On the arithmetic of the BC-system

Abstract

For each prime $p$ and each embedding $\sigma$ of the multiplicative group of an algebraic closure of $\mathbb{F}_p$ as complex roots of unity, we construct a $p$-adic indecomposable representation $\pi_\sigma$ of the integral BC-system as additive endomorphisms of the big Witt ring of $\bar{\mathbb{F}}_p$. The obtained representations are the $p$-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 $\mathbb{C}$ 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 $\mathbb{C}_p$ of an algebraic closure of $\mathbb{Q}_p$. 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 $\mathbb{Q}$ 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 $\bar{\mathbb{F}}_p$ which singles out the subsystem associated to the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$.