Computations of topological Jacobi forms

TL;DR

This paper computes the homotopy groups of spectra of topological Jacobi forms at prime 2 by analyzing a spectral sequence and a cellular decomposition, providing detailed differential identifications.

Contribution

It provides the first complete computation of the descent spectral sequence for topological Jacobi forms spectra at prime 2, including explicit differential analysis.

Findings

Complete descent spectral sequence at prime 2 for TJF spectra

Explicit cellular decomposition of TJF spectra

Identification of all differentials in the spectral sequence

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.