Blueprints - towards absolute arithmetic?

TL;DR

This paper explores the concept of $ar{ ext{Spec}} ext{ extbf{Z}}$ within the framework of blueprints, aiming to bridge number fields and function fields to potentially approach the Riemann hypothesis.

Contribution

It introduces a blueprint-based construction of $ar{ ext{Spec}} ext{ extbf{Z}}$ that mimics properties of curves over finite fields, advancing the foundation for arithmetic geometry over $ ext{ extbf{F}_1}$.

Findings

Defined an object $ar{ ext{Spec}} ext{ extbf{Z}}$ with properties similar to positive characteristic analogs.

Proposed a blueprint-based approach to interpret $ ext{ extbf{Spec}} ext{ extbf{Z}}$ as a geometric object over $ ext{ extbf{F}_1}$.

Summarized in a note based on a talk at the Max Planck Institute in 2012.

Abstract

One of the driving motivations to develop $\F_1$-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields, which might result in a proof of the classical Riemann hypothesis. The underlying idea is that the spectrum of $\Z$ should find an interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$ analogous to a curve over a finite field. The hope is that intersection theory for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$ will allow to mimic Weil's proof. It turns out that it is possible to define an object $\bar{\Spec\Z}$ from the viewpoint of blueprints that has certain properties, which come close to the properties of its analogs in positive characteristic. This shall be explained in the following note, which is a summary of a talk given at the Max Planck Institute in March, 2012.