{"id":"8d6812d4-4a11-438b-9321-95bbadbbf7c4","arxiv_id":"2506.24089","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For GL2, cuspidal functions on the ordinary Caraiani-Scholze Igusa variety form a completed Kirillov model of classical cusp forms, and their coinvariants recover Hida-ordinary forms and yield a weak local-global compatibility theorem.","lead":"The paper identifies functions on an ordinary Igusa variety for GL2 with a completed Kirillov model of classical cusp forms, and shows Hida-ordinary p-adic modular forms are the coinvariants of this space. It then proves a weak local-global compatibility result for eigenspaces and conjectures a general framework for PEL Shimura varieties.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central argument is internally coherent, and its main external dependencies (Hida finiteness, Breuil-Emerton) are invoked in ways that match their stated hypotheses.","rationale":"The reader's weakest assumption correctly identifies Hida finiteness as the main external pillar, and I agree that this is the least self-contained step in the proof of the upper bound in Theorem B. However, the paper's application of [15, Theorem 2.2] is explicit and standard: Proposition 6.4.1 reduces the admissibility of the coinvariants to the finite-rank statement for the ordinary Hecke algebra, and the twisted identification via Lemma 4.4.1 does not appear to change the relevant ordinary algebra in a way that would invalidate the rank bound. The split-reducible lower bound depends on Breuil-Emerton [1, Theorem 1.1.3], but the hypotheses (ordinary principal series, split reducible Galois representation, weight k >= 2) match the theorem's setting, and the theta-lift construction in Section 7.4 is coherent with the Kirillov model. I also checked the potentially delicate equality W_pi/S = W_pi,tildemu in Section 7.3: because continuous functions on Qp to Cp are locally constant, any element of S vanishes in a neighborhood of 0 and can be written as h*f with h vanishing at 0, so I0 W_pi is saturated inside W_pi; the step is justified. The main residual concerns are the in-preparation citation [12] for the p-adic Fourier isomorphism and the uncomputed exponent M in Theorem B. These are genuine limitations of presentation and precision, but neither contradicts the existence statement of Theorem B nor the proof strategy. A self-contained proof of Proposition 3.2.1 would remove the only unverifiable dependency, and a direct check of the twisted Hida finiteness application would further de-risk Proposition 6.4.1. Since I found no internal inconsistency or mis-cited theorem, the reader's conditional-accept verdict should remain unchanged.","tokens_in":34775,"tokens_out":34291,"duration_ms":425748,"concrete_test":"Re-derive Proposition 3.2.1 and the restriction/extension compatibility used in Lemma 3.5.4 from the explicit Hopf algebra of mu_{p^infinity} via the Mahler basis; if the resulting C(Qp,Zp)-action differs from the one used in Section 3.5, the coinvariant formalism in Theorem B would need re-examination. Separately, verify that the identification eV_Mant = C(Zp^*, eV_Katz') in Proposition 6.4.1 remains compatible with Hida's [15, Theorem 2.2] after the |det| twist of Lemma 4.4.1, e.g. by checking that the twist only changes the actions at primes away from p and does not alter the finite rank of the ordinary Hecke algebra over Zp[[Zp^*]][1/p].","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the argument from Theorem A through Corollary D to the upper and lower bounds in Theorem B. Theorem A is a bootstrap from Hida/Shimura density plus a flat-cokernel q-expansion principle; Theorem C is an explicit q-expansion computation; Proposition 6.4.1 correctly derives admissibility of the coinvariants from Hida's finite-rank ordinary Hecke algebra; the upper bound in Section 7.3 follows from the structure theory of admissible Banach representations of 1+2pZp; and the split-reducible lower bound in Section 7.4 transfers Breuil-Emerton Theorem 1.1.3 into the missing indicator vectors. The weakest genuinely load-bearing dependency is Hida's finiteness theorem in the twisted setting of Proposition 6.4.1 (with the |det| twist of Lemma 4.4.1), and, secondarily, [1, Theorem 1.1.3] for the split-reducible companion vector. Both are published and the paper uses them in ways I do not see as misapplications. The in-preparation citation [12] supports Proposition 3.2.1, a standard p-adic Fourier isomorphism; this is a verification gap rather than a detected error. I therefore do not find a concern that would change the reader's conditional verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the space of cuspidal functions on the ordinary Caraiani–Scholze Igusa formal scheme for GL2 and relates it to classical and p-adic automorphic forms. Theorem A (5.3.1) identifies V_{b,C_p}^{cusp} with the completion, in the sup-norm Banach space of bounded functions on A_f^×, of the smooth Kirillov model of classical cuspidal forms of weight k ≥ 2. Theorem C (6.3.1) identifies Hida-ordinary p-adic modular forms with the topological coinvariants of the μ_{p^∞}-action, and Corollary D derives admissibility of these coinvariants from Hida's finiteness theorem. Theorem B then gives a weak local-global compatibility statement: for an irreducible π ⊆ S_{k,C_p}, the space W_π = Hom_{GL_2(A_f^{(p)})}(π^{(p)}, V_{b,C_p}^{cusp}) contains S(Q_p^×, C_p) plus one indicator character for each Galois character appearing in ρ_p, and is contained in that lower bound up to finitely many logarithmic twists. The paper concludes with a conjectural generalization to arbitrary Caraiani–Scholze Igusa varieties.","tokens_in":34891,"tokens_out":20747,"duration_ms":246090,"significance":"The paper's main theorems are substantive and largely proved by detailed, explicit q-expansion computations. Theorem A gives a clean structural description of a natural p-adic automorphic space, Theorem C exposes a new relation between Hida theory and topological coinvariants, and Theorem B is the first local-global compatibility statement of this kind for completed Igusa-variety functions, with explicit and uniform Banach-space bounds. The paper is careful about normalizations, especially the Kirillov and Galois normalizations in §2.6. The external dependencies (Hida finiteness, Kisin's theorem, Breuil–Emerton) are invoked in ways that appear consistent with their stated hypotheses; the stress-test concern about those dependencies does not land as a substantive objection. The conjectural section is clearly labeled and provides a concrete testable framework for future work.","major_comments":[{"comment":"The integral p-adic Fourier isomorphisms are cited to [12, Theorem 7.0.1], which is listed as 'In preparation'. This proposition is load-bearing: it is the mechanism by which the geometric actions of Z_p(1), μ_{p^∞}, and tilde-μ_{p^∞} are converted into the C(Q_p/Z_p, Z_p), C(Z_p, Z_p), and C(Q_p, Z_p) actions used in Definitions 3.4.2 and 3.5.3, in Lemma 3.5.4, and in the proof of Theorem C. The manuscript should either include a proof of Proposition 3.2.1 or replace the reference by a published source before the paper can be considered complete.","section":"Section 3, Proposition 3.2.1"},{"comment":"This equality requires that the relation submodule of W_π for the C(Q_p, C_p)-action is exactly S(Q_p^×, C_p). For a general closed submodule of C^{bdd}_∞ containing S, this is not automatic. Since the admissibility argument for W_π/S depends on Corollary D applied to the coinvariants of the ambient space, the proof should justify the equality, for instance by showing that compactly supported elements of S are relations in W_π via multiplication by indicator functions of clopen sets not containing 0.","section":"Section 7.3, the identity W_π/S(Q_p^×, C_p) = W_{π, tilde-μ_{p^∞}}"}],"minor_comments":[{"comment":"The running header contains a typo: 'COMP A TIBILITY' should read 'COMPATIBILITY'.","section":"Title/header"},{"comment":"In the proof, the phrase 'we may replace π with tilde-π' is terse; the twist tilde-π should be defined explicitly so that the normalization of χ_ord is unambiguous.","section":"Lemma 2.6.7"},{"comment":"The phrase 'p-torsion free (i.e. flat over Z_p)' is not an equivalence for general modules; either add a finiteness hypothesis or rephrase to avoid a misleading implication.","section":"Section 4.5"},{"comment":"The notation C^{bdd}_∞(Q_p^×, C_p) is introduced informally, and the transition to functions on Q_p used in §7.3 is not fully spelled out; clarifying the extension-by-zero convention would remove ambiguity.","section":"Sections 7.2–7.3"},{"comment":"The ordinary projector e = lim_n U_p^{n!} is stated to exist in Lemma 6.2.1, but the text could state explicitly that the limit is taken in the strong operator topology; this is implied by the argument but would be clearer if said directly.","section":"Section 6.1–6.3"},{"comment":"Reference [13] is cited only in a remark and is also listed as 'In preparation'; since it is not used in the proofs, it could be removed or explicitly marked as speculative.","section":"Remark 3.2.4 and reference [13]"}],"recommendation":"major_revision","confidential_remarks":"The two major comments are local and fixable; they do not attack the central claims. I would be willing to accept after the manuscript supplies a proof or published reference for Proposition 3.2.1 and justifies the coinvariant identity in §7.3. The use of in-preparation references [12] and [13] should also be resolved before publication, since [12] currently supports a foundational proposition. The paper's fit with the journal is good, and the main theorems appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the paper earns a serious referee. The genuinely new pieces are Theorem A, identifying cuspidal functions on the GL2 ordinary Caraiani-Scholze Igusa variety with a completion of the smooth Kirillov model, and Theorem C, identifying a Hida-ordinary variant with topological coinvariants. Theorem B then uses these to give a two-sided containment for W_pi, with the ordinary Galois characters appearing up to finitely many logarithmic twists, and it proves the clean conjecture in the irreducible case. These are not repackaged versions of [17]; the completed-Kirillov description and the coinvariants story are new statements, and the proof has real machinery in it.\n\nWhat the paper does well: the q-expansion embedding Kir is treated carefully, with flat cokernel and a bootstrap from Hida-Shimura density; Theorem C is an honest kernel/fiber-at-zero comparison; and the upper bound in Section 7 is a real argument through admissible Banach representations and Hida's finiteness theorem. I traced the external dependencies and agree with the stress test: Hida's finiteness and Breuil-Emerton [1] are used under their stated hypotheses, and I found no circularity. The paper is self-aware about what it does not prove.\n\nSoft spots, in proportion: Theorem B's upper bound contains an undetermined M and generalized-eigenspace terms chi_{a,b} with a,b <= M. The author says directly that the argument cannot exclude them; this is a weakness relative to Conjecture 1.2.1, not a hidden error. Proposition 3.2.1 depends on [12], which is in preparation; this is a real verification gap in a foundational Fourier statement, though the content is standard. Section 8 is openly provisional and should stay labeled as such; it is a conjecture section with motivation, not a load-bearing part of the main results. The citation pattern is fine: [17] is the natural predecessor, and the in-preparation citations should be flagged to the editor but are not a reason to reject.\n\nWho it is for: p-adic automorphic forms researchers, especially those working on Hida theory, Igusa varieties, and local-global compatibility. A serious referee should engage; I recommend sending it to review, asking the author to provide a published substitute or proof for Proposition 3.2.1 and to discuss the dependence of M in Theorem B more explicitly. Conditional accept seems right.","headline":"Genuinely new structural results on p-adic automorphic forms; the proof is coherent and the main soft spots are explicit gaps, not hidden flaws.","tokens_in":35538,"tokens_out":3055,"would_cite":true,"duration_ms":34393,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","11F33","11F80","11F85","11G18","22E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the ordinary GL2 Igusa variety, the local p-adic automorphic piece attached to a modular form is pinned, up to finitely many logarithmic twists, between two Galois-determined bounds, and is exact when the Galois representation is…","keywords":["Kirillov model","Igusa varieties","local-global compatibility","p-adic automorphic forms","ordinary p-adic modular forms","Galois representations","coinvariants","p-adic Fourier theory"],"falsifier":"Compute $W_\\pi$ for a classical cuspidal eigenform of weight at least $2$ whose local component $\\pi_p$ is supercuspidal (equivalently, whose Galois representation $\\rho_p$ is irreducible): the theorem predicts $W_\\pi$ is exactly $S(Q_p^\\times, C_p)$, the functions on $Q_p^\\times$ that vanish at $0$ and $\\infty$. If this computation produces any further vector, such as a non-vanishing indicator function $1_{Z_p}\\cdot\\chi$ for some character $\\chi$, the central compatibility claim fails.","tokens_in":34431,"feed_emoji":"🧩","tokens_out":23208,"duration_ms":191484,"temperature":0.7,"pith_summary":"This paper establishes a local-global compatibility statement for the space of cuspidal $p$-adic automorphic functions on the ordinary Caraiani-Scholze Igusa variety attached to $\\mathrm{GL}_2$. The main result, Theorem B, says that the local piece $W_\\pi$ attached to a classical cuspidal automorphic representation $\\pi$ of weight at least $2$ always contains the functions vanishing at $0$ and infinity together with one indicator character per character of the associated Galois representation $\\rho_p$, and is contained in that same space extended by finitely many logarithmic twists; when $\\rho_p$ is irreducible, the two bounds coincide. The proof works by describing the whole space of cuspidal functions as a completion of the smooth Kirillov model of classical cusp forms (Theorem A) and by identifying the ordinary $p$-adic modular forms of the classical theory with the topological coinvariants of the $\\tilde{\\mu}_{p^\\infty}$ action (Theorem C). These results give a concrete way to read local Galois information off a natural space of $p$-adic automorphic forms, and they suggest a conjectural analogue for more general Igusa varieties.","feed_headline":"Local p-adic eigenspaces pinned down by Galois data","feed_subtitle":"The only slack on the GL2 Igusa variety is finitely many logarithmic twists; irreducibility makes the description exact.","key_machinery":"Two identifications carry the argument. First, Theorem A: after fixing a compatible system of roots of unity, the $q$-expansion map $\\mathrm{Kir}$ composed with the evaluation map $\\mathrm{eval}_k$ identifies the cuspidal functions $V_b^{\\mathrm{cusp}}$ on the $\\mathrm{GL}_2$ ordinary Caraiani-Scholze Igusa formal scheme with the completion of the smooth Kirillov model of the classical cusp forms $S_k$ inside the bounded continuous functions on $A_f^\\times$, the Kirillov model being the classical realization of a smooth $\\mathrm{GL}_2(Q_p)$-representation on functions on $Q_p^\\times$ with the action of the mirabolic subgroup given by translations and dilations. Second, Theorem C: the ordinary projector $e$ on the Mantovan space $V_{\\mathrm{Mant}}^{\\mathrm{cusp}}$ is a $T(Q_p) \\times \\mathrm{GL}_2(A_f^{(p)})$-equivariant isomorphism onto the topological coinvariants $(V_b^{\\mathrm{cusp}})_{\\tilde{\\mu}_{p^\\infty}}$, where the action of the universal cover $\\tilde{\\mu}_{p^\\infty}$ is understood by $p$-adic Fourier duality as an action of the bounded continuous functions on $Q_p$. Theorem C turns the finiteness theorem for the ordinary Hecke algebra into the admissibility of the coinvariant space (Corollary D), and the completed Kirillov model converts the classical descriptions of Kirillov models and Jacquet modules for $\\mathrm{GL}_2(Q_p)$ into the explicit lower bound; the upper bound then follows by feeding the admissible Banach structure through the characters of $Q_p^\\times$ and using the description of Galois representations attached to ordinary $p$-adic eigenforms.","core_discovery":"The central discovery is that the eigenspace $W_\\pi = \\mathrm{Hom}_{\\mathrm{GL}_2(A_f^{(p)})}(\\pi^{(p)}, V_b^{\\mathrm{cusp}})$ attached to an irreducible subrepresentation $\\pi$ of the classical cuspidal modular forms of weight $k \\ge 2$ satisfies, for some non-negative integer $M$, $$S(Q_p^\\times, C_p) + \\sum_{\\chi \\subset \\rho_p} C_p \\cdot 1_{Z_p} \\cdot \\chi \\subseteq W_\\pi \\subseteq S(Q_p^\\times, C_p) + \\sum_{\\chi \\subset \\rho_p}\\sum_{a,b \\le M} C_p \\cdot 1_{Z_p} \\cdot \\chi_{a,b},$$ where the $\\chi$ are the characters of $Q_p^\\times$ obtained from the subrepresentations of $\\rho_p$ via local class field theory, and $\\chi_{a,b}(p^k,\\zeta,t) = \\chi(p)^k k^a \\chi(\\zeta)\\chi(t)\\log(t)^b$. In the irreducible (non-ordinary) case the two containments collapse to an equality, so $W_\\pi$ is exactly $S(Q_p^\\times, C_p)$. Thus the local representation of the Igusa variety at $p$ carries essentially the same information as the Galois representation, with the only indeterminacy being finitely many logarithmic twists in the reducible case.","pith_inferences":["The logarithmic twists $\\chi_{a,b}$ in the upper bound have the shape of powers of $p$-adic logarithms, hinting that $W_\\pi$ is a $p$-adic interpolation of the Jacquet module of $\\pi_p$; computing $W_\\pi$ for a specific Steinberg or principal-series representation could reveal whether the bound $M$ is sharp.","The equivalent mod $p$ form of Conjecture 8.2.1 suggests that the Galois characters detected by Theorem B should already be visible in the mod $p$ coinvariants, offering a characteristic-$p$ testing ground for local-global compatibility.","The local construction of §8.3, realizing the appearing characters as bounded sections on a quotient of a local Shimura variety, suggests a concrete recipe for building local representations for higher-rank Shimura varieties; a natural next test is to see which bounded sections survive the coinvariant quotient in a rank-two example."],"forward_implications":["If Theorem B is correct, then for any classical cuspidal $\\pi$ of weight $\\ge 2$ with irreducible $\\rho_p$, the local piece $W_\\pi$ is exactly the vanishing-at-infinity space $S(Q_p^\\times, C_p)$, so supercuspidal local components are read off completely from the Galois side.","In the ordinary potentially crystalline case, $W_\\pi$ determines $\\rho_p$; in the potentially semistable non-crystalline case it determines $\\rho_p^{\\mathrm{ss}}$; and under Conjecture 1.2.1 it determines $\\rho_p$ in all reducible cases.","Theorem C identifies the classical ordinary $p$-adic modular forms with the $\\tilde{\\mu}_{p^\\infty}$-coinvariants, and the paper notes this may yield a new derivation of the finiteness theorem for the ordinary Hecke algebra from the smooth admissibility of Jacquet modules.","Conjecture 8.2.1 predicts that the same coinvariant construction on any Caraiani-Scholze Igusa variety yields an admissible Banach representation of the associated Levi subgroup, which would give a systematic supply of $p$-adic automorphic representations beyond $\\mathrm{GL}_2$."],"supporting_citations":[{"why":"Constructs the Caraiani-Scholze Igusa formal scheme, the $\\tilde{\\mu}_{p^\\infty}$ action on its functions, and announces the coinvariant result proved here as Theorem C.","marker":"[17]"},{"why":"Supplies the overconvergent companion form used to produce the missing vector in the lower bound when $\\rho_p$ is split reducible (Section 7.4).","marker":"[1]"},{"why":"Supplies the finiteness theorem for the ordinary Hecke algebra, used in Proposition 6.4.1 to prove admissibility of the coinvariants (Corollary D).","marker":"[15]"},{"why":"Supplies the explicit descriptions of Kirillov models and Jacquet modules of $\\mathrm{GL}_2(Q_p)$ used to compute the completed Kirillov model $(\\pi_p^{\\mathrm{Kir}})^\\wedge$ in the lower bound.","marker":"[21]"},{"why":"Supplies the description of the Galois representation attached to an ordinary $p$-adic eigenform, used in the upper bound argument to identify the allowed characters of $Q_p^\\times$.","marker":"[25]"},{"why":"Supplies the $q$-expansion principle used to prove that the Kirillov map is injective with flat cokernel (Proposition 5.1.5).","marker":"[23]"},{"why":"Supplies the theorem identifying the ordinary line in the local representation with a Galois subrepresentation, used in Lemma 2.6.7 for the lower bound.","marker":"[33]"},{"why":"Supplies the density of classical cusp forms in the $p$-adic cuspidal space, a key input to the completed Kirillov model (Lemma 5.2.4).","marker":"[14]"}],"fun_headline_variants":["Cuspidal functions on Igusa varieties tied to Galois reps","Igusa eigenspaces match Galois data up to log twists","Kirillov model completes ordinary p-adic forms on Igusa","Finite log twists only gap between Igusa and Galois"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound leans on a published finiteness theorem for a certain algebra of Hecke operators, and the lower bound in one case leans on an external theorem producing an extra modular form; if either external result failed, the two-sided bound on $W_\\pi$ would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Cuspidal functions on Igusa varieties tied to Galois reps","Igusa eigenspaces match Galois data up to log twists","Kirillov model completes ordinary p-adic forms on Igusa","Finite log twists only gap between Igusa and Galois"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001048,"raw_usage":{"total_tokens":4444,"prompt_tokens":1023,"completion_tokens":3421,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":3346}},"tokens_in":639,"tokens_out":3421,"duration_ms":25195,"temperature":1.0,"reasoning_tokens":3346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:25:09.897795+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $W_\\pi$ for a classical cuspidal eigenform of weight at least $2$ whose local component $\\pi_p$ is supercuspidal (equivalently, whose Galois representation $\\rho_p$ is irreducible): the theorem predicts $W_\\pi$ is exactly $S(Q_p^\\times, C_p)$, the functions on $Q_p^\\times$ that vanish at $0$ and $\\infty$. If this computation produces any further vector, such as a non-vanishing indicator function $1_{Z_p}\\cdot\\chi$ for some character $\\chi$, the central compatibility claim fails.","supporting_citations":[{"cited_title":"A unipotent circle action on p-adic modular forms","cited_arxiv_id":null,"evidence_quote":"Constructs the Caraiani-Scholze Igusa formal scheme, the $\\tilde{\\mu}_{p^\\infty}$ action on its functions, and announces the coinvariant result proved here as Theorem C."},{"cited_title":"Repr´ esentationsp-adiques ordinaires de GL 2(Qp) et compatibilit´ e local-global.Ast´ erisque, (331):255–315, 2010","cited_arxiv_id":null,"evidence_quote":"Supplies the overconvergent companion form used to produce the missing vector in the lower bound when $\\rho_p$ is split reducible (Section 7.4)."},{"cited_title":"On p-adic Hecke algebras for GL2","cited_arxiv_id":null,"evidence_quote":"Supplies the finiteness theorem for the ordinary Hecke algebra, used in Proposition 6.4.1 to prove admissibility of the coinvariants (Corollary D)."},{"cited_title":"Jacquet and R","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit descriptions of Kirillov models and Jacquet modules of $\\mathrm{GL}_2(Q_p)$ used to compute the completed Kirillov model $(\\pi_p^{\\mathrm{Kir}})^\\wedge$ in the lower bound."},{"cited_title":"Overconvergent modular forms and the Fontaine-Mazur conjecture","cited_arxiv_id":null,"evidence_quote":"Supplies the description of the Galois representation attached to an ordinary $p$-adic eigenform, used in the upper bound argument to identify the allowed characters of $Q_p^\\times$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $q$-expansion principle used to prove that the Kirillov map is injective with flat cokernel (Proposition 5.1.5)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem identifying the ordinary line in the local representation with a Galois subrepresentation, used in Lemma 2.6.7 for the lower bound."},{"cited_title":"Galois representations into GL 2(Zp[[X]]) attached to ordinary cusp forms","cited_arxiv_id":null,"evidence_quote":"Supplies the density of classical cusp forms in the $p$-adic cuspidal space, a key input to the completed Kirillov model (Lemma 5.2.4)."}],"review_version":1}