# An arithmetic site of Connes-Consani type for imaginary quadratic fields with class number 1

## TL;DR

This paper constructs an arithmetic site for imaginary quadratic fields with class number 1, linking its points to zeros of Dedekind zeta and Hecke L functions, extending Connes-Consani's framework beyond real numbers.

## Contribution

It introduces a new arithmetic site for imaginary quadratic fields with class number 1, adapting Connes-Consani's approach without relying on the natural order of real numbers.

## Key findings

- Points of the arithmetic site relate to zeros of Dedekind zeta functions.
- Points are expressed via adèles class space and spectral interpretation.
- Constructed the square of the arithmetic site.

## 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 ad\`eles 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.