# On the arithmetic of the BC-system

### Alain Connes

Collège de France### Caterina Consani

The Johns Hopkins University, Baltimore, USA

## Abstract

For each prime $p$ and each embedding $σ$ of the multiplicative group of an algebraic closure of $F_{p}$ as complex roots of unity, we construct a $p$-adic indecomposable representation $π_{σ}$ of the integral BC-system as additive endomorphisms of the big Witt ring of $Fˉ_{p}$. 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 $C_{p}$ of an algebraic closure of $Q_{p}$. 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 $Fˉ_{p}$ which singles out the subsystem associated to the $Z^$-extension of $Q$.

## Cite this article

Alain Connes, Caterina Consani, On the arithmetic of the BC-system. J. Noncommut. Geom. 8 (2014), no. 3, pp. 873–945

DOI 10.4171/JNCG/173