On the Drinfeld formal group

We identify Drinfeld’s formal group on the prismatization of $\mathrm{S}\mathrm{p}\mathrm{f}{\mathbb{Z}}_p$ with a formal group arising from homotopy theory, given locally by the Quillen formal group of a decompleted variant of topological periodic cyclic homology. We also prove that this formal group extends to the syntomization of $\mathrm{S}\mathrm{p}\mathrm{f}{\mathbb{Z}}_p$, and that it admits a certain algebraization conjectured by Drinfeld.