$\mathrm{H}{\mathbb{F}}_2$-synthetic homotopy groups of topological modular forms

Abstract

To any Adams-type spectrum $E$, Pstrągowski produced a symmetric monoidal stable $\infty$-category $Sy{n}_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Sy{n}_E$ comes equipped with a lax symmetric monoidal functor ${\nu }_E:Sp\to Sy{n}_E$ from classical spectra, which embeds $Sp$ fully and faithfully in $Sy{n}_E$, and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups ${\pi }_{\ast ,\ast }{\nu }_{E}X$ systematically record information about the homotopy groups ${\pi }_{\ast }X$ and the $E$-Adams spectral sequence of $X$. In this paper, we compute the ${\nu }_{\mathrm{H}{\mathbb{F}}_2}{\mathbb{F}}_2$-Adams spectral sequence of ${\nu }_{\mathrm{H}{\mathbb{F}}_2}tm{f}_2^{\wedge }$, synthetic versions of hidden $2$-, $\eta$-, $\nu$-, and $\overline{\kappa }$-extensions, and use this to deduce information about the homotopy ring structure of ${\pi }_{\ast ,\ast }{\nu }_{\mathrm{H}{\mathbb{F}}_2}tm{f}_2^{\wedge }$.

arXiv comment73 pages, 8 figures. v2 includes several fixed mistakes, improved exposition, computation of synthetic Adams E_r pages, new multiplicative results, addition of hidden kappabar extensions, and reworked charts and tables. Comments welcome!