On the Arithmetic of the Bc-system

For each prime p and each embedding σ of the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation πσ of the integral BC-system as additive endomorphisms of the big Witt ring of F̄p. The obtained representations are the p-adic analogues of the complex, extremal KMS∞ states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over C is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion Cp of an algebraic closure of Qp. We show that our previous work on the hyperring structure of the adèle class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of F̄p which singles out the subsystem associated to the Ẑ-extension of Q.