On the arithmetic of the BC-system

  • Alain Connes

    Collège de France
  • Caterina Consani

    The Johns Hopkins University, Baltimore, USA


For each prime pp and each embedding σ\sigma of the multiplicative group of an algebraic closure of Fp\mathbb{F}_p as complex roots of unity, we construct a pp-adic indecomposable representation πσ\pi_\sigma of the integral BC-system as additive endomorphisms of the big Witt ring of Fˉp\bar{\mathbb{F}}_p. The obtained representations are the pp-adic analogues of the complex, extremal KMS_\infty states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over C\mathbb{C} is replaced, in the pp-adic case, by the pp-adic LL-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\mathbb{C}_p of an algebraic closure of Qp\mathbb{Q}_p. We show that our previous work on the hyperring structure of the adèle class space, combines with pp-adic analysis to refine the space of valuations on the cyclotomic extension of Q\mathbb{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 Fˉp\bar{\mathbb{F}}_p which singles out the subsystem associated to the Z^\hat{\mathbb{Z}}-extension of Q\mathbb{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