Schemes over $\F_1$ and zeta functions

TL;DR

This paper explores the hypothetical geometric structure of the 'curve' over the field with one element, $ ext{F}_1$, linking it to the Riemann zeta function, and develops a functorial theory of $ ext{F}_1$-schemes that unifies previous approaches.

Contribution

It introduces a new functorial framework for $ ext{F}_1$-schemes that generalizes prior models and connects to the spectral interpretation of zeros of $L$-functions.

Findings

Determines the real counting function $N(q)$ for the hypothetical $ ext{F}_1$-curve.

Develops a unified theory of $ ext{F}_1$-schemes compatible with Kato's monoids.

Provides a conceptual interpretation of zeros of $L$-functions using $ ext{F}_1$-geometry.

Abstract

We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $\F_1$-schemes which reconciles the previous attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of ad\`ele classes over an arbitrary global field, how to apply our functorial theory of $\Mo$-schemes to interpret conceptually the spectral realization of zeros of $L$-functions.