of the connective image of $J$ spectrum. We use the model $K(\mathbb{F}_q)_p$, which is easily seen to be a commutative ring spectrum and we write $j$ throughout for a cofibrant replacement of this ring spectrum in $\Comm \mathfrak{S}$.

Throughout, we will write $\dot{=}$ to mean equivalence up to multiplication by a unit in $\mathbb{F}_p$. We write $E(x_1,x_2, ...)$ for an exterior algebra over $\mathbb{F}_p$, $P(x_1,x_2,...)$ for a polynomial algebra over $\mathbb{F}_p$, $P_p(x_1,x_2, ...)$ for a truncated polynomial algebra over $\mathbb{F}_p$ truncated at the $p$-th power, and $\Gamma(x_1,x_2,...)$ for a divided power algebra over $\mathbb{F}_p$ on generators $x_i$ for $i\ge 1$. We recall that a divided power algebra generated by $x$ has generators $\gamma_i(x)$ for $i\ge 0$ with relations $\gamma_0(x)=1$, $\gamma_1(x)=x$ and $\gamma_i(x)\gamma_j(x)=(i,j)\gamma_{i+j}(x)$ where $$(i,j)=\left ( \begin{array}{c} i+j \\ j\end{array} \right ).$$ In particular, over the finite field $\mathbb{F}_p$, there is an isomorphism $$\Gamma(z)\cong P_{p}(z, \gamma_p(z) ,\gamma_{p^2}(z) , ...).$$ When not otherwise indicated, $\otimes$ will represent $\otimes_{\mathbb{F}_p}$ and $HH_*(R)$ will indicate $HH_*^{\mathbb{F}_p}(R)$ when $R$ is a $\mathbb{F}_p$-algebra.

Acknowledgements
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It is a pleasure to thank Andrew Salch for countless conversations about material related to this paper. The author also benefited from conversations with Bjorn Dundas, Robert Bruner and Dan Isaksen. The 