{"id":"341f9a59-dcb2-491d-9922-d05ec79821bf","arxiv_id":"2507.15279","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Co-period integrals of GL(2) forms with two cubic exceptional theta series are locally one-dimensional, and an explicit unramified identity matches them to the central symmetric cube L-factor ratio.","lead":"This paper studies co-period integrals on the cubic metaplectic cover of GL(2), proving that the relevant local Hom-space is one-dimensional and relating the integrals to the central symmetric cube L-function. It proposes an Ichino-Ikeda type conjecture and shows the existence of cuspidal forms with nonvanishing central symmetric cube L-values.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unramified L-value identity rests on an unproved adaptation of Hsieh's decomposition (Props. 2.22–2.23); if that adaptation fails, Theorem 2.24 and the Eisenstein verification collapse.","rationale":"The reader's weakest-assumption analysis already identifies the unproved adaptation of [9, Proposition 5.2] in Propositions 2.22, 2.23, and 2.27 as the place where the argument is least secure. I agree with that assessment. The paper contains substantial independent material — the restriction computations in Section 2.3, the local multiplicity-one arguments in Propositions 2.14–2.17 and Theorem 2.20, and the explicit Eisenstein computation in Section 3 — so the manuscript is not obviously wrong. But the unramified calculation and the global Ichino-Ikeda verification both depend on the factorization identity with a precise constant. A one-sentence 'adapting the argument' is not a proof for a covering group: the Kubota cocycle changes the model pairings, the quotient Z(c)N\\G is not the usual triple-product quotient, and the truncation and support arguments in [9] may not transfer verbatim. This is a missing proof of a key technical lemma rather than a demonstrated contradiction, so it does not warrant rejection. It does warrant making acceptance conditional on a complete proof of the decomposition formulas, which is exactly the reader's verdict. I also note the second reader concern, the Burger-Sarnak-type step in Theorem 1.3 for the diagonal covering-group embedding, is similarly asserted with a 'carries over verbatim' footnote; it is important for the nonvanishing application but is downstream of the local multiplicity-one result and less central to the L-value identity. The concrete test above is designed to settle the factorization question directly.","tokens_in":37354,"tokens_out":4753,"duration_ms":58020,"concrete_test":"Independently derive Proposition 2.23 from first principles for the nonarchimedean case: choose σ = an induced principal series and σ± = exceptional representations, write both sides explicitly in the Whittaker and induced models, and evaluate them on a small set of depth-zero non-spherical test vectors whose supports are contained in a sufficiently small open subgroup. Verify that the equality I = ζ_F(1) Ψ Ψ∨ holds with exactly the stated constant and that Ψ_t vanishes unless t ∈ eT'. If the constant or the support set differs, recompute the spherical ratio in Theorem 2.24 and compare it with L(1/2, σ, Sym3)/L(1, σ, ad); the check settles whether the central L-value formula is correct as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central unramified identity in Theorem 2.24 and the global Eisenstein verification in Theorem 3.6 both pass through the factorization identities in Proposition 2.23 (and the archimedean analogue in Proposition 2.27). The proof of Proposition 2.23 is one sentence: 'Adapting the argument in [9, Proposition 5.2]' — and Proposition 2.22 has the same one-line justification. The argument being adapted is a reductive GL(2) triple-product decomposition in Hsieh's Hida-family setting; the present setting is the cubic metaplectic cover, with pairings realized in Whittaker and induced models and integration over Z(c)N\\G. The covering group introduces cocycle and model-specific normalizations not present in [9]. In particular, the asserted constant ζ_F(1), the summation over fZ*\\eT (or its truncation to fT*\\eT'), and the support restrictions Ψ_t = 0 unless t ∈ eT' all require independent verification. Without this, the identity I = ζ_F(1) Ψ Ψ∨ is an assumption, and the equality for spherical vectors, I = L(1/2, σ, Sym3)/L(1, σ, ad), is not established. Since the global conjecture and its Eisenstein check rely on this local factorization, this is the load-bearing joint of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies co-period integrals attached to an automorphic form on GL(2) and two exceptional theta series on the cubic Kazhdan-Patterson metaplectic cover of GL(2). In the local aspect, it proves multiplicity one for the relevant Hom-space under a caveat (v not dividing 3, or sigma_v non-supercuspidal), constructs a distinguished local trilinear period as an integral of matrix coefficients, and computes this integral for unramified spherical data, obtaining the ratio L(1/2, sigma_v, Sym^3)/L(1, sigma_v, ad). In the global aspect, it proposes an Ichino-Ikeda type conjecture for cuspidal forms, proves an Euler-type decomposition for Eisenstein series, verifies the conjectural formula for Eisenstein data in a range of parameters, and derives the existence of cuspidal automorphic representations with prescribed supercuspidal local components and nonvanishing central symmetric cube L-values.","tokens_in":37719,"tokens_out":4985,"duration_ms":56305,"significance":"If correct, the paper would be the first systematic local and global treatment of co-periods for the cubic metaplectic cover, with an explicit unramified local identity that links the metaplectic co-period to the Langlands-Shahidi symmetric cube L-factor. The detailed restriction calculations for exceptional representations in Section 2.3, the explicit computation of spherical Whittaker functions, and the Eisenstein verification are substantial and go substantially beyond a mere conjecture. The paper also explicitly names its main conditional steps, which is helpful. However, the central local decomposition formulas are not proved in the text; they are asserted as adaptations of a result in the reductive GL(2) setting, and the unramified identity and the global Eisenstein check depend directly on them. The global existence theorem similarly relies on an unproved extension of Prasad's Burger-Sarnak lemma to a covering-group pair. These points are load-bearing, and the manuscript needs a major revision to close them.","major_comments":[{"comment":"The proofs of Propositions 2.22 and 2.23 consist of the single sentence 'Adapting the argument in [9, Proposition 5.2]' (pp. 21-22). This is the load-bearing step for Theorems 2.21 and 2.24: the asserted identity I = ζ_F(1) ∑_t Ψ_t Ψ^∨_t (and its analogue I = ζ_F(1) Ψ Ψ^∨) involves the covering group, the summation over fZ*\\eT', the support restriction Ψ_t = 0 unless t ∈ eT', the constant ζ_F(1), and the specific pairings in Whittaker and induced models. None of these features is present in the reductive GL(2) triple-product setting of [9], so the adaptation is not routine. Since the unramified calculation and the Eisenstein verification both pass through this identity, the central claim of the paper is conditional on this decomposition. The authors should either prove it directly or give a detailed verification of the adaptation.","section":"§2.5, Propositions 2.22 and 2.23"},{"comment":"The archimedean analogue is treated either by the same one-line 'adapting [9, Proposition 5.2]' or by a deformation argument whose steps are only sketched. In particular, the claim that the ratio I/(ΨΨ^∨) is holomorphic in a small strip and constant because it equals ζ_F(1) when χδ^s is unitary requires control over possible zeros and poles of ΨΨ^∨; the sentence 'Choose test vectors' does not establish this. Since Theorem 3.6 uses Proposition 2.27 at archimedean places, the global Eisenstein verification inherits this gap.","section":"§2.6, Proposition 2.27"},{"comment":"The proof of Theorem 1.3 invokes Prasad's Burger-Sarnak principle [21, Lemma 1] for the pair (∆µ3\\(eG(A)×eG(A)), GL2(A)), with only the footnote that the proof carries over verbatim. This is not a proof: the cited lemma is stated for reductive groups, and the metaplectic cover and the diagonal embedding change the relevant invariant functionals, period integrals, and the relation between global nonvanishing and local Hom-spaces. Because Theorem 1.3 is one of the main advertised applications (nonvanishing of central symmetric cube L-values with prescribed supercuspidal components), the lemma must be proved or a reference containing the metaplectic case must be supplied.","section":"§1, Theorem 1.3"},{"comment":"There is a discrepancy between the statement and the proof. The theorem statement displays factors involving χ_{+,1}χ_{-,2}(ϖ^3), while the computation in the proof (p. 23) uses χ_{+,1}χ_{-,1}(ϖ^3) in the same positions and the final dichotomy is stated for χ_{+,1}χ_{-,1}(ϖ^3) = |ϖ| or −|ϖ|. If the indices in the theorem are not a typo, the derivation does not prove the displayed formula; if they are a typo, it should be corrected. This point matters because the theorem is the local manifestation of the symmetric cube L-value ratio.","section":"Theorem 2.24"},{"comment":"The abstract and introduction state that the Hom-space 'is always of one dimension', but Theorem 1.1 is proved only when v ∤ 3 or σ_v is non-supercuspidal, and the text says the supercuspidal case at v | 3 is still believed but not proved. The advertised claim should be qualified to match the theorem; otherwise the abstract overstates the result.","section":"Abstract and Theorem 1.1"}],"minor_comments":[{"comment":"There is a typo: 'Kutota' should be 'Kubota' in the description of the Kubota 2-cocycle.","section":"§2.1"},{"comment":"In the discussion of the regularized period, 'the digonal G(A)' should be 'the diagonal G(A)'.","section":"§3.2"},{"comment":"In reference [3], the page span '413-339' appears to be a typo; it should likely be corrected to the actual page range of the paper.","section":"References"},{"comment":"The additive characters ψ± used in the Whittaker-Fourier expansion of φ± are introduced only informally; for clarity, specify their relation to the fixed additive character ψ and to the exceptional characters χ±.","section":"§3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains substantial and interesting work, and the main structure is plausible, but the Euler decomposition formulas (Propositions 2.22, 2.23, 2.27) are not actually proved and are load-bearing for both the local unramified identity and the global Eisenstein check. The same holds for the adaptation of Prasad's lemma used in Theorem 1.3. These are fixable issues if the authors can supply the missing arguments, but they are not merely presentational. The abstract and theorem statements should also be made consistent with the proved cases."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper with a load-bearing gap. The genuinely new content is the local multiplicity-one theorem for the cubic metaplectic cover of GL(2) (Theorem 1.1/2.20) and the explicit unramified calculation (Theorem 2.24) relating the co-period integral to L(1/2, sigma, Sym3)/L(1, sigma, ad). The global Ichino-Ikeda-type conjecture and the Eisenstein verification are natural follow-ups, and the nonvanishing consequence for symmetric cube L-values with prescribed local components is a real result. If the local computations stand, this is the first systematic co-period theory for n=3, r=2, and that matters.\n\nThe soft spot is exactly where the stress-test puts it. Propositions 2.22, 2.23, and 2.27 are the hinges: they decompose the co-period integral into a sum of products of local trilinear forms. The unramified computation of Theorem 2.24 and the global Eisenstein check in Theorem 3.6 both pass through these identities. But the proofs of 2.22 and 2.23 are one-sentence references to Hsieh's Proposition 5.2, which is a reductive GL(2) triple-product decomposition in a Hida-family setting. The present setting involves the cubic cover, with Whittaker and induced models and a different integration domain. The adaptation may be true, but it is not demonstrated. This is not a cosmetic omission; if the adaptation fails, the equality I = zeta_F(1) Psi Psi^vee falls apart, and with it the L-value identity.\n\nThere are also two smaller issues. The abstract says the Hom-space is always one-dimensional, but Theorem 1.1 explicitly leaves open the supercuspidal case at places dividing 3; that overstatement should be fixed. Theorem 1.3 uses a Burger-Sarnak-type principle for the covering group, citing Prasad's lemma for reductive groups and saying the proof carries over verbatim. That may be fine, but it is another unproved extension that needs spelling out.\n\nThe rest of the paper is careful. The Mackey-theoretic restriction analysis to K and J is worked out in detail and looks sound. The Langlands-Shahidi L-factors are defined independently of the co-period fitting, so there is no circularity. The citation pattern is reasonable.\n\nBottom line: the paper deserves a serious referee, but the referee should be instructed to demand full proofs of the decomposition formulas, not accept 'adapting [9, Prop 5.2]' as a substitute. I would not cite the unramified identity in my own work until that gap is filled, but the local multiplicity theorem is likely to be citable on its own.","headline":"Strong new local results on co-periods for the cubic metaplectic cover, but the unramified L-value identity rests on an unproved adaptation of Hsieh's decomposition.","tokens_in":38192,"tokens_out":3395,"would_cite":false,"duration_ms":35080,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F70","22E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that co-periods on the cubic metaplectic cover of GL(2) control the central value of the symmetric cube L-function, with local multiplicity one and an exact unramified formula.","keywords":["co-period","symmetric cube L-function","Kazhdan-Patterson metaplectic cover","exceptional theta series","multiplicity one","Ichino-Ikeda conjecture","Eisenstein series","unramified calculation"],"falsifier":"Take an unramified principal series over a nonarchimedean local field with $v \\nmid 3$ and with the exceptional characters satisfying $\\chi_{+,1}\\chi_{-,1}(\\varpi^3) = -|\\varpi|$, compute the matrix-coefficient integral $I$ directly for spherical vectors, and compare it with the asserted second branch $L(1,\\sigma_v,\\mathrm{Sym}^3)/(L(1/2,\\sigma_v,\\mathrm{Sym}^3)L(1,\\sigma_v,\\mathrm{ad}))$; a mismatch would falsify Theorem 2.24 and the conjecture it supports.","tokens_in":37145,"feed_emoji":"🔢","tokens_out":9709,"duration_ms":89534,"temperature":0.7,"pith_summary":"The paper studies the co-period integral of an automorphic form on $\\mathrm{GL}(2)$ against two exceptional $\\theta$ series on the cubic Kazhdan-Patterson metaplectic cover of $\\mathrm{GL}(2)$. Its aim is to establish that these co-periods are the period-theoretic control of the central value of the symmetric cube $L$-function $L(1/2, \\sigma, \\mathrm{Sym}^3)$, in analogy with how triple-product periods control triple-product $L$-values. Locally it proves that the space of invariant trilinear forms is one-dimensional, and at unramified places it computes the local integral exactly as $L(1/2, \\sigma_v, \\mathrm{Sym}^3)/L(1, \\sigma_v, \\mathrm{ad})$. Globally it gives an Euler decomposition for Eisenstein co-periods and proposes an Ichino-Ikeda type conjecture for cuspidal forms, together with a first verification in the Eisenstein case. If correct, the paper supplies a new way to detect nonvanishing of central symmetric cube $L$-values and yields many cuspidal representations with prescribed local components and nonvanishing $L(1/2, \\sigma, \\mathrm{Sym}^3)$.","feed_headline":"Co-periods pin down symmetric cube L-values","feed_subtitle":"Local multiplicity one plus an unramified computation tie the cubic metaplectic period to L(1/2, Sym3).","key_machinery":"The carrying object is the cubic Kazhdan-Patterson metaplectic cover $\\widetilde{G}$ of $\\mathrm{GL}_2$, a central extension by the third roots of unity built from a Kubota 2-cocycle. On it live the exceptional representations $\\sigma_{\\pm} = V_0(\\chi_{\\pm})$, realized as residues of Eisenstein series, whose Whittaker models and spherical Whittaker functions are known explicitly. The proof passes through three linked mechanisms: the restriction theory of these representations to maximal compact-modulo-center subgroups, which produces the local multiplicity one; the Euler decomposition of the trilinear matrix-coefficient integral into a product of local co-period integrals, adapted from the analogous decomposition in the triple-product setting; and the substitution of the explicit spherical Whittaker function, which converts the unramified local integral into the ratio $L(1/2,\\sigma_v,\\mathrm{Sym}^3)/L(1,\\sigma_v,\\mathrm{ad})$.","core_discovery":"The central discovery is that for the cubic cover, the local co-period functional is one-dimensional: at each place $v$ with $v \\nmid 3$ or $\\sigma_v$ non-supercuspidal, $\\dim \\mathrm{Hom}_{G(F_v)}(\\sigma_v \\otimes \\sigma_{+,v} \\otimes \\sigma_{-,v}, \\mathbb{C}) = 1$. For unramified data, the integration of matrix coefficients equals $L(1/2,\\sigma_v,\\mathrm{Sym}^3)/L(1,\\sigma_v,\\mathrm{ad})$ up to a fixed normalization of the measure. Globally, the paper decomposes the regularized Eisenstein co-period into local factors and conjectures an Ichino-Ikeda type formula for cuspidal $\\sigma$: the product of the co-period and its dual equals a rational constant times $L(1/2,\\sigma,\\mathrm{Sym}^3)/L(1,\\sigma,\\mathrm{ad})$ times the product of normalized local integrals. The Eisenstein case verifies this formula in the range $-1/6 < \\lambda(\\theta) < 1/6$.","pith_inferences":["The same restriction and decomposition techniques should extend to higher covers of $\\mathrm{GL}_2$ and to $\\mathrm{GL}_r$, where analogous co-periods would govern higher symmetric-power $L$-values; the paper already notes the method applies to general $n$-th covers.","The conjecture leaves the rational constant $C$ undetermined; the Eisenstein computation is a natural place to fix $C$ explicitly, which would turn the conjectural period relation into a sharp arithmetic identity.","The missing supercuspidal case at places dividing 3 is the next test of the method; the paper's upper bound reduces it to a Mackey analysis of the exceptional representations restricted to the pro-$p$ Iwahori subgroup.","The relation between nonvanishing of co-periods and central symmetric cube values suggests that metaplectic periods fit into the general conjectural description of period integrals for covering-group Hamiltonian spaces, rather than being isolated examples."],"forward_implications":["At every local place away from 3, and for all non-supercuspidal components, the co-period functional is the unique invariant trilinear form, so there is no hidden multiplicity when passing from local to global periods.","At unramified places, nonvanishing of the normalized local integral is equivalent to nonvanishing of the local central symmetric cube factor, so the local ratio directly detects whether $L(1/2,\\sigma_v,\\mathrm{Sym}^3) \\neq 0$.","The Eisenstein verification in the range $-1/6 < \\lambda(\\theta) < 1/6$ gives the first complete instance of the conjectured Euler-product formula, including the exact rational constant.","The local multiplicity one theorem implies that any prescribed finite set of supercuspidal local components away from 3 occurs in some cuspidal automorphic representation $\\sigma$ with $L(1/2,\\sigma,\\mathrm{Sym}^3) \\neq 0$, and hence there are infinitely many such $\\sigma$.","If Conjecture 1.5 is true, the co-period $P$ is nonzero exactly when the central symmetric cube value is nonzero, giving a period-integral criterion for nonvanishing of $L(1/2,\\sigma,\\mathrm{Sym}^3)$ for all cuspidal $\\sigma$."],"supporting_citations":[{"why":"Supplies the Kubota cover construction, the exceptional representations $V_0(\\chi)$, their Jacquet modules and Whittaker models, and the explicit spherical Whittaker function used in the unramified evaluation.","marker":"[13]"},{"why":"Its Proposition 5.2 is the Euler-decomposition argument adapted here to express the trilinear matrix-coefficient integral as a product of local co-period integrals.","marker":"[9]"},{"why":"Establishes that $L(s,\\sigma,\\mathrm{Sym}^3)$ is entire and provides the bound $|\\lambda(\\sigma)|<1/6$ used in the convergence of local integrals and in the Eisenstein range.","marker":"[14]"},{"why":"Proves that nonvanishing of the global co-period implies $L(1/2,\\sigma,\\mathrm{Sym}^3)\\neq 0$, the direction used in Theorem 1.3 and in Conjecture 1.5.","marker":"[7]"},{"why":"Provides the Burger-Sarnak type lemma, applied here to the diagonal embedding of the cover, that lifts local supercuspidal components to a global cuspidal representation.","marker":"[21]"},{"why":"Serves as the model for Conjecture 1.5 and supplies the archimedean gauge estimates used in Proposition 2.26.","marker":"[10]"},{"why":"Supplies the mixed truncation operator used to regularize the divergent Eisenstein co-period integral in Theorem 3.3.","marker":"[33]"},{"why":"Supplies the higher-extension and Schneider-Stuhler duality tools used to prove local multiplicity one in the non-supercuspidal case.","marker":"[22]"}],"fun_headline_variants":["Co-periods on cubic cover pin down Sym3 L-values","Cubic metaplectic co-periods yield exact Sym3 values","Local multiplicity one proves Sym3 L-value formula","From co-periods to central symmetric cube L-values","Unramified co-period matches Sym3 L-factor"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that two results proved for reductive groups carry over verbatim to the covering group: the Euler decomposition of the trilinear period into local integrals (used for the unramified computation and the Eisenstein formula) and the Burger-Sarnak type lemma used to lift local components to global cuspidal forms; if either transfer fails, the corresponding conclusion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Co-periods on cubic cover pin down Sym3 L-values","Cubic metaplectic co-periods yield exact Sym3 values","Local multiplicity one proves Sym3 L-value formula","From co-periods to central symmetric cube L-values","Unramified co-period matches Sym3 L-factor"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000578,"raw_usage":{"total_tokens":2708,"prompt_tokens":908,"completion_tokens":1800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":1720}},"tokens_in":524,"tokens_out":1800,"duration_ms":13268,"temperature":1.0,"reasoning_tokens":1720,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:35:34.059251+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an unramified principal series over a nonarchimedean local field with $v \\nmid 3$ and with the exceptional characters satisfying $\\chi_{+,1}\\chi_{-,1}(\\varpi^3) = -|\\varpi|$, compute the matrix-coefficient integral $I$ directly for spherical vectors, and compare it with the asserted second branch $L(1,\\sigma_v,\\mathrm{Sym}^3)/(L(1/2,\\sigma_v,\\mathrm{Sym}^3)L(1,\\sigma_v,\\mathrm{ad}))$; a mismatch would falsify Theorem 2.24 and the conjecture it supports.","supporting_citations":[{"cited_title":"Kazhdan and S.J","cited_arxiv_id":null,"evidence_quote":"Supplies the Kubota cover construction, the exceptional representations $V_0(\\chi)$, their Jacquet modules and Whittaker models, and the explicit spherical Whittaker function used in the unramified evaluation."},{"cited_title":"Hsieh,Hida families andp-adic triple productL-functions","cited_arxiv_id":null,"evidence_quote":"Its Proposition 5.2 is the Euler-decomposition argument adapted here to express the trilinear matrix-coefficient integral as a product of local co-period integrals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that $L(s,\\sigma,\\mathrm{Sym}^3)$ is entire and provides the bound $|\\lambda(\\sigma)|<1/6$ used in the convergence of local integrals and in the Eisenstein range."},{"cited_title":"Ginzburg, D","cited_arxiv_id":null,"evidence_quote":"Proves that nonvanishing of the global co-period implies $L(1/2,\\sigma,\\mathrm{Sym}^3)\\neq 0$, the direction used in Theorem 1.3 and in Conjecture 1.5."},{"cited_title":"Prasad,Relating invariant linear form and local epsilon factors via global methods","cited_arxiv_id":null,"evidence_quote":"Provides the Burger-Sarnak type lemma, applied here to the diagonal embedding of the cover, that lifts local supercuspidal components to a global cuspidal representation."},{"cited_title":"Ichino,Trilinear forms and the central values of triple product L-functions","cited_arxiv_id":null,"evidence_quote":"Serves as the model for Conjecture 1.5 and supplies the archimedean gauge estimates used in Proposition 2.26."},{"cited_title":"Yamana, Periods of automorphic forms: the trilinear case","cited_arxiv_id":null,"evidence_quote":"Supplies the mixed truncation operator used to regularize the divergent Eisenstein co-period integral in Theorem 3.3."},{"cited_title":"Prasad, Ext-analogues of branching laws","cited_arxiv_id":null,"evidence_quote":"Supplies the higher-extension and Schneider-Stuhler duality tools used to prove local multiplicity one in the non-supercuspidal case."}],"review_version":1}