Knots, Primes and the adele class space
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arXiv
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| Format: | Preprint |
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2024
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| author | Connes, Alain Consani, Caterina |
| author_facet | Connes, Alain Consani, Caterina |
| contents | 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 ${\rm Spec} \,\mathbb Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Knots, Primes and the adele class space Connes, Alain Consani, Caterina Number Theory Algebraic Geometry Algebraic Topology Quantum Algebra 11R37, 11M06, 11M55, 14A15, 14F20, 57K10 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 ${\rm Spec} \,\mathbb Z$. |
| title | Knots, Primes and the adele class space |
| topic | Number Theory Algebraic Geometry Algebraic Topology Quantum Algebra 11R37, 11M06, 11M55, 14A15, 14F20, 57K10 |
| url | https://arxiv.org/abs/2401.08401 |