In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function field over a new object $\mathbb{F}_1$.
I am wondering what the étale cohomology of $\mathbb{F}_1$ should be.
Given the function field analogy, do we expect finite covers of $\text{Spec}(\mathbb{F}_1)$ to be goverened by the frobenius map, so that $H^1(\mathbb{F}_1) \cong \pi_1^{ab}(\mathbb{F}_1) \cong \hat{\mathbb{Z}}$, with the frobenius element appearing as a generator? If this thinking is true, then maybe the étale homotopy content of $\mathbb{F}_1$ is $S^1$. But this could be completely wrong, since I don't have a good enough sense of the combinatorial and cohomological analogies that should hold.
