{"id":"b1101e85-429b-4acb-b963-cf29beb1fd82","arxiv_id":"2411.12609","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinitely many GL(3) cusp forms have both central L-values L(1/2, pi x chi) and L(1/2, pi x sigma) non-zero simultaneously.","lead":"The paper proves that for any fixed Dirichlet character and any fixed cuspidal GL(2) automorphic form, there are infinitely many GL(3) cusp forms for which both the GL(3) x GL(1) and GL(3) x GL(2) L-functions are non-zero at the central point. It uses Jacquet's relative trace formula to express a weighted average of such products as a dominant main term plus two terms that vanish at large level.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (3.9) restricts the spectral expansion to I(N)-invariant vectors, although I(N) contains GL3(O_v0) and the supercuspidal component π_v0 has no such fixed vectors; the spectral side is therefore empty.","rationale":"The reader's weakest assumption was close but missed a stronger obstruction. The issue is not only that non-invariant vectors might contribute; under the literal definition of I(N), there are no invariant vectors at all, because a supercuspidal local component has no GL3(O_v0)-fixed vector. This is load-bearing because (3.9) and (3.10) are the only bridge from the geometric main term JSmall to the GL3 spectral average; the later estimates (7.1) and (7.2) inherit the break. The borrowed bound (7.3) from [MRY23, §13] is also a gap, but it is secondary to the empty spectral expansion. The theorem itself may well be true and the gap may be fixable by replacing K_v0 with a suitable open subgroup and proving the corresponding orthogonality statement. As written, however, the central proof is incomplete, so I keep the verdict conditional rather than accepting the paper in its present form.","tokens_in":23048,"tokens_out":14328,"duration_ms":154192,"concrete_test":"Compute the local dimension dim π_v0^{GL3(O_v0)} in the Kirillov model of a depth-zero supercuspidal representation of PGL3(F_v0). Standard local representation theory gives zero; if confirmed, (3.9) is vacuous and the RTF identity cannot hold as stated. If the authors respond by redefining I(N) to use K'_v0[1] at v0, the decisive check becomes: evaluate the omitted cross terms ∑_{φ∈(V^{I(N)})^⊥} P1(ρ(f_v0)φ,χ_v0) P2(φ,φσ) for a local supercuspidal model and verify they vanish by Schur orthogonality; a nonzero value would invalidate (3.10).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equation (3.9) is the interface between the geometric side and the spectral averages, so every later identity (3.10), (7.1), and (7.2) depends on it. As written it is not merely unproved; it is inconsistent. I(N) is defined in (3.1) as I_N[1]·∏_{v<∞, v∤N} K_v, and since v0∤N it contains K_v0 = GL3(O_v0). For a supercuspidal representation π_v0 of PGL3(F_v0), the space of GL3(O_v0)-fixed vectors is zero. Hence the orthonormal basis B_π^{I(N)} in (3.9) is empty for every π with local component π_v0, and the purported spectral identity gives zero, while Proposition 4.1 gives JSmall(f,φ,0) ≫ N_F(N)^3. If the authors intended a smaller open subgroup at v0, as suggested by the K'_v0[1] appearing in Lemma 3.1, they still need to prove that basis vectors outside that invariant subspace do not contribute to J(f,φ,s); right convolution with a local matrix coefficient does not automatically kill all non-invariant isotypic components, and the period P2(φ,φσ) can be nonzero on such components. Without this, the connection between JSmall and the restricted spectral sum is broken.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: for a unitary Hecke character χ of GL(1) and a unitary cuspidal automorphic representation σ of GL(2)/F, there are infinitely many unitary cuspidal automorphic representations π of PGL(3)/F such that the central values L(1/2, π×χ) and L(1/2, π×σ) are simultaneously nonzero. The proof uses Jacquet's relative trace formula with a test function that is a supercuspidal matrix coefficient at a fixed finite split place v0, an Iwahori-level factor at a large prime ideal N, and local factors matching the ramification of χ and σ. The geometric side is split into a small-cell main term, which is shown to grow like N_F(N)^3, and two regular orbital integrals, which are shown to vanish for sufficiently large N_F(N). Comparing with the spectral side yields a lower bound for a sum of products of period integrals, from which the existence of infinitely many π is inferred.","tokens_in":23337,"tokens_out":11849,"duration_ms":121802,"significance":"If the proof is correct, the result is a valuable and relatively clean application of the relative trace formula to simultaneous nonvanishing of central L-values in two families attached to the same GL(3) representation. The test function is explicit and has no free parameters, and the main term is obtained as a product of local factors rather than by fitting constants. The paper also states its scope carefully, noting where positivity is lacking and where the method would become harder. However, the current draft contains a load-bearing gap in the spectral expansion and one essential estimate is imported from the authors' earlier work rather than proved here; these issues need to be fixed before the argument is complete.","major_comments":[{"comment":"The spectral expansion (3.9) is restricted to an orthonormal basis B^{I(N)}_π of I(N)-invariant vectors, where I(N) is defined in (3.1) to contain GL3(O_{v0}) at the supercuspidal place v0. Since π_{v0} is supercuspidal, every cuspidal π occurring in the spectral side has V^{GL3(O_{v0})}=0, so B^{I(N)}_π is empty. The displayed identity (3.9) would then make the spectral side (3.10) identically zero, which contradicts the lower bound JSmall(f,φ,0) ≫ N_F(N)^3 from Proposition 4.1 via (3.12). The sentence preceding (3.9) explains why only the cuspidal spectrum appears, but it does not justify restricting to I(N)-invariant vectors: such a restriction requires the test function to be bi-I(N)-invariant, whereas the local factor f_{v0} is a matrix coefficient that is only bi-invariant under the smaller group K'^◦_{v0}[1] appearing in Lemma 3.1. Please replace I(N) by the correct compact open subgroup at v0 and prove that the non-invariant vectors do not contribute to the periods, or otherwise justify (3.9).","section":"§3, Eqs. (3.1), (3.9), (3.10)"},{"comment":"The bound (7.3), asserting that the sum over the K_fin-invariant basis is O(1), is stated with the justification 'similarly to the calculation in [MRY23, §13]' and is not proved in this paper. This estimate is load-bearing: it is exactly what forces the I(N)-level contribution to be the only part that can grow like N_F(N)^3. Since [MRY23] treats the different group U(2,1)×U(1,1), the transfer of that calculation to the present GL(3)×GL(2) setting is not automatic. The authors should either include the necessary calculation or state the precise external theorem and verify its hypotheses, especially after the subgroup in (3.9) is corrected.","section":"§7, Eq. (7.3)"},{"comment":"The step from the lower bound (7.2) to the existence of a single π with L(1/2,π×χ)L(1/2,π×σ)≠0 is too terse. One needs to justify that the number of terms in the sum is O(N_F(N)^3) (or otherwise dominated by the growth rate) so that a lower bound on the absolute value of the sum implies a lower bound on some individual term, and one needs to spell out that a nonzero value of the projected period P1(π(f)φ,χ) implies the corresponding Rankin-Selberg L-value is nonzero. These points are standard in the RTF literature, but they are part of the logical chain and should be stated explicitly.","section":"§7, final deduction"}],"minor_comments":[{"comment":"The sentence 'Therefore, (4.2) follows from (4.2), (4.3), (4.4) and (4.6)' should refer to (4.1) as the conclusion, not (4.2).","section":"§4, end of proof of Proposition 4.1"},{"comment":"The compact open subgroup I(N) is defined twice: once in (3.1) as I_N[1]∏_{v<∞, v∤N} K_v and once in (3.9) as ∏_{v<∞} K_v[e_v(N)]. These agree only if e_v(N) is interpreted as the valuation of N, but the notation should be unified and the equality made explicit.","section":"§3, notation"},{"comment":"There is a wording error: 'One may then asked for how such π satisfy (2.1)' should be 'One may then ask how many such π satisfy (2.1)' or similar.","section":"§2, Remark 2.2"},{"comment":"The embedding of K'^◦_{v0}[1] into GL3(F_{v0}) via g ↦ diag(g,1) should be stated explicitly, since the invariance properties of W_{v0} and f_{v0} with respect to this embedded subgroup are used implicitly in the proof of (3.3).","section":"§3, Lemma 3.1 and (3.2)"}],"recommendation":"major_revision","confidential_remarks":"The core idea of the paper is plausible and the geometric-side computation appears to give the desired main term. The main blocker is the spectral expansion at (3.9): as written, the invariant subspace is empty for every supercuspidal component, so the displayed identity cannot be correct. This is likely repairable by using a smaller compact open subgroup at v0, but the correction will propagate to the local orbital integral analysis and to the borrowed bound (7.3). I would ask the authors to revise the manuscript with a precise definition of the level subgroup, a proof of the restricted spectral expansion, and a self-contained or clearly transferred proof of (7.3)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this about arXiv:2411.12609: it proves a new simultaneous nonvanishing result for GL(3) x GL(1) and GL(3) x GL(2) L-functions, and the main geometric analysis is done carefully. But there is a load-bearing mistake in the spectral expansion as written: equation (3.9) restricts the sum to I(N)-invariant vectors, and I(N) contains GL3(O_v0) at the supercuspidal place v0. A supercuspidal representation of PGL3(F_v0) has no GL3(O_v0)-fixed vectors, so that basis is empty and the identity gives zero on the spectral side, contradicting the nonzero main term from Proposition 4.1. This is not just an omitted proof; it is inconsistent with the definitions. The likely fix is to use a smaller open compact subgroup at v0, one that actually fixes the Whittaker vector W_v0, and then prove the spectral expansion restricts to vectors invariant under that subgroup. The authors seem aware of the relevant subgroup, since K'^o_v0[1] appears in Lemma 3.1, but they never make the choice explicit.\n\nWhat is genuinely new: for fixed Hecke character chi and cuspidal automorphic sigma on GL(2), there are infinitely many cuspidal pi on PGL(3) with L(1/2, pi x chi) and L(1/2, pi x sigma) simultaneously nonzero. The earlier U(2,1) work of the same authors is a different group, and Yang's GL(n+1) x GL(n) paper does not give this combination. The local orbital integral analysis, especially the small cell main term and the support arguments for the regular orbital integrals, is substantial and appears correct in outline. The paper ships no code or machine-checked proofs, but the trace formula derivation is explicit enough to be checked place by place.\n\nOther soft spots are minor by comparison: some local integrals are dismissed as 'straightforward calculation', and the bound (7.3) is imported from the companion paper MRY23 rather than proved here. The final step only needs infinitely many pi, not a quantitative count, so the absence of a power saving is not a flaw. The introduction is clear, and the authors honestly state what is new and what remains open.\n\nOverall: the intended argument is coherent and the flaw looks repairable, but the printed proof is not rigorous at a central point. A good referee will need to ask for a corrected definition of the level subgroup and a proof that the restricted spectral sum accounts for all contributions. This paper deserves serious refereeing, not desk rejection, and I would want to see a revised version before trusting the theorem. Worth bringing to reading group if you want a concrete example of how a small open-compact subtlety can break a trace formula identity.","headline":"A genuinely new nonvanishing theorem for GL(3), with a real but fixable gap in the spectral-side expansion that a referee should flag.","tokens_in":835,"tokens_out":798,"would_cite":true,"duration_ms":77624,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F66","11F70","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any fixed Hecke character and GL(2) cusp form with coprime conductors, infinitely many GL(3) cusp forms have both central L-values nonzero.","keywords":["relative trace formula","simultaneous nonvanishing","central L-values","GL(3) automorphic forms","Rankin-Selberg L-functions","Iwahori level","supercuspidal representations","orbital integrals"],"falsifier":"A concrete check would be to fix a coprime pair $(\\chi,\\sigma)$, implement the paper's test function, and compute the level-$N$ spectral average in (7.2); the proof predicts growth of order $N_F(N)^3$, so observing growth like $N_F(N)^{3-\\varepsilon}$ rather than $N_F(N)^3$ would break the main-term estimate.","tokens_in":22857,"feed_emoji":"🔢","tokens_out":18005,"duration_ms":154386,"temperature":0.7,"pith_summary":"Over a number field $F$, the paper proves that for any fixed unitary Hecke character $\\chi$ and any fixed unitary cuspidal automorphic representation $\\sigma$ of $\\mathrm{GL}(2)/F$ with coprime conductors, there are infinitely many unitary cuspidal automorphic representations $\\pi$ of $\\mathrm{PGL}(3)/F$ for which both central values $L(1/2,\\pi\\times\\chi)$ and $L(1/2,\\pi\\times\\sigma)$ are nonzero. The proof uses the relative trace formula with a test function engineered so that the geometric side splits into one main term and two subsidiary orbital integrals, and the subsidiary integrals vanish once the level prime $N$ has sufficiently large norm. The main term is shown to grow like the cube of the norm of $N$, so the average of the product of the two central L-values over the level-$N$ cuspidal spectrum is forced to be large. The interest is that this gives simultaneous nonvanishing in two different L-function families attached to the same $\\mathrm{GL}(3)$ form, without relying on positivity of the individual periods.","feed_headline":"Infinitely many GL(3) forms have both central L-values nonzero at once","feed_subtitle":"The relative trace formula isolates a growing main term, so the level-N average of both central L-values scales like N^3","key_machinery":"The central object is the relative trace formula identity specialized to the two period integrals attached to the $\\mathrm{GL}(1)$ and $\\mathrm{GL}(2)$ subgroups, which by Rankin-Selberg theory represent the two central L-values. The test function is assembled place by place: a supercuspidal matrix coefficient at a fixed split place kills the continuous spectrum, Iwahori-level components at a large prime $N$ set the level, and the ramified components match the conductors of $\\chi$ and $\\sigma$. The Bruhat decomposition of $\\mathrm{GL}(3)$ splits the geometric side into a small-cell term and two regular orbital-integral terms; the proofs of Sections 5 and 6 show the regular terms vanish for large level, while the small-cell term factors into local integrals that grow like $N_F(N)^3$. This factorization is what converts a lower bound on a single geometric integral into a lower bound on the spectral average of L-values.","core_discovery":"The paper establishes Theorem A: given a number field $F$, a unitary Hecke character $\\chi$ of $F^\\times\\backslash\\mathbb{A}_F^\\times$, and a unitary cuspidal automorphic representation $\\sigma$ of $\\mathrm{GL}(2)/F$ with relatively prime conductors, infinitely many unitary cuspidal automorphic representations $\\pi$ of $\\mathrm{PGL}(3)/F$ satisfy $L(1/2,\\pi\\times\\chi)L(1/2,\\pi\\times\\sigma)\\neq 0$. The proof constructs a global test function $f$ with a fixed supercuspidal component at a split place $v_0$, an Iwahori-level component at a large prime ideal $N$, and components at the ramified places of $\\chi$ and $\\sigma$. In the relative trace formula, the geometric side decomposes by Bruhat cells into the small-cell term $J_{\\mathrm{Small}}$ and two regular orbital integrals $J_{\\mathrm{Reg}}^{\\mathrm{I}}$ and $J_{\\mathrm{Reg}}^{\\mathrm{II}}$; the latter two vanish for large $N_F(N)$, while $J_{\\mathrm{Small}}(f,\\varphi,(0,0))$ is bounded below by a positive constant times $N_F(N)^3$. The spectral side therefore has an average of products of the two period integrals, hence of the two central L-values, of size at least $N_F(N)^3$, which forces infinitely many individual products to be nonzero.","pith_inferences":["Beyond the paper: the same Bruhat-cell split suggests an analogous simultaneous-nonvanishing statement for $\\mathrm{GL}(n+1)\\times\\mathrm{GL}(1)$ and $\\mathrm{GL}(n+1)\\times\\mathrm{GL}(n)$, with the geometric side expected to be harder as the rank grows.","Beyond the paper: because the argument does not use positivity of the individual periods, a quantitative strengthening would require extra bounds on individual period integrals rather than following from the averaged identity alone.","Beyond the paper: the fixed supercuspidal component at $v_0$ is a simplifying device, so a natural test is whether the result persists when the level condition is imposed without that component, at the cost of handling the continuous spectrum."],"forward_implications":["For any fixed coprime pair $(\\chi,\\sigma)$, the average of $L(1/2,\\pi\\times\\chi)L(1/2,\\pi\\times\\sigma)$ over the level-$N$ $\\mathrm{GL}(3)$ cuspidal spectrum is at least proportional to $N_F(N)^3$, so the product cannot vanish for all but finitely many $\\pi$.","The type I and type II regular orbital integrals vanish identically for large level, so the small-cell term alone carries the arithmetic information in the relative trace formula.","Every nonvanishing $\\pi$ produced has a fixed supercuspidal component at $v_0$, an Iwahori-invariant vector at the large prime $N$, and is unramified at all other finite places, so the result holds within a sparse, explicitly described family.","Replacing $\\sigma$ by its contragredient and invoking the functional equation converts the spectral identity into the exact form of Theorem A, with $L(1/2,\\pi\\times\\sigma)$ rather than $L(1/2,\\tilde{\\pi}\\times\\sigma)$."],"supporting_citations":[{"why":"It supplies the integral representation that identifies the GL(3) × GL(2) period with the central L-value L(1/2, π × σ).","marker":"[JPSS83]"},{"why":"It supplies the integral representation that identifies the GL(3) × GL(1) period with the central L-value L(1/2, π × χ).","marker":"[JPSS79]"},{"why":"It is the template for the relative trace formula argument, including the bound on the unramified-level contribution used to pass from the averaged identity to an individual nonvanishing π.","marker":"[MRY23]"}],"fun_headline_variants":["Infinitely many GL(3) forms pass both L-value nonvanishing tests","Trace formula shows infinite simultaneous nonvanishing for GL(3)","GL(3) cusp forms with two nonzero central L-values: infinitely many","Relative trace formula forces both L-factors nonzero for infinite GL(3)","Double nonvanishing: infinitely many GL(3) representations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the kernel's spectral expansion can be restricted to an orthonormal basis of forms fixed by the chosen level subgroup, with all other forms contributing nothing; if that restricted expansion is not exact, the geometric main term no longer equals the L-value average and the conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many GL(3) forms pass both L-value nonvanishing tests","Trace formula shows infinite simultaneous nonvanishing for GL(3)","GL(3) cusp forms with two nonzero central L-values: infinitely many","Relative trace formula forces both L-factors nonzero for infinite GL(3)","Double nonvanishing: infinitely many GL(3) representations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00047,"raw_usage":{"total_tokens":2373,"prompt_tokens":1012,"completion_tokens":1361,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":1275}},"tokens_in":628,"tokens_out":1361,"duration_ms":9977,"temperature":1.0,"reasoning_tokens":1275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:23:37.691775+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to fix a coprime pair $(\\chi,\\sigma)$, implement the paper's test function, and compute the level-$N$ spectral average in (7.2); the proof predicts growth of order $N_F(N)^3$, so observing growth like $N_F(N)^{3-\\varepsilon}$ rather than $N_F(N)^3$ would break the main-term estimate.","supporting_citations":[],"review_version":1}