Spoke topological Hochschild homology

Fix primes $p$ and $\mathrm{\ell }$, and let $C_p$ be the cyclic group of order $p$. We compute the $C_p$-equivariant spoke topological Hochschild homology of ${\underset{‾}{\mathbb{F}}}_{\mathrm{\ell }}$ and prove it exhibits a form of Bökstedt periodicity. Here spoke topological Hochschild homology is a variant of topological Hochschild homology where one replaces the circle in the construction with the unreduced suspension of $C_p$. As an application, we use this result to give a new proof of the Segal conjecture for the cyclic group of order an odd prime $p$.