For each prime and each embedding of the multiplicative group of an algebraic closure of as complex roots of unity, we construct a -adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of . The obtained representations are the -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 is replaced, in the -adic case, by the -adic -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 of an algebraic closure of . We show that our previous work on the hyperring structure of the adèle class space, combines with -adic analysis to refine the space of valuations on the cyclotomic extension of 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 which singles out the subsystem associated to the -extension of .
Cite this article
Alain Connes, Caterina Consani, On the arithmetic of the BC-system. J. Noncommut. Geom. 8 (2014), no. 3, pp. 873–945