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Knots, Primes and the adele class space

Alain Connes, Caterina Consani

Abstract

We show that the scaling site $X_{\mathbb Q}$ and its periodic orbits $C_p$ of length $\log p$ offer a geometric framework for the well-known analogy between primes and knots. The role of the maximal abelian cover of $X_{\mathbb Q}$ is played by the quotient map $π:X_{\mathbb Q}^{ab}\to X_{\mathbb Q}$ from the adele class space $X_{\mathbb Q}^{ab}:={\mathbb Q}^\times \backslash {\mathbb A}_{\mathbb Q}$ to $X_{\mathbb Q}=X_{\mathbb Q}^{ab}/{\hat{\mathbb Z}^*}$. The inverse image $π^{-1}(C_p)\subset X_{\mathbb Q}^{ab}$ of the periodic orbit $C_p$ is canonically isomorphic to the mapping torus of the multiplication by the Frobenius at $p$ in the abelianized étale fundamental group $π_1^{e t}({\rm Spec} \, {\mathbb Z}_{(p)})^{ab}$ of the spectrum of the local ring ${\mathbb Z}_{(p)}$, thus exhibiting the linking of $p$ with all other primes. In the same way as the Grothendieck theory of the étale fundamental group of schemes is an extension of Galois theory to schemes, the adele class space gives, as a covering of the scaling site, the corresponding extension of the class field isomorphism for $\mathbb Q$ to schemes related to ${\rm Spec} \,\mathbb Z$.

Knots, Primes and the adele class space

Abstract

We show that the scaling site and its periodic orbits of length offer a geometric framework for the well-known analogy between primes and knots. The role of the maximal abelian cover of is played by the quotient map from the adele class space to . The inverse image of the periodic orbit is canonically isomorphic to the mapping torus of the multiplication by the Frobenius at in the abelianized étale fundamental group of the spectrum of the local ring , thus exhibiting the linking of with all other primes. In the same way as the Grothendieck theory of the étale fundamental group of schemes is an extension of Galois theory to schemes, the adele class space gives, as a covering of the scaling site, the corresponding extension of the class field isomorphism for to schemes related to .
Paper Structure (4 sections, 2 theorems, 13 equations, 2 figures)

This paper contains 4 sections, 2 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainglobalintro']}
  3. The semilocal space $\Gamma\backslash \left({\mathbb Q}_p\times {\mathbb Q}_q\times \mathbb{R}\right)$
  4. The classifying space $\Gamma\backslash\left({\mathbb A}_{{\mathbb Q},S}\times\underline{E \Gamma}\right)$ and its codimension $1$ foliation

Key Result

Theorem 1.1

Let $p$ be a prime and $\left\{F r o b_{p}\right\}\in \pi_1^{e t}({\rm Spec\,}\left({\mathbb F}_p\right))$ be the canonical generator. The inverse image $\pi^{-1}(C_p)\subset X_{\mathbb Q}^{ab}$ of the periodic orbit $C_p$ is canonically isomorphic to the mapping torusin the ordinary topological sen

Figures (2)

  • Figure 1: Lifting the periodic orbit $C_p$
  • Figure 2: (A) shows how the generic orbit (in green) for the action of $\mathbb{R}_+^*$ on $X_{\mathbb Q}$ is dense in the periodic orbit $C_p$. This holds for any $p$. (B) shows how the density of the generic orbit in every $C_p$ imitates the density of the generic point of ${\rm Spec\,}\,\mathbb{Z}$ (compare with Manin, page 16).

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 4.1