The descent spectral sequence for topological modular forms

Abstract

We prove the Gap Theorem for the spectrum of topological modular forms $\mathrm{Tmf}$. This removes a longstanding circularity in the literature, thereby confirming the computation of $\ddot{I}{€}_{\ast }\mathrm{tmf}$ from over two decades ago by Hopkins and Mahowald. Our approach is crucially a modern one, developing and refining many techniques in synthetic spectra.

v2: Added historical overview, fixed minor typos. Comments welcome!