2017/03/30 by Aurélien Sagnier, Sagnier, Aurélien
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1703.10521
openalex publication_date 2017/03/30 · openalex created_date 2017/04/14 · arxiv created 2017/05/09 · arxiv updated 2017/05/10 · openalex updated_date 2026/07/28
We construct, for imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that the constructions of Connes and Consani and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. Of course nothing of this sort exists in the case of imaginary quadratic number fields with class number 1. We first define what we call arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the adèles class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get therefore that for a fixed imaginary quadratic number field with class number 1, that the points of our arithmetic site are related to the zeroes of the Dedekind zeta function of the number field considered and to the zeroes of some Hecke L functions. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. Finally we construct the square of the arithmetic site.