Segal’s Gamma rings and universal arithmetic

Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger
${\mathbb F}_1$
-geometry and compute the Witt functor-ring
${\mathbb W}_0({\mathbb S})$
of the simplest Γ-ring
${\mathbb S}$
. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification
${\overline{{\rm Spec\,}{\mathbb Z}}}$
which acquires a structure sheaf of
${\mathbb S}$
-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on
${\overline{{\rm Spec\,}{\mathbb Z}}}$
, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann–Roch problem for D.