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Topological Jacobi Forms

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Pith's one-line read 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.

desk verdict TJF_m ≃ TMF∧P_m is new and likely right, but the load-bearing Grothendieck duality is cited to an unpublished paper—worth refereeing, not desk rejecting. read the letter →

arxiv 2508.08010 v1 pith:RA7KCJ6U submitted 2025-08-11 math.AT

Tilman Bauer , Lennart Meier This is my paper · ORCID
classification math.AT MSC 55N3455P9111F5055T99
keywords topologicalJacobiformsmodularequivarianthomotopytheoryellipticgeneraAtiyah-HirzebruchspectralsequencedescentE∞-ringspectra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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],

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [§3.1, Prop. 3.4 and Eq. (3.5)] 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.
  2. [Thm. 4.7 / Thm. 1.9 vs. §6] 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.
minor comments (4)
  1. [Thm. 1.11 and Thm. 4.6] 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).
  2. [Thm. 4.6 display] 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)".
  3. [Introduction, Definition of TJF] 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.
  4. [Prop. 3.14] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Central equivalence TJF_m ≃ TMF∧P_m is made to rest on an unpublished Grothendieck-duality identity (3.5) from the coauthor's in-preparation paper [GKMP]; the homotopy calculations themselves are not fitted.

  1. self citation load bearing [Section 3, Proposition 3.4, Eq. (3.5); used in proof of Theorem 3.7 (Theorem 1.7)]
    "For F ∈ QCoh(E or), [ GKMP, Theorem 6.5] gives a natural equivalence (3.5) Hom_{O_{M or}} (p_*F, O_{M or}) ≃ Σ^{-1}Hom_{O_{E or}}(F, O_{E or})."

    The proof of Theorem 3.7 identifies TJF_m with TMF ∧ P_m by applying Proposition 3.4 to F = TMF^T(S(mρ)_+). The only nontrivial input in Proposition 3.4 is Eq. (3.5), quoted from [GKMP, Theorem 6.5]. That paper is unpublished, in preparation, and has overlapping authorship (L. Meier). No proof, statement of hypotheses, or independent verification of (3.5) is supplied; in particular, the paper does not check that the duality holds for the sheaves obtained from equivariant TMF. Thus the central structural equivalence of the paper is not derived from first principles here but is handed over from an overlapping unpublished self-citation. The later homotopy computations are independent of this citation, but the P_m model on which they run is loaded by it.

full rationale

The homotopy computations are not circular: DJF is computed from Hopf algebroid Ext groups, and the differentials are fixed by external constraints such as η^4 = 0 in the sphere, Mosher's stable homotopy of complex projective space, and convergence to known TMF_*(P_m). None of these are fitted to the stated π_*TJF_m. The definition of TJF as global sections of L^top_m is independent of the computational target. The only genuine circularity-pattern issue is the load-bearing citation of [GKMP, Theorem 6.5] in Proposition 3.4: this is an in-preparation paper with author overlap (Meier), and the central equivalence TJF_m ≃ TMF ∧ P_m in Theorem 3.7 is obtained by applying that cited duality to F = TMF^T(X). No proof or independent verification of (3.5) appears in this paper. I score this 4 rather than higher because the algebraic and 2-primary/3-primary computations do not reduce to the cited identity and retain independent content. There is also a local internal inconsistency (Thm 1.9 and Thm 4.7 omit the relation b4 h1 = 0 that appears in the Bockstein computation and in the E_4 page of Section 6), but that is a correctness issue, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on the equivariant-TMF framework of Gepner-Meier (published), Lurie's spectral elliptic cohomology (unpublished but standard in the field), and one unpublished in-preparation duality theorem with author overlap [GKMP]. The algebraic computations are built on Hopf algebroid presentations from Bauer's computation of tmf and Eichler-Zagier's classical structure theory. No empirically fitted parameters appear; the only chosen data are conventional generator scalings. The main invented object, TJF_m, is a construction with an explicit cellular model and computed homotopy, so it carries abundant independent handles.

assumptions (6)
  • standard math Goerss-Hopkins-Miller sheaf O^top of E_∞-ring spectra on M_ell exists, with global sections TMF.
    Sections 1 and 3.1; established in [DFHH14].
  • standard math Lurie's spectral Deligne-Mumford stack M_ell^or of oriented spectral elliptic curves exists, with underlying stack M_ell, structure sheaf O^top, and universal oriented curve E^or.
    Section 3.1; [Lur18], a widely relied upon but unpublished manuscript.
  • standard math Gepner-Meier's functor TMFT exists with the properties of Thm 3.1, including TMFT^T ≃ TMF ⊕ ΣTMF ([GM23, Thm 10.1]).
    Section 3.1, Eq. (3.8); published in Compos. Math. 159 (2023).
  • ad hoc to paper Grothendieck duality on spectral stacks: Hom_{O_M}(p_*F, O_M) ≃ Σ^{-1}Hom_{O_E}(F, O_E) for F ∈ QCoh(E^or).
    Prop 3.4, Eq. (3.5), cited to [GKMP, Thm 6.5], unpublished, in preparation, with author overlap (Meier). Load-bearing for Thm 1.7.
  • standard math The Hopf algebroid (A, Γ) = (Z[a1,a2,a3,a4,a6], A[r,s,t]) presents M_Weier, and the universal Weierstrass curve W has the stated presentations, with cohomology computed as Ext over (A, Γ).
    Section 4; follows [Bau08, MO20].
  • standard math Classical Eichler-Zagier structure theory of weak Jacobi forms ([EZ85, Thm 9.4]) and the 2m-zero theorem for index m, plus standard stable homotopy facts (η⁴ = 0 in π_4(S^0); 2ν attaching map for CP³, [Mos68]).
    Section 2 (Thm 2.7, Lemma 2.5), Section 6 (d3(b2) = h1³ argument), and Lemma 5.7.
invented entities (1)
  • TJF_m and TJF_∞ = colim_m TJF_m independent evidence
    purpose: Topological ring spectra whose complexified homotopy is the ring of weakly holomorphic Jacobi forms of index m/2; target of the 2-variable elliptic genus; connect RO(S^1)-graded equivariant TMF to stunted projective spaces.
    Explicitly constructed via equivariant TMF with a concrete model TMF ∧ P_m (Thm 3.7) whose homotopy is computed (Thms 1.9, 1.11, Cor 6.1); independently used by Lin-Yamashita [LY24] to lift the 2-variable elliptic genus.

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Pith. "Pith review of Topological Jacobi Forms." pith.science (2026). https://pith.science/paper/RA7KCJ6U

@misc{pith2026250808010,
  author       = {Pith},
  title        = {Pith review of: Topological Jacobi Forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RA7KCJ6U}},
  note         = {Machine review of arXiv:2508.08010}
}
abstract

As a generalization of the ring spectrum of topological modular forms, we construct a graded ring spectrum of topological Jacobi forms, $\operatorname{TJF}_*$. This is constructed as the global sections of a sheaf of $E_\infty$-ring spectra on the stacky universal elliptic curve using circle-equivariant $\operatorname{TMF}$. Complete calculations of its homotopy at odd primes and partial results at $p=2$ are given.

Figures

Figures reproduced from arXiv: 2508.08010 by the authors.

Figure 5.1
Figure 5.1. The bigraded ring dmf∗∗[ 1 2 ] [PITH_FULL_IMAGE:figures/full_fig_p028_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. The E2 -term of the UAAHSS for djF∗∗∗[ 1 2 ] In that chart, the horizontal axis is the total dimension n = t − s, the vertical axis is the cohomological degree s, and RMA color code modulo 9 is used for the indices u and m (black = 0, brown = 1, red = 2, orange = 3, yellow = 4, green = 5, blue = 6, purple = 7, gray = 8). More precisely, a symbol with color code i represents  = Z[ 1 2 , a] = Z[ 1 2 ] • = Z/3Z[a] ◦ =… view at source ↗
Figure 5.3
Figure 5.3. The E∞ term of the unstable algebraic Atiyah￾Hirzebruch spectral sequence for H∗ (W, L∗,∗)[ 1 2 ] The class b3 is a permanent cycle, and no more differentials are possible for degree reasons. Proof of Thm. 4.6. We read off from this chart that (djF∗∗∗)[ 1 2 ] is generated by the classes α, β, c4, c6, ∆ that generate (dmf∗∗)[ 1 2 ] as well as the classes b2, b3, b4 in tridegrees |bi | = (2i, 0, i), the class a in tri… view at source ↗
Figures from the paper (1 more)
Figure 5.4
Figure 5.4. Figure 5.4: The descent spectral sequence converging to π∗ tjF∗ [ 1 2 ] which shows that the target is a free module over the source on the basis {s i}, thus is faithfully flat. Hence so is the quotient by the (invariant) ideal generated by E(a1, 0, a3, a4, a6, x, y). Thus H˜W,∞…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory

    math.AT 2025-08 conditional novelty 7.0 of 10

    Derived level structures in spectral algebraic geometry are representable, yielding Jacquet-Langlands spectra and a proposed Jacquet-Langlands dual of Morava E-theory.

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