An arithmetic site at the complex place

Abstract

TextWe construct, for specific imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that their constructions and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. We first define what we call an arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the adèle class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get that the points of our arithmetic site are related not only to the zeroes of the Dedekind zeta function of the number field considered but also to other Hecke L functions involving non-trivial characters at the archimedean place. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. An appendix by Alain Connes shows that the compatibility with the Frobenius yields the correct notion of morphisms of semirings in this context. VideoFor a video summary of this paper, please visit https://www.youtube.com/watch?v=lWF9gbfvcYY&feature=youtu.be.