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Computations of topological Jacobi forms

Akira Tominaga

TL;DR

This work provides a complete 2-local computation of the descent spectral sequences converging to the homotopy groups of topological Jacobi forms TJF_m for all m ≥ 1, establishing the TMF-module equivalence TJF_m ≃ TMF ⊗ P_m with P_m built from a transfer-based cofiber of stunted projective spaces. The E2-term is described via a Hopf algebroid arising from a GL2(Z/3)-Galois cover of the universal elliptic curve, and a horizontal vanishing line at s = 24 is proved; the authors determine all higher differentials for m = 2,3,4,5,6,7 using the cellular structure and a synthetic Leibniz rule. A central structural insight is the transfer-driven cell decomposition TJF_m ≃ TMF ⊗ P_m, where the attaching maps are governed by the Hopf maps η, ν, and 2ν, and the ∞-case TJF_∞ exhibits KO/KU-like behavior with a KO-module decomposition and a K(2)-local vanishing phenomenon. The results deepen the interaction between equivariant elliptic cohomology, topological modular forms, and Jacobi forms, providing concrete computational tools and revealing links to stunted complex projective spaces and transfer maps in equivariant homotopy theory.

Abstract

We compute, at the prime $2$, the entire descent spectral sequence converging to the homotopy groups of the spectra of topological Jacobi forms $\mathrm{TJF}_m$ for every index $m \geq 1$. An explicit $\mathrm{TMF}$-cellular decomposition $\mathrm{TJF}_m \simeq \mathrm{TMF} \otimes P_m$ reduces the problem to analyzing a finite complex $P_m$ with one even cell in each dimension $\leq 2m$ except $2$. We identify all differentials using the cell structure.

Computations of topological Jacobi forms

TL;DR

This work provides a complete 2-local computation of the descent spectral sequences converging to the homotopy groups of topological Jacobi forms TJF_m for all m ≥ 1, establishing the TMF-module equivalence TJF_m ≃ TMF ⊗ P_m with P_m built from a transfer-based cofiber of stunted projective spaces. The E2-term is described via a Hopf algebroid arising from a GL2(Z/3)-Galois cover of the universal elliptic curve, and a horizontal vanishing line at s = 24 is proved; the authors determine all higher differentials for m = 2,3,4,5,6,7 using the cellular structure and a synthetic Leibniz rule. A central structural insight is the transfer-driven cell decomposition TJF_m ≃ TMF ⊗ P_m, where the attaching maps are governed by the Hopf maps η, ν, and 2ν, and the ∞-case TJF_∞ exhibits KO/KU-like behavior with a KO-module decomposition and a K(2)-local vanishing phenomenon. The results deepen the interaction between equivariant elliptic cohomology, topological modular forms, and Jacobi forms, providing concrete computational tools and revealing links to stunted complex projective spaces and transfer maps in equivariant homotopy theory.

Abstract

We compute, at the prime , the entire descent spectral sequence converging to the homotopy groups of the spectra of topological Jacobi forms for every index . An explicit -cellular decomposition reduces the problem to analyzing a finite complex with one even cell in each dimension except . We identify all differentials using the cell structure.
Paper Structure (37 sections, 75 theorems, 98 equations, 37 figures)

This paper contains 37 sections, 75 theorems, 98 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the argument
  4. Organization of the paper
  5. Conventions
  6. Acknowledgements
  7. Preliminary
  8. The Descent Spectral Sequence: Setup
  9. E2-term of DSS
  10. DSS for TJFinfty
  11. Transfers and Cell Structures of TJF
  12. TJF2
  13. E2-term
  14. Higher Differentials
  15. TJF3
  16. ...and 22 more sections

Key Result

Theorem 1.1

For each integer $m\ge1$, there exists an equivalence of $\TMF$-modules where $P_m$ is obtained from the stunted complex projective space $\Sigma^2\bC P^{m-1}_{-1}$ by deleting its $2$-cell (Definition def:Pm). In particular, $\TJF_m$ has exactly one $\TMF$-cell in each even dimension $2d \le 2m$, $d\ne1$, and the stable attaching maps are detected by the Hopf fibration

Figures (37)

  • Figure 1: The $E_3$-page of DSS for $\TJF_{\infty}$ and differentials
  • Figure 2: The $E_2$-page of DSS for the 2-local $\TMF$
  • Figure 3: $E_1$-term of algebraic AHSS for $\TJF_2$ and differentials
  • Figure 4: The $E_{2}$-page of DSS for $\TJF_{2}$
  • Figure 5: $E_2$-term of ASS for $\TJF_3$
  • ...and 32 more figures

Theorems & Definitions (137)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Linskens_2025
  • Proposition 2.5
  • Theorem 2.6: GepnerMeierNEW
  • Theorem 2.7
  • Theorem 2.8
  • ...and 127 more