{"id":"d878e345-22fa-414f-a274-a55e15e33500","arxiv_id":"2501.15886","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper establishes a new simultaneous level-aspect subconvexity range for self-dual GL(3)xGL(2) L-functions and a Lindelof average bound in the weight aspect.","lead":"This paper proves new average estimates for central values of GL(3)xGL(2) L-functions, including a subconvexity bound in two varying levels and a Lindelof-on-average bound in the weight aspect. It matters because such bounds are core tools in analytic number theory and feed into equidistribution and nonvanishing problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transition from (4.13) to (1.6) identifies an unweighted ℓ-sum with L(1/2,f); without an approximate functional equation or dual side, Theorem 1.4 is not established as written.","rationale":"The reader's weakest_assumption identifies the same load-bearing step. My reading confirms it: the proof of Theorem 1.4 proves (4.13), an upper bound for the ℓ-twisted moment, and then claims that a crude sum over ℓ recovers the first moment with L(1/2,f). This is not a matter of a missing epsilon or a typo; it is the central mechanism. The bound (4.13) has the correct order for each fixed ℓ (about K^{1+ε}/√ℓ), so a heuristic 'sum over ℓ and divide by √ℓ' gives the right size, but an upper bound cannot be converted into an identity. An approximate functional equation would introduce a smooth weight V(n/K) and a dual side; neither appears in the text. The displayed equation (4.6) also contains a spurious λ_f(ℓ) on the right-hand side, suggesting the Petersson diagonal and Hecke eigenvalues are being conflated. These issues do not prove the result false, but they mean Theorem 1.4 is not proven as written. Since this is exactly the gap the reader flagged, and the requested remedy is a substantial addition, the conditional verdict remains appropriate: the paper should not be accepted until this step is supplied or corrected. I do not see a reason to move to REJECT, as the gap may be repairable and the other results (Corollary 1.3, Theorem 1.5) may stand independently.","tokens_in":29605,"tokens_out":11807,"duration_ms":110550,"concrete_test":"In §4.1, replace the final ℓ-summation by the standard approximate functional equation L(1/2,f)=∑_{n}λ_f(n)/√n V(n/K)+ε(f)∑_{n}λ_f(n)/√n \\bar V(n/K), with V smooth and supported on n≪K^{1+ε}. Swap the n-sum with the sum over f and check whether each of the two resulting sums (including the dual side and the initial segment n≤K^{1-δ}) is bounded by the analogue of (4.13). If either sum cannot be controlled by (4.13) or contributes O(K), the step fails. For a concrete numerical illustration, take a level-1 newform of a large even weight k (e.g., k=1000) and compare L(1/2,f) with ∑_{K^{1-δ}≤ℓ≤K^{1+ε}} λ_f(ℓ)/√ℓ for δ=0.1; the difference should be of the same order as the L-value, not negligible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After (4.13), the proof says: 'Now, after divided by a factor √ℓ on both sides of (4.13), we sum over K^{1-δ}≪ℓ≪K^{1+ε} ... obtaining' (1.6). This step asserts that ∑_{K^{1-δ}≤ℓ≤K^{1+ε}} λ_f(ℓ)/√ℓ is L(1/2,f). For a full-level weight-k newform, L(1/2,f) is a smooth-weighted Dirichlet sum of length about k together with a dual sum from the functional equation. The unweighted partial sum over ℓ≥K^{1-δ} omits the initial segment n≪K^{1-δ} and the entire dual side. Moreover, (4.13) is an upper bound for one fixed ℓ; summing it over ℓ just forms ∑_f ω^{-1}_f L(F⊗f) (∑_ℓ λ_f(ℓ)/√ℓ), which is not the left side of (1.6). No approximate functional equation, smooth partition of unity, or supplementary argument is supplied. Since this is the only step connecting the twisted moment estimate to the Lindelöf average bound, Theorem 1.4 rests on an unproved identity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies first moments of GL(3)×GL(2) Rankin–Selberg L-functions, with the GL(3) form taken to be a self-dual symmetric square lift. The main results are: (i) a twisted first-moment estimate (Theorem 1.2) which, via the amplifier method and Lapid's nonnegativity, yields a hybrid level-aspect subconvexity bound for L(1/2, F⊗f) in a simultaneous range M^{13/64+ε} ≤ q ≤ M^{11/40−ε} (Corollary 1.3); (ii) a Lindelöf-on-average bound for the first moment of L(1/2,f)L(1/2,F⊗f) in the weight aspect (Theorem 1.4); and (iii) an asymptotic formula for the first moment of L(1/2,F⊗f) with error O(K^{-1/4+ε}) (Theorem 1.5). The paper also includes a detailed root-number computation and several auxiliary lemmas on Voronoĭ summation, bilinear forms, and Bessel functions.","tokens_in":29828,"tokens_out":10823,"duration_ms":94852,"significance":"If correct, the subconvexity result in the simultaneous level aspects would be a significant breakthrough: it appears to be the first such result for self-dual GL(3)×GL(2) L-functions, and the method overcomes the ramified-Voronoĭ obstruction by a reciprocity step. The weight-aspect results are also of interest, since a Lindelöf average bound of the type (1.6) is stronger than what the classical large sieve and Cauchy–Schwarz give. The paper ships no code, but the proof is built on standard tools (Petersson formula, Voronoĭ summation, exponential-sum bounds) and cites prior art appropriately. The claimed results are falsifiable and, if established, would be a substantial contribution.","major_comments":[{"comment":"The transition from (4.13) to the Lindelöf bound (1.6) is not justified. Equation (4.13) is an upper bound for the fixed-ℓ twisted sum Σ_f ω_f^{-1} λ_f(ℓ) L(1/2,F⊗f). After dividing by √ℓ and summing over ℓ in [K^{1−δ}, K^{1+ε}], the proof asserts that one obtains (1.6), i.e. Σ_f ω_f^{-1} L(1/2,f) L(1/2,F⊗f). However, this would require the identity Σ_{ℓ} λ_f(ℓ)/√ℓ ≈ L(1/2,f) for each f (or on average). No approximate functional equation, smooth partition of unity, or dual-side argument is supplied. The short interval [K^{1−δ}, K^{1+ε}] omits the range n ≪ K^{1−δ} and completely ignores the dual sum from the functional equation of L(s,f). Moreover, summing an upper bound for the twisted moment does not produce an upper bound for the untwisted first moment; the order of the ℓ-sum and the f-sum cannot be interchanged in the asserted way. As written, Theorem 1.4 does not follow from the given proof.","section":"§4.1, Eqs. (4.13) and (1.6)"},{"comment":"The diagonal contribution from the Petersson trace formula appears to be miscomputed. In (4.4)–(4.5), the sum over f is Σ_f ω_f^{-1} λ_f(ℓ) λ_f(n). After applying the Petersson formula, the diagonal in (n,ℓ) should give δ(n,ℓ) (or an appropriate normalization of it), so that the n-sum collapses to a single term A_F(ℓ,1)/√ℓ V*(ℓ/Y) multiplied by the weight sum over k—with no additional factor λ_f(ℓ). Yet (4.6) retains λ_f(ℓ). This suggests either a different normalization of the Hecke operators (which is not stated) or an error in the handling of the trace formula. If the factor is spurious, the claimed asymptotic (1.6) is not established even if the ℓ-sum step were valid.","section":"§4.1, Eq. (4.6)"},{"comment":"The proof of the asymptotic formula (1.7) relies on the claim that the contribution (4.12) is O(Y^{-100}) when ℓ=1. Earlier in §4.1 the same quantity is only estimated by O(Y^{1/6+ε}) (with no indication of a saving in the ℓ=1 special case). The transition from a generic O(Y^{1/6+ε}) bound to a negligible bound for ℓ=1 is not justified. A separate analysis of the phase and the range of C for ℓ=1 is needed before Theorem 1.5 can be accepted.","section":"§4.2, Theorem 1.5"}],"minor_comments":[{"comment":"The Langlands parameters are written with three occurrences of the same symbol α1; the second and third should be α2 and α3 (the displayed formula reads α1 = −ν1−2ν2+1, α1 = −ν1+ν2, α1 = 2ν1+ν1+1, which is inconsistent). The third expression also appears to contain a typo (ν1 instead of ν2).","section":"§2.4, Eq. (2.4)"},{"comment":"There are numerous typos and infelicities: 'completly', 'quannity', 'theses', 'bar ely', 'Hek e', 'Maaß' vs 'Maass', and inconsistent spelling of 'Voronoĭ'. The notation {1,ℓ1ℓ2,ℓ1^2ℓ2^2} is not defined in a single place; the meaning of ℓ as a set element should be clarified early.","section":"General presentation"},{"comment":"The expansion of the amplifier A_f in (3.7) is not fully explained; in particular, the term x(ℓ^2)^2 1_{ℓ1=ℓ2} appears without a defining comment. A reader needs to infer the combinatorics of the square of the sum over primes in P_L.","section":"§3.1, around Eq. (3.7)"},{"comment":"The root-number formula ε(F⊗f) = −i^k λ_f(M)/√M ∏_{p|N} ε(Π_{F,p})^2 is stated after a local computation that gives ε(Π_{F,p}⊗π_{f,p}) = -λ_f(p)√p for p|M, but the passage from the local factors to the global product is not spelled out. It is presumably standard, but a brief explanation would help.","section":"§2.3, Eq. (2.6)"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a potentially significant subconvexity result, but the weight-aspect Lindelöf bound (Theorem 1.4) is not proved as written; the ℓ-sum step is a genuine gap, not a mere presentation issue. The diagonal-term issue in (4.6) compounds the concern. I would recommend a careful revision that either supplies the missing approximate-functional-equation argument or removes Theorem 1.4 from the main results, and that fixes the diagonal computation. The subconvexity part (Theorem 1.2 and Corollary 1.3) may survive the revision, and I would encourage the author to state its proof more clearly as a standalone result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing you should know: this is not a routine repackaging. The level-aspect result for self-dual GL(3) x GL(2) with q in [M^(13/64+eps), M^(11/40-eps)] looks new, and the reciprocity maneuver to bypass the collusion problem in the GL(3) Voronoi step is the real contribution. The root-number discussion in Section 2.3 is also useful and could be of independent value. For Corollary 1.3, I did not see a circular fit or an invented input; the optimizer is standard, if intricate. But I read the proof of Theorem 1.4 with a red flag. After (4.13), the text divides by sqrt(ell) and sums over K^(1-delta) << ell << K^(1+eps), saying this gives (1.6). That is exactly where L(1/2,f) is supposed to appear. What has been estimated is the twisted first moment sum_f omega_f^{-1} lambda_f(ell) L(F tensor f). Summing it against 1/sqrt(ell) produces sum_f omega_f^{-1} L(F tensor f) times sum_ell lambda_f(ell)/sqrt(ell), and that bracketed factor is a bare Dirichlet polynomial over a short interval. It is not L(1/2,f) unless one supplies an approximate functional equation with a smooth test function and accounts for the dual side. No such equation is given. This is not a cosmetic omission; it is the only bridge from the twisted moment to the Lindelof average bound. As written, Theorem 1.4 is not established. Elsewhere, the paper is compressed but recognizable. The 'minus' root-number case is dismissed as analogous; that is probably true, but it should be written down. The statement of Theorem 1.4 mentions the level of f, while the proof treats full-level newforms only. There are also many typos and some inconsistent references, for example 'Theorem 1.6' where Corollary 1.6 is meant. These are minor relative to the (4.13)-to-(1.6) gap. If the author can supply a genuine approximate-functional-equation argument for Theorem 1.4, the paper becomes a strong contribution. Without it, the weight-aspect Lindelof average should be treated as unproved, although Corollary 1.3 and Theorem 1.5 may still be salvageable. This deserves a serious referee: send it out, but require the missing step to be repaired and the sign cases expanded.","headline":"This paper has a serious and genuinely interesting level-aspect subconvexity package, but Theorem 1.4 as written rests on an unproved approximate-functional-equation step; it deserves refereeing, not desk rejection.","tokens_in":807,"tokens_out":846,"would_cite":false,"duration_ms":70248,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F66","11L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A first-moment method yields simultaneous level-aspect subconvexity for self-dual GL(3)×GL(2) L-functions, plus weight-aspect Lindelöf averages.","keywords":["automorphic forms","first moment","Rankin-Selberg L-functions","subconvexity","GL(3)×GL(2) L-functions","level aspect","weight aspect","Voronoi summation"],"falsifier":"Compute, for a fixed newform f of weight k ~ K, the difference between ∑_{$K^{{1−δ}}$ ≤ ℓ ≤ $K^{{1+ε}}$} λ_f(ℓ)/√ℓ and the main term of the standard approximate functional equation for L(1/2,f); if the ratio is not uniformly bounded below by a positive constant, the ℓ-summation step in the proof of Theorem 1.4 is invalid.","tokens_in":29356,"feed_emoji":"🧮","tokens_out":9086,"duration_ms":73694,"temperature":0.7,"pith_summary":"This paper proves new estimates for central values of automorphic L-functions attached to a self-dual GL(3) form (a symmetric-square lift) and a GL(2) newform. The main level-aspect result is a subconvex bound for L(1/2,F⊗f) that saves a power of the conductor simultaneously in both level parameters, in the range $M^{{13/64+ε}}$ ≤ q ≤ $M^{{11/40−ε}}$, which the paper states is the first such instance for self-dual GL(3)×GL(2) L-functions. In the weight aspect, it establishes a Lindelöf-on-average bound for the first moment of the degree-8 product L(1/2,f)L(1/2,F⊗f) of size $K^{{1+ε}}$, and an asymptotic formula for the first moment of L(1/2,F⊗f) with power-saving error O($K^{{−1/4+ε}}$). If correct, these results give the expected non-vanishing of these central values and improve on what the spectral large sieve alone provides.","feed_headline":"First simultaneous level-aspect subconvexity for GL(3)×GL(2)","feed_subtitle":"First-moment estimates with Voronoĭ reciprocity break the convex barrier in both levels and give weight-aspect Lindelöf averages.","key_machinery":"The engine is a harmonic first-moment average: a Petersson trace formula with weights $ω_f^{{-1}}$ turns sums over f into a diagonal term plus averages of Kloosterman sums. On the off-diagonal, a GL(3) Voronoĭ summation formula in the level aspect (Lemma 2.1) is applied; the proof switches between its unramified case (c,M)=1 and its ramified case N|c by a reciprocity law taken from reference [6], which lets the argument use Voronoĭ twice even when the additive twist shares a factor with the level $q^{2}$. The resulting exponential sums are bounded by bilinear-form estimates for Kloosterman sums (Lemmas 2.6–2.7) and by a square-root cancellation estimate for sums of GL(3) Fourier coefficients against √n-phases (Lemma 2.3). In the weight aspect, the first-moment average is fused with an average over even weights k ~ K; the Bessel function J_{k−1} is summed over k using Lemmas 2.8–2.9, and the exact 2π in the exponential phase (Remark 2.10) makes the phase α = 2√ℓ/c rational, which is what allows Lemma 2.3 to apply.","core_discovery":"The paper's central claim is that the first moment of the central value L(1/2,F⊗f), where F is the symmetric-square lift of a GL(2) newform of square-free level q and f runs over holomorphic newforms of prime level M, can be estimated with enough uniformity to yield a simultaneous level-aspect subconvex bound: for $M^{{13/64+ε}}$ ≤ q ≤ $M^{{11/40−ε}}$ and M > q^δ, one has L(1/2,F⊗f) ≪ $X^{{1/2−η}}$ for an explicit power saving η, with X ≍ $M^{{3/2}}$$q^{2}$ the conductor size; the paper states this is the first such simultaneous level-aspect subconvexity for self-dual GL(3)×GL(2) L-functions. In the weight aspect, it proves a Lindelöf-on-average bound of size $K^{{1+ε}}$ for the first moment of the degree-8 product L(1/2,f)L(1/2,F⊗f) over even weights K ≤ k ≤ 2K, and an asymptotic formula L(1,F)K/4 \\hat{W}(0) + O($K^{{−1/4+ε}}$) for the first moment of L(1/2,F⊗f). A corollary is that L(1/2,F⊗f) ≠ 0 for some newform f of sufficiently large weight.","pith_inferences":["If the short-ℓ recovery step in §4.1 is made explicit with a partition of unity, the same framework should yield a subconvex bound in the weight aspect for L(1/2,f)L(1/2,F⊗f), since the Lindelöf average already saves a full power of K relative to the large sieve.","The reciprocity–Voronoĭ switching mechanism is a general recipe for level-aspect subconvexity: whenever the additive twist shares a prime with the level, a one-dimensional reciprocity law can restore the unramified case, so similar hybrid bounds may hold for other GL(3)×GL(2) families, including non-self-dual F.","The sensitivity to the 2π phase in the Bessel average suggests the weight-aspect result is arithmetic rather than analytic: only this normalization makes the exponential phase rational, so the same proof would not work for a differently scaled Bessel average."],"forward_implications":["Corollary 1.3 gives a power saving over the convexity bound for L(1/2,F⊗f) simultaneously in both level aspects in the range M^{13/64+ε} ≤ q ≤ M^{11/40−ε}, the first such result for self-dual GL(3)×GL(2) L-functions.","Theorem 1.4 provides a Lindelöf-on-average bound for the first moment of the degree-8 L-function L(1/2,f)L(1/2,F⊗f), beating the O(K^{2+ε}) bound that the spectral large sieve alone supplies.","Theorem 1.5 gives an asymptotic formula with power-saving error O(K^{−1/4+ε}) for the first moment of L(1/2,F⊗f).","Corollary 1.6 states that for any self-dual GL(3) Maaß form F there exists a newform f of sufficiently large weight with L(1/2,F⊗f) ≠ 0.","The paper indicates the method extends in principle to square-free level M with additional bookkeeping, and identifies the reciprocity–Voronoĭ switch as the way the level-aspect collusion deadlock is overcome."],"supporting_citations":[{"why":"Supplies both the unramified and ramified GL(3) Voronoĭ summation formulae used in Lemma 2.1.","marker":"[50]"},{"why":"Supplies the reciprocity law and the amplifier construction used in the level-aspect argument.","marker":"[6]"},{"why":"Pioneered the first-moment method for these L-functions that the paper follows.","marker":"[35]"},{"why":"Provides the general collusion case that the paper avoids via the reciprocity switch.","marker":"[9]"},{"why":"Gives the bilinear-form estimates with Kloosterman sums used as Lemmas 2.6 and 2.7.","marker":"[26]"},{"why":"Supplies the square-root cancellation estimate for nonlinear exponential sums used as Lemma 2.3.","marker":"[45]"},{"why":"Provides the Petersson trace formula variants and the Bessel-function averages in Lemmas 2.8 and 2.9.","marker":"[24]"},{"why":"Guarantees the central values are nonnegative, turning the moment bound into the subconvex bound in Corollary 1.3.","marker":"[33]"}],"fun_headline_variants":["First simultaneous level-aspect subconvexity for GL(3)×GL(2)","First subconvexity for GL(3)×GL(2) in both level aspects","First moments yield subconvexity for GL(3)×GL(2)","First simultaneous subconvexity for GL(3)×GL(2) L-functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weight-aspect proof relies on summing over the short window $K^{{1−δ}}$ ≤ ℓ ≤ $K^{{1+ε}}$ to recover the L(1/2,f) factor for each newform, but it supplies no explicit partition of unity or approximate functional equation showing this short window reproduces every coefficient; if that recovery fails at the level of individual weights, the Lindelöf bound in Theorem 1.4 does not follow.","fun_headline_variants_meta":{"raw":{"variants":["First simultaneous level-aspect subconvexity for GL(3)×GL(2)","First subconvexity for GL(3)×GL(2) in both level aspects","First moments yield subconvexity for GL(3)×GL(2)","First simultaneous subconvexity for GL(3)×GL(2) L-functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001652,"raw_usage":{"total_tokens":6702,"prompt_tokens":1229,"completion_tokens":5473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":845,"completion_tokens_details":{"reasoning_tokens":5382}},"tokens_in":845,"tokens_out":5473,"duration_ms":37632,"temperature":1.0,"reasoning_tokens":5382,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T13:50:37.795014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a fixed newform f of weight k ~ K, the difference between ∑_{$K^{{1−δ}}$ ≤ ℓ ≤ $K^{{1+ε}}$} λ_f(ℓ)/√ℓ and the main term of the standard approximate functional equation for L(1/2,f); if the ratio is not uniformly bounded below by a positive constant, the ℓ-summation step in the proof of Theorem 1.4 is invalid.","supporting_citations":[{"cited_title":"The Voronoi formula on GL(3) with ramification","cited_arxiv_id":"1806.10786","evidence_quote":"Supplies both the unramified and ramified GL(3) Voronoĭ summation formulae used in Lemma 2.1."},{"cited_title":"Blomer and R","cited_arxiv_id":null,"evidence_quote":"Supplies the reciprocity law and the amplifier construction used in the level-aspect argument."},{"cited_title":"Li, Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions, Ann","cited_arxiv_id":null,"evidence_quote":"Pioneered the first-moment method for these L-functions that the paper follows."},{"cited_title":"Corbett, Vorono ˘ ı summation forGLn: collusion between level and modulus , Amer","cited_arxiv_id":null,"evidence_quote":"Provides the general collusion case that the paper avoids via the reciprocity switch."},{"cited_title":"Bounds on bilinear forms with Kloosterman sums","cited_arxiv_id":"2204.05038","evidence_quote":"Gives the bilinear-form estimates with Kloosterman sums used as Lemmas 2.6 and 2.7."},{"cited_title":"Ren and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the square-root cancellation estimate for nonlinear exponential sums used as Lemma 2.3."},{"cited_title":"Iwaniec, W","cited_arxiv_id":null,"evidence_quote":"Provides the Petersson trace formula variants and the Bessel-function averages in Lemmas 2.8 and 2.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Guarantees the central values are nonnegative, turning the moment bound into the subconvex bound in Corollary 1.3."}],"review_version":1}