On the arithmetic of the BC-system

Abstract

For each prime p and each embeddingof the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation �� of the integral BC-system as additive endo- morphisms of the big Witt ring of ¯ Fp. The obtained representations are the p-adic analogues of the complex, extremal KMS1 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 poly- logarithms 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 hyper- ring structure of the adele class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommuta- tive space intimately related to the integral BC-system and whose geometry comes close to fulfill the expectations of the arithmetic site. Fi- nally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model ofp which singles out the subsystem associated to the ˆ-extension of Q.