Topological Jacobi Forms

TL;DR

This paper constructs a new graded ring spectrum of topological Jacobi forms as a generalization of topological modular forms, providing explicit homotopy calculations at odd primes and partial results at 2.

Contribution

It introduces a novel spectrum of topological Jacobi forms as a sheaf of $E_$-ring spectra on the universal elliptic curve, extending the framework of topological modular forms.

Findings

Complete homotopy calculations at odd primes.

Partial homotopy results at prime 2.

Establishment of the sheaf of $E_$-ring spectra on the elliptic curve.

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.