{"id":"1b012ed2-3b72-4273-b066-b1291a0e73bc","arxiv_id":"2509.00690","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Derived level structures in spectral algebraic geometry are representable, yielding Jacquet-Langlands spectra and a proposed Jacquet-Langlands dual of Morava E-theory.","lead":"Using Cartier divisors, this paper constructs derived level structures on spectral elliptic curves and p-divisible groups, and proves that the corresponding moduli problems are representable. It then builds Jacquet-Langlands spectra, higher-homotopical versions of the Lubin-Tate tower, and proposes a Jacquet-Langlands dual of Morava E-theory with new spectral sequences for chromatic computations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 5.9's abutment S_{K(h)} is unsupported: the cited fixed-point spectral sequence only converges to π_{t-s}(LEh^{hGLh(Zp)}), and the equality with S_{K(h)} needs an integral descent along (5.2) the paper does not provide; Thm 5.8's Galois chain also clashes with Rem. 4.18.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: Proposition 5.9's spectral sequence is not shown to abut to the K(h)-local sphere, because the known isomorphism (5.2) is over the generic fiber and no spectral/integral descent is supplied. I agree this is the central weak point of the paper's headline application. The reader also noted the terseness of Theorem 5.8's Galois-extension chain; I have sharpened this into a concrete internal tension with Remark 4.18. That tension does not change the verdict: the representability theorems in Sections 2–4 may well be correct, and the Section 5 claims should be conditioned on a genuine descent argument. The paper is honest about some of these gaps in Remarks 4.18, 4.19, and 5.6, so a conditional verdict remains appropriate rather than outright rejection.","tokens_in":38343,"tokens_out":19020,"duration_ms":241304,"concrete_test":"Independently derive from the paper's own data an E∞-ring equivalence (JL^{hGh})^{hGLh(Zp)} ≃ S_{K(h)} and test it on π0: compute H^0_c(GLh(Zp)×Gh, π0 JL) using π0 JL = lim_r π0 JLr and the Drinfeld-tower action; the claimed abutment requires this to be Z_p and the relevant H^1 to vanish. A sharper check is the height-1 case, where JLr ≃ E_1: run the H^s_c(Z_p^×, π_t LE_1) spectral sequence with its differentials and compare with π_*S_{K(1)}; a mismatch would disprove the descent identification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Let E = LEh = JL^{hGh}. The proof of Prop. 5.9 invokes the profinite homotopy-fixed-point spectral sequence for G = GLh(Zp), which converges to π_{t-s}(E^{hG}). The proposition's abutment π_{t-s}S_{K(h)} therefore requires an equivalence E^{hG} ≃ S_{K(h)}. The only evidence cited is (5.2), an isomorphism of perfectoid rigid spaces over the generic fiber of the Lubin–Tate/Drinfeld towers. No argument lifts this to an equivalence of spectral DM stacks or E∞-ring spectra carrying the GLh(Zp)×Gh actions, and Remark 5.6 explicitly defers the spectral realization of LEh to future work. Thus the target of (5.3) is not identified. The difficulty is compounded in Thm 5.8: it asserts 'Galois extensions S_{K(h)} → Eh → JLr' on the basis of Prop. 5.7 and [Rog08, Thm 5.4.4(d)]. But Prop. 5.7 says JLr is Galois over Ror. If Ror is Eh (as Definition 5.5 and §5.3.1 imply by JL0 = Eh), then JLr is an Eh-algebra and hence K(h)-local, contradicting Remark 4.18, which explicitly says JLr for r>0 is not K(h)-local. If Ror is instead the ordinary Lubin–Tate ring, the middle map Eh → JLr has not been constructed. Either way, the Galois-based justification for the fixed-point identification is unsound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a spectral-algebraic-geometric theory of relative effective Cartier divisors and derived level structures, and applies it to construct higher-homotopical refinements of Lubin–Tate towers and to propose an integral Jacquet–Langlands dual of Morava E-theory. Theorems 2.17, 3.6, 3.19, 4.6, 4.12, and 4.17 form the technical core: they assert representability of moduli functors for divisors and level structures on spectral elliptic curves and p-divisible groups, corepresentability of oriented deformation functors with level structure by E-infinity rings JLr, and topological lifts of Strickland's power-operation rings. Section 5 then defines an infinite-level Jacquet–Langlands spectrum JL, sets LEh = JL^{hGh}, claims Galois towers involving S_{K(h)}, Eh, and JLr, and derives a Devinatz–Hopkins-dual homotopy fixed point spectral sequence (Prop. 5.9) converging to the K(h)-local sphere.","tokens_in":38657,"tokens_out":14663,"duration_ms":193639,"significance":"If the representability results hold, they constitute a substantial contribution to spectral algebraic geometry and chromatic homotopy theory: they provide a genuine derived approach to level structures that does not invert the level, and they offer a new method for constructing E-infinity ring spectra from Lubin–Tate-type moduli problems. The relative effective Cartier divisor representability theorem (Thm. 2.17) is a useful structural result in its own right and would generalize Lurie's spectral Artin representability applications. The proposed spectral sequence of Prop. 5.9, if its abutment were established, would be a valuable new computational tool. However, the paper's advertised Jacquet–Langlands dual is not actually proved: the passage from the perfectoid rigid-space isomorphism (5.2) to a spectral or integral descent is missing, and the Galois arguments in Section 5 sit in tension with Remark 4.18. The significance of the paper is therefore conditional on substantial repair of the Section 5 claims.","major_comments":[{"comment":"The proof invokes the profinite homotopy fixed point spectral sequence, which converges to π_{t-s}(LEh^{hGLh(Zp)}). The asserted abutment π_{t-s} S_{K(h)} therefore requires an equivalence (LEh)^{hGLh(Zp)} ≃ S_{K(h)}. The only cited evidence is (5.2), an isomorphism of perfectoid rigid spaces over the generic fiber; Remark 5.6 explicitly defers the spectral realization of LEh to future work. No argument lifts this isomorphism to an equivalence of spectral DM stacks or E∞-ring spectra carrying the GLh(Zp) × Gh actions. Thus the target of (5.3) is not identified as the K(h)-local sphere.","section":"§5.2, Proposition 5.9 (esp. eq. (5.3))"},{"comment":"Proposition 5.7 states that JLr is a GLh(Z/prZ)-Galois extension of Ror_{G0}. By Definition 5.5 and §5.3.1, JL0 = Eh = Ror_{G0}. Then JLr is an Eh-algebra, and in a finite Galois extension it is a dualizable (finite) Eh-module, hence K(h)-local. Remark 4.18, however, explicitly asserts that the JLr of Theorem 4.12 with r > 0 are not K(h)-local and are not finite algebras over Eh. If 'Ror' in Proposition 5.7 is instead meant to be the underlying discrete Lubin–Tate ring, then the E∞-map Eh → JLr has not been constructed. Either reading leaves the Galois tower S_{K(h)} → Eh → JLr and the proof of Theorem 5.8 unsupported.","section":"§5.2, Theorem 5.8 and Proposition 5.7, contrasted with Remark 4.18"},{"comment":"The nilcompleteness proof is compressed at the essential surjectivity step. After obtaining D from the system {Dn} via [Lur18c, Prop. 19.4.1.2], the proof asserts, rather than proves, that D → X is a closed immersion and that its ideal sheaf is a line bundle. The sentence 'Since τ≤n+1S → B′_{n+1} is flat' does not have a clear meaning as written, and the construction of the spectrum B′ with Spét τ≤nB′ ≃ SpétB′_n is not justified in detail. The subsequent appeal to nilcompleteness of the Picard functor also presupposes that the limit ideal is almost perfect. This step is load-bearing for criterion (3) of Theorem 2.10 and hence for Theorem 2.17.","section":"§2.2, Lemma 2.20"},{"comment":"The proof that the cotangent complex LCDivE/R is connective reduces to an assertion in classical algebraic geometry about automorphisms of a line bundle over a square-zero extension, which is dismissed with 'This can be proved, mutatis mutandis, as in the last part of [Lur18a, proof of Proposition 2.2.6].' No proof is supplied, and the statement is not a formal consequence of the cited reference as written. The preceding identification Fη(M) ≃ Map(Σ^{-1}p+(η+L_{D/(E×R S)}), M) also needs a check of the pushforward and base-change compatibilities. Since Lemma 2.24 supplies criterion (4) of Theorem 2.10, the proof should be completed.","section":"§2.2, Lemma 2.24"},{"comment":"The functor Def^r_{G0} (and Def^{or,r}_{G0}) is defined on CAlgad_cpl, the ∞-category of adically complete E∞-rings. Theorem 3.19 gives a representing object in connective R-algebras, but the proof does not show that the resulting E∞-ring P^r lies in CAlgad_cpl, nor that corepresentability on the full subcategory CAlgad_cpl follows from representability on CAlgcn_R. The finiteness of π0 over the complete local ring is not obviously sufficient for adic completeness of the E∞-ring. This affects the definition of JLr and the later applications that depend on its adic structure.","section":"§4.2, Theorems 4.10 and 4.12"}],"minor_comments":[{"comment":"There are several LaTeX/typographical artifacts, e.g. 'Sp´ etR' in the abstract and the garbled display 'X GLh(OK) || O×D' in Section 1. Please re-typeset these.","section":"Abstract and Introduction"},{"comment":"The diagram names 'LT K H' and 'MLT∞ ≃ MDr∞' are hard to parse; labels for the arrows and objects would improve readability.","section":"§1, diagram after Proposition 5.3"},{"comment":"The notation U^0_i and U_i is confusing; please define the étale cover more explicitly and check the superscript formatting.","section":"§2.1, Lemma 2.9"},{"comment":"The notation Eh,r is dangerously close to Eh; consider a distinct symbol to avoid confusion, especially because these spectra are not K(h)-local.","section":"§4.3, Theorem 4.17"}],"recommendation":"major_revision","confidential_remarks":"The representability results in Sections 2–4 are substantial and likely salvageable, but the advertised Jacquet–Langlands-dual package in Section 5 is not established: the integral descent from (5.2) is missing, and Prop. 5.7/Thm. 5.8 appear inconsistent with Remark 4.18. I would recommend that the editor require either a complete proof of the Section 5 claims or a substantial restatement of the paper's scope (e.g., presenting Section 5 as conjectural/future work), before considering publication in a top journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the representability core is real and deserves a careful referee; the Section 5 applications are not ready, and the paper contains an internal contradiction it never flags.\n\nWhat's new and good. The relative effective Cartier divisor functor (Definition 2.12) and its representability (Theorem 2.17), plus the derived level structures on spectral elliptic curves and p-divisible groups (Theorems 3.6, 3.19) and the absolute moduli stack with level (Theorem 4.6), are genuine extensions of Lurie's spectral Artin representability machinery. They handle primes dividing the level, which the earlier étale and log-étale routes could not. The spectra JLr and Eh,r (Theorems 4.12, 4.17) are new objects, and it is a real plus that their π0 recover the finite Lubin–Tate levels and Strickland's rings Ar by construction, so there is no circularity: the paper anchors those identifications to classical rings rather than fitting data.\n\nSoft spots, in proportion. The proof of Theorem 2.17 has several load-bearing compressed steps—\"it is not hard to prove\" in Lemmas 2.9 and 2.20, \"mutatis mutandis\" in Lemma 2.24. They read as fillable, not fatal, but a referee should check them. The bigger problem is Section 5. The reader and the stress-test are right: Proposition 5.9's spectral sequence converges, by the cited fixed-point results, to π_{t−s}((LEh)^{hGL_h(Z_p)}). Identifying that target with the K(h)-local sphere requires an integral descent along (5.2), and (5.2) is only an isomorphism of perfectoid rigid spaces over the generic fiber. No spectral descent is given, and Remark 5.6 explicitly defers the spectral realization of LEh. So the abutment of (5.3) is unsupported as stated. Worse, there is an actual contradiction: Proposition 5.7 says JLr is a GL_h(Z/p^rZ)-Galois extension of Ror_{Ĝ0}, which in this section is Eh. Every Eh-algebra is K(h)-local, since any module over a K(h)-local ring is K(h)-local; Remark 4.18 asserts JLr is not K(h)-local for r>0. Both cannot hold. Either Ror is not Eh and the chain S_{K(h)} → Eh → JLr in Theorem 5.8 is missing a construction, or Remark 4.18 is wrong. The authors do not acknowledge this.\n\nWho this is for: chromatic homotopy theorists and spectral algebraic geometers. The representability part is the contribution; Section 5 should be read as conjecture. This deserves a serious referee, with instructions to focus on the details behind Theorem 2.17 and on the Section 5 inconsistency.","headline":"The representability core is a genuine step forward, but Section 5's descent claims—the abutment of (5.3) and the Galois chain in Theorem 5.8—are unsupported, and Remark 4.18 contradicts Proposition 5.7.","tokens_in":39341,"tokens_out":14271,"would_cite":true,"duration_ms":153720,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P43","14L05","14A20","11S37"],"pacs":[],"model":"deepseek-v4-flash","headline":"Spectral level structures build an integral Jacquet-Langlands dual of Morava E-theory.","keywords":["derived level structures","spectral algebraic geometry","Cartier divisors","Lubin-Tate tower","Jacquet-Langlands","Morava E-theory","homotopy fixed point spectral sequence","p-divisible groups"],"falsifier":"Compute the spectral sequence (5.3) at height h=1, where the K(1)-local sphere and the group GL_1(Z_p) are completely understood; if the abutment is not the known homotopy of the K(1)-local sphere, the descent claim is false. A sharper check: verify that π_0 of the first level JL_1 is finite flat over the oriented deformation ring with the rank predicted by the classical Lubin-Tate tower.","tokens_in":38019,"feed_emoji":"♾️","tokens_out":8849,"duration_ms":90795,"temperature":0.7,"pith_summary":"This paper builds a spectral (higher-categorical) version of the arithmetic theory of level structures and uses it to construct new structured ring spectra. It defines relative effective Cartier divisors for spectral Deligne-Mumford stacks and proves the functor of such divisors is representable, which makes it possible to define derived level structures on spectral elliptic curves and on spectral p-divisible groups and to prove the associated moduli problems are representable. In particular, the oriented deformation problem with level-(Z/p^r Z)^h structure is corepresented by an E∞-ring JL_r whose π_0 is finite over the oriented deformation ring; these are the finite levels of a Lubin-Tate tower in spectra. Passing to the infinite level and descending along the Drinfeld tower, the paper defines a Jacquet-Langlands dual of Morava E-theory and a strongly convergent homotopy fixed point spectral sequence converging to the homotopy of the K(h)-local sphere. The point of the package is a new, integral route from p-adic group cohomology to higher-periodic stable homotopy.","feed_headline":"Finite Lubin-Tate levels become E∞-ring spectra","feed_subtitle":"New level-structure moduli give a spectral sequence to the K(h)-local sphere.","key_machinery":"The central object is a relative effective Cartier divisor in spectral algebraic geometry: a closed immersion D → X that is flat, proper, and locally almost of finite presentation, with ideal sheaf a line bundle. The proof of its representability checks the five criteria of the spectral Artin representability criterion; the delicate part is the cotangent complex computation, which pins first-order deformations to square-zero extensions. Derived level structures are then divisors whose underlying classical part is an A-structure; this formulation is what lets the level divide the prime, since it avoids étaleness. At the top level, the mechanism is a tower of E∞-rings JL_r corepresenting orien","core_discovery":"Classical level-structure moduli are claimed to survive integrally in spectral algebraic geometry: a level structure is a relative effective Cartier divisor (a flat, proper closed immersion with line-bundle ideal sheaf) plus a classical level structure on the underlying heart. The paper proves these divisor functors, the derived level-structure functors, and the absolute moduli of spectral elliptic curves with level structure are representable. For deformations of a fixed p-divisible group, the oriented functor with derived level structure is corepresented by an E∞-ring JL_r, with π_0 finite over the oriented deformation ring; the limit JL carries a GL_h(Z_p)-action, and homotopy fixed point","pith_inferences":["If the descent in Section 5 is given a fully spectral proof, the same package would produce an integral refinement of the classical Lubin-Tate/Drinfeld equivalence, not just a generic-fiber one.","The divisor method should adapt to oriented elliptic curves, yielding topological modular forms with level structure without inverting the level; the paper notes this variant but leaves the details out.","At height 1, the spectral sequence (5.3) should be explicitly computable and would provide a sharp test of the descent claim; the paper does not perform this check.","The proposed Serre-type spectral sequence in Section 5.3.1 could convert the abstract E∞-rings JL_r into explicit higher homotopy groups, but it is left as a conjecture."],"forward_implications":["Finite levels of the Lubin-Tate tower exist as E∞-ring spectra, refining the known spectral realization of the ground-level deformation ring.","The moduli stack of spectral elliptic curves with derived level structure exists as a spectral Deligne-Mumford stack, so level structures can be studied without inverting the level.","Non-full (Γ_1/Γ_0-type) derived level structures produce E∞-spectra whose π_0 recovers the power operation rings of Morava E-theory.","The Jacquet-Langlands dual LE_h is an E∞-ring, and the resulting homotopy fixed point spectral sequence is a computational route to the homotopy of the K(h)-local sphere.","These spectra suggest a topological realization of the Jacquet-Langlands correspondence and of categorical local Langlands phenomena in stable homotopy."],"supporting_citations":[{"why":"Supplies the spectral Artin representability criterion and the cotangent-complex technology used to prove representability of the Cartier divisor functor.","marker":"[Lur18c]"},{"why":"Defines spectral elliptic curves and the spectral Deligne-Mumford stack of spectral elliptic curves that the new level-structure moduli generalize.","marker":"[Lur18a]"},{"why":"Provides the oriented deformation ring and the representability of oriented deformations of p-divisible groups that Theorem 4.12 refines with level structures.","marker":"[Lur18b]"},{"why":"Provides the classical arithmetic theory of level structures, full sets of sections, and Drinfeld divisors that the derived definitions reduce to on the heart.","marker":"[KM85]"},{"why":"Establishes the isomorphism between the Lubin-Tate and Drinfeld towers invoked to construct the Jacquet-Langlands dual.","marker":"[FGL08]"},{"why":"Gives the infinite-level perfectoid moduli space and the p-divisible group moduli used in the tower isomorphism.","marker":"[SW13]"},{"why":"Supplies the classical homotopy fixed point spectral sequence for the Morava stabilizer group that the new spectral sequence (5.3) is dual to.","marker":"[DH04]"},{"why":"Supplies the theory of Galois extensions of E∞-rings used to prove that the homotopy fixed points JL_r^{hG_h} are E∞-rings.","marker":"[Rog08]"}],"fun_headline_variants":["Derived Cartier divisors refine Lubin-Tate towers as E∞ spectra","Jacquet-Langlands dual of Morava E-theory from level structures","Spectral moduli of level structures: E∞ refinement at all primes","Integral level structures via Cartier divisors on spectral stacks","Derived level structures: finite Lubin-Tate levels become E∞ rings"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The tower isomorphism used to identify homotopy fixed points with the K(h)-local sphere is only known on generic fibers of the two towers as rigid spaces, not as an integral statement about spectra; if that identification fails, the new spectral sequence does not converge to the sphere.","fun_headline_variants_meta":{"raw":{"variants":["Derived Cartier divisors refine Lubin-Tate towers as E∞ spectra","Jacquet-Langlands dual of Morava E-theory from level structures","Spectral moduli of level structures: E∞ refinement at all primes","Integral level structures via Cartier divisors on spectral stacks","Derived level structures: finite Lubin-Tate levels become E∞ rings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000841,"raw_usage":{"total_tokens":3499,"prompt_tokens":743,"completion_tokens":2756,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2660}},"tokens_in":487,"tokens_out":2756,"duration_ms":25342,"temperature":1.0,"reasoning_tokens":2660,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:22:03.484200+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the spectral sequence (5.3) at height h=1, where the K(1)-local sphere and the group GL_1(Z_p) are completely understood; if the abutment is not the known homotopy of the K(1)-local sphere, the descent claim is false. A sharper check: verify that π_0 of the first level JL_1 is finite flat over the oriented deformation ring with the rank predicted by the classical Lubin-Tate tower.","supporting_citations":[],"review_version":1}