2011/03/24 by Alain Connes, Connes, Alain, Caterina Consani +1 · 1 citation
Mathematics · #11M55 #46L55 #58B34 #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #math.NT #math.QA #msc:11M55 #msc:46L55 #msc:58B34
paper · pdf · doi:10.48550/arxiv.1103.4672
61 pages
arxiv created 2011/03/24 · arxiv updated 2011/03/25
For each prime p and each embedding of the multiplicative group of an algebraic closure of Fp 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 Fp. 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 Fp which singles out the subsystem associated to the Z-extension of Q.