On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910025447899136 |
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| author | Connes, Alain Consani, Caterina |
| author_facet | Connes, Alain Consani, Caterina |
| contents | We interpret the structure of the adele class space of the rationals--and specifically its Riemann sector--as the natural monoidal extension of the Picard group of the arithmetic curve $\overline{\operatorname{Spec} \mathbb Z}$. We identify the elements of this space with torsion-free rank-1 abelian groups $L$ endowed with rigidifying data. In the Riemann sector, this data corresponds to a norm, extending the classical notion of metrized line bundles in Arakelov geometry. For the full adele class space, we replace the norm with a group morphism to $\mathbb R$ and a combinatorial datum: a parametrization of the roots of unity associated with the character dual of $L$. We show that the product of adeles is represented geometrically by the tensor product of these rank-1 groups and their rigidifying structures. The resulting monoid space generalizes the Picard group to the full adelic context by incorporating the singular strata required for the spectral realization of $L$-functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_15941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$ Connes, Alain Consani, Caterina Number Theory Algebraic Geometry Quantum Algebra 14G40, 11M55, 14C20, 11R42, 06F05 We interpret the structure of the adele class space of the rationals--and specifically its Riemann sector--as the natural monoidal extension of the Picard group of the arithmetic curve $\overline{\operatorname{Spec} \mathbb Z}$. We identify the elements of this space with torsion-free rank-1 abelian groups $L$ endowed with rigidifying data. In the Riemann sector, this data corresponds to a norm, extending the classical notion of metrized line bundles in Arakelov geometry. For the full adele class space, we replace the norm with a group morphism to $\mathbb R$ and a combinatorial datum: a parametrization of the roots of unity associated with the character dual of $L$. We show that the product of adeles is represented geometrically by the tensor product of these rank-1 groups and their rigidifying structures. The resulting monoid space generalizes the Picard group to the full adelic context by incorporating the singular strata required for the spectral realization of $L$-functions. |
| title | On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$ |
| topic | Number Theory Algebraic Geometry Quantum Algebra 14G40, 11M55, 14C20, 11R42, 06F05 |
| url | https://arxiv.org/abs/2602.15941 |