Topological Hochschild homology of the image of j

Abstract

We compute the mod $(p,v_1)$ and mod $(2,\eta,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_{\zeta}$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_{\zeta} \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_{\zeta}$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$.