A modern interpretation of Quillen's computation of the K theory of finite fields In his beautiful paper On the cohomology and K theory of the general linear group over a finite field, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory $K(\mathbb{F}_q)\mid_{\ge 1} \cong \text{Fib}(1-\Psi^q:ku\to ku).$ His construction, while very cool, seems to be based on a miracle: that the Brauer lift produces complex representations from mod-$p$ ones. It also uses the +-construction for defining K-theory (though perhaps it's not that hard to update his construction to work with Segal formalism)?
I'm curious whether there is a more modern perspective on this equivalence of spectra, maybe using cyclic structure on topological Hochschild homology.
 A: Firstly, there is a proof using the motivic spectral sequence (the Atiyah-Hirzerbruch style spectral sequence from motivic cohomology to algebraic $K$-theory). This is written in the master's thesis of Gabe Angelini-Knoll.
Gabe and Andrew Salch are also working to answer this question and a paper is apparently due. From Andrew's website:
"My student Gabe Angelini-Knoll and I have been working on the problem of computing the Waldhausen algebraic K-groups of the algebraic K-theory spectra of certain finite fields. This is an example of "iterated K-theory" and Rognes' redshift conjecture is not known in these cases. Thus far, Gabe and I have (with the aid of a new "THH-May" spectral sequence for computing topological Hochschild homology) computed the homotopy groups of THH(K(F_q)) smashed with the p-primary Smith-Toda complex V(1), for p > 3 and for many (but not all) prime powers q. Gabe is working on the computations of the homotopy groups of the C_p fixed points of this spectrum (this will probably be Gabe's thesis), with the goal of using trace methods to recover the K-groups of K(F_q).
We expect to post and submit our first two papers on this topic before the end of summer 2016. "
