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

TL;DR

The paper extends the classical Picard group of $\operatorname{Spec} \mathbf{Z}$ to an arithmetic Picard monoid $\operatorname{Pic}(\overline{\operatorname{Spec} \mathbf{Z}})$ by incorporating archimedean data as a possibly degenerate norm, thereby addressing the paradox of infinite-genus in the arithmetic setting. It establishes a uniformization of the adelic monoid $X_{\mathbf{Q}}=\mathbf{Q}^\times \backslash \mathbb{A}/\widehat{\mathbf{Z}}^\times$ by framed arithmetic divisors $\operatorname{Pic}_{Fr}(\overline{\operatorname{Spec} \mathbf{Z}})$, with a canonical dual rooted description $\operatorname{Pic}_{Rt}(\overline{\operatorname{Spec} \mathbf{Z}})$, and shows that $X_{\mathbf{Q}} \cong \operatorname{Pic}(\overline{\operatorname{Spec} \mathbf{Z}})$. The arithmetic Jacobian is then the quotient by the archimedean scaling group $\mathbf{R}_+^\times$, organizing divisors into finite-type and infinite-type strata; this sets the stage for a spectral realization of zeros of $L$-functions via a Lefschetz-type trace formula driven by the idele class group action. The framework unifies Arakelov-style metrized divisors with adelic geometry, provides intrinsic and adelic descriptions of divisors, and yields a semi-local trace formula that isolates local and global contributions to Weil’s explicit formula, thereby linking boundary and singular phenomena to spectral data of $L$-functions. Overall, the work proposes a robust geometric and categorical apparatus—grounded in framed/rooted Picard monoids, a universal covering by class-field theory, and a spectral trace perspective—for understanding the arithmetic of $\overline{\operatorname{Spec} \mathbf{Z}}$ and the zeros of $L$-functions. The results bear significance for noncommutative geometry approaches to number theory and Langlands-type representations by encoding arithmetic in moduli spaces with explicit tensor and duality operations.

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.

Theorems & Definitions (99)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Geometric uniformization of the adelic monoid $Y_\mathbf{Q}$
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof