norm of g and two main terms. The proof follows the Sarnak--Lau--Liu--Ye approach: Kuznetsov trace formula, Voronoi summation, uniform asymptotics of J-Bessel functions, a shifted convolution sum, spectral decomposition, and large sieve estimates. The advertised consequences are Burgess-type subconvexity bounds L(1/2, f x g) << |t_f+t_g|^{3/4+epsilon} when one of the forms is dihedral (Corollary 1.2) or when one form has level 1 (Corollary 1.3), together with a shrinking-set QUE result for dihedral Maass forms (Theorem 1.5).","tokens_in":33222,"tokens_out":3548,"duration_ms":40563,"significance":"If correct, the paper would give a genuinely uniform spectral-aspect subconvexity bound of Burgess strength for GL(2)xGL(2), a major open-type problem that has seen substantial recent progress (Nelson, Blomer--Jana--Nelson, etc.). The method is a natural and ambitious extension of Sarnak's and Lau--Liu--Ye's work, and the use of L4-norm bounds to control triple products is a plausible and interesting idea. The paper also gives a detailed treatment of the Bessel and hypergeometric phases, which is a substantial technical contribution. The promised shrinking-set QUE result for dihedral forms would also be new. However, the announced results rely on several external or only footnoted statements: the level-q extension of [18, Prop. 6.1], Ki's announced L4 bound, and an extension of the Watson--Ichino formula to non-primitive nebentypus. The proof of Corollary 1.2 is not supplied at all. These are load-bearing gaps rather than cosmetic issues.","major_comments":[{"comment":"Lemma 5.1 states a second-moment bound for L(1/2, phi x ad g) and L(1/2, phi) at level q_1, quoting [18, Proposition 6.1]. The original proposition is stated for q=1, and the paper only says, in a footnote, that 'the tools there ... work well in our case'. This extension is not proved. The bound is used in (5.42)-(5.43) to estimate C_2, leading to the new term N T^{3/2} in (1.2), and Theorem 1.1-(1.2) is then used in Corollaries 1.2 and 1.3. Since the level-q spectral large sieve and Li's result [34, Theorem 2] may require additional arguments at squarefree level with nebentypus, this is a nontrivial gap in a load-bearing step. A complete proof or a precise reduction to [18] with all level dependencies tracked is needed.","section":"§5.4.1, Lemma 5.1 (footnote on p. 29-30)"},{"comment":"Corollary 1.2 is stated as the advertised dihedral Burgess-type bound, but no proof is given. Section 7 says 'We only state a sketch proof of Corollary 1.3' and then proves (1.7) and (1.8); Corollary 1.2 is not mentioned again. The paper needs a real proof of Corollary 1.2, or an explicit statement that it follows from the proof of Corollary 1.3 by symmetry and the dihedral L4-norm results of Luo and Humphries--Khan, with the necessary conditions on D, q, chi, and psi checked. As written, one of the two central claims in the abstract is unproved.","section":"§7, after Corollary 1.2"},{"comment":"The level-1 Burgess bound (1.8) in Corollary 1.3 uses Ki's announced L4 bound ||g||_4 << t_g^epsilon, which is not proved in this manuscript. Additionally, the paragraph before Corollary 1.3 states that the assumption that psi is primitive can be dropped and that the Watson--Ichino formula holds in a general setting with level-dependence left implicit, but no precise theorem or proof is given. This matters because Theorem 1.1 is stated for primitive real psi, while Corollary 1.3 concerns trivial nebentypus of arbitrary squarefree level. The application of (1.1) to that setting requires the non-primitive version of the Watson--Ichino formula, and this extension is load-bearing for the corollary. Please supply the missing statement and proof, or state the corollary conditionally on these external results.","section":"§7 and §1 (Corollary 1.3, Ki's bound (1.5), Watson--Ichino extension)"},{"comment":"In the case t_g << T^{1+epsilon}/H, the estimate (5.42)-(5.43) for C_2 is the key point that gives the unconditional T^{3/2} bound. The passage from (5.42) to (5.43) uses Lemma 5.1 in the form 'V_0^3(t_g^2+V_0^2)'. Since V_0 can be as large as T/H, the final N T^{3/2} is obtained only for t_g << T/H. This is consistent with the statement, but the proof of (5.42) also uses the upper bound (5.33) for the triple product, whose uniformity in q_1 and t_phi is only sketched. Please verify that the implied constants are uniform in all parameters when Lemma 5.1 is applied at level q_1.","section":"§5.4.1, equations (5.42)-(5.45)"}],"minor_comments":[{"comment":"Typo: 'spectral parameterst f' needs a space. Also 'thd functional equation' in §2.1 and 'a a' in §2.3 should be corrected.","section":"§1, p. 2"},{"comment":"The notation L_{q_2}(1, sym^2 phi) and L_{q_2}(1/2, phi) is introduced only in (2.5) and is used without being explicitly recalled in Lemma 2.5. A short reminder would improve readability.","section":"§2.3, Lemma 2.5"},{"comment":"The sentence 'we may enlarge B*(q, chi) to B([q,D], chi)' should use the spectral basis notation consistently; as written it is confusing because B*(D, chi) was defined for newforms.","section":"§4, p. 15"},{"comment":"In the proof of (1.8), the choices H = T_0^{4/3+epsilon} T^{-1} and H = T^{1/3+epsilon} should be checked explicitly against the hypotheses of Theorem 1.1, in particular H <= T^{1-epsilon} and T_0 << (T H)^{3/4-epsilon}. The argument is plausible but the verification is omitted.","section":"§7, proof of Corollary 1.3"},{"comment":"The proof is only a sketch, and the role of the dihedral assumption in the equality L(1/2, phi x ad g_xi) = L(1/2, phi x psi) L(1/2, phi x g_xi^2) should be stated more explicitly. Also, the reference to Jutila--Motohashi for a subconvexity bound is marked with a footnote that itself asserts a level extension; this should be either proved or made conditional.","section":"§8, Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious analytic-number-theory contribution and the main moment proof is long and detailed. However, the advertised corollaries rest on several unproved or only footnoted ingredients. The most important is the level-q extension of [18, Prop. 6.1]; if that extension fails, the C_2 bound in §5.4.1 and the full strength of Theorem 1.1-(1.2) collapse. The complete omission of a proof of Corollary 1.2 is also a problem for the paper's central claim. These gaps are potentially fixable within the manuscript's scope, but the paper should not be accepted in its current state."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline take: this is a real paper with a genuinely new uniform second-moment theorem, but the advertised Burgess subconvexity bounds rest on an unproved level extension and an announced L4 bound. Send it to a good referee; do not accept as is.\n\nWhat is actually new: Theorem 1.1 is not a repackaging of Lau-Liu-Ye or Jutila-Motohashi. It handles both spectral parameters uniformly and puts an L4 norm of g on the right-hand side. The proof is a long, serious calculation. The treatment of the J-Bessel and hypergeometric terms in Sections 5.1-5.3 contains real work, and the author is transparent about what is borrowed and what is new. If the corollaries hold, the Burgess-type bound uniform in both spectral parameters and the shrinking QUE result with delta < 1/12 are substantial.\n\nThe soft spots are exactly where the reader and the stress-test point. Lemma 5.1 is [18, Prop 6.1] for q=1, and the footnote says it is clear the tools work for general squarefree level. That is not a proof. Lemma 5.1 is not decorative: it produces the C2 bound in the t_g << T^{1+eps}/H regime, and that bound feeds into the second assertion of Theorem 1.1 and, through (1.1), into the corollaries. If the level extension fails, the advertised results lose their support. This is a load-bearing gap, not an aesthetic one. The stress-test gets this right.\n\nTwo more gaps, in decreasing order of severity. Corollary 1.2 is promised but the proof section only sketches Corollary 1.3, and that sketch uses Ki's announced L4-norm bound [28] as if it were available. The extension of Watson-Ichino to non-primitive characters is also asserted. None of these are internal contradictions, and each may be routine, but together they make the paper conditional.\n\nI believe the main moment calculation is not circular: the target subconvexity bound never appears as an input, and the L4 bounds and large sieve estimates are external. The paper is honest about its dependencies.\n\nWho this is for: analytic number theorists working on subconvexity and moments. It deserves a serious referee, and I would want to see it in a refereed venue after revision. My recommendation: invite a revision with the level extension of Lemma 5.1 proved, a real proof of Corollary 1.2, and a status check on Ki's bound or a replacement.","headline":"Genuinely new uniform second-moment theorem with a long real proof, but the advertised Burgess bounds rest on an unproved level extension and an announced L4 bound; send it to a serious referee rather than accepting it as is.","tokens_in":33731,"tokens_out":7320,"would_cite":false,"duration_ms":74458,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F12","11F66","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a uniform spectral-aspect subconvexity bound of Burgess type, exponent 3/4, for GL(2)×GL(2) Rankin–Selberg L-functions when one form is dihedral or has level 1, and derives a quantum-ergodicity shrinking result for dihedral","keywords":["Rankin–Selberg L-functions","spectral aspect","subconvexity","Maass newforms","dihedral Maass forms","second moment","quantum unique ergodicity","L4-norm estimates"],"falsifier":"Run the claimed second-moment estimate in the regime t_g ≪ T^{1+ε}/H for a level-q family with q>1 and compare with a direct computation of the mixed moment Σ L(1/2,φ)L(1/2,φ×ad g); if that mixed moment exceeds the stated bound at some squarefree q, the proof of Theorem 1.1 fails. Separately, compute the L4-norm of a level-one Maass form at large t_g; if ||g||_4 is not t_g^ε, the Burgess bound for level-one inputs lacks its required input.","tokens_in":32708,"feed_emoji":"🔢","tokens_out":7462,"duration_ms":76272,"temperature":0.7,"pith_summary":"The paper aims to prove that the central value of a Rankin–Selberg L-function attached to two Maass forms can be bounded by a power that is uniform in both spectral parameters, not just one. Its main theorem is a second-moment estimate over a spectral window: the average size of |L(1/2,f×g)|^2 is controlled by the L4-norm of g plus an error in the window width. Combining that estimate with known L4-norm results yields a Burgess-type exponent 3/4 when one form is dihedral or has level 1, where prior bounds were either non-uniform in the two parameters or weaker. An application shrinks the known equidistribution domain for dihedral Maass forms, proving mass equidistribution on hyperbolic balls whose radius decays like a small power of the spectral parameter.","feed_headline":"Rankin–Selberg L-functions get a 3/4-exponent spectral bound","feed_subtitle":"Uniform in both spectral parameters for dihedral or level-one forms—sharper than convexity.","key_machinery":"The workhorse is a spectral second-moment identity. After the Kuznetsov trace formula and Voronoi summation, the problem reduces to a shifted convolution sum D_g(s,1,1;r)—a Dirichlet series over pairs (m,n) with n−m=r weighted by Fourier coefficients of g—whose spectral decomposition expresses it as sums over Maass forms φ of triple products ⟨|g|^2,φ⟩. The Watson–Ichino formula converts those triple products into central values of triple-product L-functions, and the L4-norm of g enters through the diagonal. A hypergeometric function F arising from the Bessel integral is handled by a Pochhammer-series expansion, and the uniform asymptotic expansion of J_{2it} controls the oscillatory phase. T","core_discovery":"The central claim is Theorem 1.1: for squarefree levels and a real primitive nebentypus, the second moment of L(1/2,f×g) over f in a spectral window T−H≤t_f≤T+H is bounded by O(T^{1+ε}(T+t_g)|T−t_g|^{1/2}||g||_4^2 + T^{1+ε}H), under conditions that keep H smaller than |T−t_g| and T+t_g ≪ (TH)^{3/4−ε}. In the special case t_g ≪ T^{1+ε}/H, the bound improves to O(T^{3/2+ε}+T^{1+ε}H). Feeding in known L4-norm bounds for dihedral forms, and an announced L4 bound for level-one forms, the paper derives uniform Burgess-type subconvexity bounds |L(1/2,f×g)| ≪ |t_f+t_g|^{3/4+ε}. It also proves that for dihedral Maass newforms, quantum unique ergodicity holds on hyperbolic balls of radius t_g^{−δ} for","pith_inferences":["If the asserted level extension of the mixed-moment bound is proved carefully, the main theorem becomes unconditional except for the L4 input; this is the first spot a referee would send the author back to.","The same second-moment framework may apply to holomorphic forms or to higher-rank Rankin–Selberg convolutions, replacing the L4-norm of g by an appropriate L^p norm.","A proof of the announced level-1 L4 bound would make the level-1 corollary unconditional and would likely improve the QUE shrinking exponent beyond 1/12.","The balance between H and L in Theorem 1.1 suggests a wider hybrid regime where the subconvexity exponent should interpolate between 3/4 and the Weyl exponent 2/3; the paper does not explore this."],"forward_implications":["Uniform Burgess-type subconvexity bounds for dihedral Maass forms: |L(1/2,f×g)| ≪ |t_f+t_g|^{3/4+ε}.","The same 3/4 exponent holds when one of f,g has level 1, conditional on the announced L4 bound.","The second-moment theorem yields a shrinking QUE result for dihedral Maass forms on hyperbolic balls of radius t_g^{−δ} for any δ<1/12.","The method converts any future improvement of ||g||_4 bounds directly into improved spectral-aspect subconvexity exponents."],"supporting_citations":[{"why":"Supplies the shifted-convolution-sum and spectral-decomposition framework, and the Weyl-type 2/3 bound that this paper refines.","marker":"[33]"},{"why":"Supplies the uniform J-Bessel expansion, the treatment of oscillatory integrals after Kuznetsov and Voronoi, and the stationary-phase lemmas used.","marker":"[2]"},{"why":"Provides the explicit Watson–Ichino formula and the dihedral L4 asymptotic, key inputs turning triple products into L-functions.","marker":"[19]"},{"why":"Provides Proposition 6.1, the mixed moment bound for L(1/2,φ)L(1/2,φ×ad g) used in the t_g ≪ T^{1+ε}/H case.","marker":"[18]"},{"why":"Announced L4 bound ||g||_4 << t_g^ε for level-one Maass forms, needed for the level-1 Burgess corollary.","marker":"[28]"},{"why":"Gives the spectral large sieve inequality used to control the C1 term.","marker":"[8]"},{"why":"Supplies the approximate functional equation reducing L-values to smoothed Dirichlet sums.","marker":"[36]"},{"why":"Originates the shifted-convolution-sum method and the first GL(2)×GL(2) subconvexity result whose framework is followed.","marker":"[47]"}],"fun_headline_variants":["3/4-exponent spectral bound for Rankin–Selberg L-functions","Uniform subconvexity: Burgess-type for GL(2)×GL(2) in spectral aspect","Dihedral Maass forms: subconvexity and shrinking QUE","L(1/2, f×g) ≪ |t_f+t_g|^{3/4+ε} uniformly in spectral parameters","Subconvexity for Rankin–Selberg L-functions: a 3/4 exponent"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that a level-one mixed moment estimate extends to every squarefree level with no loss, together with an announced L4-norm bound for level-one Maass forms; the paper states the former is 'clear' without proof and cites the latter as announced.","fun_headline_variants_meta":{"raw":{"variants":["3/4-exponent spectral bound for Rankin–Selberg L-functions","Uniform subconvexity: Burgess-type for GL(2)×GL(2) in spectral aspect","Dihedral Maass forms: subconvexity and shrinking QUE","L(1/2, f×g) ≪ |t_f+t_g|^{3/4+ε} uniformly in spectral parameters","Subconvexity for Rankin–Selberg L-functions: a 3/4 exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1342,"prompt_tokens":764,"completion_tokens":578,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":465}},"tokens_in":508,"tokens_out":578,"duration_ms":6068,"temperature":1.0,"reasoning_tokens":465,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:44:56.443185+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the claimed second-moment estimate in the regime t_g ≪ T^{1+ε}/H for a level-q family with q>1 and compare with a direct computation of the mixed moment Σ L(1/2,φ)L(1/2,φ×ad g); if that mixed moment exceeds the stated bound at some squarefree q, the proof of Theorem 1.1 fails. Separately, compute the L4-norm of a level-one Maass form at large t_g; if ||g||_4 is not t_g^ε, the Burgess bound for level-one inputs lacks its required input.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the shifted-convolution-sum and spectral-decomposition framework, and the Weyl-type 2/3 bound that this paper refines."},{"cited_title":"Blomer, S","cited_arxiv_id":null,"evidence_quote":"Supplies the uniform J-Bessel expansion, the treatment of oscillatory integrals after Kuznetsov and Voronoi, and the stationary-phase lemmas used."},{"cited_title":"Humphries and R","cited_arxiv_id":null,"evidence_quote":"Provides the explicit Watson–Ichino formula and the dihedral L4 asymptotic, key inputs turning triple products into L-functions."},{"cited_title":"Humphries and R","cited_arxiv_id":null,"evidence_quote":"Provides Proposition 6.1, the mixed moment bound for L(1/2,φ)L(1/2,φ×ad g) used in the t_g ≪ T^{1+ε}/H case."},{"cited_title":"Deshouillers, H","cited_arxiv_id":null,"evidence_quote":"Gives the spectral large sieve inequality used to control the C1 term."},{"cited_title":"Li and M","cited_arxiv_id":null,"evidence_quote":"Supplies the approximate functional equation reducing L-values to smoothed Dirichlet sums."},{"cited_title":"Sarnak, Estimates for Rankin-SelbergL-functions and quantum unique ergodicity.J","cited_arxiv_id":null,"evidence_quote":"Originates the shifted-convolution-sum method and the first GL(2)×GL(2) subconvexity result whose framework is followed."}],"review_version":1}