On the arithmetic of the BC-system

TL;DR

This paper constructs p-adic representations of the BC-system using Witt rings and p-adic L-functions, linking noncommutative geometry, Iwasawa theory, and arithmetic geometry to extend the understanding of KMS states in a p-adic setting.

Contribution

It introduces p-adic indecomposable representations of the BC-system, connecting p-adic L-functions and polylogarithms to KMS states, and relates these to the arithmetic site and noncommutative geometry.

Findings

Constructed p-adic representations of the BC-system.

Linked KMS states to p-adic L-functions and polylogarithms.

Connected the BC-system to the arithmetic site and noncommutative space.

Abstract

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, extremal KMS states at zero temperature of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over complex numbers 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 of an algebraic closure of the p-adic field. We show that our previous work on the hyperring structure of the adeles 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 an algebraic closure of F_p which singles out the subsystem associated to the Z^-extension of Q.