{"id":"4b8eb596-e1f8-4fdf-9217-437ec0edaf03","arxiv_id":"2504.18774","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The symplectic period of certain Eisenstein series on GL(2n+1) matches the L-function ratio predicted by relative Langlands duality, confirming the dual pair Sp(2n)\\GL(2n+1) and GL(n)xGL(n+1)\\GL(2n+1) in the tested cases.","lead":"This paper proves a predicted period identity for the symplectic period on GL(2n+1), a concrete case of Ben-Zvi, Sakellaridis and Venkatesh's relative Langlands duality. It also proves a matching geometric vanishing result in the function-field setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1.3 is proven only for char(F) ≠ 2: the orbit classification in §5 relies on Lemma 5.1.1, which is stated for char(F) ≠ 2, but the theorem claims all global fields including characteristic-2 function fields.","rationale":"The reader's weakest assumption identifies a concrete scope gap: the main theorem is stated for all global fields, including characteristic-2 function fields, but the orbit classification used in the proof is proved only under the explicit hypothesis char(F) ≠ 2 in Lemma 5.1.1. I checked the surrounding numerical derivation and found no competing objection of comparable weight. The zeta-constant normalization is internally consistent if Δ_H(s) is interpreted as ζ(s+2)ζ(s+4)···ζ(s+2n), which is the standard Artin-Tate L-function for Sp_{2n} and yields the stated ratio ζ*(1)ζ(3)···ζ(2n+1)/(ζ(2)···ζ(2n)). The local unramified Rankin-Selberg step is asserted with a citation rather than proved, but it is a standard computation and not the place where the argument is most exposed. The 'one checks easily' stabilizer computations in §5.2 are a verification burden, but the reader's specific concern about characteristic 2 is sharper and testable. I therefore agree with the reader's conditional verdict and recommend no change.","tokens_in":34828,"tokens_out":39062,"duration_ms":354926,"concrete_test":"Fix q ∈ {2,4} and n ∈ {2,3}. In Sage or GAP, construct Sp_{2n}(F_q) acting on Gr(n,2n+1)(F_q), enumerate orbits, and record for each orbit its type (I-IV) and stabilizer order. Compare with the complete list implied by §5.1–5.2. If the characteristic-2 orbit counts and stabilizer sizes match the char ≠ 2 pattern with the same half-rank parameter, the concern is benign; any mismatch shows the stated global-field scope is unsupported and a separate characteristic-2 orbit argument is needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical identity (Theorem 1.1.3(2)) is obtained by unfolding the Sp_{2n}-period through the double-coset decomposition P(F)\\G(F)/H(F) in Proposition 6.2.1. That decomposition is the content of Section 5, and its proof uses Lemma 5.1.1 (existence of a Darboux basis adapted to a flag) to show that the Type III/IV orbits are single orbits and to justify the representative and stabilizer lists. Lemma 5.1.1 is explicitly stated only for char(F) ≠ 2. The theorem and abstract, however, claim all global fields, including function fields of characteristic 2. In characteristic 2 the symplectic form is symmetric as well as alternating, the standard Darboux flag lemma can fail or require a different set of invariants, and even the text's invariant in §5.1 ('Witt index r' with r ∈ {0,...,[n/2]}) is really the half-rank of the restricted form: the actual Witt index of an n-dimensional subspace ranges in {ceil(n/2),...,n}. So the classification is not stated precisely enough to extend to char 2 without a separate argument. A missed orbit or a different stabilizer in char 2 would change the right side of Proposition 6.2.1 and invalidate formula (1.1.2) for such fields. The geometric appendix (Theorem 1.2.1) works with ℓ ≠ char(K), so it is also affected when char(K) = 2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the BZSV relative Langlands duality conjecture for the hyperspherical Hamiltonian variety T*(Sp_{2n}\\GL_{2n+1}). Its main numerical result, Theorem 1.1.3, asserts that Sp_{2n}-periods of cuspidal automorphic forms on GL_{2n+1} vanish, and that for Eisenstein series attached to cuspidal data (Π_n, Π_{n+1}) the normalized square period equals an explicit product of zeta values, a ratio of L-values, and a product of local normalized periods. The proof structure is: absolute convergence via Harder--Narasimhan truncation in the function-field case and Zydor truncation in the number-field case (§3); vanishing of certain Klingen-mirabolic and symplectic periods (§4); classification of Sp_{2n}-orbits on the relevant Grassmannian (§5); and unfolding of the period integral to the main orbit, followed by a local Rankin--Selberg computation (§6). An appendix by Zeyu Wang formulates and proves étale geometric analogues: cuspidal sheaves on Bun_{GL_{2n+1}} or Bun_{GL_{2n}} have zero integral over the corresponding Sp_{2n}-stack.","tokens_in":35142,"tokens_out":25146,"duration_ms":251578,"significance":"If the main theorem is correct, this is a substantial, parameter-free verification of the BZSV duality in a new hyperspherical family, with exact global constants and local normalizations. The derivation is structural rather than fitted: the local L-factor appears from an unramified Rankin--Selberg computation, the zeta constants arise from Tamagawa measures, and the BZSV prediction is used only at the end to name the L-factor L(1,Π,ρ0). The geometric appendix is a valuable independent piece of evidence. However, the proof as written does not cover characteristic 2 function fields, despite the theorem being stated without such a restriction, and one displayed zeta constant is at best ambiguous. These issues are local and repairable, which is why I recommend major revision rather than rejection.","major_comments":[{"comment":"Theorem 1.1.3 is stated for a global field F with no characteristic restriction, and the abstract advertises evidence over function fields generally. However, the orbit classification used in Proposition 6.2.1 relies on Lemma 5.1.1, which is explicitly stated only for char(F)≠2, and the proof of Proposition 6.2.1 depends on the complete list of representatives and stabilizers in §5.2. A missed orbit or a different stabilizer in characteristic 2 would change the period identity (1.1.2). The geometric appendix is also affected when the base field has characteristic 2. Please either give a characteristic-2 proof of Lemma 5.1.1 and of the orbit classification, or restrict the main theorem and the appendix to char(F)≠2 and state this restriction in the abstract and introduction. The sentence in §6 saying that the function-field case follows by a similar, easier argument should also be expanded, since the theorem claims all global fields.","section":"§5.1, Lemma 5.1.1; Theorem 1.1.3; Appendix A"},{"comment":"The zeta constant displayed in (1.1.2) is not the one obtained from the Tamagawa computation in §6.4. From Δ*_G=ζ*(1)ζ(2)⋯ζ(2n+1) and Δ*_H=ζ(2)ζ(4)⋯ζ(2n), the ratio Δ*_G/(Δ*_H)^2 simplifies to ζ*(1)ζ(3)ζ(5)⋯ζ(2n+1)/(ζ(2)ζ(4)⋯ζ(2n)), with the denominator running over even integers. As printed, \"ζ(2)⋯ζ(2n)\" is naturally read as the product over all integers 2,...,2n, which is different for n≥2. Please correct the denominator, or explicitly define the product as running over even indices, and check the identity for n=2.","section":"Theorem 1.1.3(2) and §6.4"},{"comment":"The parameter r is called the \"Witt index\" of W, but for an n-dimensional subspace W of a 2n-dimensional symplectic space the usual Witt index lies in {ceil(n/2),...,n}, never in {0,...,[n/2]}. The quantity actually used in the representatives is the half-rank of the restricted symplectic form, i.e. the number of hyperbolic pairs. The classification statements should be phrased in terms of half-rank, and the resulting ranges for types I–IV should be restated accordingly. This is more than a terminological point, because the orbit parametrization is load-bearing for Proposition 6.2.1, and the distinction is especially important in characteristic 2.","section":"§5.1"}],"minor_comments":[{"comment":"The product for L_{Sp_{2n}}(s) should be written as ζ(s+2)ζ(s+4)⋯ζ(s+2n); in the current text \"ζ(s+2)⋯ζ(s+2n)\" is ambiguous, and the same ambiguity propagates to the zeta denominator in (1.1.2).","section":"§2.2.1"},{"comment":"The displayed matrices for γhγ^{-1} in the Type I–IV stabilizer computations are difficult to parse and several blocks are misaligned in the submitted text; please typeset them with explicit block indices and sizes.","section":"§5.2"},{"comment":"After listing the subgroups Sγ·Uγ, the proof says \"Using cuspidality or Corollary 4.3.3, the integral on this normal subgroup already vanishes\" without explicitly identifying, for each orbit type, the flag data that matches Sγ·Uγ to the subgroups appearing in Corollary 4.3.3. A table or a short case-by-case verification would make the unfolding argument checkable.","section":"Proposition 6.2.1"},{"comment":"The local normalized inner product ⟨φ_v,φ_v⟩♮ is introduced only in §6.4, after Theorem 1.1.3 is stated; adding a forward reference in the theorem would help the reader.","section":"Theorem 1.1.3 and §6.4"},{"comment":"The remark makes a strong claim that a local vanishing statement in [AGR93] is incorrect. The paper's own computation supports the intended local nonvanishing, but the remark should either give a precise local Hom statement with a reference to the local BZSV conjecture or be trimmed.","section":"Remark 4.3.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nWhat you should know: this paper proves an explicit symplectic period identity for cuspidal Eisenstein series on GL(2n+1) attached to GL(n) x GL(n+1) data, matching the BZSV prediction for the dual of T*(Sp_{2n}\\GL_{2n+1}). The derivation is real: absolute convergence via Zydor truncation, vanishing via Klingen-mirabolic cuspidality, orbit unfolding, and a local Rankin-Selberg computation. The BZSV conjecture is not used to build the identity; it only names the L-functions at the end. That is a meaningful, new check of [BSV24] in a concrete family, not a framework breakthrough.\n\nThe geometric appendix by Wang adds cuspidal vanishing for Sp_{2n} periods on Bun_{2n+1} and Bun_{2n}, in the etale setting. It is terse and follows Lysenko's template. The main soft spot there is Lemma A.4.4, whose proof is omitted (\"same as A.4.2\"); an editor should ask for the details before publication.\n\nThe real problem is characteristic 2. Lemma 5.1.1, the Darboux-basis lemma that drives the orbit classification in Section 5, is explicitly stated for char(F) not 2. The theorem and abstract claim all global fields. The typing of the type III/IV orbits in Section 5.1 uses precisely that lemma, so the numerical identity (1.1.2) is not established for function fields of characteristic 2. I do not see an immediate reason the BZSV statement fails there, but the proof as written does not cover it. The fix is straightforward: restrict the main theorem to characteristic not 2, or supply a separate char-2 orbit classification. As the paper stands, the abstract's \"verify\" is stronger than the proven coverage.\n\nMinor point: the stress-test's note that the Witt index invariant is really the half-rank is fair. In char not 2 that is harmless terminology; in char 2 it is part of the missing argument.\n\nVerdict: this deserves a serious referee. The core computation is sound and the significance for BZSV is real. I would ask for the characteristic assumption to be made precise and the omitted geometric proof to be included. Who benefits: anyone working on relative Langlands, symplectic periods, or BZSV duality. I would cite it for the explicit period identity.","headline":"A concrete, largely sound numerical and geometric check of a BZSV dual pair, held back by a genuine characteristic-2 gap in the orbit classification and a slightly overstated abstract.","tokens_in":35697,"tokens_out":5000,"would_cite":true,"duration_ms":49969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","22E55","11F66","14D24"],"pacs":[],"model":"deepseek-v4-flash","headline":"A period identity verifies the predicted Langlands duality for Sp(2n)\\GL(2n+1).","keywords":["relative Langlands duality","BZSV conjecture","symplectic periods","Eisenstein series","hyperspherical varieties","Grassmannian orbit classification","Rankin-Selberg L-functions","geometric Langlands"],"falsifier":"For $n=2$ over a global field of characteristic 2, explicitly list the $\\mathrm{Sp}_4(F)$-orbits on $\\mathrm{Gr}(2,5)(F)$; if any orbit beyond the four types I--IV exists, or any listed stabilizer lacks a normal unipotent subgroup whose period integral vanishes, then Proposition 6.2.1 fails. Alternatively, compute both sides of (1.1.2) for one explicit unramified datum in that setting and check whether the equality holds.","tokens_in":34603,"feed_emoji":"🧮","tokens_out":12270,"duration_ms":107213,"temperature":0.7,"pith_summary":"This paper establishes a period identity for symplectic periods of Eisenstein series on $\\mathrm{GL}_{2n+1}$ and reads it as numerical and geometric evidence for the relative Langlands duality conjecture (BZSV duality) in the case of the hyperspherical variety $T^*(\\mathrm{Sp}_{2n}\\backslash \\mathrm{GL}_{2n+1})$. The main theorem says that for the Eisenstein representation attached to cuspidal data $(\\Pi_n,\\Pi_{n+1})$ on $\\mathrm{GL}_n\\times \\mathrm{GL}_{n+1}$, the normalized symplectic period squared equals a ratio of completed zeta values, a ratio of $L$-functions, and a product of local normalized periods, with the global $L$-factor exactly the one attached to the conjectured dual variety $T^*(\\mathrm{GL}_n\\times \\mathrm{GL}_{n+1}\\backslash \\mathrm{GL}_{2n+1})$. The paper also proves vanishing of symplectic periods for cuspidal representations and for Eisenstein series attached to other maximal parabolics, and a geometric \\'etale analogue in the appendix: the cuspidal part of the corresponding period functor vanishes. If correct, this supplies one of the few explicit, global checks of a duality conjecture that organizes many period--$L$-function relations.","feed_headline":"Symplectic periods confirm a Langlands duality prediction","feed_subtitle":"For GL(2n+1) Eisenstein series, the symplectic period equals the zeta/L-value ratio BZSV duality predicts.","key_machinery":"The machinery is the orbit decomposition of the Grassmannian $\\mathrm{Gr}(n,2n+1)(F)$ under $\\mathrm{Sp}_{2n}(F)$, which indexes the double cosets $P_{n,n+1}(F)\\backslash \\mathrm{GL}_{2n+1}(F)/\\mathrm{Sp}_{2n}(F)$. The orbits split into four types (I--IV) according to the Witt index of the subspace and the position of the line spanned by $e_{2n+1}$; for every orbit except the isotropic type-I orbit, whose stabilizer is the Siegel parabolic, the stabilizer contains a normal unipotent subgroup $S_\\gamma\\cdot U_\\gamma$ whose adelic quotient integral vanishes by cuspidality or by a Klingen-mirabolic vanishing lemma. The remaining main-orbit contribution is the intertwining period $J(\\varphi,\\lambda)$, which unfolds to a Whittaker-model integral; at unramified places that integral evaluates to the Rankin--Selberg factor $L(s+1,\\Pi_n^\\vee\\times \\Pi_{n+1})$, which supplies the $L$-function in the main identity.","core_discovery":"The central discovery is the exact formula\n$$ \\frac{|P(E(\\cdot,\\varphi))|^2}{\\langle f,f\\rangle_{\\mathrm{Pet}}} = \\frac{\\zeta^*(1)\\zeta(3)\\cdots\\zeta(2n+1)}{\\zeta(2)\\cdots\\zeta(2n)} \\cdot \\frac{L^*(1,\\Pi,\\hat\\rho_0)}{L^*(1,\\Pi,\\mathrm{Ad})^2} \\cdot \\prod_v \\frac{|P^\\natural_v(f_v)|^2}{\\langle f_v,f_v\\rangle^\\natural} $$\nfor the $\\mathrm{Sp}_{2n}$-period of the Eisenstein series attached to a cuspidal representation $\\Pi = \\Pi_n \\boxtimes \\Pi_{n+1}$. The paper interprets the $L$-ratio as the BZSV-predicted spectral factor for the dual pair $(\\mathrm{GL}_{2n+1},\\mathrm{GL}_n\\times \\mathrm{GL}_{n+1})$, and it proves that all non-main orbits in the unfolding contribute zero, leaving only the term attached to the Siegel parabolic. It further shows that symplectic periods of genuine cusp forms on $\\mathrm{GL}_{2n+1}$ vanish, and that the same vanishing holds in the \\'etale geometric setting for the cuspidal part of the functor $\\Gamma_c(\\pi_*F)$.","pith_inferences":["If the period identity extends from this single-Eisenstein family to a full spectral expansion, the same $L$-ratio should control the symplectic-period Plancherel formula for $\\mathrm{GL}_{2n+1}$, giving a spectral interpretation of the duality beyond the cuspidal case.","The characteristic-2 gap is testable in small rank: a direct enumeration of $\\mathrm{Sp}_4(F)$-orbits on $\\mathrm{Gr}(2,5)(F)$ over $\\mathbb{F}_2(t)$ would either locate a missing orbit or confirm that the Darboux-basis lemma has a characteristic-2 analogue, and either outcome sharpens the range of the theorem.","The same unfolding mechanism---one main orbit plus vanishing normal subgroups on all other orbits---should apply to other hyperspherical pairs $(G,H)$, yielding explicit period identities of the same shape for orthogonal or unitary periods."],"forward_implications":["The global identity (1.1.2) confirms, in the tested Eisenstein family, the BZSV prediction that $T^*(\\mathrm{Sp}_{2n}\\backslash \\mathrm{GL}_{2n+1})$ and $T^*(\\mathrm{GL}_n\\times \\mathrm{GL}_{n+1}\\backslash \\mathrm{GL}_{2n+1})$ are dual Hamiltonian varieties.","Symplectic periods of cuspidal automorphic forms on $\\mathrm{GL}_{2n+1}$, and of Eisenstein series attached to maximal parabolics of type other than $(n,n+1)$, vanish identically.","Because the local normalized periods are 1 at unramified places, the unramified form of the identity is a pure equality of completed zeta values and $L$-functions.","In the geometric setting, the cuspidal part of the period functor $\\Gamma_c(\\pi_*F)$ vanishes for $X=\\mathrm{GL}_{2n+1}/\\mathrm{Sp}_{2n}$ and $X=\\mathrm{GL}_{2n}/\\mathrm{Sp}_{2n}$, matching the geometric BZSV conjecture on the irreducible locus.","Together with the companion linear-period results for the dual pair, the duality is verified in both directions: the linear period computes an $L$-value, and the symplectic period computes the dual $L$-ratio."],"supporting_citations":[{"why":"Supplies the duality conjecture being tested and the predicted dual variety $T^*(\\mathrm{GL}_n\\times \\mathrm{GL}_{n+1}\\backslash \\mathrm{GL}_{2n+1})$ with its $L$-function factor.","marker":"[BSV24]"},{"why":"Establishes the linear-period side of the duality: vanishing for cuspidal representations and the identity of the regularized linear period with $L(1,\\pi)$.","marker":"[FJ93]"},{"why":"Prior treatment of the symplectic period for $\\mathrm{Sp}_{2n}\\backslash \\mathrm{GL}_{2n}$ that this paper extends to $\\mathrm{GL}_{2n+1}$.","marker":"[JR92a]"},{"why":"Provides the local Rankin--Selberg integral whose unramified evaluation gives the local $L$-factor in the main identity.","marker":"[JS81]"},{"why":"Gives the truncation theorem used to prove that the Eisenstein series is of rapid decay, hence that the symplectic period converges absolutely.","marker":"[Zyd19]"},{"why":"Supplies the constant-term formulas for pseudo-Eisenstein and Eisenstein series used in the vanishing arguments.","marker":"[MW95]"},{"why":"The source of the Darboux-basis lemma used to classify the $\\mathrm{Sp}_{2n}$-orbits on the Grassmannian.","marker":"[Shm94]"}],"fun_headline_variants":["Symplectic period equals zeta/L-ratio, verifying BZSV duality","Exact symplectic period formula confirms Langlands duality","Relative Langlands duality verified for Sp(2n)\\GL(2n+1)","BZSV duality confirmed by exact symplectic period formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the completeness of the Section 5 orbit classification: the Darboux-basis lemma used there is proved only in characteristic not 2, while the main theorem is stated for all global fields, so in characteristic 2 a missed orbit or miscomputed stabilizer would invalidate the unfolding and change the period identity.","fun_headline_variants_meta":{"raw":{"variants":["Symplectic period equals zeta/L-ratio, verifying BZSV duality","Exact symplectic period formula confirms Langlands duality","Relative Langlands duality verified for Sp(2n)\\GL(2n+1)","BZSV duality confirmed by exact symplectic period formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000522,"raw_usage":{"total_tokens":2540,"prompt_tokens":974,"completion_tokens":1566,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":1487}},"tokens_in":590,"tokens_out":1566,"duration_ms":12172,"temperature":1.0,"reasoning_tokens":1487,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:10:41.054931+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $n=2$ over a global field of characteristic 2, explicitly list the $\\mathrm{Sp}_4(F)$-orbits on $\\mathrm{Gr}(2,5)(F)$; if any orbit beyond the four types I--IV exists, or any listed stabilizer lacks a normal unipotent subgroup whose period integral vanishes, then Proposition 6.2.1 fails. Alternatively, compute both sides of (1.1.2) for one explicit unramified datum in that setting and check whether the equality holds.","supporting_citations":[],"review_version":1}