{"id":"04df6550-ac1d-41ff-b58f-45649a8bdc08","arxiv_id":"2508.08010","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Topological Jacobi forms TJF_m are constructed via circle-equivariant TMF, identified with TMF ∧ P_m for stunted projective spaces P_m, and their homotopy is computed completely at odd primes and partially at 2.","lead":"A new family of ring spectra, topological Jacobi forms TJF_m, generalizes topological modular forms and carries the homotopy-theoretic version of Jacobi forms. Each TJF_m is identified with TMF smashed with a stunted projective space spectrum, and its homotopy groups are computed completely at odd primes.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central equivalence TJF_m ≃ TMF∧P_m rests on unpublished Grothendieck-duality identity (Eq 3.5) from [GKMP]; if that identity fails, Prop 3.4 and Thm 3.7 collapse.","rationale":"The reader's verdict of CONDITIONAL is appropriate. The paper constructs a new object and computes its homotopy, but the central structural identification depends on an unpublished and author-overlapping duality theorem. This is a genuine correctness risk, not a matter of taste: Prop 3.4 is used exactly at the point where the sheaf-theoretic definition is replaced by the cellular model TMF∧P_m, and every subsequent computation is performed on that cellular model. If Eq. 3.5 is false or inapplicable, the computational results do not concern TJF_m as defined. The rest of the paper contains substantial and plausible algebraic computation, and the internal inconsistency around b4h1=0 in Thm 4.7 versus its proof and Section 6 is a concrete error that should be corrected but does not threaten the main construction. A focused verification of Eq. 3.5 in the two specific cases used would settle the concern without needing the full unpublished [GKMP] framework. Until then, CONDITIONAL remains the right verdict: accept with the condition that the duality is supplied or replaced by a proof.","tokens_in":35490,"tokens_out":15560,"duration_ms":172364,"concrete_test":"Independently verify Eq. 3.5 for the two cases needed in Thm 3.7: F = TMF^T(S(mρ)_+) and F = TMF^T(S^0), using only published ingredients: the cofiber sequence (3.2), (TMF^T)^T ≃ TMF∨ΣTMF [GM23], and the Mathew–Meier affineness theorem. Compute both sides of Prop 3.4 for X = S(mρ)_+ and X = S^0 and check the resulting TMF-module equivalence. If the two sides fail to be Σ-shifted duals for any m≥1, Prop 3.4 is false and Thm 3.7 must be revised; if they agree, the unpublished citation is not blocking.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.7 (Thm 1.7) identifies the sheaf-theoretic TJF_m with TMF∧P_m by applying Proposition 3.4, whose proof uses Eq. 3.5, a Grothendieck-duality equivalence Hom_{O_M}(p_*F,O_M) ≃ Σ^{-1}Hom_{O_E}(F,O_E), cited to [GKMP, Thm 6.5] (in preparation, author overlap). This duality is not proved or sketched in the paper, and it is exactly what converts the fill-in map t into the TMF-smash of the reduced transfer. Without (3.5), the cofiber diagram in Thm 3.7 does not identify TJF_m with TMF∧P_m, and the entire computational strategy (AHSS/descent on the P_m model, Theorems 1.9/1.11/4.7) loses its target. The Gepner–Meier equivalence [GM23, Thm 10.1] is published and less risky, but (3.5) is not. A secondary internal inconsistency (Thm 4.7 and Thm 1.9 omit the relation b4h1=0 that the proof and Section 6's E4 page include) is real but localized; the [GKMP] dependency is the load-bearing one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs spectra TJF_m of topological Jacobi forms as the global sections of a sheaf L^{top}_m of O^{top}_E-module spectra on the spectral universal elliptic curve, using T-equivariant topological modular forms. The central structural theorem (Thm 1.7) identifies TJF_m with TMF ∧ P_m, where P_m is the cofiber of the reduced S^1-transfer, and TJF_∞ with TMF ∧ P_∞. The authors then carry out extensive Hopf algebroid Ext computations to determine the derived Jacobi forms DJF and the homotopy of TJF. They obtain complete calculations at odd primes, a complete description of (π_*TJF_∞)(2), and a partial 2-primary description of TJF_m, together with the ring structure of π_*TJF_0. The paper also establishes connections with classical Jacobi forms and with the two-variable elliptic genus.","tokens_in":35745,"tokens_out":5750,"duration_ms":67954,"significance":"If the main theorem is correct, the paper introduces a canonical, explicitly computable spectrum whose rational homotopy is the ring of weakly holomorphic Jacobi forms, thereby providing a topological counterpart to the Ochanine–Witten genus. The identification with TMF ∧ P_m gives a concrete cellular model and makes the spectrum amenable to Atiyah–Hirzebruch and descent spectral sequence computations. The odd-primary calculations are detailed and internally consistent, and the 2-primary computation is supported by independent constraints such as η^4 = 0 in the sphere and comparison with ko_*(X). The reliance on an unpublished, same-author Grothendieck-duality identity, however, leaves the central structural result conditional, and a missing relation in the stated 2-local ring must be corrected.","major_comments":[{"comment":"The proof of Proposition 3.4 uses the Grothendieck-duality equivalence Hom_{O_{M^or}}(p_*F,O_{M^or}) ≃ Σ^{-1}Hom_{O_E}(F,O_E), cited to [GKMP, Theorem 6.5], an unpublished manuscript with author overlap. This equivalence is load-bearing: in Theorem 3.7 it is exactly what identifies the fill-in map t with TMF smashed with the reduced transfer, establishing TJF_m ≃ TMF ∧ P_m. If (3.5) is unavailable or incorrect, the cofiber-diagram argument in Theorem 3.7 collapses, and the subsequent computations on the P_m model lose their target. The manuscript gives no proof or even a sketch of (3.5), so the central theorem is currently conditional on an external unpublished result. Please provide a proof in an appendix, replace the citation by a published reference, or state Theorem 1.7 as conditional.","section":"§3.1, Prop. 3.4 and Eq. (3.5)"},{"comment":"The displayed ring in Theorem 4.7 (and Theorem 1.9) is Z_(2)[b2,b3,b4,b8,h1]/(2h1, b3h1, 4b8 + b4^2 - b2b3^2). However, the proof of Theorem 4.7 ends with the presentation Z_(2)[h1,b2,b3,b4,b8]/(2h1, b3h1, b4h1, b4^2 - b2b3^2 - 4b8), and the E4-page in Section 6 also contains the relation b4h1 = 0. As stated, Theorems 4.7 and 1.9 omit a necessary relation, so they do not describe the computed ring. This is a localized but real inconsistency in statements advertised as complete calculations. Please add b4h1 = 0 and adjust Corollary 6.1 if necessary.","section":"Thm. 4.7 / Thm. 1.9 vs. §6"}],"minor_comments":[{"comment":"The relation involving τb3 is listed as τb3 − aγ in Theorem 1.11 but as τb3 − 2aγ in Theorem 4.6. Since both statements are localized away from 2 these are equivalent, but the statements should be consistent. In the proof of Theorem 4.6, the line \"τb3 = 2aα\" appears to be a typo, likely for 2aγ; as written the degrees do not match (τb3 has index 3, while 2aα has index 1).","section":"Thm. 1.11 and Thm. 4.6"},{"comment":"In the displayed tridegrees of Theorem 4.6, \"|τ = (1,1,0)\" is missing a closing vertical bar; it should read \"|τ| = (1,1,0)\".","section":"Thm. 4.6 display"},{"comment":"The abstract and introduction call TJF_* a \"graded ring spectrum\"; the body makes precise that TJF_m is an E_2-spectrum for finite m and TJF_∞ is E_∞. It would help the reader if this distinction were stated in the introduction.","section":"Introduction, Definition of TJF"},{"comment":"The E1-term display π_*TMF[z,τ]/(τ^2−τη, zτ−zη) is used immediately but the relation zτ = zη is only established in the proof. This is fine, but the display could be annotated to prevent the appearance of an unstated relation.","section":"Prop. 3.14"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unpublished, author-overlapping citation [GKMP] at the load-bearing point of the proof of Theorem 1.7. The b4h1 omission is a clear typo-level error but appears in a central theorem. Both are fixable within the manuscript's scope: the first by supplying a proof or a published reference, the second by correcting the relation. The paper is otherwise substantial and well within the scope of a topology journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper with real content. The construction of TJF_* as global sections of a sheaf of E_2-ring spectra on the universal elliptic curve, landing in a graded spectrum whose homotopy is Jacobi forms, is new and natural. The clean identification TJF_m ≃ TMF ∧ P_m for all m (only m = 0 was known) is the kind of result that makes people sit up, and the subsequent computations of derived Jacobi forms and of the odd-primary homotopy are detailed and convincing. The algebraic Ext computations and the new unstable algebraic Atiyah–Hirzebruch spectral sequence are explicit, with internal consistency checks; the authors also honestly flag what they could not do, such as the ring structure on the connective version and the full 2-primary picture.\n\nThe soft spots are real but localized. The main one is Proposition 3.4, where the proof of the central equivalence uses a Grothendieck duality identity (Eq. 3.5) cited to [GKMP, Theorem 6.5], an in-preparation paper with author overlap. This is not a minor tool—it is exactly what converts the fill-in map t into the TMF-smash of the reduced transfer. If that identity fails, the identification TJF_m ≃ TMF ∧ P_m and everything downstream in the spectral sequence computations loses its target. I have not seen the proof of [GKMP, Thm 6.5], and the paper does not sketch it, so this is a genuine dependency, not a stylistic quibble. That said, the rest of the algebraic section (computing DJF) is independent, and the consistency checks in the homotopy computations make it plausible that the duality is correct.\n\nThere are also smaller issues. Some differentials and an extension (Prop 5.5, Lemma 5.7, the d_4({x^2α}) differential) are asserted in compressed form, which makes verification slow but not impossible. And Theorem 4.7's statement omits the relation b_4 h_1 = 0 that its own proof and the displayed E_4 page include; that looks like a typo rather than a substantive gap. The paper would be stronger if the [GKMP] result were either proved or made available as a preprint before the referee report is due.\n\nWho is this for? Anyone working in chromatic homotopy theory, equivariant TMF, or elliptic genera. It deserves a serious referee, and I would send it out rather than desk reject it. My recommendation: conditional accept after the authors either supply a proof of Eq. 3.5 or cite a publicly available version of [GKMP]. The math is too well-executed and the central claim too natural to let the unpublished citation quietly kill it.","headline":"TJF_m ≃ TMF∧P_m is new and likely right, but the load-bearing Grothendieck duality is cited to an unpublished paper—worth refereeing, not desk rejecting.","tokens_in":36443,"tokens_out":1660,"would_cite":true,"duration_ms":22080,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N34","55P91","11F50","55T99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a graded E2-ring spectrum TJF_* of topological Jacobi forms, proves each index piece is equivalent to TMF ∧ P_m, and computes its homotopy at odd primes completely and at 2 partially.","keywords":["topological Jacobi forms","topological modular forms","equivariant homotopy theory","Jacobi forms","elliptic genera","Atiyah-Hirzebruch spectral sequence","descent spectral sequence","E∞-ring spectra"],"falsifier":"Compute the composite pr_2 ∘ tr_m : TMF ∧ Σ $CP^{{m-1}}$ → TMF ∨ Σ TMF → Σ TMF; the proof of Theorem 3.7 claims it is trivial, and any nonzero value would force the fill-in map t to differ from TMF smashed with the reduced transfer. Alternatively, verify Eq. (3.5) directly for F = O_{E^or}(-e); if the duality does not hold, Proposition 3.4 and hence the central identification collapses.","tokens_in":35213,"feed_emoji":"🧮","tokens_out":6126,"duration_ms":72011,"temperature":0.7,"pith_summary":"The paper aims to give Jacobi forms a spectrum-level home in the same way that topological modular forms (TMF) gave modular forms a home. It constructs a graded ring spectrum TJF_* whose complexified homotopy groups are the classical weakly holomorphic Jacobi forms of index m/2, and proves the central identity TJF_m ≃ TMF ∧ P_m, where P_m is the cofiber of the reduced $S^{1}$-transfer from Σ $CP^{{m-1}}$ to $S^{0}$. This cellular description turns a sheaf-theoretic construction into an explicitly computable TMF-module spectrum. From it the paper obtains the complete odd-primary homotopy of TJF_m and a partial 2-primary description, including rings of derived Jacobi forms and the relevant descent spectral sequences. A reader should care because this provides a canonical, torsion-sensitive target for the two-variable elliptic genus, directly parallel to the classical TMF story.","feed_headline":"Topological Jacobi forms equal TMF smashed with a projective cofiber","feed_subtitle":"The new spectrum gives the 2-variable elliptic genus a ring-spectrum home, with complete odd-primary homotopy.","key_machinery":"The machinery is circle-equivariant TMF: a genuine $S^{1}$-spectrum TMFT whose underlying nonequivariant spectrum is TMF, built from the spectral universal elliptic curve and from sheaves O_{E^or}(me) = TMFT($S^{{-mρ}}$). These sheaves assemble into an E2-algebra, and TJF_m is their global sections. The load-bearing mechanism is the equivalence TJF_m ≃ TMF ∧ P_m, where P_m = cofib(Σ $CP^{{m-1}}$ → $S^{0}$) is the cofiber of the reduced $S^{1}$-transfer; this converts a spectral stack computation into an Atiyah–Hirzebruch spectral sequence for stunted projective spaces. The key identity used to prove it is the Grothendieck duality equivalence Hom_{O_M}(p_*F,O_M) ≃ $Σ^{{-1}}$Hom_{O_E}(F,O_E) from [GKMP, Theorem 6.5],","core_discovery":"The central claim is Theorem 3.7: for every m ≥ 0, the topological Jacobi form spectrum TJF_m = Γ(E, $L^{{top}}$_m), defined as global sections of a sheaf on the spectral universal elliptic curve, is equivalent as a TMF-module spectrum to TMF ∧ P_m, with P_m = cofib(Σ $CP^{{m-1}}$ → $S^{0}$). Under this equivalence the inclusion a : TJF_m → TJF_{m+1} corresponds to the skeletal inclusion P_m → P_{m+1}, and TJF_∞ ≃ TMF ∧ P_∞. The proof interprets TJF_m as T-fixed points of T-equivariant TMF smashed with $S^{{mρ}}$, uses the known identification of the T-fixed points as TMF ⊕ Σ TMF, and applies a Grothendieck duality statement to show the fill-in map is exactly TMF smashed with the reduced transfer. The paper","pith_inferences":["If the identification TJF_m ≃ TMF ∧ P_m holds integrally, the same cellular model should govern twisted or RO(S^1)-graded TMF for other stunted projective spectra, potentially giving a uniform machine for computing equivariant TMF in finite cyclic groups.","The connective spectra tmf ∧ P_m are natural candidates for a 'topological weak Jacobi forms' theory; the paper leaves open whether the missing ring structure can be constructed, and the algebraic computations here suggest exactly what obstruction such a construction would face.","The unpublished Grothendieck duality input, once available, would likely imply a general self-duality for equivariant TMF sheaves; checking it directly for the sheaf O_{E^or}(-e) would sharpen the 2-primary computations.","The explicit p = 2 presentation of π_*(TJF_∞)(2) may serve as a test case for hidden multiplicative extensions or for comparing TJF with forthcoming constructions of Cn-equivariant TMF."],"forward_implications":["TJF_* is a graded E2-ring spectrum and TJF_∞ is an E∞-ring spectrum, giving the two-variable elliptic genus a genuine ring-spectrum target rather than only a formal lift.","TJF_0 ≃ TMF ∨ Σ TMF with π_*TJF_0 = π_*TMF[τ]/(τ^2 - τη), recovering and refining the known T-fixed-point computation of T-equivariant TMF.","Away from 6, π_*TJF_m[1/6] ≅ JF_{*,m}[1/6] for m > 0, so the classical Jacobi forms appear as the edge of the descent spectral sequence with no higher derived contributions.","At p = 3 the homotopy of TJF_m is computed completely, and at p = 2 the paper gives an explicit ring presentation for π_*(TJF_∞)(2) together with an additive ko/ku decomposition.","The Pm model gives a connective analogue tjF_m = tmf ∧ P_m, whose descent spectral sequence the paper computes to the same algebraic input, even though a ring structure on tjF_∞ is not established."],"supporting_citations":[{"why":"Supplies the equivalence TMFT ≃ TMF ⊕ Σ TMF that is the starting point for identifying the fill-in map in Theorem 3.7.","marker":"[GM23, Theorem 10.1]"},{"why":"Supplies the Grothendieck duality identity used in Proposition 3.4 to show the fill-in map is TMF smashed with the reduced transfer.","marker":"[GKMP, Theorem 6.5]"},{"why":"Provides the symmetric monoidal equivalence Γ : QCoh(M^or_ell) → ModTMF used to pass from sheaf cohomology to TMF-module spectra in Proposition 3.4.","marker":"[MM15]"},{"why":"Gives the computation of derived modular forms dmf_* that serves as the input and comparison ring for the algebraic computations of Sections 4 and 5.","marker":"[Bau08]"},{"why":"Provides the classical theory of Jacobi forms, including the structure theorem used for the presentation of weak Jacobi forms in Theorem 2.7.","marker":"[EZ85]"},{"why":"Gives the spectral algebraic geometry construction of the sheaf of E∞-ring spectra on the universal elliptic curve underlying the definition of TJF_m.","marker":"[Lur18]"}],"fun_headline_variants":["TJF equals TMF smashed with a projective cofiber","Topological Jacobi forms: TMF ∧ P∞, odd primes complete","Jacobi spectrum TJF = TMF ∧ P_m, full odd-primary homotopy","TJF: TMF with a projective cofiber, p=2 partial"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The proof leans on the unpublished Grothendieck duality identity Hom_{O_M}(p_*F,O_M) ≃ $Σ^{{-1}}$Hom_{O_E}(F,O_E) quoted from [GKMP, Theorem 6.5]; if that identity fails for the sheaves arising from finite T-spectra, the equivalence TJF_m ≃ TMF ∧ P_m is not established.","fun_headline_variants_meta":{"raw":{"variants":["TJF equals TMF smashed with a projective cofiber","Topological Jacobi forms: TMF ∧ P∞, odd primes complete","Jacobi spectrum TJF = TMF ∧ P_m, full odd-primary homotopy","TJF: TMF with a projective cofiber, p=2 partial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1423,"prompt_tokens":645,"completion_tokens":778,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":694}},"tokens_in":389,"tokens_out":778,"duration_ms":8342,"temperature":1.0,"reasoning_tokens":694,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T21:46:34.220422+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the composite pr_2 ∘ tr_m : TMF ∧ Σ $CP^{{m-1}}$ → TMF ∨ Σ TMF → Σ TMF; the proof of Theorem 3.7 claims it is trivial, and any nonzero value would force the fill-in map t to differ from TMF smashed with the reduced transfer. Alternatively, verify Eq. (3.5) directly for F = O_{E^or}(-e); if the duality does not hold, Proposition 3.4 and hence the central identification collapses.","supporting_citations":[],"review_version":1}