On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$

TL;DR

This paper explores the structure of the adele class space of the rationals, interpreting it as an extension of the Picard group of the arithmetic curve, and connects it to the spectral realization of L-functions.

Contribution

It introduces a geometric interpretation of the adele class space as a monoidal extension of the Picard group, incorporating rigidifying data and generalizing Arakelov geometry.

Findings

Identifies elements of the adele class space with torsion-free rank-1 abelian groups with additional data.

Shows the product of adeles corresponds to tensor product of these groups and structures.

Provides a geometric framework for the spectral realization of L-functions.

Abstract

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.