{"id":"386b361f-151e-4ec5-b022-efec1744da66","arxiv_id":"2608.12722","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The archimedean Asai L-factor for GL_n(C) and the exterior-square L-factor for GL_m(F) are finite sums of Flicker and Jacquet-Shalika local zeta integrals, respectively.","lead":"Mathematicians proved that two archimedean L-factors, the Asai and exterior-square L-factors, can be written as finite sums of standard local zeta integrals. This gives a concrete local realization of these L-factors and yields applications to distinction and to global L-function properties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The missing proof of c A^s-invariance for I_As and J_\\wedge^2 in Section 2.4 is the key gap; it is true by central scaling, so the verdict should remain CONDITIONAL pending that proof.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the unproved invariance of the integral spaces under multiplication by c A^s. I agree with the reader that this invariance is true but needs proof, and that it is essential for the functional-equation identities (2.4.2) and (2.4.4) used in the inductive reductions. I checked the rest of the argument: the L-space decomposition (Proposition 3.6) is coherent, the pole-disjointness lemmas (4.1 and 5.1) are correct, the base cases are justified by explicit computations, and the inductive inclusions in Propositions 4.6 and 5.4 follow once the invariance is supplied. The paper has substantive independent support: the proofs are parameter-free, the main theorems resolve an expectation of Beuzart-Plessis, and the global applications are straightforward consequences of the local finite-sum identities. I do not find a more serious flaw. I also considered the repeated notation 'V_n(F)/M_n(F)' in the displayed integral definitions, which appears to be a typographical issue since the stated definitions make V_n a subspace of M_n; however, the integrals are explicitly attributed to [JS90] and all later manipulations use the standard forms, so I do not treat this as a load-bearing mathematical gap. The verdict should remain CONDITIONAL: the paper is broadly sound, but the stated proof has a small gap that should be repaired before final acceptance.","tokens_in":29240,"tokens_out":36197,"duration_ms":401340,"concrete_test":"Add an explicit lemma in Section 2.4 proving the cA^s-invariance: for f=I(s,W,\\phi), show A^s f(s)=I(s,R(a^{-1})W,\\phi_a) with a=A^{1/n}I_n, and verify the analogous identities for even and odd Jacquet\\textendash Shalika integrals, using the R(\\phi)W spanning identity in the odd case. Then re-derive (2.4.2) and (2.4.4) from the functional equations; if the derivation goes through without any further unstated assumption, the gap is closed and the conditional acceptance can stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.4 asserts, without proof, that the integral spaces I_As(\\sigma) and J_{\\wedge^2}(\\sigma) are invariant under multiplication by functions c A^s (c\\in C^\\times, A>0). This assertion is load-bearing: it is exactly what converts the functional equations of Theorems 2.4.1 and 2.4.3 into the identities (2.4.2) and (2.4.4), which Propositions 4.6 and 5.4 then use to strip the \\varepsilon-factors and obtain the required inclusions. The invariance is true: for the Flicker integral, with a=A^{1/n}I_n and \\phi_a(x)=\\phi(A^{-1/n}x), one has A^s I(s,W,\\phi)=I(s,R(a^{-1})W,\\phi_a), and R(a^{-1})W\\in W(\\pi_\\sigma,\\psi_C). The Jacquet\\textendash Shalika case is analogous, with the odd-rank case handled through the spanning identity J_{\\wedge^2}(\\sigma)=\\mathrm{span}\\{J(s,R(\\phi)W)\\} noted earlier in Section 2.4. But because the assertion is unproved, the derivation of the two functional-equation identities is incomplete as written. This is a minor, fixable gap rather than a fatal error; the inductive L-space decomposition and pole-disjointness arguments otherwise check out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that archimedean Asai L-factors of irreducible generic representations of GL_n(C) are finite sums of Flicker local zeta integrals (Theorem 1.1), and that archimedean exterior-square L-factors of irreducible generic representations of GL_m(F), F=R or C, are finite sums of Jacquet–Shalika local zeta integrals (Theorem 1.2). The proofs use an L-space decomposition for Weil-group representations (Proposition 3.6), an induction on the dimension of the Langlands parameter, and technical lemmas showing that the relevant integral spaces contain the corresponding L-spaces. Applications include a characterization of GL_n(R)-distinguished representations via exceptional poles (Theorem 6.7) and finite-sum expressions for global Asai and exterior-square L-functions (Theorems 7.2.2 and 7.4.2).","tokens_in":29496,"tokens_out":19886,"duration_ms":212180,"significance":"If correct, Theorems 1.1 and 1.2 give an exact finite-sum realization of archimedean Asai and exterior-square L-factors by local zeta integrals, resolving an expectation explicitly recorded in Beuzart-Plessis's paper [BP21] and extending non-archimedean results of Matringe, Kewat–Raghunathan, and Jo. The L-space decomposition and the pole-disjointness arguments are original and well suited to the inductive strategy. The paper is transparent about its reliance on prior work (BP21, JLST25, Jac09), and the main proofs are detailed and largely coherent. The local and global applications are natural and would be valuable.","major_comments":[{"comment":"The assertion that the integral spaces I_As(σ) and J_∧^2(σ) are invariant under multiplication by functions of the form c A^s (c ∈ C^×, A>0) is stated without proof. This invariance is load-bearing: it is used to absorb the exponential factors c A^s in the functional equations of Theorems 2.4.1 and 2.4.3, and it is exactly what converts those functional equations into the identities (2.4.2) and (2.4.4). These identities are then used in Propositions 4.6 and 5.4 to strip the epsilon factors in the inductive reductions, so Theorems 1.1 and 1.2 depend on this unproved assertion. I believe the invariance is true: in the Flicker case, taking a = A^{1/n} I_n and φ_a(x) = φ(A^{-1/n} x) gives A^s I(s,W,φ) = I(s, R(a^{-1})W, φ_a), with R(a^{-1})W ∈ W(π_σ, ψ_C), and the Jacquet–Shalika case follows similarly by central scaling, with the odd-rank case reduced via the spanning identity J_∧^2(σ) = span{J(s,R(φ)W)}. However, the proof must be supplied; as written, the derivation of (2.4.2) and (2.4.4) is incomplete.","section":"Section 2.4, preceding (2.4.2) and (2.4.4)"},{"comment":"The notation V_n(F)/M_n(F) in the definition of the Jacquet–Shalika integrals is problematic. Since M_n(F) is defined as the space of all n×n matrices and V_n(F) as its subspace of upper triangular matrices, the quotient V_n/M_n is not a meaningful space (V_n is a subspace of M_n, not a quotient), and if interpreted literally the integral would ignore the X-variable entirely. The standard Jacquet–Shalika integral (as in [JS90] and as used in the global integral I^even_JS in Proposition 7.3.1, where the X-integration is over M_n(K)\\M_n(A_K)) integrates X over all of M_n(F). The same typo appears in the partial integrals J_{2r}(s,W,ϕ) and J_{2r+1}(s,W) in Section 5 and in the global Eulerian integrals in Section 7.3. Please replace V_n(F)/M_n(F) by M_n(F) (and similarly for adelic versions) throughout; with this correction the matrix calculations in Lemma 5.3 and Proposition 5.4 are consistent.","section":"Section 1 (definitions of J(s,W,ϕ) and J(s,W)), Section 5 (partial integrals), Section 7.3"}],"minor_comments":[{"comment":"The phrase 'does not vanish at s=0' for I(s,W_1,φ_1) is ambiguous when L(s,π,As) has a pole at s=0; please clarify that the integral has a nonvanishing leading Laurent coefficient at that point.","section":"Section 6, proof of Theorem 6.7"},{"comment":"The sentence 'The factor preceding the epsilon factor in Theorem 2.4.1 is of the form cA^s' is correct, but the value of A (namely A=|τ|_C^N with N=n(n−1)/2) could be stated explicitly for clarity.","section":"Section 2.4, after Theorem 2.4.1"},{"comment":"The sentence 'By Theorem 1.1 and [Jac09, Theorem 2.7], this holds for every v ∈ S_∞ as well' is terse; since split archimedean places use the Rankin–Selberg integral, a sentence explaining which result applies at split versus inert real places would help the reader.","section":"Section 7.2, proof of Theorem 7.2.2"},{"comment":"The reference [CPS94] is cited as unpublished and appears only in the acknowledgments; please confirm that no mathematical statement in the paper depends on that manuscript.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically strong and the main results are likely correct, but the proof as written has two issues that must be addressed: the unproved cA^s-invariance in Section 2.4, and the misdefined V_n/M_n quotient in the Jacquet–Shalika integral definitions. Both are fixable and do not appear to affect the overall strategy, but they make the current text incomplete. I recommend major revision with a request to add the missing invariance proof and to correct the integral notation throughout."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the main theorems deliver exactly what the title promises: finite-sum realizations of archimedean Asai and exterior-square L-factors, with the Asai case resolving the expectation Beuzart-Plessis states on p.5 of BP21. Second, the paper is essentially right but has one unproved assertion the authors should fix before I'd sign off: the claim in Section 2.4 that the integral spaces I_As(σ) and J_∧2(σ) are invariant under multiplication by cA^s. It is used to pass from functional equations to (2.4.2) and (2.4.4), and again in Propositions 4.6 and 5.4. The stress-test note says it is true by central scaling, and I agree: for the Flicker integral, scaling by λI_n gives the factor λ^{ns}, and the Whittaker and Schwartz data transform accordingly; the Jacquet–Shalika case works the same way. But the paper states it in passing, and a referee should ask them to write it out.\n\nWhat is genuinely new: the L-space decomposition (Proposition 3.6) using two overlapping subparameters. It avoids comparing Flicker and Rankin–Selberg integrals directly, and it is clean. The inductive reductions in Sections 4 and 5 are well organized, and the base cases are handled explicitly. The local distinction application (Theorem 6.7) and the global finite-sum consequences (Theorems 7.2.2 and 7.4.2) are natural and appear to follow. I checked the pole-disjointness arguments in Lemmas 4.1 and 5.1; they work. The citation pattern is fine. No circularity: the L-factors are defined via local Langlands, not via the integrals. The self-citation to [Yad24] in Theorem 6.7 is for a specific nonvanishing fact, not for the main identity.\n\nSoft spots beyond the invariance gap are minor. The arXiv text has OCR-type artifacts and the quotient notation N_n(R)/GL_n(R) should be a backslash quotient. There are also places where the paper says 'the proof is identical' or 'we omit the proof'—acceptable for a paper of this length, but the omitted proof in Theorem 6.8 is a bit more substantial than the others. None of this shakes the central argument.\n\nWho this is for: anyone working on archimedean Rankin–Selberg theory, local L-factors, or distinction. It deserves a serious referee; I would send it out. The fix is small and the main theorems are likely correct.","headline":"A solid, important paper: the archimedean finite-sum realizations are new and mostly well-proved, with one true but unproved invariance claim in Section 2.4 that needs a written proof.","tokens_in":30106,"tokens_out":6283,"would_cite":true,"duration_ms":71092,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","11F66","22E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every archimedean Asai and exterior-square $L$-factor is a finite sum of the local zeta integrals built to represent it.","keywords":["Archimedean Asai L-factor","Archimedean exterior-square L-factor","Flicker integral","Jacquet-Shalika integral","finite-sum realization","local zeta integrals","exceptional poles","distinguished representations"],"falsifier":"A direct check for a small parameter would settle the load-bearing step: take $\\sigma=\\chi_1\\oplus\\chi_2$ with ordered exponents, compute the space of Flicker integrals explicitly for one-dimensional constituents, and test whether multiplication by $A^s$ preserves the space. For the theorem itself, one can compare pole orders at a point $s_0$: if for some irreducible generic $\\pi$ the Asai $L$-factor $L(s,\\pi,\\mathrm{As})$ has a pole of order $d$ at $s_0$ while every Flicker integral $I(s,W,\\phi)$ in the family has smaller pole order there, then Theorem 1.1 would be false; the analogous residue check applies to $J(s,W,\\phi)$ and $L(s,\\pi,\\wedge^2)$.","tokens_in":28957,"feed_emoji":"∑","tokens_out":14721,"duration_ms":154402,"temperature":0.7,"pith_summary":"The paper establishes the archimedean analogue of the known non-archimedean finite-sum theorems: for every irreducible generic representation $\\pi$ of $\\mathrm{GL}_n(\\mathbb{C})$, the Asai $L$-factor $L(s,\\pi,\\mathrm{As})$ is a finite sum of Flicker local zeta integrals $I(s,W_i,\\phi_i)$, and for every archimedean local field $F$ and every irreducible generic representation of $\\mathrm{GL}_m(F)$, the exterior-square $L$-factor $L(s,\\pi,\\wedge^2)$ is a finite sum of Jacquet--Shalika local zeta integrals. Earlier archimedean theory showed that each local integral converges, continues meromorphically, satisfies a functional equation, and has poles no worse than the corresponding $L$-factor, but it did not show that the $L$-factor itself lies in the span of the integrals. The paper closes that gap by induction on dimension, cutting the Langlands parameter into two overlapping subparameters at each step. If the theorems are correct, the archimedean $L$-factors are exactly the objects those local integrals compute, which yields a local distinction criterion and expresses the global Asai and exterior-square $L$-functions as finite sums of global integrals with the resulting analytic properties.","feed_headline":"Asai and exterior-square L-factors are finite sums of local integrals","feed_subtitle":"Exact identities upgrade prior analytic-control results and power distinction and global applications.","key_machinery":"The load-bearing construction is the $L$-space $\\mathcal{L}(\\rho)$: the space of meromorphic functions of the form $L(s,\\rho)h(s)$, where $h$ is entire and the product is rapidly decreasing in vertical strips. Proposition 3.6 is the key identity: if an admissible representation $\\theta$ of the Weil group has two decompositions $\\theta=\\rho_1\\oplus\\rho_2=\\eta_1\\oplus\\eta_2$ and the pole sets of $L(1-s,\\rho_1^{\\vee})$ and $L(s,\\eta_2)$ are disjoint, then $\\mathcal{L}(\\theta)=\\mathcal{L}(\\eta_1)+\\frac{L(s,\\rho_1)}{L(1-s,\\rho_1^{\\vee})}\\mathcal{L}(\\rho_2)$. Propositions 4.6 and 5.4 translate this into integral language, showing that the span of Flicker integrals (respectively Jacquet--Shalika integrals) for a direct-sum parameter contains the span of each subparameter multiplied by the corresponding ratio of $L$-factors. Running those inclusions in both directions forces the span to contain $\\mathcal{L}(\\mathrm{As}(\\sigma))$ (respectively $\\mathcal{L}(\\wedge^2\\sigma)$), which is exactly the finite-sum realization.","core_discovery":"On its own terms the paper's discovery is Theorem 1.1 and Theorem 1.2: the archimedean Asai $L$-factor of any irreducible generic representation of $\\mathrm{GL}_n(\\mathbb{C})$ and the archimedean exterior-square $L$-factor of any irreducible generic representation of $\\mathrm{GL}_m(F)$ are equal, as meromorphic functions, to finite sums of the corresponding local zeta integrals. In the Asai case the summands are $I(s,W_i,\\phi_i)$ with $W_i$ in the Whittaker model and $\\phi_i$ Schwartz functions; in the exterior-square case they are $J(s,W_i,\\phi_i)$ for even $m$ and $J(s,W_i)$ for odd $m$. The proof associates to each Langlands parameter $\\sigma$ an $L$-space $\\mathcal{L}(\\mathrm{As}(\\sigma))$ of functions $L(s,\\mathrm{As}(\\sigma))h(s)$ with $h$ entire and rapidly decreasing in vertical strips, proves a decomposition identity for this space under two overlapping decompositions of $\\sigma$, and then shows by induction that the span of the local integrals contains the whole $L$-space. The same template proves both theorems.","pith_inferences":["Beyond the paper, the two-overlapping-subparameter strategy should apply to other functorial $L$-factors whose archimedean integral representations have functional equations but no direct reduction to Rankin--Selberg integrals.","Beyond the paper, the absence of an analogue of the $\\mathrm{GL}_n(\\mathbb{R})$-invariance statement for the Shalika subgroup is the only step separating Theorem 6.8 from an exceptional-pole classification of Shalika-distinguished representations; supplying that analogue would complete the classification.","Beyond the paper, extracting an explicit bound on the number of summands from the induction would turn the finite-sum realization into a quantitative local formula and provide a direct check for small $n$."],"forward_implications":["The normalized Flicker functionals $\\Lambda_{s,\\phi}(W)=I(s,W,\\phi)/L(s,\\pi,\\mathrm{As})$ are continuous on the Whittaker model, and for every $s_0$ some $W,\\phi$ gives a nonzero value.","An irreducible generic representation of $\\mathrm{GL}_n(\\mathbb{C})$ is $\\mathrm{GL}_n(\\mathbb{R})$-distinguished exactly when $s=0$ is an exceptional pole of level $0$ of its Asai $L$-factor.","The global Asai $L$-function is a finite sum of global Flicker integrals, hence has meromorphic continuation, is bounded in vertical strips, satisfies the standard functional equation, and is entire except for at most simple poles in the exceptional central-character case.","The global exterior-square $L$-function is a finite sum of global Jacquet--Shalika integrals, with the same analytic consequences; it is entire in odd rank and in even rank except for at most simple poles in the same exceptional case.","The normalized Jacquet--Shalika functionals are continuous and nonzero at every point, giving the archimedean starting point for exceptional-pole studies of exterior-square $L$-factors."],"supporting_citations":[{"why":"Supplies the archimedean Flicker-integral analytic theory and functional equation from which the span identity (2.4.2) and Theorem 1.1 proceed.","marker":"[BP21]"},{"why":"Supplies the archimedean analytic theory and functional equations for Jacquet--Shalika integrals used to obtain (2.4.4) and prove Theorem 1.2.","marker":"[JLST25]"},{"why":"Provides the L-space lemmas, Dixmier--Malliavin propagation, and the inductive method that the paper adapts to Asai and exterior-square integrals.","marker":"[Jac09]"},{"why":"Gives the non-archimedean finite-sum theorem for Asai L-factors that Theorem 1.1 mirrors and that the global Asai application uses at finite places.","marker":"[Mat11]"},{"why":"Defines the exterior-square local zeta integrals and supplies the unramified computation relied on in global Theorem 7.4.2.","marker":"[JS90]"},{"why":"Defines the Flicker integral representation of the Asai L-function that Theorem 1.1 realizes as a finite sum.","marker":"[Fli93]"},{"why":"Provides the archimedean local Langlands correspondence through which the L-factors in Theorems 1.1 and 1.2 are defined.","marker":"[Kna94]"},{"why":"Supplies the Rankin--Selberg local computations for split and unramified places used in the global Asai applications.","marker":"[JPSS83]"}],"fun_headline_variants":["Asai and exterior L-factors as finite sums of zeta integrals","Finite-sum realization of Archimedean L-factors","L-factors expressed as finite integral combos","Asai and ext-square L-factors: finite local zeta sums","Generic reps: Asai and exterior L-factors are finite sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper states without proof that each span of local zeta integrals is invariant under multiplication by every exponential factor of the form $cA^s$ with $c\\in\\mathbb{C}^{\\times}$ and $A>0$; this invariance is what converts the functional equations into the span identities (2.4.2) and (2.4.4), and those identities feed directly into the inductive reductions in Propositions 4.6 and 5.4. If the invariance failed, the proof's central reductions would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Asai and exterior L-factors as finite sums of zeta integrals","Finite-sum realization of Archimedean L-factors","L-factors expressed as finite integral combos","Asai and ext-square L-factors: finite local zeta sums","Generic reps: Asai and exterior L-factors are finite sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1533,"prompt_tokens":878,"completion_tokens":655,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":567}},"tokens_in":494,"tokens_out":655,"duration_ms":7173,"temperature":1.0,"reasoning_tokens":567,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:48:26.440394+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check for a small parameter would settle the load-bearing step: take $\\sigma=\\chi_1\\oplus\\chi_2$ with ordered exponents, compute the space of Flicker integrals explicitly for one-dimensional constituents, and test whether multiplication by $A^s$ preserves the space. For the theorem itself, one can compare pole orders at a point $s_0$: if for some irreducible generic $\\pi$ the Asai $L$-factor $L(s,\\pi,\\mathrm{As})$ has a pole of order $d$ at $s_0$ while every Flicker integral $I(s,W,\\phi)$ in the family has smaller pole order there, then Theorem 1.1 would be false; the analogous residue check applies to $J(s,W,\\phi)$ and $L(s,\\pi,\\wedge^2)$.","supporting_citations":[],"review_version":1}