What does Spec ${\mathbb Z}$ look like?

Mazur observed that the etale cohomology groups of Spec ${\mathbb Z}$ indicate that it looks like a 3-manifold, and for each prime p the closed subset Spec ${\mathbb F}_p$ of Spec ${\mathbb Z}$ looks like a circle. Deninger suggested that moreover, there should be an action of the reals on Spec ${\mathbb Z}$, with the periodic orbits being precisely those Spec ${\mathbb F}_p$'s, each one becoming an orbit of length $\log p$. More recently, there is growing evidence that there should be a version for Spec $\mathbb Z$ of the function field theory of shtukas, which should relate to the Langlands correspondence and yield a cohomology theory close to the theory of motives. While all of this remains unrealized, I will indicate a geometric framework which at least has the potential to allow for a rather faithful realization of this picture; in particular, Deninger's expected action of the real numbers is realized naturally. Restricted to the p-adic or real part of Spec ${\mathbb Z}$, it can be made rather precise, and I will indicate relations to the Fargues-Fontaine curve, the twistor projective line, as well as p-adic and complex Hodge (or twistor) theory, and related theories of local shtukas and geometric versions of local Langlands correspondences.

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