The cyclotomic structure on is given by maps as Borel--equivariant spectra.
This is constructed as follows:
There’s a map where the first map is the Tate diagonal, the second is applied to the map , which comes from being the initial object in . Informally it multiplies an element with -copies of itself around the circle. The map comes from restricting along .
The -frobenius then comes from the fact that this map lands in a Borel--equivariant spectrum (a local system on ): so this canonically defines a map out of -cycles on , by universal property of .
The Frobenius map (as defined for any cyclotomic spectrum) in this case can be unpacked as follows: the maps defining the cyclotomic structure are -equivariant, hence we can take fixed points to get a map
}$$ Now the final step has hypotheses: what's true is that for any bounded-below $\mathbb T$-spectrum $X$ we have $X^{t\mathbb T} \simeq (X^{tC_p})^{h \mathbb T}$, so we need conditions on $R$ to ensure $THH(R)$ is bounded below. This equivalence is induced by the restriction map $X^{t \mathbb T} \to X^{tC_p}$: namely this map is $\mathbb T$-equivariant, hence factors through the homotopy fixed-points. This induces $p$-completion. **Remark** look at how this structure is present on the paracyclic stacks from [[2-categorical G_m reps]]) **Remark** Now the syntomification/cyclotomic fixed points"trivializes" this data: write out what this means by viewing the residual action by $\mathbb T/C_p \simeq \mathbb T$ as a $p$-th power map.