Title: An arithmetic site of Connes-Consani type for imaginary quadratic fields with class number 1,
Abstract: 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.